ITP001 Axioms: ITP007^7.ax


%------------------------------------------------------------------------------
% File     : ITP007^7 : TPTP v9.0.0. Bugfixed v7.5.0.
% Domain   : Interactive Theorem Proving
% Axioms   : HOL4 syntactic export, chainy 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    : sat.ax [Gau19]
%          : HL4007^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   43 (   7 unt;  12 typ;   0 def)
%            Number of atoms       :   79 (   5 equ;  52 cnn)
%            Maximal formula atoms :    9 (   1 avg)
%            Number of connectives :  241 (  52   ~;  39   |;  20   &;  72   @)
%                                         (  25 <=>;  33  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   13 (   7 avg;  72 nst)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :   20 (  20   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   13 (  11 usr;   3 con; 0-4 aty)
%            Number of variables   :   67 (   0   ^  62   !;   1   ?;  67   :)
%                                         (   4  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

thf(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: $tType > $tType > $tType ).

thf(c_2Ebool_2E_21,type,
    c_2Ebool_2E_21: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Ebool_2E_2F_5C,type,
    c_2Ebool_2E_2F_5C: $o > $o > $o ).

thf(c_2Emin_2E_3D,type,
    c_2Emin_2E_3D: 
      !>[A_27a: $tType] : ( A_27a > A_27a > $o ) ).

thf(c_2Emin_2E_3D_3D_3E,type,
    c_2Emin_2E_3D_3D_3E: $o > $o > $o ).

thf(c_2Ebool_2E_3F,type,
    c_2Ebool_2E_3F: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).

thf(c_2Ebool_2ECOND,type,
    c_2Ebool_2ECOND: 
      !>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).

thf(c_2Ebool_2EF,type,
    c_2Ebool_2EF: $o ).

thf(c_2Ebool_2ET,type,
    c_2Ebool_2ET: $o ).

thf(c_2Ebool_2E_5C_2F,type,
    c_2Ebool_2E_5C_2F: $o > $o > $o ).

thf(c_2Ebool_2E_7E,type,
    c_2Ebool_2E_7E: $o > $o ).

thf(logicdef_2E_2F_5C,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_2F_5C @ V0 @ V1 )
    <=> ( V0
        & V1 ) ) ).

thf(logicdef_2E_5C_2F,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Ebool_2E_5C_2F @ V0 @ V1 )
    <=> ( V0
        | V1 ) ) ).

thf(logicdef_2E_7E,axiom,
    ! [V0: $o] :
      ( ( c_2Ebool_2E_7E @ V0 )
    <=> ( (~) @ V0 ) ) ).

thf(logicdef_2E_3D_3D_3E,axiom,
    ! [V0: $o,V1: $o] :
      ( ( c_2Emin_2E_3D_3D_3E @ V0 @ V1 )
    <=> ( V0
       => V1 ) ) ).

thf(logicdef_2E_3D,axiom,
    ! [A_27a: $tType,V0: A_27a,V1: A_27a] :
      ( ( c_2Emin_2E_3D @ A_27a @ V0 @ V1 )
    <=> ( V0 = V1 ) ) ).

thf(quantdef_2E_21,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_21 @ A_27a @ V0f )
    <=> ! [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(quantdef_2E_3F,axiom,
    ! [A_27a: $tType,V0f: A_27a > $o] :
      ( ( c_2Ebool_2E_3F @ A_27a @ V0f )
    <=> ? [V1x: A_27a] : ( V0f @ V1x ) ) ).

thf(thm_2Esat_2EAND__IMP,axiom,
    ! [V0A: $o,V1B: $o,V2C: $o] :
      ( ( ( V0A
          & V1B )
       => V2C )
    <=> ( V0A
       => ( V1B
         => V2C ) ) ) ).

thf(thm_2Esat_2ENOT__NOT,axiom,
    ! [V0t: $o] :
      ( ( (~) @ ( (~) @ V0t ) )
    <=> V0t ) ).

thf(thm_2Esat_2EAND__INV,axiom,
    ! [V0A: $o] :
      ( ( ( (~) @ V0A )
        & V0A )
    <=> c_2Ebool_2EF ) ).

thf(thm_2Esat_2EAND__INV__IMP,axiom,
    ! [V0A: $o] :
      ( V0A
     => ( ( (~) @ V0A )
       => c_2Ebool_2EF ) ) ).

thf(thm_2Esat_2EOR__DUAL,axiom,
    ! [V0B: $o,V1A: $o] :
      ( ( ( (~)
          @ ( V1A
            | V0B ) )
       => c_2Ebool_2EF )
    <=> ( ( (~) @ V1A )
       => ( ( (~) @ V0B )
         => c_2Ebool_2EF ) ) ) ).

thf(thm_2Esat_2EOR__DUAL2,axiom,
    ! [V0B: $o,V1A: $o] :
      ( ( ( (~)
          @ ( V1A
            | V0B ) )
       => c_2Ebool_2EF )
    <=> ( ( V1A
         => c_2Ebool_2EF )
       => ( ( (~) @ V0B )
         => c_2Ebool_2EF ) ) ) ).

thf(thm_2Esat_2EOR__DUAL3,axiom,
    ! [V0B: $o,V1A: $o] :
      ( ( ( (~)
          @ ( ( (~) @ V1A )
            | V0B ) )
       => c_2Ebool_2EF )
    <=> ( V1A
       => ( ( (~) @ V0B )
         => c_2Ebool_2EF ) ) ) ).

thf(thm_2Esat_2EAND__INV2,axiom,
    ! [V0A: $o] :
      ( ( ( (~) @ V0A )
       => c_2Ebool_2EF )
     => ( ( V0A
         => c_2Ebool_2EF )
       => c_2Ebool_2EF ) ) ).

thf(thm_2Esat_2ENOT__ELIM2,axiom,
    ! [V0A: $o] :
      ( ( ( (~) @ V0A )
       => c_2Ebool_2EF )
    <=> V0A ) ).

thf(thm_2Esat_2EEQT__Imp1,axiom,
    ! [V0b: $o] :
      ( V0b
     => ( V0b = c_2Ebool_2ET ) ) ).

thf(thm_2Esat_2EEQF__Imp1,axiom,
    ! [V0b: $o] :
      ( ( (~) @ V0b )
     => ( V0b = c_2Ebool_2EF ) ) ).

thf(thm_2Esat_2Edc__eq,axiom,
    ! [V0r: $o,V1q: $o,V2p: $o] :
      ( ( V2p
      <=> ( V1q = V0r ) )
    <=> ( ( V2p
          | V1q
          | V0r )
        & ( V2p
          | ( (~) @ V0r )
          | ( (~) @ V1q ) )
        & ( V1q
          | ( (~) @ V0r )
          | ( (~) @ V2p ) )
        & ( V0r
          | ( (~) @ V1q )
          | ( (~) @ V2p ) ) ) ) ).

thf(thm_2Esat_2Edc__conj,axiom,
    ! [V0r: $o,V1q: $o,V2p: $o] :
      ( ( V2p
      <=> ( V1q
          & V0r ) )
    <=> ( ( V2p
          | ( (~) @ V1q )
          | ( (~) @ V0r ) )
        & ( V1q
          | ( (~) @ V2p ) )
        & ( V0r
          | ( (~) @ V2p ) ) ) ) ).

thf(thm_2Esat_2Edc__disj,axiom,
    ! [V0r: $o,V1q: $o,V2p: $o] :
      ( ( V2p
      <=> ( V1q
          | V0r ) )
    <=> ( ( V2p
          | ( (~) @ V1q ) )
        & ( V2p
          | ( (~) @ V0r ) )
        & ( V1q
          | V0r
          | ( (~) @ V2p ) ) ) ) ).

thf(thm_2Esat_2Edc__imp,axiom,
    ! [V0r: $o,V1q: $o,V2p: $o] :
      ( ( V2p
      <=> ( V1q
         => V0r ) )
    <=> ( ( V2p
          | V1q )
        & ( V2p
          | ( (~) @ V0r ) )
        & ( ( (~) @ V1q )
          | V0r
          | ( (~) @ V2p ) ) ) ) ).

thf(thm_2Esat_2Edc__neg,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( V1p
      <=> ( (~) @ V0q ) )
    <=> ( ( V1p
          | V0q )
        & ( ( (~) @ V0q )
          | ( (~) @ V1p ) ) ) ) ).

thf(thm_2Esat_2Edc__cond,axiom,
    ! [V0s: $o,V1r: $o,V2q: $o,V3p: $o] :
      ( ( V3p
        = ( c_2Ebool_2ECOND @ $o @ V2q @ V1r @ V0s ) )
    <=> ( ( V3p
          | V2q
          | ( (~) @ V0s ) )
        & ( V3p
          | ( (~) @ V1r )
          | ( (~) @ V2q ) )
        & ( V3p
          | ( (~) @ V1r )
          | ( (~) @ V0s ) )
        & ( ( (~) @ V2q )
          | V1r
          | ( (~) @ V3p ) )
        & ( V2q
          | V0s
          | ( (~) @ V3p ) ) ) ) ).

thf(thm_2Esat_2Epth__ni1,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( (~)
        @ ( V1p
         => V0q ) )
     => V1p ) ).

thf(thm_2Esat_2Epth__ni2,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( (~)
        @ ( V1p
         => V0q ) )
     => ( (~) @ V0q ) ) ).

thf(thm_2Esat_2Epth__no1,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( (~)
        @ ( V1p
          | V0q ) )
     => ( (~) @ V1p ) ) ).

thf(thm_2Esat_2Epth__no2,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( (~)
        @ ( V1p
          | V0q ) )
     => ( (~) @ V0q ) ) ).

thf(thm_2Esat_2Epth__an1,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( V1p
        & V0q )
     => V1p ) ).

thf(thm_2Esat_2Epth__an2,axiom,
    ! [V0q: $o,V1p: $o] :
      ( ( V1p
        & V0q )
     => V0q ) ).

thf(thm_2Esat_2Epth__nn,axiom,
    ! [V0p: $o] :
      ( ( (~) @ ( (~) @ V0p ) )
     => V0p ) ).

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