ITP001 Axioms: ITP040^5.ax
%------------------------------------------------------------------------------
% File : ITP040^5 : 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 : indexedLists^2.ax [Gau20]
% : HL4040^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 67 ( 1 unt; 9 typ; 0 def)
% Number of atoms : 2007 ( 72 equ; 0 cnn)
% Maximal formula atoms : 98 ( 29 avg)
% Number of connectives : 3463 ( 0 ~; 2 |; 49 &;3158 @)
% ( 8 <=>; 246 =>; 0 <=; 0 <~>)
% Maximal formula depth : 36 ( 16 avg;3158 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 17 ( 17 >; 0 *; 0 +; 0 <<)
% Number of symbols : 56 ( 55 usr; 46 con; 0-3 aty)
% Number of variables : 366 ( 10 ^ 348 !; 8 ?; 366 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2EindexedLists_2EFOLDRi,type,
c_2EindexedLists_2EFOLDRi: del > del > $i ).
thf(mem_c_2EindexedLists_2EFOLDRi,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EindexedLists_2EFOLDRi @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ A_27a ) ) ) ) ).
thf(tp_c_2EindexedLists_2ELIST__RELi,type,
c_2EindexedLists_2ELIST__RELi: del > del > $i ).
thf(mem_c_2EindexedLists_2ELIST__RELi,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) ) ) ) ).
thf(tp_c_2EindexedLists_2EMAP2i,type,
c_2EindexedLists_2EMAP2i: del > del > del > $i ).
thf(mem_c_2EindexedLists_2EMAP2i,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EindexedLists_2EMAP2i @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2EindexedLists_2EMAP2ia,type,
c_2EindexedLists_2EMAP2ia: del > del > del > $i ).
thf(mem_c_2EindexedLists_2EMAP2ia,axiom,
! [A_27a: del,A_27b: del,A_27c: del] : ( mem @ ( c_2EindexedLists_2EMAP2ia @ A_27a @ A_27b @ A_27c ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2EindexedLists_2EMAPi,type,
c_2EindexedLists_2EMAPi: del > del > $i ).
thf(mem_c_2EindexedLists_2EMAPi,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2EindexedLists_2EMAPi__ACC,type,
c_2EindexedLists_2EMAPi__ACC: del > del > $i ).
thf(mem_c_2EindexedLists_2EMAPi__ACC,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EindexedLists_2EMAPi__ACC @ A_27a @ A_27b ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ) ).
thf(tp_c_2EindexedLists_2EdelN,type,
c_2EindexedLists_2EdelN: del > $i ).
thf(mem_c_2EindexedLists_2EdelN,axiom,
! [A_27a: del] : ( mem @ ( c_2EindexedLists_2EdelN @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(tp_c_2EindexedLists_2Efindi,type,
c_2EindexedLists_2Efindi: del > $i ).
thf(mem_c_2EindexedLists_2Efindi,axiom,
! [A_27a: del] : ( mem @ ( c_2EindexedLists_2Efindi @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ty_2Enum_2Enum ) ) ) ).
thf(tp_c_2EindexedLists_2EfupdLast,type,
c_2EindexedLists_2EfupdLast: del > $i ).
thf(mem_c_2EindexedLists_2EfupdLast,axiom,
! [A_27a: del] : ( mem @ ( c_2EindexedLists_2EfupdLast @ A_27a ) @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ).
thf(ax_thm_2EindexedLists_2EMAPi__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27b ) @ V0f ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27b )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h ) @ V3t ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V2h ) ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) @ ty_2Enum_2Enum ) @ V1f ) @ c_2Enum_2ESUC ) ) @ V3t ) ) ) ) ) ) ) ).
thf(ax_thm_2EindexedLists_2EMAPi__ACC__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1i: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi__ACC @ A_27a @ A_27b ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V2a ) @ ( c_2Elist_2ENIL @ A_27b ) )
= ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V2a ) ) ) )
& ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V4i: tp__ty_2Enum_2Enum,V5a: $i] :
( ( mem @ V5a @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V6h: $i] :
( ( mem @ V6h @ A_27b )
=> ! [V7t: $i] :
( ( mem @ V7t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi__ACC @ A_27a @ A_27b ) @ V3f ) @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V5a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V6h ) @ V7t ) )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi__ACC @ A_27a @ A_27b ) @ V3f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( ap @ V3f @ ( inj__ty_2Enum_2Enum @ V4i ) ) @ V6h ) ) @ V5a ) ) @ V7t ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAPi__ACC__MAPi,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2a: $i] :
( ( mem @ V2a @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi__ACC @ A_27a @ A_27b ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2a ) @ V3l )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( c_2Elist_2EREVERSE @ A_27a ) @ V2a ) ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) @ ty_2Enum_2Enum ) @ V0f ) @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) @ V3l ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAPi__compute,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27b ) @ V0f ) @ V1l )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi__ACC @ A_27a @ A_27b ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1l ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELT__SUC,axiom,
! [V0n: tp__ty_2Enum_2Enum,V1m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) )
<=> ( ( V0n = fo__c_2Enum_2E0 )
| ? [V2n0: tp__ty_2Enum_2Enum] :
( ( V0n
= ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n0 ) ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2n0 ) ) @ ( inj__ty_2Enum_2Enum @ V1m ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMEM__MAPi,axiom,
! [A_27a: del,A_27b: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ A_27a ) ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27b ) @ V1f ) @ V2l ) ) ) )
<=> ? [V3n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l ) ) )
& ( V0x
= ( ap @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V2l ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAPi__CONG,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( ( V0l1 = V1l2 )
& ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5n: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V4x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) )
=> ( ( ap @ ( ap @ V2f1 @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V4x )
= ( ap @ ( ap @ V3f2 @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V4x ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V2f1 ) @ V0l1 )
= ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V3f2 ) @ V1l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAPi__CONG_27,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( V0l1 = V1l2 )
=> ( ! [V4x: $i] :
( ( mem @ V4x @ A_27a )
=> ! [V5n: tp__ty_2Enum_2Enum] :
( ( V4x
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V1l2 ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l2 ) ) )
=> ( ( ap @ ( ap @ V2f1 @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V4x )
= ( ap @ ( ap @ V3f2 @ ( inj__ty_2Enum_2Enum @ V5n ) ) @ V4x ) ) ) ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V2f1 ) @ V0l1 )
= ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V3f2 ) @ V1l2 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELENGTH__MAPi,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V0f ) @ V1l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAP__MAPi,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27b ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EMAP @ A_27a @ A_27b ) @ V0f ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27c ) @ V1g ) @ V2l ) )
= ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27c @ A_27b ) @ ( arr @ A_27c @ A_27a ) ) @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ A_27b @ A_27a ) @ V0f ) ) @ V1g ) ) @ V2l ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EEL__MAPi,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ! [V1n: tp__ty_2Enum_2Enum,V2l: $i] :
( ( mem @ V2l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V2l ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V0f ) @ V2l ) )
= ( ap @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V2l ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAPi__APPEND,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V0l1 ) @ V1l2 ) )
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V2f ) @ V0l1 ) ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) @ ty_2Enum_2Enum ) @ V2f ) @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l1 ) ) ) ) @ V1l2 ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAPi__GENLIST,axiom,
! [A_27a: del,A_27b: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27b ) ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27b @ A_27a ) @ V1f ) @ V0l )
= ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2ES @ ty_2Enum_2Enum @ A_27a @ A_27b ) @ V1f ) @ ( ap @ ( ap @ ( c_2Ecombin_2EC @ ty_2Enum_2Enum @ ( ty_2Elist_2Elist @ A_27a ) @ A_27a ) @ ( c_2Elist_2EEL @ A_27a ) ) @ V0l ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EGENLIST__CONG,axiom,
! [A_27a: del,V0n: tp__ty_2Enum_2Enum,V1f1: $i] :
( ( mem @ V1f1 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ! [V2f2: $i] :
( ( mem @ V2f2 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
=> ( ! [V3m: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
=> ( ( ap @ V1f1 @ ( inj__ty_2Enum_2Enum @ V3m ) )
= ( ap @ V2f2 @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V1f1 ) @ ( inj__ty_2Enum_2Enum @ V0n ) )
= ( ap @ ( ap @ ( c_2Elist_2EGENLIST @ A_27a ) @ V2f2 ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ) ) ).
thf(ax_thm_2EindexedLists_2EFOLDRi__def,axiom,
! [A_27a: del,A_27b: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V1a: $i] :
( ( mem @ V1a @ A_27a )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27a @ A_27b ) @ V0f ) @ V1a ) @ ( c_2Elist_2ENIL @ A_27b ) )
= V1a ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) ) )
=> ! [V3a: $i] :
( ( mem @ V3a @ A_27a )
=> ! [V4h: $i] :
( ( mem @ V4h @ A_27b )
=> ! [V5t: $i] :
( ( mem @ V5t @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27a @ A_27b ) @ V2f ) @ V3a ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4h ) @ V5t ) )
= ( ap @ ( ap @ ( ap @ V2f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V4h ) @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27a @ A_27a ) ) @ ty_2Enum_2Enum ) @ V2f ) @ c_2Enum_2ESUC ) ) @ V3a ) @ V5t ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EFOLDR__MAPi,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) )
=> ! [V1g: $i] :
( ( mem @ V1g @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27c @ A_27a ) ) )
=> ! [V2a: $i] :
( ( mem @ V2a @ A_27b )
=> ! [V3l: $i] :
( ( mem @ V3l @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2Elist_2EFOLDR @ A_27a @ A_27b ) @ V0f ) @ V2a ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAPi @ A_27a @ A_27c ) @ V1g ) @ V3l ) )
= ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27c @ ( arr @ A_27b @ A_27b ) ) @ ( arr @ A_27c @ A_27a ) ) @ ( ap @ ( c_2Ecombin_2Eo @ A_27c @ ( arr @ A_27b @ A_27b ) @ A_27a ) @ V0f ) ) @ V1g ) ) @ V2a ) @ V3l ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EFOLDRi__APPEND,axiom,
! [A_27a: del,A_27b: del,V0a: $i] :
( ( mem @ V0a @ A_27b )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27b @ A_27a ) @ V3f ) @ V0a ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V2l2 ) )
= ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27b @ A_27a ) @ V3f ) @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) @ ty_2Enum_2Enum ) @ V3f ) @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) ) ) @ V0a ) @ V2l2 ) ) @ V1l1 ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EFOLDRi__CONG,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) ) )
=> ! [V4a1: $i] :
( ( mem @ V4a1 @ A_27b )
=> ! [V5a2: $i] :
( ( mem @ V5a2 @ A_27b )
=> ( ( V0l1 = V1l2 )
=> ( ! [V6n: tp__ty_2Enum_2Enum,V7e: $i] :
( ( mem @ V7e @ A_27a )
=> ! [V8a: $i] :
( ( mem @ V8a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l2 ) ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V7e ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l2 ) ) )
=> ( ( ap @ ( ap @ ( ap @ V2f1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V7e ) @ V8a )
= ( ap @ ( ap @ ( ap @ V3f2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V7e ) @ V8a ) ) ) ) ) )
=> ( ( V4a1 = V5a2 )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27b @ A_27a ) @ V2f1 ) @ V4a1 ) @ V0l1 )
= ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27b @ A_27a ) @ V3f2 ) @ V5a2 ) @ V1l2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EFOLDRi__CONG_27,axiom,
! [A_27a: del,A_27b: del,V0l1: $i] :
( ( mem @ V0l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1l2: $i] :
( ( mem @ V1l2 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2f1: $i] :
( ( mem @ V2f1 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) ) )
=> ! [V3f2: $i] :
( ( mem @ V3f2 @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27b ) ) ) )
=> ! [V4a1: $i] :
( ( mem @ V4a1 @ A_27b )
=> ! [V5a2: $i] :
( ( mem @ V5a2 @ A_27b )
=> ( ( ( V0l1 = V1l2 )
& ! [V6n: tp__ty_2Enum_2Enum,V7a: $i] :
( ( mem @ V7a @ A_27b )
=> ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l2 ) ) )
=> ( ( ap @ ( ap @ ( ap @ V2f1 @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V1l2 ) ) @ V7a )
= ( ap @ ( ap @ ( ap @ V3f2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ V1l2 ) ) @ V7a ) ) ) )
& ( V4a1 = V5a2 ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27b @ A_27a ) @ V2f1 ) @ V4a1 ) @ V0l1 )
= ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EFOLDRi @ A_27b @ A_27a ) @ V3f2 ) @ V5a2 ) @ V1l2 ) ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EindexedLists_2Efindi__def,axiom,
! [A_27a: del] :
( ! [V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2EindexedLists_2Efindi @ A_27a ) @ V0x ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= fo__c_2Enum_2E0 ) )
& ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2EindexedLists_2Efindi @ A_27a ) @ V1x ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V1x ) @ V2h ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) @ ( ap @ ( ap @ ( c_2EindexedLists_2Efindi @ A_27a ) @ V1x ) @ V3t ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMEM__findi,axiom,
! [A_27a: del,V0x: $i] :
( ( mem @ V0x @ A_27a )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1l ) ) )
=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ ( c_2EindexedLists_2Efindi @ A_27a ) @ V0x ) @ V1l ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2Efindi__EL,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
& ( p @ ( ap @ ( c_2Elist_2EALL__DISTINCT @ A_27a ) @ V0l ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ ( c_2EindexedLists_2Efindi @ A_27a ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) @ V0l ) ) @ V0l ) )
= V1n ) ) ) ).
thf(conj_thm_2EindexedLists_2EEL__findi,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V1x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V0l ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ ( ap @ ( c_2EindexedLists_2Efindi @ A_27a ) @ V1x ) @ V0l ) ) @ V0l )
= V1x ) ) ) ) ).
thf(ax_thm_2EindexedLists_2EdelN__def,axiom,
! [A_27a: del] :
( ! [V0i: tp__ty_2Enum_2Enum] :
( ( ap @ ( ap @ ( c_2EindexedLists_2EdelN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V0i ) ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) )
& ! [V1i: tp__ty_2Enum_2Enum,V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EdelN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V3t ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EdelN @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V3t ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EdelN__shortens,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EdelN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V0l ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2E_2D @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EEL__delN__BEFORE,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1i: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EdelN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V0l ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V0l ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EEL__delN__AFTER,axiom,
! [A_27a: del,V0l: $i] :
( ( mem @ V0l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V1i: tp__ty_2Enum_2Enum,V2j: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Earithmetic_2E_3C_3D @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V0l ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EdelN @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V2j ) ) @ V0l ) )
= ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V0l ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EfupdLast__ind,axiom,
! [A_27a: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ bool ) ) )
=> ( ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ V1f ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ! [V3h: $i] :
( ( mem @ V3h @ A_27a )
=> ( p @ ( ap @ ( ap @ V0P @ V2f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ A_27a @ A_27a ) )
=> ! [V5h: $i] :
( ( mem @ V5h @ A_27a )
=> ! [V6v4: $i] :
( ( mem @ V6v4 @ A_27a )
=> ! [V7v5: $i] :
( ( mem @ V7v5 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( p @ ( ap @ ( ap @ V0P @ V4f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6v4 ) @ V7v5 ) ) )
=> ( p @ ( ap @ ( ap @ V0P @ V4f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5h ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6v4 ) @ V7v5 ) ) ) ) ) ) ) ) ) )
=> ! [V8v: $i] :
( ( mem @ V8v @ ( arr @ A_27a @ A_27a ) )
=> ! [V9v1: $i] :
( ( mem @ V9v1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( p @ ( ap @ ( ap @ V0P @ V8v ) @ V9v1 ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EfupdLast__def,axiom,
! [A_27a: del] :
( ! [V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EfupdLast @ A_27a ) @ V0f ) @ ( c_2Elist_2ENIL @ A_27a ) )
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ! [V1h: $i] :
( ( mem @ V1h @ A_27a )
=> ! [V2f: $i] :
( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EfupdLast @ A_27a ) @ V2f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h ) @ ( c_2Elist_2ENIL @ A_27a ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ V2f @ V1h ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V3v5: $i] :
( ( mem @ V3v5 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4v4: $i] :
( ( mem @ V4v4 @ A_27a )
=> ! [V5h: $i] :
( ( mem @ V5h @ A_27a )
=> ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ A_27a @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EfupdLast @ A_27a ) @ V6f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5h ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4v4 ) @ V3v5 ) ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V5h ) @ ( ap @ ( ap @ ( c_2EindexedLists_2EfupdLast @ A_27a ) @ V6f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V4v4 ) @ V3v5 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EfupdLast__EQ__NIL,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ( ( ap @ ( ap @ ( c_2EindexedLists_2EfupdLast @ A_27a ) @ V0f ) @ V1x )
= ( c_2Elist_2ENIL @ A_27a ) )
<=> ( V1x
= ( c_2Elist_2ENIL @ A_27a ) ) )
& ( ( ( c_2Elist_2ENIL @ A_27a )
= ( ap @ ( ap @ ( c_2EindexedLists_2EfupdLast @ A_27a ) @ V0f ) @ V1x ) )
<=> ( V1x
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EfupdLast__FRONT__LAST,axiom,
! [A_27a: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
=> ! [V1l: $i] :
( ( mem @ V1l @ ( ty_2Elist_2Elist @ A_27a ) )
=> ( ( ap @ ( ap @ ( c_2EindexedLists_2EfupdLast @ A_27a ) @ V0f ) @ V1l )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V1l ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ ( ap @ ( c_2Elist_2EFRONT @ A_27a ) @ V1l ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ V0f @ ( ap @ ( c_2Elist_2ELAST @ A_27a ) @ V1l ) ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ) ).
thf(ax_thm_2EindexedLists_2ELIST__RELi__def,axiom,
! [A_27a: del,A_27b: del] :
( ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b )
= ( lam @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) )
@ ^ [V0R: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V1a0: $i] :
( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V2a1: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) ) )
@ ( lam @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) )
@ ^ [V3LIST__RELi_27: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V4a0: $i] :
( ap @ ( c_2Ebool_2E_21 @ ( ty_2Elist_2Elist @ A_27b ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V5a1: $i] :
( ap
@ ( ap @ c_2Emin_2E_3D_3D_3E
@ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V4a0 ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27b ) ) @ V5a1 ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) )
@ ( ap @ ( c_2Ebool_2E_3F @ A_27a )
@ ( lam @ A_27a
@ ^ [V6h1: $i] :
( ap @ ( c_2Ebool_2E_3F @ A_27b )
@ ( lam @ A_27b
@ ^ [V7h2: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elist_2Elist @ A_27a ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27a )
@ ^ [V8l1: $i] :
( ap @ ( c_2Ebool_2E_3F @ ( ty_2Elist_2Elist @ A_27b ) )
@ ( lam @ ( ty_2Elist_2Elist @ A_27b )
@ ^ [V9l2: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27a ) ) @ V4a0 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V8l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V6h1 ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27b ) ) @ V5a1 ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V9l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V7h2 ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V8l1 ) ) @ V6h1 ) @ V7h2 ) ) @ ( ap @ ( ap @ V3LIST__RELi_27 @ V8l1 ) @ V9l2 ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( ap @ ( ap @ V3LIST__RELi_27 @ V4a0 ) @ V5a1 ) ) ) ) ) ) )
@ ( ap @ ( ap @ V3LIST__RELi_27 @ V1a0 ) @ V2a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELIST__RELi__rules,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
& ! [V1h1: $i] :
( ( mem @ V1h1 @ A_27a )
=> ! [V2h2: $i] :
( ( mem @ V2h2 @ A_27b )
=> ! [V3l1: $i] :
( ( mem @ V3l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l2: $i] :
( ( mem @ V4l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V3l1 ) ) @ V1h1 ) @ V2h2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V3l1 ) @ V4l2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V3l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V1h1 ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V4l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V2h2 ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELIST__RELi__ind,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V1LIST__RELi_27: $i] :
( ( mem @ V1LIST__RELi_27 @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V1LIST__RELi_27 @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
& ! [V2h1: $i] :
( ( mem @ V2h1 @ A_27a )
=> ! [V3h2: $i] :
( ( mem @ V3h2 @ A_27b )
=> ! [V4l1: $i] :
( ( mem @ V4l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4l1 ) ) @ V2h1 ) @ V3h2 ) )
& ( p @ ( ap @ ( ap @ V1LIST__RELi_27 @ V4l1 ) @ V5l2 ) ) )
=> ( p @ ( ap @ ( ap @ V1LIST__RELi_27 @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h1 ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V5l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3h2 ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) ) ) ) ) ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V6a0 ) @ V7a1 ) )
=> ( p @ ( ap @ ( ap @ V1LIST__RELi_27 @ V6a0 ) @ V7a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELIST__RELi__strongind,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V1LIST__RELi_27: $i] :
( ( mem @ V1LIST__RELi_27 @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ bool ) ) )
=> ( ( ( p @ ( ap @ ( ap @ V1LIST__RELi_27 @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) )
& ! [V2h1: $i] :
( ( mem @ V2h1 @ A_27a )
=> ! [V3h2: $i] :
( ( mem @ V3h2 @ A_27b )
=> ! [V4l1: $i] :
( ( mem @ V4l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V5l2: $i] :
( ( mem @ V5l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V4l1 ) ) @ V2h1 ) @ V3h2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V4l1 ) @ V5l2 ) )
& ( p @ ( ap @ ( ap @ V1LIST__RELi_27 @ V4l1 ) @ V5l2 ) ) )
=> ( p @ ( ap @ ( ap @ V1LIST__RELi_27 @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V4l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h1 ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V5l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3h2 ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) ) ) ) ) ) )
=> ! [V6a0: $i] :
( ( mem @ V6a0 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V7a1: $i] :
( ( mem @ V7a1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V6a0 ) @ V7a1 ) )
=> ( p @ ( ap @ ( ap @ V1LIST__RELi_27 @ V6a0 ) @ V7a1 ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELIST__RELi__cases,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V1a0: $i] :
( ( mem @ V1a0 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2a1: $i] :
( ( mem @ V2a1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V1a0 ) @ V2a1 ) )
<=> ( ( ( V1a0
= ( c_2Elist_2ENIL @ A_27a ) )
& ( V2a1
= ( c_2Elist_2ENIL @ A_27b ) ) )
| ? [V3h1: $i] :
( ( mem @ V3h1 @ A_27a )
& ? [V4h2: $i] :
( ( mem @ V4h2 @ A_27b )
& ? [V5l1: $i] :
( ( mem @ V5l1 @ ( ty_2Elist_2Elist @ A_27a ) )
& ? [V6l2: $i] :
( ( mem @ V6l2 @ ( ty_2Elist_2Elist @ A_27b ) )
& ( V1a0
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V5l1 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V3h1 ) @ ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ( V2a1
= ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V6l2 ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4h2 ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) )
& ( p @ ( ap @ ( ap @ ( ap @ V0R @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V5l1 ) ) @ V3h1 ) @ V4h2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V5l1 ) @ V6l2 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELIST__RELi__LENGTH,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V1l1 ) @ V2l2 ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELIST__RELi__EL__EQN,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V1l1 ) @ V2l2 ) )
<=> ( ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l2 ) ) )
& ! [V3i: tp__ty_2Enum_2Enum] :
( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3i ) ) @ V2l2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELIST__RELi__thm,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V2h: $i] :
( ( mem @ V2h @ A_27a )
=> ! [V3t: $i] :
( ( mem @ V3t @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4l: $i] :
( ( mem @ V4l @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ V1x ) )
<=> ( V1x
= ( c_2Elist_2ENIL @ A_27b ) ) )
& ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ V2h ) @ V3t ) ) @ V4l ) )
<=> ? [V5h_27: $i] :
( ( mem @ V5h_27 @ A_27b )
& ? [V6t_27: $i] :
( ( mem @ V6t_27 @ ( ty_2Elist_2Elist @ A_27b ) )
& ( V4l
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V5h_27 ) @ V6t_27 ) )
& ( p @ ( ap @ ( ap @ ( ap @ V0R @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V2h ) @ V5h_27 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ty_2Enum_2Enum ) @ V0R ) @ c_2Enum_2ESUC ) ) @ V3t ) @ V6t_27 ) ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELIST__RELi__APPEND__I,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( ( mem @ V0R @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3m1: $i] :
( ( mem @ V3m1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V4m2: $i] :
( ( mem @ V4m2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ V1l1 ) @ V2l2 ) )
& ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ bool ) ) @ ty_2Enum_2Enum ) @ V0R ) @ ( ap @ c_2Earithmetic_2E_2B @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) ) ) @ V3m1 ) @ V4m2 ) ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2ELIST__RELi @ A_27a @ A_27b ) @ V0R ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27a ) @ V1l1 ) @ V3m1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EAPPEND @ A_27b ) @ V2l2 ) @ V4m2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAP2i__ind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ bool ) ) ) )
=> ( ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V2v0: $i] :
( ( mem @ V2v0 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V1f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V2v0 ) ) ) )
& ! [V3f: $i] :
( ( mem @ V3f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V4v5: $i] :
( ( mem @ V4v5 @ A_27b )
=> ! [V5v6: $i] :
( ( mem @ V5v6 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V3f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4v5 ) @ V5v6 ) ) @ ( c_2Elist_2ENIL @ A_27c ) ) ) ) ) )
& ! [V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V7h1: $i] :
( ( mem @ V7h1 @ A_27b )
=> ! [V8t1: $i] :
( ( mem @ V8t1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V9h2: $i] :
( ( mem @ V9h2 @ A_27c )
=> ! [V10t2: $i] :
( ( mem @ V10t2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ V0P @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ ty_2Enum_2Enum ) @ V6f ) @ c_2Enum_2ESUC ) ) @ V8t1 ) @ V10t2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V6f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V7h1 ) @ V8t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V9h2 ) @ V10t2 ) ) ) ) ) ) ) ) ) )
=> ! [V11v: $i] :
( ( mem @ V11v @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V12v1: $i] :
( ( mem @ V12v1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V13v2: $i] :
( ( mem @ V13v2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( p @ ( ap @ ( ap @ ( ap @ V0P @ V11v ) @ V12v1 ) @ V13v2 ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAP2i__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0v0: $i] :
( ( mem @ V0v0 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2i @ A_27a @ A_27b @ A_27c ) @ V1f ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V0v0 )
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V2v6: $i] :
( ( mem @ V2v6 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3v5: $i] :
( ( mem @ V3v5 @ A_27b )
=> ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2i @ A_27a @ A_27b @ A_27c ) @ V4f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V3v5 ) @ V2v6 ) ) @ ( c_2Elist_2ENIL @ A_27c ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V5t2: $i] :
( ( mem @ V5t2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V6t1: $i] :
( ( mem @ V6t1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V7h2: $i] :
( ( mem @ V7h2 @ A_27c )
=> ! [V8h1: $i] :
( ( mem @ V8h1 @ A_27b )
=> ! [V9f: $i] :
( ( mem @ V9f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2i @ A_27a @ A_27b @ A_27c ) @ V9f ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V8h1 ) @ V6t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V7h2 ) @ V5t2 ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( ap @ ( ap @ V9f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V8h1 ) @ V7h2 ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2i @ A_27a @ A_27b @ A_27c ) @ ( ap @ ( ap @ ( c_2Ecombin_2Eo @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) @ ty_2Enum_2Enum ) @ V9f ) @ c_2Enum_2ESUC ) ) @ V6t1 ) @ V5t2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAP2i__NIL2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2i @ A_27a @ A_27b @ A_27c ) @ V0f ) @ V1l1 ) @ ( c_2Elist_2ENIL @ A_27c ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2EindexedLists_2ELENGTH__MAP2i,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( surj__ty_2Enum_2Enum @ ( ap @ ( c_2Elist_2ELENGTH @ A_27c ) @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2i @ A_27c @ A_27a @ A_27b ) @ V0f ) @ V1l1 ) @ V2l2 ) ) )
= ( surj__ty_2Enum_2Enum @ ( ap @ ( ap @ c_2Earithmetic_2EMIN @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EEL__MAP2i,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V3n: tp__ty_2Enum_2Enum] :
( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27a ) @ V1l1 ) ) )
& ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( c_2Elist_2ELENGTH @ A_27b ) @ V2l2 ) ) ) )
=> ( ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27c ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2i @ A_27c @ A_27a @ A_27b ) @ V0f ) @ V1l1 ) @ V2l2 ) )
= ( ap @ ( ap @ ( ap @ V0f @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27a ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V1l1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2EEL @ A_27b ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ V2l2 ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAP2ia__ind,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0P: $i] :
( ( mem @ V0P @ ( arr @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ( ty_2Elist_2Elist @ A_27b ) @ ( arr @ ( ty_2Elist_2Elist @ A_27c ) @ bool ) ) ) ) )
=> ( ( ! [V1f: $i] :
( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V2i: tp__ty_2Enum_2Enum,V3v0: $i] :
( ( mem @ V3v0 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2i ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V3v0 ) ) ) )
& ! [V4f: $i] :
( ( mem @ V4f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V5i: tp__ty_2Enum_2Enum,V6v7: $i] :
( ( mem @ V6v7 @ A_27b )
=> ! [V7v8: $i] :
( ( mem @ V7v8 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V4f ) @ ( inj__ty_2Enum_2Enum @ V5i ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V6v7 ) @ V7v8 ) ) @ ( c_2Elist_2ENIL @ A_27c ) ) ) ) ) )
& ! [V8f: $i] :
( ( mem @ V8f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V9i: tp__ty_2Enum_2Enum,V10h1: $i] :
( ( mem @ V10h1 @ A_27b )
=> ! [V11t1: $i] :
( ( mem @ V11t1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V12h2: $i] :
( ( mem @ V12h2 @ A_27c )
=> ! [V13t2: $i] :
( ( mem @ V13t2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V8f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V11t1 ) @ V13t2 ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V8f ) @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V10h1 ) @ V11t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V12h2 ) @ V13t2 ) ) ) ) ) ) ) ) ) )
=> ! [V14v: $i] :
( ( mem @ V14v @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V15v1: tp__ty_2Enum_2Enum,V16v2: $i] :
( ( mem @ V16v2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V17v3: $i] :
( ( mem @ V17v3 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( ap @ V0P @ V14v ) @ ( inj__ty_2Enum_2Enum @ V15v1 ) ) @ V16v2 ) @ V17v3 ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAP2ia__def,axiom,
! [A_27a: del,A_27b: del,A_27c: del] :
( ! [V0v0: $i] :
( ( mem @ V0v0 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V1i: tp__ty_2Enum_2Enum,V2f: $i] :
( ( mem @ V2f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2ia @ A_27a @ A_27b @ A_27c ) @ V2f ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) @ V0v0 )
= ( c_2Elist_2ENIL @ A_27a ) ) ) )
& ! [V3v8: $i] :
( ( mem @ V3v8 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V4v7: $i] :
( ( mem @ V4v7 @ A_27b )
=> ! [V5i: tp__ty_2Enum_2Enum,V6f: $i] :
( ( mem @ V6f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2ia @ A_27a @ A_27b @ A_27c ) @ V6f ) @ ( inj__ty_2Enum_2Enum @ V5i ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V4v7 ) @ V3v8 ) ) @ ( c_2Elist_2ENIL @ A_27c ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) )
& ! [V7t2: $i] :
( ( mem @ V7t2 @ ( ty_2Elist_2Elist @ A_27c ) )
=> ! [V8t1: $i] :
( ( mem @ V8t1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ! [V9i: tp__ty_2Enum_2Enum,V10h2: $i] :
( ( mem @ V10h2 @ A_27c )
=> ! [V11h1: $i] :
( ( mem @ V11h1 @ A_27b )
=> ! [V12f: $i] :
( ( mem @ V12f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2ia @ A_27a @ A_27b @ A_27c ) @ V12f ) @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27b ) @ V11h1 ) @ V8t1 ) ) @ ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27c ) @ V10h2 ) @ V7t2 ) )
= ( ap @ ( ap @ ( c_2Elist_2ECONS @ A_27a ) @ ( ap @ ( ap @ ( ap @ V12f @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ V11h1 ) @ V10h2 ) ) @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2ia @ A_27a @ A_27b @ A_27c ) @ V12f ) @ ( ap @ ( ap @ c_2Earithmetic_2E_2B @ ( inj__ty_2Enum_2Enum @ V9i ) ) @ ( ap @ c_2Earithmetic_2ENUMERAL @ ( ap @ c_2Earithmetic_2EBIT1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Earithmetic_2EZERO ) ) ) ) ) @ V8t1 ) @ V7t2 ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAP2ia__NIL2,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27b @ ( arr @ A_27c @ A_27a ) ) ) )
=> ! [V1i: tp__ty_2Enum_2Enum,V2l1: $i] :
( ( mem @ V2l1 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2ia @ A_27a @ A_27b @ A_27c ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ V1i ) ) @ V2l1 ) @ ( c_2Elist_2ENIL @ A_27c ) )
= ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(conj_thm_2EindexedLists_2EMAP2i__compute,axiom,
! [A_27a: del,A_27b: del,A_27c: del,V0f: $i] :
( ( mem @ V0f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ ( arr @ A_27b @ A_27c ) ) ) )
=> ! [V1l1: $i] :
( ( mem @ V1l1 @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2l2: $i] :
( ( mem @ V2l2 @ ( ty_2Elist_2Elist @ A_27b ) )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2i @ A_27c @ A_27a @ A_27b ) @ V0f ) @ V1l1 ) @ V2l2 )
= ( ap @ ( ap @ ( ap @ ( ap @ ( c_2EindexedLists_2EMAP2ia @ A_27c @ A_27a @ A_27b ) @ V0f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ V1l1 ) @ V2l2 ) ) ) ) ) ).
%------------------------------------------------------------------------------