ITP001 Axioms: ITP047^5.ax
%------------------------------------------------------------------------------
% File : ITP047^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : fcp^2.ax [Gau20]
% : HL4047^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 146 ( 8 unt; 35 typ; 0 def)
% Number of atoms : 1646 ( 111 equ; 0 cnn)
% Maximal formula atoms : 94 ( 11 avg)
% Number of connectives : 3243 ( 7 ~; 7 |; 47 &;2981 @)
% ( 18 <=>; 183 =>; 0 <=; 0 <~>)
% Maximal formula depth : 41 ( 11 avg;2981 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 62 ( 62 >; 0 *; 0 +; 0 <<)
% Number of symbols : 97 ( 96 usr; 61 con; 0-3 aty)
% Number of variables : 435 ( 39 ^ 380 !; 16 ?; 435 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_ty_2Efcp_2Ebit0,type,
ty_2Efcp_2Ebit0: del > del ).
thf(tp_ty_2Efcp_2Ebit1,type,
ty_2Efcp_2Ebit1: del > del ).
thf(tp_ty_2Efcp_2Ecart,type,
ty_2Efcp_2Ecart: del > del > del ).
thf(tp_ty_2Efcp_2Efinite__image,type,
ty_2Efcp_2Efinite__image: del > del ).
thf(tp_c_2Efcp_2E_3A_2B,type,
c_2Efcp_2E_3A_2B: del > del > $i ).
thf(mem_c_2Efcp_2E_3A_2B,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) ) ) ) ) ).
thf(tp_c_2Efcp_2EBIT0A,type,
c_2Efcp_2EBIT0A: del > $i ).
thf(mem_c_2Efcp_2EBIT0A,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2EBIT0A @ A_27a ) @ ( arr @ A_27a @ ( ty_2Efcp_2Ebit0 @ A_27a ) ) ) ).
thf(tp_c_2Efcp_2EBIT0B,type,
c_2Efcp_2EBIT0B: del > $i ).
thf(mem_c_2Efcp_2EBIT0B,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2EBIT0B @ A_27a ) @ ( arr @ A_27a @ ( ty_2Efcp_2Ebit0 @ A_27a ) ) ) ).
thf(tp_c_2Efcp_2EBIT1A,type,
c_2Efcp_2EBIT1A: del > $i ).
thf(mem_c_2Efcp_2EBIT1A,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2EBIT1A @ A_27a ) @ ( arr @ A_27a @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) ) ).
thf(tp_c_2Efcp_2EBIT1B,type,
c_2Efcp_2EBIT1B: del > $i ).
thf(mem_c_2Efcp_2EBIT1B,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2EBIT1B @ A_27a ) @ ( arr @ A_27a @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) ) ).
thf(tp_c_2Efcp_2EBIT1C,type,
c_2Efcp_2EBIT1C: del > $i ).
thf(mem_c_2Efcp_2EBIT1C,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2EBIT1C @ A_27a ) @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) ).
thf(tp_c_2Efcp_2EFCP,type,
c_2Efcp_2EFCP: del > del > $i ).
thf(mem_c_2Efcp_2EFCP,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2EFCP @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Efcp_2EFCP__CONCAT,type,
c_2Efcp_2EFCP__CONCAT: del > del > del > $i ).
thf(mem_c_2Efcp_2EFCP__CONCAT,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efcp_2EFCP__CONCAT @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) @ ( ty_2Efcp_2Ecart @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ A_27c ) ) ) ) ) ).
thf(tp_c_2Efcp_2EFCP__CONS,type,
c_2Efcp_2EFCP__CONS: del > del > del > $i ).
thf(mem_c_2Efcp_2EFCP__CONS,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efcp_2EFCP__CONS @ A_27a @ A_27b @ A_27c ) @ ( arr @ A_27a @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) ) ) ) ).
thf(tp_c_2Efcp_2EFCP__EVERY,type,
c_2Efcp_2EFCP__EVERY: del > del > $i ).
thf(mem_c_2Efcp_2EFCP__EVERY,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2EFCP__EVERY @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27b @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Efcp_2EFCP__EXISTS,type,
c_2Efcp_2EFCP__EXISTS: del > del > $i ).
thf(mem_c_2Efcp_2EFCP__EXISTS,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2EFCP__EXISTS @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27b @ A_27a ) @ bool ) ) ) ).
thf(tp_c_2Efcp_2EFCP__FOLD,type,
c_2Efcp_2EFCP__FOLD: del > del > del > $i ).
thf(mem_c_2Efcp_2EFCP__FOLD,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efcp_2EFCP__FOLD @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27b @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) @ A_27b ) ) ) ) ).
thf(tp_c_2Efcp_2EFCP__HD,type,
c_2Efcp_2EFCP__HD: del > del > $i ).
thf(mem_c_2Efcp_2EFCP__HD,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2EFCP__HD @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ A_27a ) ) ).
thf(tp_c_2Efcp_2EFCP__MAP,type,
c_2Efcp_2EFCP__MAP: del > del > del > $i ).
thf(mem_c_2Efcp_2EFCP__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efcp_2EFCP__MAP @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) @ ( ty_2Efcp_2Ecart @ A_27b @ A_27c ) ) ) ) ).
thf(tp_c_2Efcp_2EFCP__TL,type,
c_2Efcp_2EFCP__TL: del > del > del > $i ).
thf(mem_c_2Efcp_2EFCP__TL,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efcp_2EFCP__TL @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Efcp_2EFCP__ZIP,type,
c_2Efcp_2EFCP__ZIP: del > del > del > $i ).
thf(mem_c_2Efcp_2EFCP__ZIP,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efcp_2EFCP__ZIP @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27c @ A_27b ) @ ( ty_2Efcp_2Ecart @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b ) ) ) ) ).
thf(tp_c_2Efcp_2EHAS__SIZE,type,
c_2Efcp_2EHAS__SIZE: del > $i ).
thf(mem_c_2Efcp_2EHAS__SIZE,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2EHAS__SIZE @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ).
thf(tp_c_2Efcp_2EL2V,type,
c_2Efcp_2EL2V: del > del > $i ).
thf(mem_c_2Efcp_2EL2V,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2EL2V @ A_27a @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Efcp_2EV2L,type,
c_2Efcp_2EV2L: del > del > $i ).
thf(mem_c_2Efcp_2EV2L,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ).
thf(tp_c_2Efcp_2Ebit0__CASE,type,
c_2Efcp_2Ebit0__CASE: del > del > $i ).
thf(mem_c_2Efcp_2Ebit0__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ebit0 @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ A_27b ) ) ) ) ).
thf(tp_c_2Efcp_2Ebit0__size,type,
c_2Efcp_2Ebit0__size: del > $i ).
thf(mem_c_2Efcp_2Ebit0__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2Ebit0__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Efcp_2Ebit0 @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Efcp_2Ebit1__CASE,type,
c_2Efcp_2Ebit1__CASE: del > del > $i ).
thf(mem_c_2Efcp_2Ebit1__CASE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ebit1 @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27b @ A_27b ) ) ) ) ) ).
thf(tp_c_2Efcp_2Ebit1__size,type,
c_2Efcp_2Ebit1__size: del > $i ).
thf(mem_c_2Efcp_2Ebit1__size,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2Ebit1__size @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ ( ty_2Efcp_2Ebit1 @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2Efcp_2Edest__cart,type,
c_2Efcp_2Edest__cart: del > del > $i ).
thf(mem_c_2Efcp_2Edest__cart,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2Edest__cart @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) ) ) ).
thf(tp_c_2Efcp_2Edest__finite__image,type,
c_2Efcp_2Edest__finite__image: del > $i ).
thf(mem_c_2Efcp_2Edest__finite__image,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2Edest__finite__image @ A_27a ) @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27a ) @ A_27a ) ) ).
thf(tp_c_2Efcp_2Edimindex,type,
c_2Efcp_2Edimindex: del > $i ).
thf(mem_c_2Efcp_2Edimindex,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( arr @ ( ty_2Ebool_2Eitself @ A_27a ) @ ty_2Enum_2Enum ) ) ).
thf(tp_c_2Efcp_2Efcp__CASE,type,
c_2Efcp_2Efcp__CASE: del > del > del > $i ).
thf(mem_c_2Efcp_2Efcp__CASE,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2Efcp_2Efcp__CASE @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) @ A_27c ) @ A_27c ) ) ) ).
thf(tp_c_2Efcp_2Efcp__index,type,
c_2Efcp_2Efcp__index: del > del > $i ).
thf(mem_c_2Efcp_2Efcp__index,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ).
thf(tp_c_2Efcp_2Efinite__index,type,
c_2Efcp_2Efinite__index: del > $i ).
thf(mem_c_2Efcp_2Efinite__index,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2Efinite__index @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ).
thf(tp_c_2Efcp_2Emk__cart,type,
c_2Efcp_2Emk__cart: del > del > $i ).
thf(mem_c_2Efcp_2Emk__cart,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b ) @ ( arr @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) ) ) ).
thf(tp_c_2Efcp_2Emk__finite__image,type,
c_2Efcp_2Emk__finite__image: del > $i ).
thf(mem_c_2Efcp_2Emk__finite__image,axiom,
! [A_27a: del] : ( mem @ ( c_2Efcp_2Emk__finite__image @ A_27a ) @ ( arr @ A_27a @ ( ty_2Efcp_2Efinite__image @ A_27a ) ) ) ).
thf(ax_thm_2Efcp_2EHAS__SIZE__def,axiom,
! [A_27a: del,V0s: $i] :
( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Efcp_2EHAS__SIZE @ A_27a ) @ V0s ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ V0s ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ V0s ) )
= V1n ) ) ) ) ).
thf(ax_thm_2Efcp_2Efinite__image__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27a ) @ A_27a ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ ( ty_2Efcp_2Efinite__image @ A_27a ) )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) @ ( c_2Ebool_2EARB @ A_27a ) ) ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Efcp_2Efinite__image__tybij,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Efcp_2Efinite__image @ A_27a ) )
=> ( ( ap @ ( c_2Efcp_2Emk__finite__image @ A_27a ) @ ( ap @ ( c_2Efcp_2Edest__finite__image @ A_27a ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ A_27a )
=> ( ( p
@ ( ap
@ ( lam @ A_27a
@ ^ [V2x: $i] : ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V2x ) @ ( c_2Ebool_2EARB @ A_27a ) ) ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) )
@ V1r ) )
<=> ( ( ap @ ( c_2Efcp_2Edest__finite__image @ A_27a ) @ ( ap @ ( c_2Efcp_2Emk__finite__image @ A_27a ) @ V1r ) )
= V1r ) ) ) ) ).
thf(ax_thm_2Efcp_2Edimindex__def,axiom,
! [A_27a: del] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2ENOT__FINITE__IMP__dimindex__1,axiom,
! [A_27a: del] :
( ~ ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EDIMINDEX__GE__1,axiom,
! [A_27a: del] : ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ) ).
thf(ax_thm_2Efcp_2Efinite__index__def,axiom,
! [A_27a: del] :
( ( c_2Efcp_2Efinite__index @ A_27a )
= ( ap @ ( c_2Emin_2E_40 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
@ ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
@ ^ [V0f: $i] :
( ap @ ( c_2Ebool_2E_21 @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( ap @ ( c_2Ebool_2E_3F_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2n: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V2n ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V0f @ V2n ) ) @ V1x ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2Ecart__TY__DEF,axiom,
! [A_27a: del,A_27b: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
@ ( lam @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a )
@ ^ [V1f: $i] : c_2Ebool_2ET ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Efcp_2Ecart__tybij,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2Edest__cart @ A_27a @ A_27b ) @ V0a ) )
= V0a ) )
& ! [V1r: $i] :
( ( mem @ V1r @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) )
=> ( ( p
@ ( ap
@ ( lam @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a )
@ ^ [V2f: $i] : c_2Ebool_2ET )
@ V1r ) )
<=> ( ( ap @ ( c_2Efcp_2Edest__cart @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b ) @ V1r ) )
= V1r ) ) ) ) ).
thf(ax_thm_2Efcp_2Efcp__index,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0x ) @ ( inj__ty_2Enum_2Enum @ V1i ) )
= ( ap @ ( ap @ ( c_2Efcp_2Edest__cart @ A_27a @ A_27b ) @ V0x ) @ ( ap @ ( c_2Efcp_2Efinite__index @ ( ty_2Efcp_2Efinite__image @ A_27b ) ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Efcp__Axiom,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) @ A_27c ) )
=> ? [V1g: $i] :
( ( mem @ V1g @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ A_27c ) )
& ! [V2h: $i] :
( ( mem @ V2h @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) )
=> ( ( ap @ V1g @ ( ap @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b ) @ V2h ) )
= ( ap @ V0f @ V2h ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Efcp__ind,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ bool ) )
=> ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b ) @ V1f ) ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( p @ ( ap @ V0P @ V2a ) ) ) ) ) ).
thf(ax_thm_2Efcp_2Efcp__case__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ( arr @ ( ty_2Efcp_2Efinite__image @ A_27b ) @ A_27a ) @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2Efcp__CASE @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b ) @ V0h ) ) @ V1f )
= ( ap @ V1f @ V0h ) ) ) ) ).
thf(conj_thm_2Efcp_2ECART__EQ,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1y: $i] :
( ( mem @ V1y @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( V0x = V1y )
<=> ! [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0x ) @ ( inj__ty_2Enum_2Enum @ V2i ) )
= ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V1y ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2Efcp_2EFCP @ A_27a @ A_27b )
= ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
@ ^ [V0g: $i] :
( ap @ ( c_2Emin_2E_40 @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
@ ( lam @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b )
@ ^ [V1f: $i] :
( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V2i ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V1f ) @ V2i ) ) @ ( ap @ V0g @ V2i ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__BETA,axiom,
! [A_27a: del,A_27b: del,V0g: $i] :
( ( mem @ V0g @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b ) @ V0g ) ) @ ( inj__ty_2Enum_2Enum @ V1i ) )
= ( ap @ V0g @ ( inj__ty_2Enum_2Enum @ V1i ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__UNIQUE,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ! [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V2i ) )
= ( ap @ V1g @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) )
<=> ( ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b ) @ V1g )
= V0f ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__ETA,axiom,
! [A_27a: del,A_27b: del,V0g: $i] :
( ( mem @ V0g @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] : ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0g ) @ V1i ) ) )
= V0g ) ) ).
thf(conj_thm_2Efcp_2Ecard__dimindex,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Epred__set_2ECARD @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Eindex__sum,axiom,
! [A_27a: del,A_27b: del] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Efinite__sum,axiom,
! [A_27a: del,A_27b: del] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Esum_2Esum @ A_27a @ A_27b ) ) ) )
<=> ( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
& ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27b ) @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ) ).
thf(ax_thm_2Efcp_2Ebit0__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Efcp_2Ebit0 @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ ( ty_2Efcp_2Ebit0 @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V1a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool )
@ ^ [V2_27bit0_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V3a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0 )
@ ( ap
@ ( lam @ A_27a
@ ^ [V5a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V5a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) )
@ V4a ) ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V7a: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0 )
@ ( ap
@ ( lam @ A_27a
@ ^ [V8a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V8a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) )
@ V7a ) ) ) ) ) )
@ ( ap @ V2_27bit0_27 @ V3a0 ) ) ) ) )
@ ( ap @ V2_27bit0_27 @ V1a0 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Efcp_2Ebit0__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V0a ) ) @ V1f ) @ V2f1 )
= ( ap @ V1f @ V0a ) ) ) ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V3a ) ) @ V4f ) @ V5f1 )
= ( ap @ V5f1 @ V3a ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2Ebit0__size__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Efcp_2Ebit0__size @ A_27a ) @ V0f ) @ ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V1a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ V0f @ V1a ) ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Efcp_2Ebit0__size @ A_27a ) @ V2f ) @ ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V3a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ V2f @ V3a ) ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Edatatype__bit0,axiom,
! [A_27a: del,V0bit0: $i] :
( ( mem @ V0bit0 @ ( arr @ ( arr @ A_27a @ ( ty_2Efcp_2Ebit0 @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2Efcp_2Ebit0 @ A_27a ) ) @ bool ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ V0bit0 @ ( c_2Efcp_2EBIT0A @ A_27a ) ) @ ( c_2Efcp_2EBIT0B @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit0__11,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a_27: $i] :
( ( mem @ V1a_27 @ A_27a )
=> ( ( ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V0a )
= ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V1a_27 ) )
<=> ( V0a = V1a_27 ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3a_27: $i] :
( ( mem @ V3a_27 @ A_27a )
=> ( ( ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V2a )
= ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V3a_27 ) )
<=> ( V2a = V3a_27 ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit0__distinct,axiom,
! [A_27a: del,V0a_27: $i] :
( ( mem @ V0a_27 @ A_27a )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V1a )
!= ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V0a_27 ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit0__nchotomy,axiom,
! [A_27a: del,V0bb: $i] :
( ( mem @ V0bb @ ( ty_2Efcp_2Ebit0 @ A_27a ) )
=> ( ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ( V0bb
= ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V1a ) ) )
| ? [V2a: $i] :
( ( mem @ V2a @ A_27a )
& ( V0bb
= ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V2a ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit0__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f0: $i] :
( ( mem @ V0f0 @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ A_27b ) )
=> ? [V2fn: $i] :
( ( mem @ V2fn @ ( arr @ ( ty_2Efcp_2Ebit0 @ A_27a ) @ A_27b ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( ap @ V2fn @ ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V3a ) )
= ( ap @ V0f0 @ V3a ) ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ( ( ap @ V2fn @ ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V4a ) )
= ( ap @ V1f1 @ V4a ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit0__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Efcp_2Ebit0 @ A_27a ) @ bool ) )
=> ( ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V1a ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V2a ) ) ) ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Efcp_2Ebit0 @ A_27a ) )
=> ( p @ ( ap @ V0P @ V3b ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit0__case__cong,axiom,
! [A_27a: del,A_27b: del,V0f_27: $i] :
( ( mem @ V0f_27 @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f1_27: $i] :
( ( mem @ V1f1_27 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2M: $i] :
( ( mem @ V2M @ ( ty_2Efcp_2Ebit0 @ A_27a ) )
=> ! [V3M_27: $i] :
( ( mem @ V3M_27 @ ( ty_2Efcp_2Ebit0 @ A_27a ) )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27b ) )
=> ! [V5f1: $i] :
( ( mem @ V5f1 @ ( arr @ A_27a @ A_27b ) )
=> ( ( ( V2M = V3M_27 )
& ! [V6a: $i] :
( ( mem @ V6a @ A_27a )
=> ( ( V3M_27
= ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V6a ) )
=> ( ( ap @ V4f @ V6a )
= ( ap @ V0f_27 @ V6a ) ) ) )
& ! [V7a: $i] :
( ( mem @ V7a @ A_27a )
=> ( ( V3M_27
= ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V7a ) )
=> ( ( ap @ V5f1 @ V7a )
= ( ap @ V1f1_27 @ V7a ) ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b ) @ V2M ) @ V4f ) @ V5f1 )
= ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b ) @ V3M_27 ) @ V0f_27 ) @ V1f1_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit0__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Efcp_2Ebit0 @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b ) @ V0x ) @ V1f ) @ V2f1 )
= V3v )
<=> ( ? [V4a: $i] :
( ( mem @ V4a @ A_27a )
& ( V0x
= ( ap @ ( c_2Efcp_2EBIT0A @ A_27a ) @ V4a ) )
& ( ( ap @ V1f @ V4a )
= V3v ) )
| ? [V5a: $i] :
( ( mem @ V5a @ A_27a )
& ( V0x
= ( ap @ ( c_2Efcp_2EBIT0B @ A_27a ) @ V5a ) )
& ( ( ap @ V2f1 @ V5a )
= V3v ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Eindex__bit0,axiom,
! [A_27a: del] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ ( ty_2Efcp_2Ebit0 @ A_27a ) ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Efcp_2Ebit0 @ A_27a ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Efinite__bit0,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Efcp_2Ebit0 @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Efcp_2Ebit0 @ A_27a ) ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(ax_thm_2Efcp_2Ebit1__TY__DEF,axiom,
! [A_27a: del] :
? [V0rep: $i] :
( ( mem @ V0rep @ ( arr @ ( ty_2Efcp_2Ebit1 @ A_27a ) @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) )
& ( p
@ ( ap
@ ( ap @ ( c_2Ebool_2ETYPE__DEFINITION @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ ( ty_2Efcp_2Ebit1 @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V1a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool ) )
@ ( lam @ ( arr @ ( ty_2Eind__type_2Erecspace @ A_27a ) @ bool )
@ ^ [V2_27bit1_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Eind__type_2Erecspace @ A_27a ) )
@ ( lam @ ( ty_2Eind__type_2Erecspace @ A_27a )
@ ^ [V3a0: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V4a: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0 )
@ ( ap
@ ( lam @ A_27a
@ ^ [V5a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V5a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V6n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) )
@ V4a ) ) ) ) )
@ ( ap
@ ( ap @ c_2Ebool_2E_5C_2F
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V7a: $i] :
( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0 )
@ ( ap
@ ( lam @ A_27a
@ ^ [V8a: $i] :
( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V8a )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V9n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) )
@ V7a ) ) ) ) )
@ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Eind__type_2Erecspace @ A_27a ) ) @ V3a0 )
@ ( ap @ ( ap @ ( ap @ ( c_2Eind__type_2ECONSTR @ A_27a ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) @ ( c_2Ebool_2EARB @ A_27a ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V10n: $i] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) ) ) ) )
@ ( ap @ V2_27bit1_27 @ V3a0 ) ) ) ) )
@ ( ap @ V2_27bit1_27 @ V1a0 ) ) ) ) ) )
@ V0rep ) ) ) ).
thf(ax_thm_2Efcp_2Ebit1__case__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V0a ) ) @ V1f ) @ V2f1 ) @ V3v )
= ( ap @ V1f @ V0a ) ) ) ) ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ A_27b ) )
=> ! [V6f1: $i] :
( ( mem @ V6f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V7v: $i] :
( ( mem @ V7v @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V4a ) ) @ V5f ) @ V6f1 ) @ V7v )
= ( ap @ V6f1 @ V4a ) ) ) ) ) )
& ! [V8f: $i] :
( ( mem @ V8f @ ( arr @ A_27a @ A_27b ) )
=> ! [V9f1: $i] :
( ( mem @ V9f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V10v: $i] :
( ( mem @ V10v @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b ) @ ( c_2Efcp_2EBIT1C @ A_27a ) ) @ V8f ) @ V9f1 ) @ V10v )
= V10v ) ) ) ) ) ).
thf(ax_thm_2Efcp_2Ebit1__size__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__size @ A_27a ) @ V0f ) @ ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V1a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ V0f @ V1a ) ) ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__size @ A_27a ) @ V2f ) @ ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V3a ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ V2f @ V3a ) ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__size @ A_27a ) @ V4f ) @ ( c_2Efcp_2EBIT1C @ A_27a ) ) )
= fo__c_2Enum_2E0 ) ) ) ).
thf(conj_thm_2Efcp_2Edatatype__bit1,axiom,
! [A_27a: del,V0bit1: $i] :
( ( mem @ V0bit1 @ ( arr @ ( arr @ A_27a @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) @ ( arr @ ( arr @ A_27a @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) @ ( arr @ ( ty_2Efcp_2Ebit1 @ A_27a ) @ bool ) ) ) )
=> ( p @ ( ap @ ( c_2Ebool_2EDATATYPE @ bool ) @ ( ap @ ( ap @ ( ap @ V0bit1 @ ( c_2Efcp_2EBIT1A @ A_27a ) ) @ ( c_2Efcp_2EBIT1B @ A_27a ) ) @ ( c_2Efcp_2EBIT1C @ A_27a ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit1__11,axiom,
! [A_27a: del] :
( ! [V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1a_27: $i] :
( ( mem @ V1a_27 @ A_27a )
=> ( ( ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V0a )
= ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V1a_27 ) )
<=> ( V0a = V1a_27 ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ! [V3a_27: $i] :
( ( mem @ V3a_27 @ A_27a )
=> ( ( ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V2a )
= ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V3a_27 ) )
<=> ( V2a = V3a_27 ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit1__distinct,axiom,
! [A_27a: del] :
( ! [V0a_27: $i] :
( ( mem @ V0a_27 @ A_27a )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V1a )
!= ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V0a_27 ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V2a )
!= ( c_2Efcp_2EBIT1C @ A_27a ) ) )
& ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ( ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V3a )
!= ( c_2Efcp_2EBIT1C @ A_27a ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit1__nchotomy,axiom,
! [A_27a: del,V0bb: $i] :
( ( mem @ V0bb @ ( ty_2Efcp_2Ebit1 @ A_27a ) )
=> ( ? [V1a: $i] :
( ( mem @ V1a @ A_27a )
& ( V0bb
= ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V1a ) ) )
| ? [V2a: $i] :
( ( mem @ V2a @ A_27a )
& ( V0bb
= ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V2a ) ) )
| ( V0bb
= ( c_2Efcp_2EBIT1C @ A_27a ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit1__Axiom,axiom,
! [A_27a: del,A_27b: del,V0f0: $i] :
( ( mem @ V0f0 @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ A_27b )
=> ? [V3fn: $i] :
( ( mem @ V3fn @ ( arr @ ( ty_2Efcp_2Ebit1 @ A_27a ) @ A_27b ) )
& ! [V4a: $i] :
( ( mem @ V4a @ A_27a )
=> ( ( ap @ V3fn @ ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V4a ) )
= ( ap @ V0f0 @ V4a ) ) )
& ! [V5a: $i] :
( ( mem @ V5a @ A_27a )
=> ( ( ap @ V3fn @ ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V5a ) )
= ( ap @ V1f1 @ V5a ) ) )
& ( ( ap @ V3fn @ ( c_2Efcp_2EBIT1C @ A_27a ) )
= V2f2 ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit1__induction,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( ty_2Efcp_2Ebit1 @ A_27a ) @ bool ) )
=> ( ( ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V1a ) ) ) )
& ! [V2a: $i] :
( ( mem @ V2a @ A_27a )
=> ( p @ ( ap @ V0P @ ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V2a ) ) ) )
& ( p @ ( ap @ V0P @ ( c_2Efcp_2EBIT1C @ A_27a ) ) ) )
=> ! [V3b: $i] :
( ( mem @ V3b @ ( ty_2Efcp_2Ebit1 @ A_27a ) )
=> ( p @ ( ap @ V0P @ V3b ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit1__case__cong,axiom,
! [A_27a: del,A_27b: del,V0f_27: $i] :
( ( mem @ V0f_27 @ ( arr @ A_27a @ A_27b ) )
=> ! [V1f1_27: $i] :
( ( mem @ V1f1_27 @ ( arr @ A_27a @ A_27b ) )
=> ! [V2v_27: $i] :
( ( mem @ V2v_27 @ A_27b )
=> ! [V3M: $i] :
( ( mem @ V3M @ ( ty_2Efcp_2Ebit1 @ A_27a ) )
=> ! [V4M_27: $i] :
( ( mem @ V4M_27 @ ( ty_2Efcp_2Ebit1 @ A_27a ) )
=> ! [V5f: $i] :
( ( mem @ V5f @ ( arr @ A_27a @ A_27b ) )
=> ! [V6f1: $i] :
( ( mem @ V6f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V7v: $i] :
( ( mem @ V7v @ A_27b )
=> ( ( ( V3M = V4M_27 )
& ! [V8a: $i] :
( ( mem @ V8a @ A_27a )
=> ( ( V4M_27
= ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V8a ) )
=> ( ( ap @ V5f @ V8a )
= ( ap @ V0f_27 @ V8a ) ) ) )
& ! [V9a: $i] :
( ( mem @ V9a @ A_27a )
=> ( ( V4M_27
= ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V9a ) )
=> ( ( ap @ V6f1 @ V9a )
= ( ap @ V1f1_27 @ V9a ) ) ) )
& ( ( V4M_27
= ( c_2Efcp_2EBIT1C @ A_27a ) )
=> ( V7v = V2v_27 ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b ) @ V3M ) @ V5f ) @ V6f1 ) @ V7v )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b ) @ V4M_27 ) @ V0f_27 ) @ V1f1_27 ) @ V2v_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Ebit1__case__eq,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Efcp_2Ebit1 @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27b ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ A_27a @ A_27b ) )
=> ! [V3v: $i] :
( ( mem @ V3v @ A_27b )
=> ! [V4v_27: $i] :
( ( mem @ V4v_27 @ A_27b )
=> ( ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b ) @ V0x ) @ V1f ) @ V2f1 ) @ V3v )
= V4v_27 )
<=> ( ? [V5a: $i] :
( ( mem @ V5a @ A_27a )
& ( V0x
= ( ap @ ( c_2Efcp_2EBIT1A @ A_27a ) @ V5a ) )
& ( ( ap @ V1f @ V5a )
= V4v_27 ) )
| ? [V6a: $i] :
( ( mem @ V6a @ A_27a )
& ( V0x
= ( ap @ ( c_2Efcp_2EBIT1B @ A_27a ) @ V6a ) )
& ( ( ap @ V2f1 @ V6a )
= V4v_27 ) )
| ( ( V0x
= ( c_2Efcp_2EBIT1C @ A_27a ) )
& ( V3v = V4v_27 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Eindex__bit1,axiom,
! [A_27a: del] :
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) @ ( c_2Ebool_2Ethe__value @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( ap @ c_2Earithmetic_2E_2A @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT2 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Efinite__bit1,axiom,
! [A_27a: del] :
( ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) @ ( c_2Epred__set_2EUNIV @ ( ty_2Efcp_2Ebit1 @ A_27a ) ) ) )
<=> ( p @ ( ap @ ( c_2Epred__set_2EFINITE @ A_27a ) @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ) ).
thf(conj_thm_2Efcp_2Eindex__one,axiom,
( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ ty_2Eone_2Eone ) @ ( c_2Ebool_2Ethe__value @ ty_2Eone_2Eone ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Efinite__one,axiom,
p @ ( ap @ ( c_2Epred__set_2EFINITE @ ty_2Eone_2Eone ) @ ( c_2Epred__set_2EUNIV @ ty_2Eone_2Eone ) ) ).
thf(ax_thm_2Efcp_2EFCP__UPDATE__def,axiom,
! [A_27a: del,A_27b: del,V0a: tp__ty_2Enum_2Enum,V1b: $i] :
( ( mem @ V1b @ A_27a )
=> ( ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ V1b )
= ( lam @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b )
@ ^ [V2m: $i] :
( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V3c: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0a ) ) @ V3c ) ) @ V1b ) @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V2m ) @ V3c ) ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__UPDATE__COMMUTES,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1a: tp__ty_2Enum_2Enum,V2b: tp__ty_2Enum_2Enum,V3c: $i] :
( ( mem @ V3c @ A_27a )
=> ! [V4d: $i] :
( ( mem @ V4d @ A_27a )
=> ( ( V1a != V2b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V3c ) @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) @ V4d ) @ V0m ) )
= ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2b ) ) @ V4d ) @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V3c ) @ V0m ) ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__UPDATE__EQ,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1a: tp__ty_2Enum_2Enum,V2b: $i] :
( ( mem @ V2b @ A_27a )
=> ! [V3c: $i] :
( ( mem @ V3c @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V3c ) @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V2b ) @ V0m ) )
= ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V3c ) @ V0m ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__UPDATE__IMP__ID,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1a: tp__ty_2Enum_2Enum,V2v: $i] :
( ( mem @ V2v @ A_27a )
=> ( ( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0m ) @ ( inj__ty_2Enum_2Enum @ V1a ) )
= V2v )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ V2v ) @ V0m )
= V0m ) ) ) ) ).
thf(conj_thm_2Efcp_2EAPPLY__FCP__UPDATE__ID,axiom,
! [A_27a: del,A_27b: del,V0m: $i] :
( ( mem @ V0m @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1a: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0m ) @ ( inj__ty_2Enum_2Enum @ V1a ) ) ) @ V0m )
= V0m ) ) ).
thf(conj_thm_2Efcp_2EFCP__APPLY__UPDATE__THM,axiom,
! [A_27a: del,A_27b: del,V0index_20out_20of_20range: $i] :
( ( mem @ V0index_20out_20of_20range @ bool )
=> ! [V1m: $i] :
( ( mem @ V1m @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V2a: tp__ty_2Enum_2Enum,V3w: $i] :
( ( mem @ V3w @ A_27a )
=> ! [V4b: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) @ V3w ) @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V4b ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4b ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) @ ( inj__ty_2Enum_2Enum @ V4b ) ) ) @ V3w ) @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V1m ) @ ( inj__ty_2Enum_2Enum @ V4b ) ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EFAIL @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) ) @ V0index_20out_20of_20range ) @ ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V2a ) ) @ V3w ) @ V1m ) ) @ ( inj__ty_2Enum_2Enum @ V4b ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__HD__def,axiom,
! [A_27a: del,A_27b: del,V0v: $i] :
( ( mem @ V0v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efcp_2EFCP__HD @ A_27a @ A_27b ) @ V0v )
= ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0v ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__TL__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0v: $i] :
( ( mem @ V0v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) )
=> ( ( ap @ ( c_2Efcp_2EFCP__TL @ A_27a @ A_27b @ A_27c ) @ V0v )
= ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] : ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27c ) @ V0v ) @ ( ap @ c_2Enum_2ESUC @ V1i ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__CONS__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0h: $i] :
( ( mem @ V0h @ A_27a )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2EFCP__CONS @ A_27a @ A_27b @ A_27c ) @ V0h ) @ V1v )
= ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27c ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V0h )
@ ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27c )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V1v ) @ ( ap @ c_2Eprim__rec_2EPRE @ V2i ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__MAP__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2EFCP__MAP @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1v )
= ( ap @ ( c_2Efcp_2EFCP @ A_27b @ A_27c )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ V0f @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27c ) @ V1v ) @ V2i ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__EXISTS__def,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27b @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efcp_2EFCP__EXISTS @ A_27a @ A_27b ) @ V0P ) @ V1v ) )
<=> ? [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) )
& ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27b @ A_27a ) @ V1v ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__EVERY__def,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27b @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27b @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efcp_2EFCP__EVERY @ A_27a @ A_27b ) @ V0P ) @ V1v ) )
<=> ! [V2i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( ap @ ( c_2Efcp_2Edimindex @ A_27a ) @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) )
| ( p @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27b @ A_27a ) @ V1v ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__CONCAT__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2EFCP__CONCAT @ A_27a @ A_27b @ A_27c ) @ V0a ) @ V1b )
= ( ap @ ( c_2Efcp_2EFCP @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ A_27c ) )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ V2i ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27c ) @ ( c_2Ebool_2Ethe__value @ A_27c ) ) ) ) @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27c ) @ V1b ) @ V2i ) ) @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ V2i ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27c ) @ ( c_2Ebool_2Ethe__value @ A_27c ) ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__ZIP__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0a: $i] :
( ( mem @ V0a @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ! [V1b: $i] :
( ( mem @ V1b @ ( ty_2Efcp_2Ecart @ A_27c @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2EFCP__ZIP @ A_27a @ A_27b @ A_27c ) @ V0a ) @ V1b )
= ( ap @ ( c_2Efcp_2EFCP @ ( ty_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V2i: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ A_27c ) @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0a ) @ V2i ) ) @ ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27c @ A_27b ) @ V1b ) @ V2i ) ) ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EV2L__def,axiom,
! [A_27a: del,A_27b: del,V0v: $i] :
( ( mem @ V0v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V0v )
= ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0v ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EL2V__def,axiom,
! [A_27a: del,A_27b: del,V0L: $i] :
( ( mem @ V0L @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( c_2Efcp_2EL2V @ A_27a @ A_27b ) @ V0L )
= ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V1i: $i] : ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ V1i ) @ V0L ) ) ) ) ) ).
thf(ax_thm_2Efcp_2EFCP__FOLD__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27b @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V1i: $i] :
( ( mem @ V1i @ A_27b )
=> ! [V2v: $i] :
( ( mem @ V2v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2EFCP__FOLD @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1i ) @ V2v )
= ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDL @ A_27a @ A_27b ) @ V0f ) @ V1i ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27c ) @ V2v ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2ELENGTH__V2L,axiom,
! [A_27a: del,A_27b: del,V0v: $i] :
( ( mem @ V0v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V0v ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EEL__V2L,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V1v ) )
= ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V1v ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__MAP,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2EFCP__MAP @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1v )
= ( ap @ ( c_2Efcp_2EL2V @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27c ) @ V1v ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__TL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0v: $i] :
( ( mem @ V0v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) )
& ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ A_27c ) @ ( c_2Ebool_2Ethe__value @ A_27c ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) )
=> ( ( ap @ ( c_2Efcp_2EFCP__TL @ A_27a @ A_27c @ A_27b ) @ V0v )
= ( ap @ ( c_2Efcp_2EL2V @ A_27a @ A_27c ) @ ( ap @ ( c_2Elist_2ETL @ A_27a ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V0v ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__EXISTS,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efcp_2EFCP__EXISTS @ A_27b @ A_27a ) @ V0P ) @ V1v ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEXISTS @ A_27a ) @ V0P ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V1v ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__EVERY,axiom,
! [A_27a: del,A_27b: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Efcp_2EFCP__EVERY @ A_27b @ A_27a ) @ V0P ) @ V1v ) )
<=> ( p @ ( ap @ ( ap @ ( c_2Elist_2EEVERY @ A_27a ) @ V0P ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V1v ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__HD,axiom,
! [A_27a: del,A_27b: del,V0v: $i] :
( ( mem @ V0v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( ap @ ( c_2Efcp_2EFCP__HD @ A_27a @ A_27b ) @ V0v )
= ( ap @ ( c_2Elist_2EHD @ A_27a ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V0v ) ) ) ) ).
thf(conj_thm_2Efcp_2EFCP__CONS,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27a )
=> ! [V1v: $i] :
( ( mem @ V1v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2EFCP__CONS @ A_27a @ A_27b @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ V0a ) @ V1v )
= ( ap @ ( c_2Efcp_2EL2V @ A_27a @ ( ty_2Esum_2Esum @ A_27b @ ty_2Eone_2Eone ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V0a ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V1v ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EV2L__L2V,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0x ) ) )
=> ( ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EL2V @ A_27a @ A_27b ) @ V0x ) )
= V0x ) ) ) ).
thf(conj_thm_2Efcp_2ENULL__V2L,axiom,
! [A_27a: del,A_27b: del,V0v: $i] :
( ( mem @ V0v @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) )
=> ~ ( p @ ( ap @ ( c_2Elist_2ENULL @ A_27a ) @ ( ap @ ( c_2Efcp_2EV2L @ A_27a @ A_27b ) @ V0v ) ) ) ) ).
thf(conj_thm_2Efcp_2EREAD__TL,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0i: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ ( ty_2Efcp_2Ecart @ A_27a @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EFCP__TL @ A_27a @ A_27b @ A_27c ) @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V0i ) )
= ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27c ) @ V1a ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0i ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2EREAD__L2V,axiom,
! [A_27a: del,A_27b: del,V0i: tp__ty_2Enum_2Enum,V1a: $i] :
( ( mem @ V1a @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EL2V @ A_27a @ A_27b ) @ V1a ) ) @ ( inj__ty_2Enum_2Enum @ V0i ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ V1a ) ) ) ) ).
thf(conj_thm_2Efcp_2Eindex__comp,axiom,
! [A_27a: del,A_27b: del,V0FCP_20out_20of_20bounds: $i] :
( ( mem @ V0FCP_20out_20of_20bounds @ bool )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2n: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b ) @ V1f ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n ) ) @ ( ap @ ( c_2Efcp_2Edimindex @ A_27b ) @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Ecombin_2EFAIL @ ( arr @ ( ty_2Efcp_2Ecart @ A_27a @ A_27b ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) @ bool ) @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) ) @ V0FCP_20out_20of_20bounds ) @ ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b ) @ V1f ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).
thf(conj_thm_2Efcp_2Efcp__subst__comp,axiom,
! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0x: tp__ty_2Enum_2Enum,V1y: $i] :
( ( mem @ V1y @ A_27a )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27c )
=> ! [V3b: $i] :
( ( mem @ V3b @ A_27d )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V1y ) @ ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b ) @ V4f ) )
= ( ap @ ( c_2Efcp_2EFCP @ A_27a @ A_27b )
@ ( lam @ ty_2Enum_2Enum
@ ^ [V5c: $i] : ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ A_27a ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0x ) ) @ V5c ) ) @ V1y ) @ ( ap @ V4f @ V5c ) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------