ITP001 Axioms: ITP111^7.ax


%------------------------------------------------------------------------------
% File     : ITP111^7 : 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    : intto.ax [Gau19]
%          : HL4111^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   53 (  20 unt;  29 typ;   0 def)
%            Number of atoms       :   29 (  18 equ;   3 cnn)
%            Maximal formula atoms :    2 (   0 avg)
%            Number of connectives :  115 (   3   ~;   1   |;   1   &; 102   @)
%                                         (   7 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   3 avg; 102 nst)
%            Number of types       :    5 (   4 usr)
%            Number of type conns  :   39 (  39   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   27 (  25 usr;   7 con; 0-4 aty)
%            Number of variables   :   44 (   0   ^  37   !;   1   ?;  44   :)
%                                         (   6  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Einteger_2Eint,type,
    tyop_2Einteger_2Eint: $tType ).

thf(tyop_2Emin_2Ebool,type,
    tyop_2Emin_2Ebool: $tType ).

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

thf(tyop_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

thf(tyop_2EternaryComparisons_2Eordering,type,
    tyop_2EternaryComparisons_2Eordering: $tType ).

thf(tyop_2Etoto_2Etoto,type,
    tyop_2Etoto_2Etoto: $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_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_2Earithmetic_2EBIT1,type,
    c_2Earithmetic_2EBIT1: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Earithmetic_2EBIT2,type,
    c_2Earithmetic_2EBIT2: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2EternaryComparisons_2EEQUAL,type,
    c_2EternaryComparisons_2EEQUAL: tyop_2EternaryComparisons_2Eordering ).

thf(c_2EternaryComparisons_2EGREATER,type,
    c_2EternaryComparisons_2EGREATER: tyop_2EternaryComparisons_2Eordering ).

thf(c_2EternaryComparisons_2ELESS,type,
    c_2EternaryComparisons_2ELESS: tyop_2EternaryComparisons_2Eordering ).

thf(c_2Etoto_2ETO,type,
    c_2Etoto_2ETO: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) > ( tyop_2Etoto_2Etoto @ A_27a ) ) ).

thf(c_2Etoto_2ETO__of__LinearOrder,type,
    c_2Etoto_2ETO__of__LinearOrder: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).

thf(c_2Earithmetic_2EZERO,type,
    c_2Earithmetic_2EZERO: tyop_2Enum_2Enum ).

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

thf(c_2Etoto_2Eapto,type,
    c_2Etoto_2Eapto: 
      !>[A_27a: $tType] : ( ( tyop_2Etoto_2Etoto @ A_27a ) > A_27a > A_27a > tyop_2EternaryComparisons_2Eordering ) ).

thf(c_2Eintto_2EintOrd,type,
    c_2Eintto_2EintOrd: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > tyop_2EternaryComparisons_2Eordering ).

thf(c_2Einteger_2Eint__lt,type,
    c_2Einteger_2Eint__lt: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint > $o ).

thf(c_2Einteger_2Eint__neg,type,
    c_2Einteger_2Eint__neg: tyop_2Einteger_2Eint > tyop_2Einteger_2Eint ).

thf(c_2Einteger_2Eint__of__num,type,
    c_2Einteger_2Eint__of__num: tyop_2Enum_2Enum > tyop_2Einteger_2Eint ).

thf(c_2Eintto_2Eintto,type,
    c_2Eintto_2Eintto: tyop_2Etoto_2Etoto @ tyop_2Einteger_2Eint ).

thf(c_2Etoto_2EnumOrd,type,
    c_2Etoto_2EnumOrd: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2EternaryComparisons_2Eordering ).

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_2Eintto_2EintOrd,axiom,
    ( c_2Eintto_2EintOrd
    = ( c_2Etoto_2ETO__of__LinearOrder @ tyop_2Einteger_2Eint @ c_2Einteger_2Eint__lt ) ) ).

thf(thm_2Eintto_2Eintto,axiom,
    ( c_2Eintto_2Eintto
    = ( c_2Etoto_2ETO @ tyop_2Einteger_2Eint @ c_2Eintto_2EintOrd ) ) ).

thf(thm_2Eintto_2Eapintto__thm,axiom,
    ( ( c_2Etoto_2Eapto @ tyop_2Einteger_2Eint @ c_2Eintto_2Eintto )
    = c_2Eintto_2EintOrd ) ).

thf(thm_2Eintto_2Epos__pos__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__of__num @ V0m ) @ ( c_2Einteger_2Eint__of__num @ V1n ) )
      = ( c_2Etoto_2EnumOrd @ V0m @ V1n ) ) ).

thf(thm_2Eintto_2Eneg__neg__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ V0m ) ) @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ V1n ) ) )
      = ( c_2Etoto_2EnumOrd @ V1n @ V0m ) ) ).

thf(thm_2Eintto_2EBIT1__nz,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( (~)
      @ ( ( c_2Earithmetic_2EBIT1 @ V0n )
        = c_2Enum_2E0 ) ) ).

thf(thm_2Eintto_2EBIT2__nz,axiom,
    ! [V0n: tyop_2Enum_2Enum] :
      ( (~)
      @ ( ( c_2Earithmetic_2EBIT2 @ V0n )
        = c_2Enum_2E0 ) ) ).

thf(thm_2Eintto_2Eneg__lt__BIT1__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ V0m ) ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2EBIT1 @ V1n ) ) )
      = c_2EternaryComparisons_2ELESS ) ).

thf(thm_2Eintto_2Eneg__lt__BIT2__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ V0m ) ) @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2EBIT2 @ V1n ) ) )
      = c_2EternaryComparisons_2ELESS ) ).

thf(thm_2Eintto_2Eneg__BIT1__lt__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2EBIT1 @ V0m ) ) ) @ ( c_2Einteger_2Eint__of__num @ V1n ) )
      = c_2EternaryComparisons_2ELESS ) ).

thf(thm_2Eintto_2Eneg__BIT2__lt__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2EBIT2 @ V0m ) ) ) @ ( c_2Einteger_2Eint__of__num @ V1n ) )
      = c_2EternaryComparisons_2ELESS ) ).

thf(thm_2Eintto_2Eneg__ZERO__eq__ZERO__thm,axiom,
    ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ c_2Earithmetic_2EZERO ) ) @ ( c_2Einteger_2Eint__of__num @ c_2Earithmetic_2EZERO ) )
    = c_2EternaryComparisons_2EEQUAL ) ).

thf(thm_2Eintto_2EBIT1__gt__neg__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2EBIT1 @ V0m ) ) @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ V1n ) ) )
      = c_2EternaryComparisons_2EGREATER ) ).

thf(thm_2Eintto_2EBIT2__gt__neg__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2EBIT2 @ V0m ) ) @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ V1n ) ) )
      = c_2EternaryComparisons_2EGREATER ) ).

thf(thm_2Eintto_2Egt__neg__BIT1__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__of__num @ V0m ) @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2EBIT1 @ V1n ) ) ) )
      = c_2EternaryComparisons_2EGREATER ) ).

thf(thm_2Eintto_2Egt__neg__BIT2__thm,axiom,
    ! [V0m: tyop_2Enum_2Enum,V1n: tyop_2Enum_2Enum] :
      ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__of__num @ V0m ) @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ ( c_2Earithmetic_2EBIT2 @ V1n ) ) ) )
      = c_2EternaryComparisons_2EGREATER ) ).

thf(thm_2Eintto_2EZERO__eq__neg__ZERO__thm,axiom,
    ( ( c_2Eintto_2EintOrd @ ( c_2Einteger_2Eint__of__num @ c_2Earithmetic_2EZERO ) @ ( c_2Einteger_2Eint__neg @ ( c_2Einteger_2Eint__of__num @ c_2Earithmetic_2EZERO ) ) )
    = c_2EternaryComparisons_2EEQUAL ) ).

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