ITP001 Axioms: ITP004+5.ax


%------------------------------------------------------------------------------
% File     : ITP004+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    : ConseqConv+2.ax [Gau20]
%          : HL4004+5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   42 (   2 unt;   0 def)
%            Number of atoms       :  265 (   4 equ)
%            Maximal formula atoms :   14 (   6 avg)
%            Number of connectives :  237 (  14   ~;   9   |;  23   &)
%                                         (  27 <=>; 164  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   7 avg)
%            Maximal term depth    :    5 (   1 avg)
%            Number of predicates  :    6 (   3 usr;   2 prp; 0-2 aty)
%            Number of functors    :    9 (   9 usr;   4 con; 0-2 aty)
%            Number of variables   :   91 (  89   !;   2   ?)
% SPC      : FOF_SAT_RFO_SEQ

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2EConseqConv_2EASM__MARKER,axiom,
    mem(c_2EConseqConv_2EASM__MARKER,arr(bool,arr(bool,bool))) ).

fof(conj_thm_2EConseqConv_2Eforall__eq__thm,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0P] :
          ( mem(V0P,arr(A_27a,bool))
         => ! [V1Q] :
              ( mem(V1Q,arr(A_27a,bool))
             => ( ! [V2s] :
                    ( mem(V2s,A_27a)
                   => ( p(ap(V0P,V2s))
                    <=> p(ap(V1Q,V2s)) ) )
               => ( ! [V3s] :
                      ( mem(V3s,A_27a)
                     => p(ap(V0P,V3s)) )
                <=> ! [V4s] :
                      ( mem(V4s,A_27a)
                     => p(ap(V1Q,V4s)) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2Eexists__eq__thm,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0P] :
          ( mem(V0P,arr(A_27a,bool))
         => ! [V1Q] :
              ( mem(V1Q,arr(A_27a,bool))
             => ( ! [V2s] :
                    ( mem(V2s,A_27a)
                   => ( p(ap(V0P,V2s))
                    <=> p(ap(V1Q,V2s)) ) )
               => ( ? [V3s] :
                      ( mem(V3s,A_27a)
                      & p(ap(V0P,V3s)) )
                <=> ? [V4s] :
                      ( mem(V4s,A_27a)
                      & p(ap(V1Q,V4s)) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2Etrue__imp,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( p(V0t)
       => $true ) ) ).

fof(conj_thm_2EConseqConv_2Efalse__imp,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( $false
       => p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2ENOT__CLAUSES__X,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ~ ~ p(V0t)
      <=> p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2ENOT__CLAUSES__T,axiom,
    ( ~ $true
  <=> $false ) ).

fof(conj_thm_2EConseqConv_2ENOT__CLAUSES__F,axiom,
    ( ~ $false
  <=> $true ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__conj__strengthen,axiom,
    ! [V0x] :
      ( mem(V0x,bool)
     => ! [V1x_27] :
          ( mem(V1x_27,bool)
         => ! [V2y] :
              ( mem(V2y,bool)
             => ! [V3y_27] :
                  ( mem(V3y_27,bool)
                 => ( ( ( p(V2y)
                       => ( p(V1x_27)
                         => p(V0x) ) )
                      & ( p(V1x_27)
                       => ( p(V3y_27)
                         => p(V2y) ) ) )
                   => ( ( p(V1x_27)
                        & p(V3y_27) )
                     => ( p(V0x)
                        & p(V2y) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__conj__weaken,axiom,
    ! [V0x] :
      ( mem(V0x,bool)
     => ! [V1x_27] :
          ( mem(V1x_27,bool)
         => ! [V2y] :
              ( mem(V2y,bool)
             => ! [V3y_27] :
                  ( mem(V3y_27,bool)
                 => ( ( ( p(V2y)
                       => ( p(V0x)
                         => p(V1x_27) ) )
                      & ( p(V1x_27)
                       => ( p(V2y)
                         => p(V3y_27) ) ) )
                   => ( ( p(V0x)
                        & p(V2y) )
                     => ( p(V1x_27)
                        & p(V3y_27) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EAND__CLAUSES__TX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( $true
          & p(V0t) )
      <=> p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2EAND__CLAUSES__XT,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
          & $true )
      <=> p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2EAND__CLAUSES__FX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( $false
          & p(V0t) )
      <=> $false ) ) ).

fof(conj_thm_2EConseqConv_2EAND__CLAUSES__XF,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
          & $false )
      <=> $false ) ) ).

fof(conj_thm_2EConseqConv_2EAND__CLAUSES__XX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
          & p(V0t) )
      <=> p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__disj__strengthen,axiom,
    ! [V0x] :
      ( mem(V0x,bool)
     => ! [V1x_27] :
          ( mem(V1x_27,bool)
         => ! [V2y] :
              ( mem(V2y,bool)
             => ! [V3y_27] :
                  ( mem(V3y_27,bool)
                 => ( ( ( ~ p(V2y)
                       => ( p(V1x_27)
                         => p(V0x) ) )
                      & ( ~ p(V1x_27)
                       => ( p(V3y_27)
                         => p(V2y) ) ) )
                   => ( ( p(V1x_27)
                        | p(V3y_27) )
                     => ( p(V0x)
                        | p(V2y) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__disj__weaken,axiom,
    ! [V0x] :
      ( mem(V0x,bool)
     => ! [V1x_27] :
          ( mem(V1x_27,bool)
         => ! [V2y] :
              ( mem(V2y,bool)
             => ! [V3y_27] :
                  ( mem(V3y_27,bool)
                 => ( ( ( ~ p(V2y)
                       => ( p(V0x)
                         => p(V1x_27) ) )
                      & ( ~ p(V1x_27)
                       => ( p(V2y)
                         => p(V3y_27) ) ) )
                   => ( ( p(V0x)
                        | p(V2y) )
                     => ( p(V1x_27)
                        | p(V3y_27) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EOR__CLAUSES__TX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( $true
          | p(V0t) )
      <=> $true ) ) ).

fof(conj_thm_2EConseqConv_2EOR__CLAUSES__XT,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
          | $true )
      <=> $true ) ) ).

fof(conj_thm_2EConseqConv_2EOR__CLAUSES__FX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( $false
          | p(V0t) )
      <=> p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2EOR__CLAUSES__XF,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
          | $false )
      <=> p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2EOR__CLAUSES__XX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
          | p(V0t) )
      <=> p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__imp__strengthen,axiom,
    ! [V0x] :
      ( mem(V0x,bool)
     => ! [V1x_27] :
          ( mem(V1x_27,bool)
         => ! [V2y] :
              ( mem(V2y,bool)
             => ! [V3y_27] :
                  ( mem(V3y_27,bool)
                 => ( ( ( p(V0x)
                       => ( p(V3y_27)
                         => p(V2y) ) )
                      & ( ~ p(V3y_27)
                       => ( p(V0x)
                         => p(V1x_27) ) ) )
                   => ( ( p(V1x_27)
                       => p(V3y_27) )
                     => ( p(V0x)
                       => p(V2y) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__imp__weaken,axiom,
    ! [V0x] :
      ( mem(V0x,bool)
     => ! [V1x_27] :
          ( mem(V1x_27,bool)
         => ! [V2y] :
              ( mem(V2y,bool)
             => ! [V3y_27] :
                  ( mem(V3y_27,bool)
                 => ( ( ( p(V0x)
                       => ( p(V2y)
                         => p(V3y_27) ) )
                      & ( ~ p(V3y_27)
                       => ( p(V1x_27)
                         => p(V0x) ) ) )
                   => ( ( p(V0x)
                       => p(V2y) )
                     => ( p(V1x_27)
                       => p(V3y_27) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__simple__imp__strengthen,axiom,
    ! [V0x] :
      ( mem(V0x,bool)
     => ! [V1x_27] :
          ( mem(V1x_27,bool)
         => ! [V2y] :
              ( mem(V2y,bool)
             => ! [V3y_27] :
                  ( mem(V3y_27,bool)
                 => ( ( ( p(V0x)
                       => p(V1x_27) )
                      & ( p(V1x_27)
                       => ( p(V3y_27)
                         => p(V2y) ) ) )
                   => ( ( p(V1x_27)
                       => p(V3y_27) )
                     => ( p(V0x)
                       => p(V2y) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__simple__imp__weaken,axiom,
    ! [V0x] :
      ( mem(V0x,bool)
     => ! [V1x_27] :
          ( mem(V1x_27,bool)
         => ! [V2y] :
              ( mem(V2y,bool)
             => ! [V3y_27] :
                  ( mem(V3y_27,bool)
                 => ( ( ( p(V1x_27)
                       => p(V0x) )
                      & ( p(V1x_27)
                       => ( p(V2y)
                         => p(V3y_27) ) ) )
                   => ( ( p(V0x)
                       => p(V2y) )
                     => ( p(V1x_27)
                       => p(V3y_27) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__TX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( $true
         => p(V0t) )
      <=> p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__XT,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
         => $true )
      <=> $true ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__FX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( $false
         => p(V0t) )
      <=> $true ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__XX,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
         => p(V0t) )
      <=> $true ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CLAUSES__XF,axiom,
    ! [V0t] :
      ( mem(V0t,bool)
     => ( ( p(V0t)
         => $false )
      <=> ~ p(V0t) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__cond__simple,axiom,
    ! [V0c] :
      ( mem(V0c,bool)
     => ! [V1x] :
          ( mem(V1x,bool)
         => ! [V2x_27] :
              ( mem(V2x_27,bool)
             => ! [V3y] :
                  ( mem(V3y,bool)
                 => ! [V4y_27] :
                      ( mem(V4y_27,bool)
                     => ( ( ( p(V2x_27)
                           => p(V1x) )
                          & ( p(V4y_27)
                           => p(V3y) ) )
                       => ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V2x_27),V4y_27))
                         => p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V1x),V3y)) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2EIMP__CONG__cond,axiom,
    ! [V0c] :
      ( mem(V0c,bool)
     => ! [V1x] :
          ( mem(V1x,bool)
         => ! [V2x_27] :
              ( mem(V2x_27,bool)
             => ! [V3y] :
                  ( mem(V3y,bool)
                 => ! [V4y_27] :
                      ( mem(V4y_27,bool)
                     => ( ( ( p(V0c)
                           => ( p(V2x_27)
                             => p(V1x) ) )
                          & ( ~ p(V0c)
                           => ( p(V4y_27)
                             => p(V3y) ) ) )
                       => ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V2x_27),V4y_27))
                         => p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V1x),V3y)) ) ) ) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__CT,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0t1] :
          ( mem(V0t1,A_27a)
         => ! [V1t2] :
              ( mem(V1t2,A_27a)
             => ap(ap(ap(c_2Ebool_2ECOND(A_27a),c_2Ebool_2ET),V0t1),V1t2) = V0t1 ) ) ) ).

fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__CF,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0t1] :
          ( mem(V0t1,A_27a)
         => ! [V1t2] :
              ( mem(V1t2,A_27a)
             => ap(ap(ap(c_2Ebool_2ECOND(A_27a),c_2Ebool_2EF),V0t1),V1t2) = V1t2 ) ) ) ).

fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__ID,axiom,
    ! [A_27a] :
      ( ne(A_27a)
     => ! [V0b] :
          ( mem(V0b,bool)
         => ! [V1t] :
              ( mem(V1t,A_27a)
             => ap(ap(ap(c_2Ebool_2ECOND(A_27a),V0b),V1t),V1t) = V1t ) ) ) ).

fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__TT,axiom,
    ! [V0c] :
      ( mem(V0c,bool)
     => ! [V1x] :
          ( mem(V1x,bool)
         => ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),c_2Ebool_2ET),V1x))
          <=> ( ~ p(V0c)
             => p(V1x) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__FT,axiom,
    ! [V0c] :
      ( mem(V0c,bool)
     => ! [V1x] :
          ( mem(V1x,bool)
         => ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V1x),c_2Ebool_2ET))
          <=> ( p(V0c)
             => p(V1x) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__TF,axiom,
    ! [V0c] :
      ( mem(V0c,bool)
     => ! [V1x] :
          ( mem(V1x,bool)
         => ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),c_2Ebool_2EF),V1x))
          <=> ( ~ p(V0c)
              & p(V1x) ) ) ) ) ).

fof(conj_thm_2EConseqConv_2ECOND__CLAUSES__FF,axiom,
    ! [V0c] :
      ( mem(V0c,bool)
     => ! [V1x] :
          ( mem(V1x,bool)
         => ( p(ap(ap(ap(c_2Ebool_2ECOND(bool),V0c),V1x),c_2Ebool_2EF))
          <=> ( p(V0c)
              & p(V1x) ) ) ) ) ).

fof(ax_thm_2EConseqConv_2EASM__MARKER__DEF,axiom,
    c_2EConseqConv_2EASM__MARKER = k(bool,i(bool)) ).

fof(conj_thm_2EConseqConv_2EASM__MARKER__THM,axiom,
    ! [V0y] :
      ( mem(V0y,bool)
     => ! [V1x] :
          ( mem(V1x,bool)
         => ( p(ap(ap(c_2EConseqConv_2EASM__MARKER,V0y),V1x))
          <=> p(V1x) ) ) ) ).

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