TPTP Problem File: ITP009^2.p

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%------------------------------------------------------------------------------
% File     : ITP009^2 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Problem  : HOL4 set theory export of thm_2Equotient_2EFUN__REL__EQ__REL.p, bushy mode
% Version  : [BG+19] axioms.
% English  :

% Refs     : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
%          : [Gau19] Gauthier (2019), Email to Geoff Sutcliffe
% Source   : [BG+19]
% Names    : thm_2Equotient_2EFUN__REL__EQ__REL.p [Gau19]
%          : HL404001^2.p [TPAP]

% Status   : Theorem
% Rating   : 0.75 v9.0.0, 0.80 v8.2.0, 0.92 v8.1.0, 0.82 v7.5.0
% Syntax   : Number of formulae    :   57 (   5 unt;  21 typ;   0 def)
%            Number of atoms       :  319 (  10 equ;   0 cnn)
%            Maximal formula atoms :   48 (   8 avg)
%            Number of connectives :  635 (   4   ~;   0   |;  18   &; 531   @)
%                                         (  22 <=>;  60  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   36 (   9 avg)
%            Number of types       :    3 (   1 usr)
%            Number of type conns  :   27 (  27   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   28 (  25 usr;  14 con; 0-4 aty)
%            Number of variables   :   86 (   0   ^;  86   !;   0   ?;  86   :)
% SPC      : TH0_THM_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Bugfixes in axioms and export.
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
thf(tp_c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $i ).

thf(mem_c_2Ebool_2E_7E,axiom,
    mem @ c_2Ebool_2E_7E @ ( arr @ bool @ bool ) ).

thf(ax_neg_p,axiom,
    ! [Q: $i] :
      ( ( mem @ Q @ bool )
     => ( ( p @ ( ap @ c_2Ebool_2E_7E @ Q ) )
      <=> ~ ( p @ Q ) ) ) ).

thf(tp_c_2Ebool_2EF,type,
    c_2Ebool_2EF: $i ).

thf(mem_c_2Ebool_2EF,axiom,
    mem @ c_2Ebool_2EF @ bool ).

thf(ax_false_p,axiom,
    ~ ( p @ c_2Ebool_2EF ) ).

thf(tp_c_2Ebool_2ET,type,
    c_2Ebool_2ET: $i ).

thf(mem_c_2Ebool_2ET,axiom,
    mem @ c_2Ebool_2ET @ bool ).

thf(ax_true_p,axiom,
    p @ c_2Ebool_2ET ).

thf(tp_c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $i ).

thf(mem_c_2Ebool_2E_2F_5C,axiom,
    mem @ c_2Ebool_2E_2F_5C @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).

thf(ax_and_p,axiom,
    ! [Q: $i] :
      ( ( mem @ Q @ bool )
     => ! [R: $i] :
          ( ( mem @ R @ bool )
         => ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ Q ) @ R ) )
          <=> ( ( p @ Q )
              & ( p @ R ) ) ) ) ) ).

thf(tp_c_2Equotient_2E_2D_2D_3E,type,
    c_2Equotient_2E_2D_2D_3E: del > del > del > del > $i ).

thf(mem_c_2Equotient_2E_2D_2D_3E,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del] : ( mem @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27b @ A_27c @ A_27d ) @ ( arr @ ( arr @ A_27a @ A_27c ) @ ( arr @ ( arr @ A_27b @ A_27d ) @ ( arr @ ( arr @ A_27c @ A_27b ) @ ( arr @ A_27a @ A_27d ) ) ) ) ) ).

thf(tp_c_2Equotient_2E_3D_3D_3D_3E,type,
    c_2Equotient_2E_3D_3D_3D_3E: del > del > $i ).

thf(mem_c_2Equotient_2E_3D_3D_3D_3E,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ bool ) ) ) ) ) ).

thf(tp_c_2Equotient_2EQUOTIENT,type,
    c_2Equotient_2EQUOTIENT: del > del > $i ).

thf(mem_c_2Equotient_2EQUOTIENT,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) @ ( arr @ ( arr @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ A_27a ) @ bool ) ) ) ) ).

thf(tp_c_2Emin_2E_3D_3D_3E,type,
    c_2Emin_2E_3D_3D_3E: $i ).

thf(mem_c_2Emin_2E_3D_3D_3E,axiom,
    mem @ c_2Emin_2E_3D_3D_3E @ ( arr @ bool @ ( arr @ bool @ bool ) ) ).

thf(ax_imp_p,axiom,
    ! [Q: $i] :
      ( ( mem @ Q @ bool )
     => ! [R: $i] :
          ( ( mem @ R @ bool )
         => ( ( p @ ( ap @ ( ap @ c_2Emin_2E_3D_3D_3E @ Q ) @ R ) )
          <=> ( ( p @ Q )
             => ( p @ R ) ) ) ) ) ).

thf(tp_c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: del > $i ).

thf(mem_c_2Ebool_2E_21,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Ebool_2E_21 @ A_27a ) @ ( arr @ ( arr @ A_27a @ bool ) @ bool ) ) ).

thf(ax_all_p,axiom,
    ! [A: del,Q: $i] :
      ( ( mem @ Q @ ( arr @ A @ bool ) )
     => ( ( p @ ( ap @ ( c_2Ebool_2E_21 @ A ) @ Q ) )
      <=> ! [X: $i] :
            ( ( mem @ X @ A )
           => ( p @ ( ap @ Q @ X ) ) ) ) ) ).

thf(tp_c_2Ecombin_2EW,type,
    c_2Ecombin_2EW: del > del > $i ).

thf(mem_c_2Ecombin_2EW,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ A_27b ) ) ) ).

thf(tp_c_2Equotient_2Erespects,type,
    c_2Equotient_2Erespects: del > del > $i ).

thf(mem_c_2Equotient_2Erespects,axiom,
    ! [A_27a: del,A_27b: del] : ( mem @ ( c_2Equotient_2Erespects @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) @ ( arr @ A_27a @ A_27b ) ) ) ).

thf(tp_c_2Emin_2E_3D,type,
    c_2Emin_2E_3D: del > $i ).

thf(mem_c_2Emin_2E_3D,axiom,
    ! [A_27a: del] : ( mem @ ( c_2Emin_2E_3D @ A_27a ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ).

thf(ax_eq_p,axiom,
    ! [A: del,X: $i] :
      ( ( mem @ X @ A )
     => ! [Y: $i] :
          ( ( mem @ Y @ A )
         => ( ( p @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ A ) @ X ) @ Y ) )
          <=> ( X = Y ) ) ) ) ).

thf(conj_thm_2Ebool_2ETRUTH,axiom,
    $true ).

thf(conj_thm_2Ebool_2EIMP__ANTISYM__AX,axiom,
    ! [V0t1: $i] :
      ( ( mem @ V0t1 @ bool )
     => ! [V1t2: $i] :
          ( ( mem @ V1t2 @ bool )
         => ( ( ( p @ V0t1 )
             => ( p @ V1t2 ) )
           => ( ( ( p @ V1t2 )
               => ( p @ V0t1 ) )
             => ( ( p @ V0t1 )
              <=> ( p @ V1t2 ) ) ) ) ) ) ).

thf(conj_thm_2Ebool_2EAND__CLAUSES,axiom,
    ! [V0t: $i] :
      ( ( mem @ V0t @ bool )
     => ( ( ( $true
            & ( p @ V0t ) )
        <=> ( p @ V0t ) )
        & ( ( ( p @ V0t )
            & $true )
        <=> ( p @ V0t ) )
        & ( ( $false
            & ( p @ V0t ) )
        <=> $false )
        & ( ( ( p @ V0t )
            & $false )
        <=> $false )
        & ( ( ( p @ V0t )
            & ( p @ V0t ) )
        <=> ( p @ V0t ) ) ) ) ).

thf(conj_thm_2Ebool_2EEQ__CLAUSES,axiom,
    ! [V0t: $i] :
      ( ( mem @ V0t @ bool )
     => ( ( ( $true
          <=> ( p @ V0t ) )
        <=> ( p @ V0t ) )
        & ( ( ( p @ V0t )
          <=> $true )
        <=> ( p @ V0t ) )
        & ( ( $false
          <=> ( p @ V0t ) )
        <=> ~ ( p @ V0t ) )
        & ( ( ( p @ V0t )
          <=> $false )
        <=> ~ ( p @ V0t ) ) ) ) ).

thf(conj_thm_2Ebool_2EAND__IMP__INTRO,axiom,
    ! [V0t1: $i] :
      ( ( mem @ V0t1 @ bool )
     => ! [V1t2: $i] :
          ( ( mem @ V1t2 @ bool )
         => ! [V2t3: $i] :
              ( ( mem @ V2t3 @ bool )
             => ( ( ( p @ V0t1 )
                 => ( ( p @ V1t2 )
                   => ( p @ V2t3 ) ) )
              <=> ( ( ( p @ V0t1 )
                    & ( p @ V1t2 ) )
                 => ( p @ V2t3 ) ) ) ) ) ) ).

thf(conj_thm_2Ecombin_2EW__THM,axiom,
    ! [A_27a: del,A_27b: del,V0f: $i] :
      ( ( mem @ V0f @ ( arr @ A_27a @ ( arr @ A_27a @ A_27b ) ) )
     => ! [V1x: $i] :
          ( ( mem @ V1x @ A_27a )
         => ( ( ap @ ( ap @ ( c_2Ecombin_2EW @ A_27a @ A_27b ) @ V0f ) @ V1x )
            = ( ap @ ( ap @ V0f @ V1x ) @ V1x ) ) ) ) ).

thf(conj_thm_2Equotient_2EQUOTIENT__REL,axiom,
    ! [A_27a: del,A_27b: del,V0R: $i] :
      ( ( mem @ V0R @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ! [V1abs: $i] :
          ( ( mem @ V1abs @ ( arr @ A_27a @ A_27b ) )
         => ! [V2rep: $i] :
              ( ( mem @ V2rep @ ( arr @ A_27b @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27b ) @ V0R ) @ V1abs ) @ V2rep ) )
               => ! [V3r: $i] :
                    ( ( mem @ V3r @ A_27a )
                   => ! [V4s: $i] :
                        ( ( mem @ V4s @ A_27a )
                       => ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V4s ) )
                        <=> ( ( p @ ( ap @ ( ap @ V0R @ V3r ) @ V3r ) )
                            & ( p @ ( ap @ ( ap @ V0R @ V4s ) @ V4s ) )
                            & ( ( ap @ V1abs @ V3r )
                              = ( ap @ V1abs @ V4s ) ) ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Equotient_2EFUN__QUOTIENT,axiom,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
      ( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ! [V1abs1: $i] :
          ( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
         => ! [V2rep1: $i] :
              ( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
               => ! [V3R2: $i] :
                    ( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
                   => ! [V4abs2: $i] :
                        ( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
                       => ! [V5rep2: $i] :
                            ( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
                           => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
                             => ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ ( arr @ A_27a @ A_27b ) @ ( arr @ A_27c @ A_27d ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) ) @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27a @ A_27d @ A_27c @ A_27b ) @ V1abs1 ) @ V5rep2 ) ) ) ) ) ) ) ) ) ) ) ).

thf(ax_thm_2Equotient_2Erespects__def,axiom,
    ! [A_27a: del,A_27b: del] :
      ( ( c_2Equotient_2Erespects @ A_27a @ A_27b )
      = ( c_2Ecombin_2EW @ A_27a @ A_27b ) ) ).

thf(conj_thm_2Equotient_2EFUN__REL__EQ__REL,conjecture,
    ! [A_27a: del,A_27b: del,A_27c: del,A_27d: del,V0R1: $i] :
      ( ( mem @ V0R1 @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) )
     => ! [V1abs1: $i] :
          ( ( mem @ V1abs1 @ ( arr @ A_27a @ A_27c ) )
         => ! [V2rep1: $i] :
              ( ( mem @ V2rep1 @ ( arr @ A_27c @ A_27a ) )
             => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27a @ A_27c ) @ V0R1 ) @ V1abs1 ) @ V2rep1 ) )
               => ! [V3R2: $i] :
                    ( ( mem @ V3R2 @ ( arr @ A_27b @ ( arr @ A_27b @ bool ) ) )
                   => ! [V4abs2: $i] :
                        ( ( mem @ V4abs2 @ ( arr @ A_27b @ A_27d ) )
                       => ! [V5rep2: $i] :
                            ( ( mem @ V5rep2 @ ( arr @ A_27d @ A_27b ) )
                           => ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2EQUOTIENT @ A_27b @ A_27d ) @ V3R2 ) @ V4abs2 ) @ V5rep2 ) )
                             => ! [V6f: $i] :
                                  ( ( mem @ V6f @ ( arr @ A_27a @ A_27b ) )
                                 => ! [V7g: $i] :
                                      ( ( mem @ V7g @ ( arr @ A_27a @ A_27b ) )
                                     => ( ( p @ ( ap @ ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) @ V6f ) @ V7g ) )
                                      <=> ( ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V6f ) )
                                          & ( p @ ( ap @ ( ap @ ( c_2Equotient_2Erespects @ ( arr @ A_27a @ A_27b ) @ bool ) @ ( ap @ ( ap @ ( c_2Equotient_2E_3D_3D_3D_3E @ A_27a @ A_27b ) @ V0R1 ) @ V3R2 ) ) @ V7g ) )
                                          & ( ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V6f )
                                            = ( ap @ ( ap @ ( ap @ ( c_2Equotient_2E_2D_2D_3E @ A_27c @ A_27b @ A_27a @ A_27d ) @ V2rep1 ) @ V4abs2 ) @ V7g ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

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