ITP001 Axioms: ITP009^7.ax


%------------------------------------------------------------------------------
% File     : ITP009^7 : 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    : num.ax [Gau19]
%          : HL4009^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   38 (   8 unt;  21 typ;   0 def)
%            Number of atoms       :   26 (   9 equ;   4 cnn)
%            Maximal formula atoms :    3 (   0 avg)
%            Number of connectives :   80 (   4   ~;   1   |;   5   &;  55   @)
%                                         (   9 <=>;   6  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   10 (   5 avg;  55 nst)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :   33 (  33   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   20 (  18 usr;   3 con; 0-4 aty)
%            Number of variables   :   39 (   0   ^  28   !;   2   ?;  39   :)
%                                         (   9  !>;   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(tyop_2Emin_2Eind,type,
    tyop_2Emin_2Eind: $tType ).

thf(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $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_2Enum_2E0,type,
    c_2Enum_2E0: tyop_2Enum_2Enum ).

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_2Enum_2EABS__num,type,
    c_2Enum_2EABS__num: tyop_2Emin_2Eind > tyop_2Enum_2Enum ).

thf(c_2Enum_2EIS__NUM__REP,type,
    c_2Enum_2EIS__NUM__REP: tyop_2Emin_2Eind > $o ).

thf(c_2Ebool_2EONE__ONE,type,
    c_2Ebool_2EONE__ONE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).

thf(c_2Ebool_2EONTO,type,
    c_2Ebool_2EONTO: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > $o ) ).

thf(c_2Enum_2EREP__num,type,
    c_2Enum_2EREP__num: tyop_2Enum_2Enum > tyop_2Emin_2Eind ).

thf(c_2Enum_2ESUC,type,
    c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Enum_2ESUC__REP,type,
    c_2Enum_2ESUC__REP: tyop_2Emin_2Eind > tyop_2Emin_2Eind ).

thf(c_2Ebool_2ETYPE__DEFINITION,type,
    c_2Ebool_2ETYPE__DEFINITION: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > A_27a ) > $o ) ).

thf(c_2Enum_2EZERO__REP,type,
    c_2Enum_2EZERO__REP: tyop_2Emin_2Eind ).

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_2Enum_2ESUC__REP__DEF,axiom,
    ( ( c_2Ebool_2EONE__ONE @ tyop_2Emin_2Eind @ tyop_2Emin_2Eind @ c_2Enum_2ESUC__REP )
    & ( (~) @ ( c_2Ebool_2EONTO @ tyop_2Emin_2Eind @ tyop_2Emin_2Eind @ c_2Enum_2ESUC__REP ) ) ) ).

thf(thm_2Enum_2EZERO__REP__DEF,axiom,
    ! [V0y: tyop_2Emin_2Eind] :
      ( (~)
      @ ( c_2Enum_2EZERO__REP
        = ( c_2Enum_2ESUC__REP @ V0y ) ) ) ).

thf(thm_2Enum_2EIS__NUM__REP,axiom,
    ! [V0m: tyop_2Emin_2Eind] :
      ( ( c_2Enum_2EIS__NUM__REP @ V0m )
    <=> ! [V1P: tyop_2Emin_2Eind > $o] :
          ( ( ( V1P @ c_2Enum_2EZERO__REP )
            & ! [V2n: tyop_2Emin_2Eind] :
                ( ( V1P @ V2n )
               => ( V1P @ ( c_2Enum_2ESUC__REP @ V2n ) ) ) )
         => ( V1P @ V0m ) ) ) ).

thf(thm_2Enum_2Enum__TY__DEF,axiom,
    ? [V0rep: tyop_2Enum_2Enum > tyop_2Emin_2Eind] : ( c_2Ebool_2ETYPE__DEFINITION @ tyop_2Emin_2Eind @ tyop_2Enum_2Enum @ c_2Enum_2EIS__NUM__REP @ V0rep ) ).

thf(thm_2Enum_2Enum__ISO__DEF,axiom,
    ( ! [V0a: tyop_2Enum_2Enum] :
        ( ( c_2Enum_2EABS__num @ ( c_2Enum_2EREP__num @ V0a ) )
        = V0a )
    & ! [V1r: tyop_2Emin_2Eind] :
        ( ( c_2Enum_2EIS__NUM__REP @ V1r )
      <=> ( ( c_2Enum_2EREP__num @ ( c_2Enum_2EABS__num @ V1r ) )
          = V1r ) ) ) ).

thf(thm_2Enum_2EZERO__DEF,axiom,
    ( c_2Enum_2E0
    = ( c_2Enum_2EABS__num @ c_2Enum_2EZERO__REP ) ) ).

thf(thm_2Enum_2ESUC__DEF,axiom,
    ! [V0m: tyop_2Enum_2Enum] :
      ( ( c_2Enum_2ESUC @ V0m )
      = ( c_2Enum_2EABS__num @ ( c_2Enum_2ESUC__REP @ ( c_2Enum_2EREP__num @ V0m ) ) ) ) ).

thf(thm_2Enum_2ENOT__SUC,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( (~)
      @ ( ( c_2Enum_2ESUC @ V0n )
        = c_2Enum_2E0 ) ) ).

thf(thm_2Enum_2EINV__SUC,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( ( c_2Enum_2ESUC @ V0m )
        = ( c_2Enum_2ESUC @ V1n ) )
     => ( V0m = V1n ) ) ).

thf(thm_2Enum_2EINDUCTION,axiom,
    ! [V0P: tyop_2Enum_2Enum > $o] :
      ( ( ( V0P @ c_2Enum_2E0 )
        & ! [V1n: tyop_2Enum_2Enum] :
            ( ( V0P @ V1n )
           => ( V0P @ ( c_2Enum_2ESUC @ V1n ) ) ) )
     => ! [V2n: tyop_2Enum_2Enum] : ( V0P @ V2n ) ) ).

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