ITP001 Axioms: ITP007^5.ax


%------------------------------------------------------------------------------
% File     : ITP007^5 : 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    : sat^2.ax [Gau20]
%          : HL4007^5.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   24 (   0 unt;   0 typ;   0 def)
%            Number of atoms       :  234 (   0 equ;   0 cnn)
%            Maximal formula atoms :   30 (   9 avg)
%            Number of connectives :  430 (  51   ~;  38   |;  19   &; 221   @)
%                                         (  22 <=>;  79  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   19 (  10 avg; 221 nst)
%            Number of types       :    1 (   0 usr)
%            Number of type conns  :    0 (   0   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :    7 (   5 usr;   7 con; 0-0 aty)
%            Number of variables   :   47 (   0   ^  47   !;   0   ?;  47   :)
% SPC      : TH0_SAT_NEQ_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(conj_thm_2Esat_2EAND__IMP,axiom,
    ! [V0A: $i] :
      ( ( mem @ V0A @ bool )
     => ! [V1B: $i] :
          ( ( mem @ V1B @ bool )
         => ! [V2C: $i] :
              ( ( mem @ V2C @ bool )
             => ( ( ( ( p @ V0A )
                    & ( p @ V1B ) )
                 => ( p @ V2C ) )
              <=> ( ( p @ V0A )
                 => ( ( p @ V1B )
                   => ( p @ V2C ) ) ) ) ) ) ) ).

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

thf(conj_thm_2Esat_2EAND__INV,axiom,
    ! [V0A: $i] :
      ( ( mem @ V0A @ bool )
     => ( ( ~ ( p @ V0A )
          & ( p @ V0A ) )
      <=> $false ) ) ).

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

thf(conj_thm_2Esat_2EOR__DUAL,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_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_2ENOT__ELIM2,axiom,
    ! [V0A: $i] :
      ( ( mem @ V0A @ bool )
     => ( ( ~ ( p @ V0A )
         => $false )
      <=> ( p @ V0A ) ) ) ).

thf(conj_thm_2Esat_2EEQT__Imp1,axiom,
    ! [V0b: $i] :
      ( ( mem @ V0b @ bool )
     => ( ( p @ V0b )
       => ( ( p @ V0b )
        <=> $true ) ) ) ).

thf(conj_thm_2Esat_2EEQF__Imp1,axiom,
    ! [V0b: $i] :
      ( ( mem @ V0b @ bool )
     => ( ~ ( p @ V0b )
       => ( ( p @ V0b )
        <=> $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__conj,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 @ V1q )
                    | ~ ( p @ V0p ) )
                  & ( ( p @ V2r )
                    | ~ ( p @ V0p ) ) ) ) ) ) ) ).

thf(conj_thm_2Esat_2Edc__disj,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 @ V0p )
                    | ~ ( p @ V2r ) )
                  & ( ( p @ V1q )
                    | ( p @ V2r )
                    | ~ ( p @ V0p ) ) ) ) ) ) ) ).

thf(conj_thm_2Esat_2Edc__imp,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 @ V0p )
                    | ~ ( p @ V2r ) )
                  & ( ~ ( p @ V1q )
                    | ( p @ V2r )
                    | ~ ( 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_2Esat_2Edc__cond,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ! [V2r: $i] :
              ( ( mem @ V2r @ bool )
             => ! [V3s: $i] :
                  ( ( mem @ V3s @ bool )
                 => ( ( ( p @ V0p )
                    <=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ bool ) @ V1q ) @ V2r ) @ V3s ) ) )
                  <=> ( ( ( p @ V0p )
                        | ( p @ V1q )
                        | ~ ( p @ V3s ) )
                      & ( ( p @ V0p )
                        | ~ ( p @ V2r )
                        | ~ ( p @ V1q ) )
                      & ( ( p @ V0p )
                        | ~ ( p @ V2r )
                        | ~ ( p @ V3s ) )
                      & ( ~ ( p @ V1q )
                        | ( p @ V2r )
                        | ~ ( p @ V0p ) )
                      & ( ( p @ V1q )
                        | ( p @ V3s )
                        | ~ ( p @ V0p ) ) ) ) ) ) ) ) ).

thf(conj_thm_2Esat_2Epth__ni1,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ( ~ ( ( p @ V0p )
               => ( p @ V1q ) )
           => ( p @ V0p ) ) ) ) ).

thf(conj_thm_2Esat_2Epth__ni2,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ( ~ ( ( p @ V0p )
               => ( p @ V1q ) )
           => ~ ( p @ V1q ) ) ) ) ).

thf(conj_thm_2Esat_2Epth__no1,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ( ~ ( ( p @ V0p )
                | ( p @ V1q ) )
           => ~ ( p @ V0p ) ) ) ) ).

thf(conj_thm_2Esat_2Epth__no2,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ( ~ ( ( p @ V0p )
                | ( p @ V1q ) )
           => ~ ( p @ V1q ) ) ) ) ).

thf(conj_thm_2Esat_2Epth__an1,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ( ( ( p @ V0p )
              & ( p @ V1q ) )
           => ( p @ V0p ) ) ) ) ).

thf(conj_thm_2Esat_2Epth__an2,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ! [V1q: $i] :
          ( ( mem @ V1q @ bool )
         => ( ( ( p @ V0p )
              & ( p @ V1q ) )
           => ( p @ V1q ) ) ) ) ).

thf(conj_thm_2Esat_2Epth__nn,axiom,
    ! [V0p: $i] :
      ( ( mem @ V0p @ bool )
     => ( ~ ~ ( p @ V0p )
       => ( p @ V0p ) ) ) ).

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