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 : 87 ( 11 unt; 19 typ; 0 def)
% Number of atoms : 2122 ( 82 equ)
% Maximal formula atoms : 19 ( 24 avg)
% Number of connectives : 338 ( 0 ~; 2 |; 49 &)
% ( 8 <=>; 279 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 11 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 1716 (1716 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 70 ( 19 >; 51 *; 0 +; 0 <<)
% Number of predicates : 46 ( 45 usr; 17 prp; 0-3 aty)
% Number of functors : 19 ( 19 usr; 0 con; 1-9 aty)
% Number of variables : 419 ( 411 !; 8 ?; 419 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2EindexedLists_2EFOLDRi,type,
c_2EindexedLists_2EFOLDRi: ( del * del ) > $i ).
tff(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)))) ).
tff(tp_c_2EindexedLists_2ELIST__RELi,type,
c_2EindexedLists_2ELIST__RELi: ( del * del ) > $i ).
tff(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)))) ).
tff(tp_c_2EindexedLists_2EMAP2i,type,
c_2EindexedLists_2EMAP2i: ( del * del * del ) > $i ).
tff(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))))) ).
tff(tp_c_2EindexedLists_2EMAP2ia,type,
c_2EindexedLists_2EMAP2ia: ( del * del * del ) > $i ).
tff(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)))))) ).
tff(tp_c_2EindexedLists_2EMAPi,type,
c_2EindexedLists_2EMAPi: ( del * del ) > $i ).
tff(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)))) ).
tff(tp_c_2EindexedLists_2EMAPi__ACC,type,
c_2EindexedLists_2EMAPi__ACC: ( del * del ) > $i ).
tff(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)))))) ).
tff(tp_c_2EindexedLists_2EdelN,type,
c_2EindexedLists_2EdelN: del > $i ).
tff(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)))) ).
tff(tp_c_2EindexedLists_2Efindi,type,
c_2EindexedLists_2Efindi: del > $i ).
tff(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))) ).
tff(tp_c_2EindexedLists_2EfupdLast,type,
c_2EindexedLists_2EfupdLast: del > $i ).
tff(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)))) ).
tff(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)) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ).
tff(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) ) ) ) ).
tff(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))) ) ) ) ).
tff(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)) ) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ).
tff(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) ) ) ) ) ).
tff(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)) ) ) ) ) ).
tff(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)) ) ) ) ) ).
tff(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)) ) ) ) ).
tff(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)) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ) ) ) ).
tff(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)))) ) ) ) ) ) ).
tff(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))) ) ) ) ).
tff(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 ) ) ) ).
tff(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 ) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(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))))) ) ) ) ).
tff(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) ) ) ) ).
tff(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) ) ) ) ).
tff(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)) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ).
tff(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) ) ) ) ) ) ).
tff(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)))) ) ) ) ).
tff(lamtp_f868,type,
f868: ( del * del * $i * $i * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f868,axiom,
! [A_27a: del,A_27b: del,V7h2: $i] :
( mem(V7h2,A_27b)
=> ! [V5a1: $i] :
( mem(V5a1,ty_2Elist_2Elist(A_27b))
=> ! [V0R: $i] :
( mem(V0R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27b,bool))))
=> ! [V3LIST__RELi_27: $i] :
( mem(V3LIST__RELi_27,arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),bool)))
=> ! [V4a0: $i] :
( mem(V4a0,ty_2Elist_2Elist(A_27a))
=> ! [V8l1: $i] :
( mem(V8l1,ty_2Elist_2Elist(A_27a))
=> ! [V6h1: $i] :
( mem(V6h1,A_27a)
=> ! [V9l2: $i] : ( ap(f868(A_27a,A_27b,V7h2,V5a1,V0R,V3LIST__RELi_27,V4a0,V8l1,V6h1),V9l2) = 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)))) ) ) ) ) ) ) ) ) ).
tff(lamtp_f869,type,
f869: ( del * del * $i * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f869,axiom,
! [A_27b: del,A_27a: del,V6h1: $i] :
( mem(V6h1,A_27a)
=> ! [V4a0: $i] :
( mem(V4a0,ty_2Elist_2Elist(A_27a))
=> ! [V3LIST__RELi_27: $i] :
( mem(V3LIST__RELi_27,arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),bool)))
=> ! [V0R: $i] :
( mem(V0R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27b,bool))))
=> ! [V5a1: $i] :
( mem(V5a1,ty_2Elist_2Elist(A_27b))
=> ! [V7h2: $i] :
( mem(V7h2,A_27b)
=> ! [V8l1: $i] : ( ap(f869(A_27b,A_27a,V6h1,V4a0,V3LIST__RELi_27,V0R,V5a1,V7h2),V8l1) = ap(c_2Ebool_2E_3F(ty_2Elist_2Elist(A_27b)),f868(A_27a,A_27b,V7h2,V5a1,V0R,V3LIST__RELi_27,V4a0,V8l1,V6h1)) ) ) ) ) ) ) ) ).
tff(lamtp_f870,type,
f870: ( del * del * $i * $i * $i * $i * $i ) > $i ).
tff(lameq_f870,axiom,
! [A_27a: del,A_27b: del,V5a1: $i] :
( mem(V5a1,ty_2Elist_2Elist(A_27b))
=> ! [V0R: $i] :
( mem(V0R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27b,bool))))
=> ! [V3LIST__RELi_27: $i] :
( mem(V3LIST__RELi_27,arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),bool)))
=> ! [V4a0: $i] :
( mem(V4a0,ty_2Elist_2Elist(A_27a))
=> ! [V6h1: $i] :
( mem(V6h1,A_27a)
=> ! [V7h2: $i] : ( ap(f870(A_27a,A_27b,V5a1,V0R,V3LIST__RELi_27,V4a0,V6h1),V7h2) = ap(c_2Ebool_2E_3F(ty_2Elist_2Elist(A_27a)),f869(A_27b,A_27a,V6h1,V4a0,V3LIST__RELi_27,V0R,V5a1,V7h2)) ) ) ) ) ) ) ).
tff(lamtp_f871,type,
f871: ( del * del * $i * $i * $i * $i ) > $i ).
tff(lameq_f871,axiom,
! [A_27b: del,A_27a: del,V4a0: $i] :
( mem(V4a0,ty_2Elist_2Elist(A_27a))
=> ! [V3LIST__RELi_27: $i] :
( mem(V3LIST__RELi_27,arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),bool)))
=> ! [V0R: $i] :
( mem(V0R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27b,bool))))
=> ! [V5a1: $i] :
( mem(V5a1,ty_2Elist_2Elist(A_27b))
=> ! [V6h1: $i] : ( ap(f871(A_27b,A_27a,V4a0,V3LIST__RELi_27,V0R,V5a1),V6h1) = ap(c_2Ebool_2E_3F(A_27b),f870(A_27a,A_27b,V5a1,V0R,V3LIST__RELi_27,V4a0,V6h1)) ) ) ) ) ) ).
tff(lamtp_f872,type,
f872: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f872,axiom,
! [A_27a: del,A_27b: del,V0R: $i] :
( mem(V0R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27b,bool))))
=> ! [V4a0: $i] :
( mem(V4a0,ty_2Elist_2Elist(A_27a))
=> ! [V3LIST__RELi_27: $i] :
( mem(V3LIST__RELi_27,arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),bool)))
=> ! [V5a1: $i] : ( ap(f872(A_27a,A_27b,V0R,V4a0,V3LIST__RELi_27),V5a1) = 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),f871(A_27b,A_27a,V4a0,V3LIST__RELi_27,V0R,V5a1)))),ap(ap(V3LIST__RELi_27,V4a0),V5a1)) ) ) ) ) ).
tff(lamtp_f873,type,
f873: ( del * del * $i * $i ) > $i ).
tff(lameq_f873,axiom,
! [A_27b: del,A_27a: del,V3LIST__RELi_27: $i] :
( mem(V3LIST__RELi_27,arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),bool)))
=> ! [V0R: $i] :
( mem(V0R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27b,bool))))
=> ! [V4a0: $i] : ( ap(f873(A_27b,A_27a,V3LIST__RELi_27,V0R),V4a0) = ap(c_2Ebool_2E_21(ty_2Elist_2Elist(A_27b)),f872(A_27a,A_27b,V0R,V4a0,V3LIST__RELi_27)) ) ) ) ).
tff(lamtp_f874,type,
f874: ( del * del * $i * $i * $i ) > $i ).
tff(lameq_f874,axiom,
! [A_27a: del,A_27b: del,V2a1: $i] :
( mem(V2a1,ty_2Elist_2Elist(A_27b))
=> ! [V1a0: $i] :
( mem(V1a0,ty_2Elist_2Elist(A_27a))
=> ! [V0R: $i] :
( mem(V0R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27b,bool))))
=> ! [V3LIST__RELi_27: $i] : ( ap(f874(A_27a,A_27b,V2a1,V1a0,V0R),V3LIST__RELi_27) = ap(ap(c_2Emin_2E_3D_3D_3E,ap(c_2Ebool_2E_21(ty_2Elist_2Elist(A_27a)),f873(A_27b,A_27a,V3LIST__RELi_27,V0R))),ap(ap(V3LIST__RELi_27,V1a0),V2a1)) ) ) ) ) ).
tff(lamtp_f875,type,
f875: ( del * del * $i * $i ) > $i ).
tff(lameq_f875,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] : ( ap(f875(A_27a,A_27b,V0R,V1a0),V2a1) = ap(c_2Ebool_2E_21(arr(ty_2Elist_2Elist(A_27a),arr(ty_2Elist_2Elist(A_27b),bool))),f874(A_27a,A_27b,V2a1,V1a0,V0R)) ) ) ) ).
tff(lamtp_f876,type,
f876: ( del * del * $i ) > $i ).
tff(lameq_f876,axiom,
! [A_27b: del,A_27a: del,V0R: $i] :
( mem(V0R,arr(ty_2Enum_2Enum,arr(A_27a,arr(A_27b,bool))))
=> ! [V1a0: $i] : ( ap(f876(A_27b,A_27a,V0R),V1a0) = f875(A_27a,A_27b,V0R,V1a0) ) ) ).
tff(lamtp_f877,type,
f877: ( del * del ) > $i ).
tff(lameq_f877,axiom,
! [A_27b: del,A_27a: del,V0R: $i] : ( ap(f877(A_27b,A_27a),V0R) = f876(A_27b,A_27a,V0R) ) ).
tff(ax_thm_2EindexedLists_2ELIST__RELi__def,axiom,
! [A_27a: del,A_27b: del] : ( c_2EindexedLists_2ELIST__RELi(A_27a,A_27b) = f877(A_27b,A_27a) ) ).
tff(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))))) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ) ) ).
tff(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))) ) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ).
tff(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) ) ) ) ).
tff(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))) ) ) ) ) ).
tff(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)) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ).
tff(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)) ) ) ) ) ) ) ) ).
tff(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) ) ) ) ).
tff(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) ) ) ) ) ).
%------------------------------------------------------------------------------