ITP001 Axioms: ITP040^5.ax


%------------------------------------------------------------------------------
% File     : ITP040^5 : TPTP v9.0.0. 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 ) ) ) ) ) ).

%------------------------------------------------------------------------------