TPTP Problem File: ITP010^2.p

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%------------------------------------------------------------------------------
% File     : ITP010^2 : TPTP v7.4.0. Released v7.4.0.
% Domain   : Interactive Theorem Proving
% Problem  : HOL4 set theory export of thm_2Ecardinal_2ECARD__NOT__LE.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_2Ecardinal_2ECARD__NOT__LE.p [Gau19]
%          : HL404501^2.p [TPAP]

% Status   : Theorem
% Rating   : 0.43 v7.4.0
% Syntax   : Number of formulae    :   58 (   1 unit;  18 type;   0 defn)
%            Number of atoms       :  484 (   7 equality; 194 variable)
%            Maximal formula depth :   16 (   6 average)
%            Number of connectives :  459 (  29   ~;  14   |;  14   &; 305   @)
%                                         (  33 <=>;  64  =>;   0  <=;   0 <~>)
%                                         (   0  ~|;   0  ~&)
%            Number of type conns  :   17 (  17   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   23 (  18   :;   0   =)
%            Number of variables   :   65 (   0 sgn;  65   !;   0   ?;   0   ^)
%                                         (  65   :;   0  !>;   0  ?*)
%                                         (   0  @-;   0  @+)
% SPC      : TH0_THM_EQU_NAR

% Comments :
%------------------------------------------------------------------------------
include('Axioms/ITP001/ITP001^2.ax').
%------------------------------------------------------------------------------
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_2Ecardinal_2Ecardleq,type,(
    c_2Ecardinal_2Ecardleq: del > del > $i )).

thf(mem_c_2Ecardinal_2Ecardleq,axiom,(
    ! [A_27a: del,A_27b: del] :
      ( mem @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ bool ) @ ( arr @ ( arr @ A_27b @ bool ) @ bool ) ) ) )).

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_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_5C_2F,type,(
    c_2Ebool_2E_5C_2F: $i )).

thf(mem_c_2Ebool_2E_5C_2F,axiom,(
    mem @ c_2Ebool_2E_5C_2F @ ( arr @ bool @ ( arr @ bool @ bool ) ) )).

thf(ax_or_p,axiom,(
    ! [Q: $i] :
      ( ( mem @ Q @ bool )
     => ! [R: $i] :
          ( ( mem @ R @ bool )
         => ( ( p @ ( ap @ ( ap @ c_2Ebool_2E_5C_2F @ Q ) @ R ) )
          <=> ( ( p @ Q )
              | ( p @ R ) ) ) ) ) )).

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_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_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(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(conj_thm_2Ebool_2ETRUTH,axiom,(
    $true )).

thf(conj_thm_2Ebool_2EFALSITY,axiom,(
    ! [V0t: $i] :
      ( ( mem @ V0t @ bool )
     => ( $false
       => ( p @ V0t ) ) ) )).

thf(conj_thm_2Ebool_2EFORALL__SIMP,axiom,(
    ! [A_27a: del,V0t: $i] :
      ( ( mem @ V0t @ bool )
     => ( ! [V1x: $i] :
            ( ( mem @ V1x @ A_27a )
           => ( p @ V0t ) )
      <=> ( p @ V0t ) ) ) )).

thf(conj_thm_2Ebool_2EIMP__CLAUSES,axiom,(
    ! [V0t: $i] :
      ( ( mem @ V0t @ bool )
     => ( ( ( $true
           => ( p @ V0t ) )
        <=> ( p @ V0t ) )
        & ( ( ( p @ V0t )
           => $true )
        <=> $true )
        & ( ( $false
           => ( p @ V0t ) )
        <=> $true )
        & ( ( ( p @ V0t )
           => ( p @ V0t ) )
        <=> $true )
        & ( ( ( p @ V0t )
           => $false )
        <=> ~ ( p @ V0t ) ) ) ) )).

thf(conj_thm_2Ebool_2ENOT__CLAUSES,axiom,
    ( ! [V0t: $i] :
        ( ( mem @ V0t @ bool )
       => ( ~ ~ ( p @ V0t )
        <=> ( p @ V0t ) ) )
    & ( ~ $true
    <=> $false )
    & ( ~ $false
    <=> $true ) )).

thf(conj_thm_2Ebool_2EREFL__CLAUSE,axiom,(
    ! [A_27a: del,V0x: $i] :
      ( ( mem @ V0x @ A_27a )
     => ( ( V0x = V0x )
      <=> $true ) ) )).

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_2Ecardinal_2ECARD__LE__TOTAL,axiom,(
    ! [A_27a: del,A_27b: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1t: $i] :
          ( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
         => ( ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
            | ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27b @ A_27a ) @ V1t ) @ V0s ) ) ) ) ) )).

thf(conj_thm_2Esat_2ENOT__NOT,axiom,(
    ! [V0t: $i] :
      ( ( mem @ V0t @ bool )
     => ( ~ ~ ( p @ V0t )
      <=> ( p @ V0t ) ) ) )).

thf(conj_thm_2Esat_2EAND__INV__IMP,axiom,(
    ! [V0A: $i] :
      ( ( mem @ V0A @ bool )
     => ( ( p @ V0A )
       => ( ~ ( p @ V0A )
         => $false ) ) ) )).

thf(conj_thm_2Esat_2EOR__DUAL2,axiom,(
    ! [V0A: $i] :
      ( ( mem @ V0A @ bool )
     => ! [V1B: $i] :
          ( ( mem @ V1B @ bool )
         => ( ( ~ ( ( p @ V0A )
                  | ( p @ V1B ) )
             => $false )
          <=> ( ( ( p @ V0A )
               => $false )
             => ( ~ ( p @ V1B )
               => $false ) ) ) ) ) )).

thf(conj_thm_2Esat_2EOR__DUAL3,axiom,(
    ! [V0A: $i] :
      ( ( mem @ V0A @ bool )
     => ! [V1B: $i] :
          ( ( mem @ V1B @ bool )
         => ( ( ~ ( ~ ( p @ V0A )
                  | ( p @ V1B ) )
             => $false )
          <=> ( ( p @ V0A )
             => ( ~ ( p @ V1B )
               => $false ) ) ) ) ) )).

thf(conj_thm_2Esat_2EAND__INV2,axiom,(
    ! [V0A: $i] :
      ( ( mem @ V0A @ bool )
     => ( ( ~ ( p @ V0A )
         => $false )
       => ( ( ( p @ V0A )
           => $false )
         => $false ) ) ) )).

thf(conj_thm_2Esat_2Edc__eq,axiom,(
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ! [V2r: $i] :
              ( ( mem @ V2r @ bool )
             => ( ( ( p @ V0p )
                <=> ( ( p @ V1q )
                  <=> ( p @ V2r ) ) )
              <=> ( ( ( p @ V0p )
                    | ( p @ V1q )
                    | ( p @ V2r ) )
                  & ( ( p @ V0p )
                    | ~ ( p @ V2r )
                    | ~ ( p @ V1q ) )
                  & ( ( p @ V1q )
                    | ~ ( p @ V2r )
                    | ~ ( p @ V0p ) )
                  & ( ( p @ V2r )
                    | ~ ( p @ V1q )
                    | ~ ( p @ V0p ) ) ) ) ) ) ) )).

thf(conj_thm_2Esat_2Edc__neg,axiom,(
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ( ( ( p @ V0p )
            <=> ~ ( p @ V1q ) )
          <=> ( ( ( p @ V0p )
                | ( p @ V1q ) )
              & ( ~ ( p @ V1q )
                | ~ ( p @ V0p ) ) ) ) ) ) )).

thf(conj_thm_2Ecardinal_2ECARD__NOT__LE,conjecture,(
    ! [A_27a: del,A_27b: del,V0s: $i] :
      ( ( mem @ V0s @ ( arr @ A_27a @ bool ) )
     => ! [V1t: $i] :
          ( ( mem @ V1t @ ( arr @ A_27b @ bool ) )
         => ( ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) )
          <=> ~ ( p @ ( ap @ ( ap @ ( c_2Ecardinal_2Ecardleq @ A_27a @ A_27b ) @ V0s ) @ V1t ) ) ) ) ) )).

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