ITP001 Axioms: ITP047^7.ax
%------------------------------------------------------------------------------
% File : ITP047^7 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 syntactic export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : fcp.ax [Gau19]
% : HL4047^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 176 ( 42 unt; 89 typ; 0 def)
% Number of atoms : 234 ( 116 equ; 8 cnn)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 1488 ( 8 ~; 8 |; 33 &;1395 @)
% ( 21 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 7 avg;1395 nst)
% Number of types : 4 ( 3 usr)
% Number of type conns : 247 ( 247 >; 0 *; 0 +; 0 <<)
% Number of symbols : 88 ( 86 usr; 4 con; 0-6 aty)
% Number of variables : 484 ( 39 ^ 338 !; 17 ?; 484 :)
% ( 90 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Ebool_2Eitself,type,
tyop_2Ebool_2Eitself: $tType > $tType ).
thf(tyop_2Efcp_2Ebit0,type,
tyop_2Efcp_2Ebit0: $tType > $tType ).
thf(tyop_2Efcp_2Ebit1,type,
tyop_2Efcp_2Ebit1: $tType > $tType ).
thf(tyop_2Efcp_2Ecart,type,
tyop_2Efcp_2Ecart: $tType > $tType > $tType ).
thf(tyop_2Efcp_2Efinite__image,type,
tyop_2Efcp_2Efinite__image: $tType > $tType ).
thf(tyop_2Eind__type_2Erecspace,type,
tyop_2Eind__type_2Erecspace: $tType > $tType ).
thf(tyop_2Elist_2Elist,type,
tyop_2Elist_2Elist: $tType > $tType ).
thf(tyop_2Emin_2Ebool,type,
tyop_2Emin_2Ebool: $tType ).
thf(tyop_2Emin_2Efun,type,
tyop_2Emin_2Efun: $tType > $tType > $tType ).
thf(tyop_2Enum_2Enum,type,
tyop_2Enum_2Enum: $tType ).
thf(tyop_2Eone_2Eone,type,
tyop_2Eone_2Eone: $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $tType > $tType > $tType ).
thf(c_2Ebool_2E_21,type,
c_2Ebool_2E_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Earithmetic_2E_2A,type,
c_2Earithmetic_2E_2A: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Earithmetic_2E_2B,type,
c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Epair_2E_2C,type,
c_2Epair_2E_2C:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).
thf(c_2Earithmetic_2E_2D,type,
c_2Earithmetic_2E_2D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_2F_5C,type,
c_2Ebool_2E_2F_5C: $o > $o > $o ).
thf(c_2Enum_2E0,type,
c_2Enum_2E0: tyop_2Enum_2Enum ).
thf(c_2Efcp_2E_3A_2B,type,
c_2Efcp_2E_3A_2B:
!>[A_27a: $tType,A_27b: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) ) ).
thf(c_2Eprim__rec_2E_3C,type,
c_2Eprim__rec_2E_3C: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Earithmetic_2E_3C_3D,type,
c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).
thf(c_2Emin_2E_3D,type,
c_2Emin_2E_3D:
!>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).
thf(c_2Emin_2E_3D_3D_3E,type,
c_2Emin_2E_3D_3D_3E: $o > $o > $o ).
thf(c_2Ebool_2E_3F,type,
c_2Ebool_2E_3F:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Ebool_2E_3F_21,type,
c_2Ebool_2E_3F_21:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Emin_2E_40,type,
c_2Emin_2E_40:
!>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a ) ).
thf(c_2Ebool_2EARB,type,
c_2Ebool_2EARB:
!>[A_27a: $tType] : A_27a ).
thf(c_2Efcp_2EBIT0A,type,
c_2Efcp_2EBIT0A:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Efcp_2Ebit0 @ A_27a ) ) ).
thf(c_2Efcp_2EBIT0B,type,
c_2Efcp_2EBIT0B:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Efcp_2Ebit0 @ A_27a ) ) ).
thf(c_2Earithmetic_2EBIT1,type,
c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Efcp_2EBIT1A,type,
c_2Efcp_2EBIT1A:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Efcp_2Ebit1 @ A_27a ) ) ).
thf(c_2Efcp_2EBIT1B,type,
c_2Efcp_2EBIT1B:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Efcp_2Ebit1 @ A_27a ) ) ).
thf(c_2Efcp_2EBIT1C,type,
c_2Efcp_2EBIT1C:
!>[A_27a: $tType] : ( tyop_2Efcp_2Ebit1 @ A_27a ) ).
thf(c_2Earithmetic_2EBIT2,type,
c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eind__type_2EBOTTOM,type,
c_2Eind__type_2EBOTTOM:
!>[A_27a: $tType] : ( tyop_2Eind__type_2Erecspace @ A_27a ) ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Eind__type_2ECONSTR,type,
c_2Eind__type_2ECONSTR:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a > ( tyop_2Enum_2Enum > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) > ( tyop_2Eind__type_2Erecspace @ A_27a ) ) ).
thf(c_2Ebool_2EDATATYPE,type,
c_2Ebool_2EDATATYPE:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Elist_2EEL,type,
c_2Elist_2EEL:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
thf(c_2Elist_2EEVERY,type,
c_2Elist_2EEVERY:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Elist_2EEXISTS,type,
c_2Elist_2EEXISTS:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Ecombin_2EFAIL,type,
c_2Ecombin_2EFAIL:
!>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > A_27a ) ).
thf(c_2Efcp_2EFCP,type,
c_2Efcp_2EFCP:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Enum_2Enum > A_27a ) > ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) ) ).
thf(c_2Efcp_2EFCP__CONCAT,type,
c_2Efcp_2EFCP__CONCAT:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > ( tyop_2Efcp_2Ecart @ A_27a @ A_27c ) > ( tyop_2Efcp_2Ecart @ A_27a @ ( tyop_2Esum_2Esum @ A_27b @ A_27c ) ) ) ).
thf(c_2Efcp_2EFCP__CONS,type,
c_2Efcp_2EFCP__CONS:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( A_27a > ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > ( tyop_2Efcp_2Ecart @ A_27a @ A_27c ) ) ).
thf(c_2Efcp_2EFCP__EVERY,type,
c_2Efcp_2EFCP__EVERY:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > $o ) > ( tyop_2Efcp_2Ecart @ A_27b @ A_27a ) > $o ) ).
thf(c_2Efcp_2EFCP__EXISTS,type,
c_2Efcp_2EFCP__EXISTS:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > $o ) > ( tyop_2Efcp_2Ecart @ A_27b @ A_27a ) > $o ) ).
thf(c_2Efcp_2EFCP__FOLD,type,
c_2Efcp_2EFCP__FOLD:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27b > A_27a > A_27b ) > A_27b > ( tyop_2Efcp_2Ecart @ A_27a @ A_27c ) > A_27b ) ).
thf(c_2Efcp_2EFCP__HD,type,
c_2Efcp_2EFCP__HD:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > A_27a ) ).
thf(c_2Efcp_2EFCP__MAP,type,
c_2Efcp_2EFCP__MAP:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Efcp_2Ecart @ A_27a @ A_27c ) > ( tyop_2Efcp_2Ecart @ A_27b @ A_27c ) ) ).
thf(c_2Efcp_2EFCP__TL,type,
c_2Efcp_2EFCP__TL:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27c ) > ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) ) ).
thf(c_2Efcp_2EFCP__ZIP,type,
c_2Efcp_2EFCP__ZIP:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > ( tyop_2Efcp_2Ecart @ A_27c @ A_27b ) > ( tyop_2Efcp_2Ecart @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b ) ) ).
thf(c_2Epred__set_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Elist_2EFOLDL,type,
c_2Elist_2EFOLDL:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > A_27a > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Elist_2EGENLIST,type,
c_2Elist_2EGENLIST:
!>[A_27a: $tType] : ( ( tyop_2Enum_2Enum > A_27a ) > tyop_2Enum_2Enum > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Efcp_2EHAS__SIZE,type,
c_2Efcp_2EHAS__SIZE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum > $o ) ).
thf(c_2Elist_2EHD,type,
c_2Elist_2EHD:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a ) ).
thf(c_2Efcp_2EL2V,type,
c_2Efcp_2EL2V:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Elist_2EMAP,type,
c_2Elist_2EMAP:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) ) ).
thf(c_2Elist_2ENULL,type,
c_2Elist_2ENULL:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Earithmetic_2ENUMERAL,type,
c_2Earithmetic_2ENUMERAL: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Eprim__rec_2EPRE,type,
c_2Eprim__rec_2EPRE: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $o ).
thf(c_2Elist_2ETL,type,
c_2Elist_2ETL:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Ebool_2ETYPE__DEFINITION,type,
c_2Ebool_2ETYPE__DEFINITION:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).
thf(c_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Efcp_2EV2L,type,
c_2Efcp_2EV2L:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Earithmetic_2EZERO,type,
c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Efcp_2Ebit0__CASE,type,
c_2Efcp_2Ebit0__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efcp_2Ebit0 @ A_27a ) > ( A_27a > A_27b ) > ( A_27a > A_27b ) > A_27b ) ).
thf(c_2Efcp_2Ebit0__size,type,
c_2Efcp_2Ebit0__size:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Efcp_2Ebit0 @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Efcp_2Ebit1__CASE,type,
c_2Efcp_2Ebit1__CASE:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efcp_2Ebit1 @ A_27a ) > ( A_27a > A_27b ) > ( A_27a > A_27b ) > A_27b > A_27b ) ).
thf(c_2Efcp_2Ebit1__size,type,
c_2Efcp_2Ebit1__size:
!>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Efcp_2Ebit1 @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Efcp_2Edest__cart,type,
c_2Efcp_2Edest__cart:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a ) ).
thf(c_2Efcp_2Edest__finite__image,type,
c_2Efcp_2Edest__finite__image:
!>[A_27a: $tType] : ( ( tyop_2Efcp_2Efinite__image @ A_27a ) > A_27a ) ).
thf(c_2Efcp_2Edimindex,type,
c_2Efcp_2Edimindex:
!>[A_27a: $tType] : ( ( tyop_2Ebool_2Eitself @ A_27a ) > tyop_2Enum_2Enum ) ).
thf(c_2Efcp_2Efcp__CASE,type,
c_2Efcp_2Efcp__CASE:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > ( ( ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a ) > A_27c ) > A_27c ) ).
thf(c_2Efcp_2Efcp__index,type,
c_2Efcp_2Efcp__index:
!>[A_27a: $tType,A_27b: $tType] : ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Efcp_2Efinite__index,type,
c_2Efcp_2Efinite__index:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > A_27a ) ).
thf(c_2Efcp_2Emk__cart,type,
c_2Efcp_2Emk__cart:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a ) > ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) ) ).
thf(c_2Efcp_2Emk__finite__image,type,
c_2Efcp_2Emk__finite__image:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Efcp_2Efinite__image @ A_27a ) ) ).
thf(c_2Ebool_2Ethe__value,type,
c_2Ebool_2Ethe__value:
!>[A_27a: $tType] : ( tyop_2Ebool_2Eitself @ A_27a ) ).
thf(c_2Ebool_2E_7E,type,
c_2Ebool_2E_7E: $o > $o ).
thf(logicdef_2E_2F_5C,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
<=> ( V0
& V1 ) ) ).
thf(logicdef_2E_5C_2F,axiom,
! [V0: $o,V1: $o] :
( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
<=> ( V0
| V1 ) ) ).
thf(logicdef_2E_7E,axiom,
! [V0: $o] :
( ( c_2Ebool_2E_7E @ V0 )
<=> ( (~) @ V0 ) ) ).
thf(logicdef_2E_3D_3D_3E,axiom,
! [V0: $o,V1: $o] :
( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
<=> ( V0
=> V1 ) ) ).
thf(logicdef_2E_3D,axiom,
! [A_27a: $tType,V0: A_27a,V1: A_27a] :
( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
<=> ( V0 = V1 ) ) ).
thf(quantdef_2E_21,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_21 @ A_27a @ V0f )
<=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(quantdef_2E_3F,axiom,
! [A_27a: $tType,V0f: A_27a > $o] :
( ( c_2Ebool_2E_3F @ A_27a @ V0f )
<=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).
thf(thm_2Efcp_2EHAS__SIZE__def,axiom,
! [A_27a: $tType,V0s: A_27a > $o,V1n: tyop_2Enum_2Enum] :
( ( c_2Efcp_2EHAS__SIZE @ A_27a @ V0s @ V1n )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ V0s )
& ( ( c_2Epred__set_2ECARD @ A_27a @ V0s )
= V1n ) ) ) ).
thf(thm_2Efcp_2Efinite__image__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Efcp_2Efinite__image @ A_27a ) > A_27a] :
( c_2Ebool_2ETYPE__DEFINITION @ A_27a @ ( tyop_2Efcp_2Efinite__image @ A_27a )
@ ^ [V1x: A_27a] : ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ A_27a @ V1x @ ( c_2Ebool_2EARB @ A_27a ) ) @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
@ V0rep ) ).
thf(thm_2Efcp_2Efinite__image__tybij,axiom,
! [A_27a: $tType] :
( ! [V0a: tyop_2Efcp_2Efinite__image @ A_27a] :
( ( c_2Efcp_2Emk__finite__image @ A_27a @ ( c_2Efcp_2Edest__finite__image @ A_27a @ V0a ) )
= V0a )
& ! [V1r: A_27a] :
( ( ^ [V2x: A_27a] : ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ A_27a @ V2x @ ( c_2Ebool_2EARB @ A_27a ) ) @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
@ V1r )
<=> ( ( c_2Efcp_2Edest__finite__image @ A_27a @ ( c_2Efcp_2Emk__finite__image @ A_27a @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Efcp_2Edimindex__def,axiom,
! [A_27a: $tType] :
( ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Efcp_2Efinite__index__def,axiom,
! [A_27a: $tType] :
( ( c_2Efcp_2Efinite__index @ A_27a )
= ( c_2Emin_2E_40 @ ( tyop_2Enum_2Enum > A_27a )
@ ^ [V0f: tyop_2Enum_2Enum > A_27a] :
( c_2Ebool_2E_21 @ A_27a
@ ^ [V1x: A_27a] :
( c_2Ebool_2E_3F_21 @ tyop_2Enum_2Enum
@ ^ [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2E_2F_5C @ ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) @ ( c_2Emin_2E_3D @ A_27a @ ( V0f @ V2n ) @ V1x ) ) ) ) ) ) ).
thf(thm_2Efcp_2Ecart__TY__DEF,axiom,
! [A_27a: $tType,A_27b: $tType] :
? [V0rep: ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a] :
( c_2Ebool_2ETYPE__DEFINITION @ ( ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a ) @ ( tyop_2Efcp_2Ecart @ A_27a @ A_27b )
@ ^ [V1f: ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a] : c_2Ebool_2ET
@ V0rep ) ).
thf(thm_2Efcp_2Ecart__tybij,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2Emk__cart @ A_27a @ A_27b @ ( c_2Efcp_2Edest__cart @ A_27a @ A_27b @ V0a ) )
= V0a )
& ! [V1r: ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a] :
( ( ^ [V2f: ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a] : c_2Ebool_2ET
@ V1r )
<=> ( ( c_2Efcp_2Edest__cart @ A_27a @ A_27b @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b @ V1r ) )
= V1r ) ) ) ).
thf(thm_2Efcp_2Efcp__index,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1i: tyop_2Enum_2Enum] :
( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0x @ V1i )
= ( c_2Efcp_2Edest__cart @ A_27a @ A_27b @ V0x @ ( c_2Efcp_2Efinite__index @ ( tyop_2Efcp_2Efinite__image @ A_27b ) @ V1i ) ) ) ).
thf(thm_2Efcp_2Efcp__case__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0h: ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a,V1f: ( ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a ) > A_27c] :
( ( c_2Efcp_2Efcp__CASE @ A_27a @ A_27b @ A_27c @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b @ V0h ) @ V1f )
= ( V1f @ V0h ) ) ).
thf(thm_2Efcp_2EFCP,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efcp_2EFCP @ A_27a @ A_27b )
= ( ^ [V0g: tyop_2Enum_2Enum > A_27a] :
( c_2Emin_2E_40 @ ( tyop_2Efcp_2Ecart @ A_27a @ A_27b )
@ ^ [V1f: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Eprim__rec_2E_3C @ V2i @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) @ ( c_2Emin_2E_3D @ A_27a @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V1f @ V2i ) @ ( V0g @ V2i ) ) ) ) ) ) ) ).
thf(thm_2Efcp_2Ebit0__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Efcp_2Ebit0 @ A_27a ) > ( tyop_2Eind__type_2Erecspace @ A_27a )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ ( tyop_2Efcp_2Ebit0 @ A_27a )
@ ^ [V1a0: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ A_27a ) > $o )
@ ^ [V2_27bit0_27: ( tyop_2Eind__type_2Erecspace @ A_27a ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V3a0: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V4a: A_27a] :
( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0
@ ( ^ [V5a: A_27a] :
( c_2Eind__type_2ECONSTR @ A_27a @ c_2Enum_2E0 @ V5a
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) )
@ V4a ) ) )
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V7a: A_27a] :
( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0
@ ( ^ [V8a: A_27a] :
( c_2Eind__type_2ECONSTR @ A_27a @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ V8a
@ ^ [V9n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) )
@ V7a ) ) ) )
@ ( V2_27bit0_27 @ V3a0 ) ) )
@ ( V2_27bit0_27 @ V1a0 ) ) )
@ V0rep ) ).
thf(thm_2Efcp_2Ebit0__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: A_27a,V1f: A_27a > A_27b,V2f1: A_27a > A_27b] :
( ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b @ ( c_2Efcp_2EBIT0A @ A_27a @ V0a ) @ V1f @ V2f1 )
= ( V1f @ V0a ) )
& ! [V3a: A_27a,V4f: A_27a > A_27b,V5f1: A_27a > A_27b] :
( ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b @ ( c_2Efcp_2EBIT0B @ A_27a @ V3a ) @ V4f @ V5f1 )
= ( V5f1 @ V3a ) ) ) ).
thf(thm_2Efcp_2Ebit0__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum,V1a: A_27a] :
( ( c_2Efcp_2Ebit0__size @ A_27a @ V0f @ ( c_2Efcp_2EBIT0A @ A_27a @ V1a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( V0f @ V1a ) ) )
& ! [V2f: A_27a > tyop_2Enum_2Enum,V3a: A_27a] :
( ( c_2Efcp_2Ebit0__size @ A_27a @ V2f @ ( c_2Efcp_2EBIT0B @ A_27a @ V3a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( V2f @ V3a ) ) ) ) ).
thf(thm_2Efcp_2Ebit1__TY__DEF,axiom,
! [A_27a: $tType] :
? [V0rep: ( tyop_2Efcp_2Ebit1 @ A_27a ) > ( tyop_2Eind__type_2Erecspace @ A_27a )] :
( c_2Ebool_2ETYPE__DEFINITION @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ ( tyop_2Efcp_2Ebit1 @ A_27a )
@ ^ [V1a0: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Ebool_2E_21 @ ( ( tyop_2Eind__type_2Erecspace @ A_27a ) > $o )
@ ^ [V2_27bit1_27: ( tyop_2Eind__type_2Erecspace @ A_27a ) > $o] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_21 @ ( tyop_2Eind__type_2Erecspace @ A_27a )
@ ^ [V3a0: tyop_2Eind__type_2Erecspace @ A_27a] :
( c_2Emin_2E_3D_3D_3E
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V4a: A_27a] :
( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0
@ ( ^ [V5a: A_27a] :
( c_2Eind__type_2ECONSTR @ A_27a @ c_2Enum_2E0 @ V5a
@ ^ [V6n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) )
@ V4a ) ) )
@ ( c_2Ebool_2E_5C_2F
@ ( c_2Ebool_2E_3F @ A_27a
@ ^ [V7a: A_27a] :
( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0
@ ( ^ [V8a: A_27a] :
( c_2Eind__type_2ECONSTR @ A_27a @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) @ V8a
@ ^ [V9n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) )
@ V7a ) ) )
@ ( c_2Emin_2E_3D @ ( tyop_2Eind__type_2Erecspace @ A_27a ) @ V3a0
@ ( c_2Eind__type_2ECONSTR @ A_27a @ ( c_2Enum_2ESUC @ ( c_2Enum_2ESUC @ c_2Enum_2E0 ) ) @ ( c_2Ebool_2EARB @ A_27a )
@ ^ [V10n: tyop_2Enum_2Enum] : ( c_2Eind__type_2EBOTTOM @ A_27a ) ) ) ) )
@ ( V2_27bit1_27 @ V3a0 ) ) )
@ ( V2_27bit1_27 @ V1a0 ) ) )
@ V0rep ) ).
thf(thm_2Efcp_2Ebit1__case__def,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ! [V0a: A_27a,V1f: A_27a > A_27b,V2f1: A_27a > A_27b,V3v: A_27b] :
( ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b @ ( c_2Efcp_2EBIT1A @ A_27a @ V0a ) @ V1f @ V2f1 @ V3v )
= ( V1f @ V0a ) )
& ! [V4a: A_27a,V5f: A_27a > A_27b,V6f1: A_27a > A_27b,V7v: A_27b] :
( ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b @ ( c_2Efcp_2EBIT1B @ A_27a @ V4a ) @ V5f @ V6f1 @ V7v )
= ( V6f1 @ V4a ) )
& ! [V8f: A_27a > A_27b,V9f1: A_27a > A_27b,V10v: A_27b] :
( ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b @ ( c_2Efcp_2EBIT1C @ A_27a ) @ V8f @ V9f1 @ V10v )
= V10v ) ) ).
thf(thm_2Efcp_2Ebit1__size__def,axiom,
! [A_27a: $tType] :
( ! [V0f: A_27a > tyop_2Enum_2Enum,V1a: A_27a] :
( ( c_2Efcp_2Ebit1__size @ A_27a @ V0f @ ( c_2Efcp_2EBIT1A @ A_27a @ V1a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( V0f @ V1a ) ) )
& ! [V2f: A_27a > tyop_2Enum_2Enum,V3a: A_27a] :
( ( c_2Efcp_2Ebit1__size @ A_27a @ V2f @ ( c_2Efcp_2EBIT1B @ A_27a @ V3a ) )
= ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( V2f @ V3a ) ) )
& ! [V4f: A_27a > tyop_2Enum_2Enum] :
( ( c_2Efcp_2Ebit1__size @ A_27a @ V4f @ ( c_2Efcp_2EBIT1C @ A_27a ) )
= c_2Enum_2E0 ) ) ).
thf(thm_2Efcp_2EFCP__UPDATE__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0a: tyop_2Enum_2Enum,V1b: A_27a] :
( ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V0a @ V1b )
= ( ^ [V2m: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( c_2Efcp_2EFCP @ A_27a @ A_27b
@ ^ [V3c: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ A_27a @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V0a @ V3c ) @ V1b @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V2m @ V3c ) ) ) ) ) ).
thf(thm_2Efcp_2EFCP__HD__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2EFCP__HD @ A_27a @ A_27b @ V0v )
= ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0v @ c_2Enum_2E0 ) ) ).
thf(thm_2Efcp_2EFCP__TL__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v: tyop_2Efcp_2Ecart @ A_27a @ A_27c] :
( ( c_2Efcp_2EFCP__TL @ A_27a @ A_27b @ A_27c @ V0v )
= ( c_2Efcp_2EFCP @ A_27a @ A_27b
@ ^ [V1i: tyop_2Enum_2Enum] : ( c_2Efcp_2Efcp__index @ A_27a @ A_27c @ V0v @ ( c_2Enum_2ESUC @ V1i ) ) ) ) ).
thf(thm_2Efcp_2EFCP__CONS__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0h: A_27a,V1v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2EFCP__CONS @ A_27a @ A_27b @ A_27c @ V0h @ V1v )
= ( c_2Efcp_2E_3A_2B @ A_27a @ A_27c @ c_2Enum_2E0 @ V0h
@ ( c_2Efcp_2EFCP @ A_27a @ A_27c
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V1v @ ( c_2Eprim__rec_2EPRE @ V2i ) ) ) ) ) ).
thf(thm_2Efcp_2EFCP__MAP__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1v: tyop_2Efcp_2Ecart @ A_27a @ A_27c] :
( ( c_2Efcp_2EFCP__MAP @ A_27a @ A_27b @ A_27c @ V0f @ V1v )
= ( c_2Efcp_2EFCP @ A_27b @ A_27c
@ ^ [V2i: tyop_2Enum_2Enum] : ( V0f @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27c @ V1v @ V2i ) ) ) ) ).
thf(thm_2Efcp_2EFCP__EXISTS__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27b > $o,V1v: tyop_2Efcp_2Ecart @ A_27b @ A_27a] :
( ( c_2Efcp_2EFCP__EXISTS @ A_27a @ A_27b @ V0P @ V1v )
<=> ? [V2i: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2i @ ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) ) )
& ( V0P @ ( c_2Efcp_2Efcp__index @ A_27b @ A_27a @ V1v @ V2i ) ) ) ) ).
thf(thm_2Efcp_2EFCP__EVERY__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27b > $o,V1v: tyop_2Efcp_2Ecart @ A_27b @ A_27a] :
( ( c_2Efcp_2EFCP__EVERY @ A_27a @ A_27b @ V0P @ V1v )
<=> ! [V2i: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) ) @ V2i )
| ( V0P @ ( c_2Efcp_2Efcp__index @ A_27b @ A_27a @ V1v @ V2i ) ) ) ) ).
thf(thm_2Efcp_2EFCP__CONCAT__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0a: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1b: tyop_2Efcp_2Ecart @ A_27a @ A_27c] :
( ( c_2Efcp_2EFCP__CONCAT @ A_27a @ A_27b @ A_27c @ V0a @ V1b )
= ( c_2Efcp_2EFCP @ A_27a @ ( tyop_2Esum_2Esum @ A_27b @ A_27c )
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ A_27a @ ( c_2Eprim__rec_2E_3C @ V2i @ ( c_2Efcp_2Edimindex @ A_27c @ ( c_2Ebool_2Ethe__value @ A_27c ) ) ) @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27c @ V1b @ V2i ) @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0a @ ( c_2Earithmetic_2E_2D @ V2i @ ( c_2Efcp_2Edimindex @ A_27c @ ( c_2Ebool_2Ethe__value @ A_27c ) ) ) ) ) ) ) ).
thf(thm_2Efcp_2EFCP__ZIP__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0a: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1b: tyop_2Efcp_2Ecart @ A_27c @ A_27b] :
( ( c_2Efcp_2EFCP__ZIP @ A_27a @ A_27b @ A_27c @ V0a @ V1b )
= ( c_2Efcp_2EFCP @ ( tyop_2Epair_2Eprod @ A_27a @ A_27c ) @ A_27b
@ ^ [V2i: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ A_27a @ A_27c @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0a @ V2i ) @ ( c_2Efcp_2Efcp__index @ A_27c @ A_27b @ V1b @ V2i ) ) ) ) ).
thf(thm_2Efcp_2EV2L__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V0v )
= ( c_2Elist_2EGENLIST @ A_27a @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0v ) @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) ) ).
thf(thm_2Efcp_2EL2V__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0L: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Efcp_2EL2V @ A_27a @ A_27b @ V0L )
= ( c_2Efcp_2EFCP @ A_27a @ A_27b
@ ^ [V1i: tyop_2Enum_2Enum] : ( c_2Elist_2EEL @ A_27a @ V1i @ V0L ) ) ) ).
thf(thm_2Efcp_2EFCP__FOLD__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27b > A_27a > A_27b,V1i: A_27b,V2v: tyop_2Efcp_2Ecart @ A_27a @ A_27c] :
( ( c_2Efcp_2EFCP__FOLD @ A_27a @ A_27b @ A_27c @ V0f @ V1i @ V2v )
= ( c_2Elist_2EFOLDL @ A_27a @ A_27b @ V0f @ V1i @ ( c_2Efcp_2EV2L @ A_27a @ A_27c @ V2v ) ) ) ).
thf(thm_2Efcp_2ENOT__FINITE__IMP__dimindex__1,axiom,
! [A_27a: $tType] :
( ( (~) @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) )
=> ( ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Efcp_2EDIMINDEX__GE__1,axiom,
! [A_27a: $tType] : ( c_2Earithmetic_2E_3C_3D @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ).
thf(thm_2Efcp_2Efcp__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: ( ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a ) > A_27c] :
? [V1g: ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > A_27c] :
! [V2h: ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a] :
( ( V1g @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b @ V2h ) )
= ( V0f @ V2h ) ) ).
thf(thm_2Efcp_2Efcp__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > $o] :
( ! [V1f: ( tyop_2Efcp_2Efinite__image @ A_27b ) > A_27a] : ( V0P @ ( c_2Efcp_2Emk__cart @ A_27a @ A_27b @ V1f ) )
=> ! [V2a: tyop_2Efcp_2Ecart @ A_27a @ A_27b] : ( V0P @ V2a ) ) ).
thf(thm_2Efcp_2ECART__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1y: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( V0x = V1y )
<=> ! [V2i: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2i @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) )
=> ( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0x @ V2i )
= ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V1y @ V2i ) ) ) ) ).
thf(thm_2Efcp_2EFCP__BETA,axiom,
! [A_27a: $tType,A_27b: $tType,V0g: tyop_2Enum_2Enum > A_27a,V1i: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V1i @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) )
=> ( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ ( c_2Efcp_2EFCP @ A_27a @ A_27b @ V0g ) @ V1i )
= ( V0g @ V1i ) ) ) ).
thf(thm_2Efcp_2EFCP__UNIQUE,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1g: tyop_2Enum_2Enum > A_27a] :
( ! [V2i: tyop_2Enum_2Enum] :
( ( c_2Eprim__rec_2E_3C @ V2i @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) )
=> ( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0f @ V2i )
= ( V1g @ V2i ) ) )
<=> ( ( c_2Efcp_2EFCP @ A_27a @ A_27b @ V1g )
= V0f ) ) ).
thf(thm_2Efcp_2EFCP__ETA,axiom,
! [A_27a: $tType,A_27b: $tType,V0g: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2EFCP @ A_27a @ A_27b
@ ^ [V1i: tyop_2Enum_2Enum] : ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0g @ V1i ) )
= V0g ) ).
thf(thm_2Efcp_2Ecard__dimindex,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
=> ( ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
= ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) ) ).
thf(thm_2Efcp_2Eindex__sum,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Efcp_2Edimindex @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Ebool_2Ethe__value @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2E_2F_5C @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) ) @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Efcp_2Efinite__sum,axiom,
! [A_27a: $tType,A_27b: $tType] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) )
<=> ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) )
& ( c_2Epred__set_2EFINITE @ A_27b @ ( c_2Epred__set_2EUNIV @ A_27b ) ) ) ) ).
thf(thm_2Efcp_2Edatatype__bit0,axiom,
! [A_27a: $tType,V0bit0: ( A_27a > ( tyop_2Efcp_2Ebit0 @ A_27a ) ) > ( A_27a > ( tyop_2Efcp_2Ebit0 @ A_27a ) ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0bit0 @ ( c_2Efcp_2EBIT0A @ A_27a ) @ ( c_2Efcp_2EBIT0B @ A_27a ) ) ) ).
thf(thm_2Efcp_2Ebit0__11,axiom,
! [A_27a: $tType] :
( ! [V0a: A_27a,V1a_27: A_27a] :
( ( ( c_2Efcp_2EBIT0A @ A_27a @ V0a )
= ( c_2Efcp_2EBIT0A @ A_27a @ V1a_27 ) )
<=> ( V0a = V1a_27 ) )
& ! [V2a: A_27a,V3a_27: A_27a] :
( ( ( c_2Efcp_2EBIT0B @ A_27a @ V2a )
= ( c_2Efcp_2EBIT0B @ A_27a @ V3a_27 ) )
<=> ( V2a = V3a_27 ) ) ) ).
thf(thm_2Efcp_2Ebit0__distinct,axiom,
! [A_27a: $tType,V0a_27: A_27a,V1a: A_27a] :
( (~)
@ ( ( c_2Efcp_2EBIT0A @ A_27a @ V1a )
= ( c_2Efcp_2EBIT0B @ A_27a @ V0a_27 ) ) ) ).
thf(thm_2Efcp_2Ebit0__nchotomy,axiom,
! [A_27a: $tType,V0bb: tyop_2Efcp_2Ebit0 @ A_27a] :
( ? [V1a: A_27a] :
( V0bb
= ( c_2Efcp_2EBIT0A @ A_27a @ V1a ) )
| ? [V2a: A_27a] :
( V0bb
= ( c_2Efcp_2EBIT0B @ A_27a @ V2a ) ) ) ).
thf(thm_2Efcp_2Ebit0__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0: A_27a > A_27b,V1f1: A_27a > A_27b] :
? [V2fn: ( tyop_2Efcp_2Ebit0 @ A_27a ) > A_27b] :
( ! [V3a: A_27a] :
( ( V2fn @ ( c_2Efcp_2EBIT0A @ A_27a @ V3a ) )
= ( V0f0 @ V3a ) )
& ! [V4a: A_27a] :
( ( V2fn @ ( c_2Efcp_2EBIT0B @ A_27a @ V4a ) )
= ( V1f1 @ V4a ) ) ) ).
thf(thm_2Efcp_2Ebit0__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Efcp_2Ebit0 @ A_27a ) > $o] :
( ( ! [V1a: A_27a] : ( V0P @ ( c_2Efcp_2EBIT0A @ A_27a @ V1a ) )
& ! [V2a: A_27a] : ( V0P @ ( c_2Efcp_2EBIT0B @ A_27a @ V2a ) ) )
=> ! [V3b: tyop_2Efcp_2Ebit0 @ A_27a] : ( V0P @ V3b ) ) ).
thf(thm_2Efcp_2Ebit0__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0f1_27: A_27a > A_27b,V1f_27: A_27a > A_27b,V2M: tyop_2Efcp_2Ebit0 @ A_27a,V3M_27: tyop_2Efcp_2Ebit0 @ A_27a,V4f: A_27a > A_27b,V5f1: A_27a > A_27b] :
( ( ( V2M = V3M_27 )
& ! [V6a: A_27a] :
( ( V3M_27
= ( c_2Efcp_2EBIT0A @ A_27a @ V6a ) )
=> ( ( V4f @ V6a )
= ( V1f_27 @ V6a ) ) )
& ! [V7a: A_27a] :
( ( V3M_27
= ( c_2Efcp_2EBIT0B @ A_27a @ V7a ) )
=> ( ( V5f1 @ V7a )
= ( V0f1_27 @ V7a ) ) ) )
=> ( ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b @ V2M @ V4f @ V5f1 )
= ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b @ V3M_27 @ V1f_27 @ V0f1_27 ) ) ) ).
thf(thm_2Efcp_2Ebit0__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Efcp_2Ebit0 @ A_27a,V1v: A_27b,V2f1: A_27a > A_27b,V3f: A_27a > A_27b] :
( ( ( c_2Efcp_2Ebit0__CASE @ A_27a @ A_27b @ V0x @ V3f @ V2f1 )
= V1v )
<=> ( ? [V4a: A_27a] :
( ( V0x
= ( c_2Efcp_2EBIT0A @ A_27a @ V4a ) )
& ( ( V3f @ V4a )
= V1v ) )
| ? [V5a: A_27a] :
( ( V0x
= ( c_2Efcp_2EBIT0B @ A_27a @ V5a ) )
& ( ( V2f1 @ V5a )
= V1v ) ) ) ) ).
thf(thm_2Efcp_2Eindex__bit0,axiom,
! [A_27a: $tType] :
( ( c_2Efcp_2Edimindex @ ( tyop_2Efcp_2Ebit0 @ A_27a ) @ ( c_2Ebool_2Ethe__value @ ( tyop_2Efcp_2Ebit0 @ A_27a ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Efcp_2Efinite__bit0,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Efcp_2Ebit0 @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Efcp_2Ebit0 @ A_27a ) ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Efcp_2Edatatype__bit1,axiom,
! [A_27a: $tType,V0bit1: ( A_27a > ( tyop_2Efcp_2Ebit1 @ A_27a ) ) > ( A_27a > ( tyop_2Efcp_2Ebit1 @ A_27a ) ) > ( tyop_2Efcp_2Ebit1 @ A_27a ) > $o] : ( c_2Ebool_2EDATATYPE @ $o @ ( V0bit1 @ ( c_2Efcp_2EBIT1A @ A_27a ) @ ( c_2Efcp_2EBIT1B @ A_27a ) @ ( c_2Efcp_2EBIT1C @ A_27a ) ) ) ).
thf(thm_2Efcp_2Ebit1__11,axiom,
! [A_27a: $tType] :
( ! [V0a: A_27a,V1a_27: A_27a] :
( ( ( c_2Efcp_2EBIT1A @ A_27a @ V0a )
= ( c_2Efcp_2EBIT1A @ A_27a @ V1a_27 ) )
<=> ( V0a = V1a_27 ) )
& ! [V2a: A_27a,V3a_27: A_27a] :
( ( ( c_2Efcp_2EBIT1B @ A_27a @ V2a )
= ( c_2Efcp_2EBIT1B @ A_27a @ V3a_27 ) )
<=> ( V2a = V3a_27 ) ) ) ).
thf(thm_2Efcp_2Ebit1__distinct,axiom,
! [A_27a: $tType] :
( ! [V0a_27: A_27a,V1a: A_27a] :
( (~)
@ ( ( c_2Efcp_2EBIT1A @ A_27a @ V1a )
= ( c_2Efcp_2EBIT1B @ A_27a @ V0a_27 ) ) )
& ! [V2a: A_27a] :
( (~)
@ ( ( c_2Efcp_2EBIT1A @ A_27a @ V2a )
= ( c_2Efcp_2EBIT1C @ A_27a ) ) )
& ! [V3a: A_27a] :
( (~)
@ ( ( c_2Efcp_2EBIT1B @ A_27a @ V3a )
= ( c_2Efcp_2EBIT1C @ A_27a ) ) ) ) ).
thf(thm_2Efcp_2Ebit1__nchotomy,axiom,
! [A_27a: $tType,V0bb: tyop_2Efcp_2Ebit1 @ A_27a] :
( ? [V1a: A_27a] :
( V0bb
= ( c_2Efcp_2EBIT1A @ A_27a @ V1a ) )
| ? [V2a: A_27a] :
( V0bb
= ( c_2Efcp_2EBIT1B @ A_27a @ V2a ) )
| ( V0bb
= ( c_2Efcp_2EBIT1C @ A_27a ) ) ) ).
thf(thm_2Efcp_2Ebit1__Axiom,axiom,
! [A_27a: $tType,A_27b: $tType,V0f0: A_27a > A_27b,V1f1: A_27a > A_27b,V2f2: A_27b] :
? [V3fn: ( tyop_2Efcp_2Ebit1 @ A_27a ) > A_27b] :
( ! [V4a: A_27a] :
( ( V3fn @ ( c_2Efcp_2EBIT1A @ A_27a @ V4a ) )
= ( V0f0 @ V4a ) )
& ! [V5a: A_27a] :
( ( V3fn @ ( c_2Efcp_2EBIT1B @ A_27a @ V5a ) )
= ( V1f1 @ V5a ) )
& ( ( V3fn @ ( c_2Efcp_2EBIT1C @ A_27a ) )
= V2f2 ) ) ).
thf(thm_2Efcp_2Ebit1__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Efcp_2Ebit1 @ A_27a ) > $o] :
( ( ! [V1a: A_27a] : ( V0P @ ( c_2Efcp_2EBIT1A @ A_27a @ V1a ) )
& ! [V2a: A_27a] : ( V0P @ ( c_2Efcp_2EBIT1B @ A_27a @ V2a ) )
& ( V0P @ ( c_2Efcp_2EBIT1C @ A_27a ) ) )
=> ! [V3b: tyop_2Efcp_2Ebit1 @ A_27a] : ( V0P @ V3b ) ) ).
thf(thm_2Efcp_2Ebit1__case__cong,axiom,
! [A_27a: $tType,A_27b: $tType,V0v_27: A_27b,V1f1_27: A_27a > A_27b,V2f_27: A_27a > A_27b,V3M: tyop_2Efcp_2Ebit1 @ A_27a,V4M_27: tyop_2Efcp_2Ebit1 @ A_27a,V5f: A_27a > A_27b,V6f1: A_27a > A_27b,V7v: A_27b] :
( ( ( V3M = V4M_27 )
& ! [V8a: A_27a] :
( ( V4M_27
= ( c_2Efcp_2EBIT1A @ A_27a @ V8a ) )
=> ( ( V5f @ V8a )
= ( V2f_27 @ V8a ) ) )
& ! [V9a: A_27a] :
( ( V4M_27
= ( c_2Efcp_2EBIT1B @ A_27a @ V9a ) )
=> ( ( V6f1 @ V9a )
= ( V1f1_27 @ V9a ) ) )
& ( ( V4M_27
= ( c_2Efcp_2EBIT1C @ A_27a ) )
=> ( V7v = V0v_27 ) ) )
=> ( ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b @ V3M @ V5f @ V6f1 @ V7v )
= ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b @ V4M_27 @ V2f_27 @ V1f1_27 @ V0v_27 ) ) ) ).
thf(thm_2Efcp_2Ebit1__case__eq,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Efcp_2Ebit1 @ A_27a,V1v_27: A_27b,V2v: A_27b,V3f1: A_27a > A_27b,V4f: A_27a > A_27b] :
( ( ( c_2Efcp_2Ebit1__CASE @ A_27a @ A_27b @ V0x @ V4f @ V3f1 @ V2v )
= V1v_27 )
<=> ( ? [V5a: A_27a] :
( ( V0x
= ( c_2Efcp_2EBIT1A @ A_27a @ V5a ) )
& ( ( V4f @ V5a )
= V1v_27 ) )
| ? [V6a: A_27a] :
( ( V0x
= ( c_2Efcp_2EBIT1B @ A_27a @ V6a ) )
& ( ( V3f1 @ V6a )
= V1v_27 ) )
| ( ( V0x
= ( c_2Efcp_2EBIT1C @ A_27a ) )
& ( V2v = V1v_27 ) ) ) ) ).
thf(thm_2Efcp_2Eindex__bit1,axiom,
! [A_27a: $tType] :
( ( c_2Efcp_2Edimindex @ ( tyop_2Efcp_2Ebit1 @ A_27a ) @ ( c_2Ebool_2Ethe__value @ ( tyop_2Efcp_2Ebit1 @ A_27a ) ) )
= ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) @ ( c_2Earithmetic_2E_2B @ ( c_2Earithmetic_2E_2A @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT2 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Efcp_2Edimindex @ A_27a @ ( c_2Ebool_2Ethe__value @ A_27a ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) ).
thf(thm_2Efcp_2Efinite__bit1,axiom,
! [A_27a: $tType] :
( ( c_2Epred__set_2EFINITE @ ( tyop_2Efcp_2Ebit1 @ A_27a ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Efcp_2Ebit1 @ A_27a ) ) )
= ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2Epred__set_2EUNIV @ A_27a ) ) ) ).
thf(thm_2Efcp_2Eindex__one,axiom,
( ( c_2Efcp_2Edimindex @ tyop_2Eone_2Eone @ ( c_2Ebool_2Ethe__value @ tyop_2Eone_2Eone ) )
= ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ).
thf(thm_2Efcp_2Efinite__one,axiom,
c_2Epred__set_2EFINITE @ tyop_2Eone_2Eone @ ( c_2Epred__set_2EUNIV @ tyop_2Eone_2Eone ) ).
thf(thm_2Efcp_2EFCP__UPDATE__COMMUTES,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1a: tyop_2Enum_2Enum,V2b: tyop_2Enum_2Enum,V3c: A_27a,V4d: A_27a] :
( ( (~) @ ( V1a = V2b ) )
=> ( ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V1a @ V3c @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V2b @ V4d @ V0m ) )
= ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V2b @ V4d @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V1a @ V3c @ V0m ) ) ) ) ).
thf(thm_2Efcp_2EFCP__UPDATE__EQ,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1a: tyop_2Enum_2Enum,V2b: A_27a,V3c: A_27a] :
( ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V1a @ V3c @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V1a @ V2b @ V0m ) )
= ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V1a @ V3c @ V0m ) ) ).
thf(thm_2Efcp_2EFCP__UPDATE__IMP__ID,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1a: tyop_2Enum_2Enum,V2v: A_27a] :
( ( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0m @ V1a )
= V2v )
=> ( ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V1a @ V2v @ V0m )
= V0m ) ) ).
thf(thm_2Efcp_2EAPPLY__FCP__UPDATE__ID,axiom,
! [A_27a: $tType,A_27b: $tType,V0m: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V1a: tyop_2Enum_2Enum] :
( ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V1a @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V0m @ V1a ) @ V0m )
= V0m ) ).
thf(thm_2Efcp_2EFCP__APPLY__UPDATE__THM,axiom,
! [A_27a: $tType,A_27b: $tType,V0index_20out_20of_20range: $o,V1m: tyop_2Efcp_2Ecart @ A_27a @ A_27b,V2a: tyop_2Enum_2Enum,V3w: A_27a,V4b: tyop_2Enum_2Enum] :
( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V2a @ V3w @ V1m ) @ V4b )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Eprim__rec_2E_3C @ V4b @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) @ ( c_2Ebool_2ECOND @ A_27a @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V2a @ V4b ) @ V3w @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V1m @ V4b ) ) @ ( c_2Ecombin_2EFAIL @ ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > tyop_2Enum_2Enum > A_27a ) @ $o @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0index_20out_20of_20range @ ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V2a @ V3w @ V1m ) @ V4b ) ) ) ).
thf(thm_2Efcp_2ELENGTH__V2L,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Elist_2ELENGTH @ A_27a @ ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V0v ) )
= ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) ).
thf(thm_2Efcp_2EEL__V2L,axiom,
! [A_27a: $tType,A_27b: $tType,V0i: tyop_2Enum_2Enum,V1v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Eprim__rec_2E_3C @ V0i @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) )
=> ( ( c_2Elist_2EEL @ A_27a @ V0i @ ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V1v ) )
= ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ V1v @ V0i ) ) ) ).
thf(thm_2Efcp_2EFCP__MAP,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0f: A_27a > A_27b,V1v: tyop_2Efcp_2Ecart @ A_27a @ A_27c] :
( ( c_2Efcp_2EFCP__MAP @ A_27a @ A_27b @ A_27c @ V0f @ V1v )
= ( c_2Efcp_2EL2V @ A_27b @ A_27c @ ( c_2Elist_2EMAP @ A_27a @ A_27b @ V0f @ ( c_2Efcp_2EV2L @ A_27a @ A_27c @ V1v ) ) ) ) ).
thf(thm_2Efcp_2EFCP__TL,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( ( c_2Eprim__rec_2E_3C @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) )
& ( ( c_2Efcp_2Edimindex @ A_27c @ ( c_2Ebool_2Ethe__value @ A_27c ) )
= ( c_2Earithmetic_2E_2D @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) @ ( c_2Earithmetic_2ENUMERAL @ ( c_2Earithmetic_2EBIT1 @ c_2Earithmetic_2EZERO ) ) ) ) )
=> ( ( c_2Efcp_2EFCP__TL @ A_27a @ A_27c @ A_27b @ V0v )
= ( c_2Efcp_2EL2V @ A_27a @ A_27c @ ( c_2Elist_2ETL @ A_27a @ ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V0v ) ) ) ) ) ).
thf(thm_2Efcp_2EFCP__EXISTS,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2EFCP__EXISTS @ A_27b @ A_27a @ V0P @ V1v )
= ( c_2Elist_2EEXISTS @ A_27a @ V0P @ ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V1v ) ) ) ).
thf(thm_2Efcp_2EFCP__EVERY,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: A_27a > $o,V1v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2EFCP__EVERY @ A_27b @ A_27a @ V0P @ V1v )
= ( c_2Elist_2EEVERY @ A_27a @ V0P @ ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V1v ) ) ) ).
thf(thm_2Efcp_2EFCP__HD,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2EFCP__HD @ A_27a @ A_27b @ V0v )
= ( c_2Elist_2EHD @ A_27a @ ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V0v ) ) ) ).
thf(thm_2Efcp_2EFCP__CONS,axiom,
! [A_27a: $tType,A_27b: $tType,V0a: A_27a,V1v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] :
( ( c_2Efcp_2EFCP__CONS @ A_27a @ A_27b @ ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) @ V0a @ V1v )
= ( c_2Efcp_2EL2V @ A_27a @ ( tyop_2Esum_2Esum @ A_27b @ tyop_2Eone_2Eone ) @ ( c_2Elist_2ECONS @ A_27a @ V0a @ ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V1v ) ) ) ) ).
thf(thm_2Efcp_2EV2L__L2V,axiom,
! [A_27a: $tType,A_27b: $tType,V0x: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) )
= ( c_2Elist_2ELENGTH @ A_27a @ V0x ) )
=> ( ( c_2Efcp_2EV2L @ A_27a @ A_27b @ ( c_2Efcp_2EL2V @ A_27a @ A_27b @ V0x ) )
= V0x ) ) ).
thf(thm_2Efcp_2ENULL__V2L,axiom,
! [A_27a: $tType,A_27b: $tType,V0v: tyop_2Efcp_2Ecart @ A_27a @ A_27b] : ( (~) @ ( c_2Elist_2ENULL @ A_27a @ ( c_2Efcp_2EV2L @ A_27a @ A_27b @ V0v ) ) ) ).
thf(thm_2Efcp_2EREAD__TL,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,V0i: tyop_2Enum_2Enum,V1a: tyop_2Efcp_2Ecart @ A_27a @ A_27c] :
( ( c_2Eprim__rec_2E_3C @ V0i @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) )
=> ( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ ( c_2Efcp_2EFCP__TL @ A_27a @ A_27b @ A_27c @ V1a ) @ V0i )
= ( c_2Efcp_2Efcp__index @ A_27a @ A_27c @ V1a @ ( c_2Enum_2ESUC @ V0i ) ) ) ) ).
thf(thm_2Efcp_2EREAD__L2V,axiom,
! [A_27a: $tType,A_27b: $tType,V0i: tyop_2Enum_2Enum,V1a: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Eprim__rec_2E_3C @ V0i @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) )
=> ( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ ( c_2Efcp_2EL2V @ A_27a @ A_27b @ V1a ) @ V0i )
= ( c_2Elist_2EEL @ A_27a @ V0i @ V1a ) ) ) ).
thf(thm_2Efcp_2Eindex__comp,axiom,
! [A_27a: $tType,A_27b: $tType,V0FCP_20out_20of_20bounds: $o,V1f: tyop_2Enum_2Enum > A_27a,V2n: tyop_2Enum_2Enum] :
( ( c_2Efcp_2Efcp__index @ A_27a @ A_27b @ ( c_2Efcp_2EFCP @ A_27a @ A_27b @ V1f ) @ V2n )
= ( c_2Ebool_2ECOND @ A_27a @ ( c_2Eprim__rec_2E_3C @ V2n @ ( c_2Efcp_2Edimindex @ A_27b @ ( c_2Ebool_2Ethe__value @ A_27b ) ) ) @ ( V1f @ V2n ) @ ( c_2Ecombin_2EFAIL @ ( ( tyop_2Efcp_2Ecart @ A_27a @ A_27b ) > tyop_2Enum_2Enum > A_27a ) @ $o @ ( c_2Efcp_2Efcp__index @ A_27a @ A_27b ) @ V0FCP_20out_20of_20bounds @ ( c_2Efcp_2EFCP @ A_27a @ A_27b @ V1f ) @ V2n ) ) ) ).
thf(thm_2Efcp_2Efcp__subst__comp,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,V0y: A_27a,V1x: tyop_2Enum_2Enum,V2a: A_27c,V3b: A_27d,V4f: tyop_2Enum_2Enum > A_27a] :
( ( c_2Efcp_2E_3A_2B @ A_27a @ A_27b @ V1x @ V0y @ ( c_2Efcp_2EFCP @ A_27a @ A_27b @ V4f ) )
= ( c_2Efcp_2EFCP @ A_27a @ A_27b
@ ^ [V5c: tyop_2Enum_2Enum] : ( c_2Ebool_2ECOND @ A_27a @ ( c_2Emin_2E_3D @ tyop_2Enum_2Enum @ V1x @ V5c ) @ V0y @ ( V4f @ V5c ) ) ) ) ).
%------------------------------------------------------------------------------