ITP001 Axioms: ITP015^5.ax


%------------------------------------------------------------------------------
% File     : ITP015^5 : 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    : prim_rec^2.ax [Gau20]
%          : HL4015^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   64 (   5 unt;   9 typ;   0 def)
%            Number of atoms       :  824 (  31 equ;   0 cnn)
%            Maximal formula atoms :   38 (  12 avg)
%            Number of connectives : 1236 (   8   ~;   3   |;  23   &;1135   @)
%                                         (  10 <=>;  57  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   26 (  10 avg;1135 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    7 (   7   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   34 (  33 usr;  26 con; 0-2 aty)
%            Number of variables   :  149 (  20   ^ 122   !;   7   ?; 149   :)
% SPC      : TH0_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2Eprim__rec_2E_3C,type,
    c_2Eprim__rec_2E_3C: $i ).

thf(mem_c_2Eprim__rec_2E_3C,axiom,
    mem @ c_2Eprim__rec_2E_3C @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ bool ) ) ).

thf(tp_c_2Eprim__rec_2EPRE,type,
    c_2Eprim__rec_2EPRE: $i ).

thf(mem_c_2Eprim__rec_2EPRE,axiom,
    mem @ c_2Eprim__rec_2EPRE @ ( arr @ ty_2Enum_2Enum @ ty_2Enum_2Enum ) ).

thf(stp_fo_c_2Eprim__rec_2EPRE,type,
    fo__c_2Eprim__rec_2EPRE: tp__ty_2Enum_2Enum > tp__ty_2Enum_2Enum ).

thf(stp_eq_fo_c_2Eprim__rec_2EPRE,axiom,
    ! [X0: tp__ty_2Enum_2Enum] :
      ( ( inj__ty_2Enum_2Enum @ ( fo__c_2Eprim__rec_2EPRE @ X0 ) )
      = ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ X0 ) ) ) ).

thf(tp_c_2Eprim__rec_2EPRIM__REC,type,
    c_2Eprim__rec_2EPRIM__REC: del > $i ).

thf(mem_c_2Eprim__rec_2EPRIM__REC,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).

thf(tp_c_2Eprim__rec_2EPRIM__REC__FUN,type,
    c_2Eprim__rec_2EPRIM__REC__FUN: del > $i ).

thf(mem_c_2Eprim__rec_2EPRIM__REC__FUN,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) @ ( arr @ ty_2Enum_2Enum @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ) ) ).

thf(tp_c_2Eprim__rec_2ESIMP__REC,type,
    c_2Eprim__rec_2ESIMP__REC: del > $i ).

thf(mem_c_2Eprim__rec_2ESIMP__REC,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) ) ) ).

thf(tp_c_2Eprim__rec_2ESIMP__REC__REL,type,
    c_2Eprim__rec_2ESIMP__REC__REL: del > $i ).

thf(mem_c_2Eprim__rec_2ESIMP__REC__REL,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ ( arr @ ( arr @ ty_2Enum_2Enum @ A_27a ) @ ( arr @ A_27a @ ( arr @ ( arr @ A_27a @ A_27a ) @ ( arr @ ty_2Enum_2Enum @ bool ) ) ) ) ) ).

thf(tp_c_2Eprim__rec_2Emeasure,type,
    c_2Eprim__rec_2Emeasure: del > $i ).

thf(mem_c_2Eprim__rec_2Emeasure,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2Emeasure @ A_27a ) @ ( arr @ ( arr @ A_27a @ ty_2Enum_2Enum ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).

thf(tp_c_2Eprim__rec_2Ewellfounded,type,
    c_2Eprim__rec_2Ewellfounded: del > $i ).

thf(mem_c_2Eprim__rec_2Ewellfounded,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Eprim__rec_2Ewellfounded @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ bool ) ) ).

thf(ax_thm_2Eprim__rec_2ELESS__DEF,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
    <=> ? [V2P: $i] :
          ( ( mem @ V2P @ ( arr @ ty_2Enum_2Enum @ bool ) )
          & ! [V3n: tp__ty_2Enum_2Enum] :
              ( ( p @ ( ap @ V2P @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
             => ( p @ ( ap @ V2P @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) )
          & ( p @ ( ap @ V2P @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
          & ~ ( p @ ( ap @ V2P @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EINV__SUC__EQ,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
        = ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
    <=> ( V0m = V1n ) ) ).

thf(ax_thm_2Eprim__rec_2EPRE__DEF,axiom,
    ! [V0m: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
      = ( surj__ty_2Enum_2Enum
        @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ty_2Enum_2Enum ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
          @ ( ap @ ( c_2Emin_2E_40 @ ty_2Enum_2Enum )
            @ ( lam @ ty_2Enum_2Enum
              @ ^ [V1n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ V1n ) ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EPRE,axiom,
    ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) )
      = fo__c_2Enum_2E0 )
    & ! [V0m: tp__ty_2Enum_2Enum] :
        ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Eprim__rec_2EPRE @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) )
        = V0m ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__REFL,axiom,
    ! [V0n: tp__ty_2Enum_2Enum] :
      ~ ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ).

thf(conj_thm_2Eprim__rec_2ESUC__LESS,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
     => ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ENOT__LESS__0,axiom,
    ! [V0n: tp__ty_2Enum_2Enum] :
      ~ ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0n ) ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__0,axiom,
    ! [V0n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__0__0,axiom,
    p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__MONO,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
     => ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__MONO__REV,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
     => ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__MONO__EQ,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
    <=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ETC__IM__RTC__SUC,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p
        @ ( ap
          @ ( ap
            @ ( ap @ ( c_2Erelation_2ETC @ ty_2Enum_2Enum )
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V2x: $i] :
                    ( lam @ ty_2Enum_2Enum
                    @ ^ [V3y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V3y ) @ ( ap @ c_2Enum_2ESUC @ V2x ) ) ) ) )
            @ ( inj__ty_2Enum_2Enum @ V0m ) )
          @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
    <=> ( p
        @ ( ap
          @ ( ap
            @ ( ap @ ( c_2Erelation_2ERTC @ ty_2Enum_2Enum )
              @ ( lam @ ty_2Enum_2Enum
                @ ^ [V4x: $i] :
                    ( lam @ ty_2Enum_2Enum
                    @ ^ [V5y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V5y ) @ ( ap @ c_2Enum_2ESUC @ V4x ) ) ) ) )
            @ ( inj__ty_2Enum_2Enum @ V0m ) )
          @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ERTC__IM__TC,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ A_27a ) )
     => ! [V1m: $i] :
          ( ( mem @ V1m @ A_27a )
         => ! [V2n: $i] :
              ( ( mem @ V2n @ A_27a )
             => ( ( p
                  @ ( ap
                    @ ( ap
                      @ ( ap @ ( c_2Erelation_2ERTC @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V3x: $i] :
                              ( lam @ A_27a
                              @ ^ [V4y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V4y ) @ ( ap @ V0f @ V3x ) ) ) ) )
                      @ ( ap @ V0f @ V1m ) )
                    @ V2n ) )
              <=> ( p
                  @ ( ap
                    @ ( ap
                      @ ( ap @ ( c_2Erelation_2ETC @ A_27a )
                        @ ( lam @ A_27a
                          @ ^ [V5x: $i] :
                              ( lam @ A_27a
                              @ ^ [V6y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V6y ) @ ( ap @ V0f @ V5x ) ) ) ) )
                      @ V1m )
                    @ V2n ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__ALT,axiom,
    ( c_2Eprim__rec_2E_3C
    = ( ap @ ( c_2Erelation_2ETC @ ty_2Enum_2Enum )
      @ ( lam @ ty_2Enum_2Enum
        @ ^ [V0x: $i] :
            ( lam @ ty_2Enum_2Enum
            @ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1y ) @ ( ap @ c_2Enum_2ESUC @ V0x ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__SUC__REFL,axiom,
    ! [V0n: 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 @ V0n ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__SUC,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
     => ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__LEMMA1,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
     => ( ( V0m = V1n )
        | ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__LEMMA2,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( ( V0m = V1n )
        | ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
     => ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__THM,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
    <=> ( ( V0m = V1n )
        | ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__SUC__IMP,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) )
     => ( ( V0m != V1n )
       => ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EEQ__LESS,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) )
        = V1n )
     => ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ESUC__ID,axiom,
    ! [V0n: tp__ty_2Enum_2Enum] :
      ( ( surj__ty_2Enum_2Enum @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0n ) ) )
     != V0n ) ).

thf(conj_thm_2Eprim__rec_2ENOT__LESS__EQ,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( V0m = V1n )
     => ~ ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__NOT__EQ,axiom,
    ! [V0m: tp__ty_2Enum_2Enum,V1n: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( inj__ty_2Enum_2Enum @ V1n ) ) )
     => ( V0m != V1n ) ) ).

thf(ax_thm_2Eprim__rec_2ESIMP__REC__REL,axiom,
    ! [A_27a: del,V0fun: $i] :
      ( ( mem @ V0fun @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2f: $i] :
              ( ( mem @ V2f @ ( arr @ A_27a @ A_27a ) )
             => ! [V3n: tp__ty_2Enum_2Enum] :
                  ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V0fun ) @ V1x ) @ V2f ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
                <=> ( ( ( ap @ V0fun @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
                      = V1x )
                    & ! [V4m: tp__ty_2Enum_2Enum] :
                        ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V4m ) ) @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
                       => ( ( ap @ V0fun @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V4m ) ) )
                          = ( ap @ V2f @ ( ap @ V0fun @ ( inj__ty_2Enum_2Enum @ V4m ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ESIMP__REC__EXISTS,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ! [V2n: tp__ty_2Enum_2Enum] :
            ? [V3fun: $i] :
              ( ( mem @ V3fun @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
              & ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V3fun ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ESIMP__REC__REL__UNIQUE,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ! [V2g1: $i] :
              ( ( mem @ V2g1 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
             => ! [V3g2: $i] :
                  ( ( mem @ V3g2 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
                 => ! [V4m1: tp__ty_2Enum_2Enum,V5m2: tp__ty_2Enum_2Enum] :
                      ( ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V2g1 ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V4m1 ) ) )
                        & ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V3g2 ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V5m2 ) ) ) )
                     => ! [V6n: tp__ty_2Enum_2Enum] :
                          ( ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( inj__ty_2Enum_2Enum @ V4m1 ) ) )
                            & ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V6n ) ) @ ( inj__ty_2Enum_2Enum @ V5m2 ) ) ) )
                         => ( ( ap @ V2g1 @ ( inj__ty_2Enum_2Enum @ V6n ) )
                            = ( ap @ V3g2 @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ESIMP__REC__REL__UNIQUE__RESULT,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ! [V2n: tp__ty_2Enum_2Enum] :
              ( p
              @ ( ap @ ( c_2Ebool_2E_3F_21 @ A_27a )
                @ ( lam @ A_27a
                  @ ^ [V3y: $i] :
                      ( ap @ ( c_2Ebool_2E_3F @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
                      @ ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
                        @ ^ [V4g: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V4g ) @ V0x ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ V3y ) @ ( ap @ V4g @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Eprim__rec_2ESIMP__REC,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f_27: $i] :
          ( ( mem @ V1f_27 @ ( arr @ A_27a @ A_27a ) )
         => ! [V2n: tp__ty_2Enum_2Enum] :
            ? [V3g: $i] :
              ( ( mem @ V3g @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
              & ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC__REL @ A_27a ) @ V3g ) @ V0x ) @ V1f_27 ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) )
              & ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ V0x ) @ V1f_27 ) @ ( inj__ty_2Enum_2Enum @ V2n ) )
                = ( ap @ V3g @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ELESS__SUC__SUC,axiom,
    ! [V0m: tp__ty_2Enum_2Enum] :
      ( ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) )
      & ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( inj__ty_2Enum_2Enum @ V0m ) ) @ ( ap @ c_2Enum_2ESUC @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V0m ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2ESIMP__REC__THM,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ A_27a ) )
         => ( ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
              = V0x )
            & ! [V2m: tp__ty_2Enum_2Enum] :
                ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ V0x ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
                = ( ap @ V1f @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ) ).

thf(ax_thm_2Eprim__rec_2EPRIM__REC__FUN,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
         => ( ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f )
            = ( ap
              @ ( ap @ ( c_2Eprim__rec_2ESIMP__REC @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
                @ ( lam @ ty_2Enum_2Enum
                  @ ^ [V2n: $i] : V0x ) )
              @ ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
                @ ^ [V3fun: $i] :
                    ( lam @ ty_2Enum_2Enum
                    @ ^ [V4n: $i] : ( ap @ ( ap @ V1f @ ( ap @ V3fun @ ( ap @ c_2Eprim__rec_2EPRE @ V4n ) ) ) @ V4n ) ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EPRIM__REC__EQN,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
         => ( ! [V2n: tp__ty_2Enum_2Enum] :
                ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) @ ( inj__ty_2Enum_2Enum @ V2n ) )
                = V0x )
            & ! [V3m: tp__ty_2Enum_2Enum,V4n: tp__ty_2Enum_2Enum] :
                ( ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3m ) ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) )
                = ( ap @ ( ap @ V1f @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V3m ) ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) @ ( inj__ty_2Enum_2Enum @ V4n ) ) ) ) ) ) ).

thf(ax_thm_2Eprim__rec_2EPRIM__REC,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
         => ! [V2m: tp__ty_2Enum_2Enum] :
              ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2m ) )
              = ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC__FUN @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) @ ( ap @ c_2Eprim__rec_2EPRE @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EPRIM__REC__THM,axiom,
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
         => ( ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
              = V0x )
            & ! [V2m: tp__ty_2Enum_2Enum] :
                ( ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ V0x ) @ V1f ) @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2m ) ) )
                = ( ap @ ( ap @ V1f @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2EPRIM__REC @ A_27a ) @ V0x ) @ V1f ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) @ ( inj__ty_2Enum_2Enum @ V2m ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EDC,axiom,
    ! [A_27a: del,V0P: $i] :
      ( ( mem @ V0P @ ( arr @ A_27a @ bool ) )
     => ! [V1R: $i] :
          ( ( mem @ V1R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
         => ! [V2a: $i] :
              ( ( mem @ V2a @ A_27a )
             => ( ( ( p @ ( ap @ V0P @ V2a ) )
                  & ! [V3x: $i] :
                      ( ( mem @ V3x @ A_27a )
                     => ( ( p @ ( ap @ V0P @ V3x ) )
                       => ? [V4y: $i] :
                            ( ( mem @ V4y @ A_27a )
                            & ( p @ ( ap @ V0P @ V4y ) )
                            & ( p @ ( ap @ ( ap @ V1R @ V3x ) @ V4y ) ) ) ) ) )
               => ? [V5f: $i] :
                    ( ( mem @ V5f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
                    & ( ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
                      = V2a )
                    & ! [V6n: tp__ty_2Enum_2Enum] :
                        ( ( p @ ( ap @ V0P @ ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) )
                        & ( p @ ( ap @ ( ap @ V1R @ ( ap @ V5f @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) @ ( ap @ V5f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V6n ) ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2Enum__Axiom__old,axiom,
    ! [A_27a: del,V0e: $i] :
      ( ( mem @ V0e @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ A_27a @ ( arr @ ty_2Enum_2Enum @ A_27a ) ) )
         => ( p
            @ ( ap @ ( c_2Ebool_2E_3F_21 @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
              @ ( lam @ ( arr @ ty_2Enum_2Enum @ A_27a )
                @ ^ [V2fn1: $i] :
                    ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V2fn1 @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) ) ) @ V0e ) )
                    @ ( ap @ ( c_2Ebool_2E_21 @ ty_2Enum_2Enum )
                      @ ( lam @ ty_2Enum_2Enum
                        @ ^ [V3n: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ A_27a ) @ ( ap @ V2fn1 @ ( ap @ c_2Enum_2ESUC @ V3n ) ) ) @ ( ap @ ( ap @ V1f @ ( ap @ V2fn1 @ V3n ) ) @ V3n ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2Enum__Axiom,axiom,
    ! [A_27a: del,V0e: $i] :
      ( ( mem @ V0e @ A_27a )
     => ! [V1f: $i] :
          ( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ ( arr @ A_27a @ A_27a ) ) )
         => ? [V2fn: $i] :
              ( ( mem @ V2fn @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
              & ( ( ap @ V2fn @ ( inj__ty_2Enum_2Enum @ fo__c_2Enum_2E0 ) )
                = V0e )
              & ! [V3n: tp__ty_2Enum_2Enum] :
                  ( ( ap @ V2fn @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V3n ) ) )
                  = ( ap @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V3n ) ) @ ( ap @ V2fn @ ( inj__ty_2Enum_2Enum @ V3n ) ) ) ) ) ) ) ).

thf(ax_thm_2Eprim__rec_2Ewellfounded__def,axiom,
    ! [A_27a: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ( ( p @ ( ap @ ( c_2Eprim__rec_2Ewellfounded @ A_27a ) @ V0R ) )
      <=> ~ ? [V1f: $i] :
              ( ( mem @ V1f @ ( arr @ ty_2Enum_2Enum @ A_27a ) )
              & ! [V2n: tp__ty_2Enum_2Enum] : ( p @ ( ap @ ( ap @ V0R @ ( ap @ V1f @ ( ap @ c_2Enum_2ESUC @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) @ ( ap @ V1f @ ( inj__ty_2Enum_2Enum @ V2n ) ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EWF__IFF__WELLFOUNDED,axiom,
    ! [A_27a: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ( ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ V0R ) )
      <=> ( p @ ( ap @ ( c_2Eprim__rec_2Ewellfounded @ A_27a ) @ V0R ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EWF__PRED,axiom,
    ( p
    @ ( ap @ ( c_2Erelation_2EWF @ ty_2Enum_2Enum )
      @ ( lam @ ty_2Enum_2Enum
        @ ^ [V0x: $i] :
            ( lam @ ty_2Enum_2Enum
            @ ^ [V1y: $i] : ( ap @ ( ap @ ( c_2Emin_2E_3D @ ty_2Enum_2Enum ) @ V1y ) @ ( ap @ c_2Enum_2ESUC @ V0x ) ) ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2EWF__LESS,axiom,
    p @ ( ap @ ( c_2Erelation_2EWF @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) ).

thf(ax_thm_2Eprim__rec_2Emeasure__def,axiom,
    ! [A_27a: del] :
      ( ( c_2Eprim__rec_2Emeasure @ A_27a )
      = ( ap @ ( c_2Erelation_2Einv__image @ A_27a @ ty_2Enum_2Enum ) @ c_2Eprim__rec_2E_3C ) ) ).

thf(conj_thm_2Eprim__rec_2EWF__measure,axiom,
    ! [A_27a: del,V0m: $i] :
      ( ( mem @ V0m @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
     => ( p @ ( ap @ ( c_2Erelation_2EWF @ A_27a ) @ ( ap @ ( c_2Eprim__rec_2Emeasure @ A_27a ) @ V0m ) ) ) ) ).

thf(conj_thm_2Eprim__rec_2Emeasure__thm,axiom,
    ! [A_27a: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ty_2Enum_2Enum ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ! [V2y: $i] :
              ( ( mem @ V2y @ A_27a )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Eprim__rec_2Emeasure @ A_27a ) @ V0f ) @ V1x ) @ V2y ) )
              <=> ( p @ ( ap @ ( ap @ c_2Eprim__rec_2E_3C @ ( ap @ V0f @ V1x ) ) @ ( ap @ V0f @ V2y ) ) ) ) ) ) ) ).

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