ITP001 Axioms: ITP020^7.ax


%------------------------------------------------------------------------------
% File     : ITP020^7 : 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    : basicSize.ax [Gau19]
%          : HL4020^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   39 (   8 unt;  27 typ;   0 def)
%            Number of atoms       :   16 (   8 equ;   1 cnn)
%            Maximal formula atoms :    2 (   0 avg)
%            Number of connectives :   86 (   1   ~;   1   |;   3   &;  73   @)
%                                         (   7 <=>;   1  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    8 (   5 avg;  73 nst)
%            Number of types       :    4 (   3 usr)
%            Number of type conns  :   55 (  55   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   26 (  24 usr;   2 con; 0-5 aty)
%            Number of variables   :   53 (   2   ^  33   !;   1   ?;  53   :)
%                                         (  17  !>;   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_2Enum_2Enum,type,
    tyop_2Enum_2Enum: $tType ).

thf(tyop_2Eone_2Eone,type,
    tyop_2Eone_2Eone: $tType ).

thf(tyop_2Eoption_2Eoption,type,
    tyop_2Eoption_2Eoption: $tType > $tType ).

thf(tyop_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: $tType > $tType > $tType ).

thf(tyop_2Esum_2Esum,type,
    tyop_2Esum_2Esum: $tType > $tType > $tType ).

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

thf(c_2Earithmetic_2E_2B,type,
    c_2Earithmetic_2E_2B: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

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_2Esum_2EINL,type,
    c_2Esum_2EINL: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).

thf(c_2Esum_2EINR,type,
    c_2Esum_2EINR: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27b > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) ) ).

thf(c_2Eoption_2ENONE,type,
    c_2Eoption_2ENONE: 
      !>[A_27a: $tType] : ( tyop_2Eoption_2Eoption @ A_27a ) ).

thf(c_2Eoption_2ESOME,type,
    c_2Eoption_2ESOME: 
      !>[A_27a: $tType] : ( A_27a > ( tyop_2Eoption_2Eoption @ A_27a ) ) ).

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

thf(c_2Epair_2EUNCURRY,type,
    c_2Epair_2EUNCURRY: 
      !>[A_27a: $tType,A_27b: $tType,A_27c: $tType] : ( ( A_27a > A_27b > A_27c ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > A_27c ) ).

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

thf(c_2EbasicSize_2Ebool__size,type,
    c_2EbasicSize_2Ebool__size: $o > tyop_2Enum_2Enum ).

thf(c_2EbasicSize_2Eone__size,type,
    c_2EbasicSize_2Eone__size: tyop_2Eone_2Eone > tyop_2Enum_2Enum ).

thf(c_2EbasicSize_2Eoption__size,type,
    c_2EbasicSize_2Eoption__size: 
      !>[A_27a: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( tyop_2Eoption_2Eoption @ A_27a ) > tyop_2Enum_2Enum ) ).

thf(c_2EbasicSize_2Epair__size,type,
    c_2EbasicSize_2Epair__size: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27b > tyop_2Enum_2Enum ) > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) > tyop_2Enum_2Enum ) ).

thf(c_2EbasicSize_2Esum__size,type,
    c_2EbasicSize_2Esum__size: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > tyop_2Enum_2Enum ) > ( A_27b > tyop_2Enum_2Enum ) > ( tyop_2Esum_2Esum @ A_27a @ A_27b ) > tyop_2Enum_2Enum ) ).

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_2EbasicSize_2Ebool__size__def,axiom,
    ! [V0b: $o] :
      ( ( c_2EbasicSize_2Ebool__size @ V0b )
      = c_2Enum_2E0 ) ).

thf(thm_2EbasicSize_2Epair__size__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27b > tyop_2Enum_2Enum] :
      ( ( c_2EbasicSize_2Epair__size @ A_27a @ A_27b @ V0f @ V1g )
      = ( c_2Epair_2EUNCURRY @ A_27a @ A_27b @ tyop_2Enum_2Enum
        @ ^ [V2x: A_27a,V3y: A_27b] : ( c_2Earithmetic_2E_2B @ ( V0f @ V2x ) @ ( V1g @ V3y ) ) ) ) ).

thf(thm_2EbasicSize_2Eone__size__def,axiom,
    ! [V0x: tyop_2Eone_2Eone] :
      ( ( c_2EbasicSize_2Eone__size @ V0x )
      = c_2Enum_2E0 ) ).

thf(thm_2EbasicSize_2Esum__size__def,axiom,
    ! [A_27a: $tType,A_27b: $tType] :
      ( ! [V0f: A_27a > tyop_2Enum_2Enum,V1g: A_27b > tyop_2Enum_2Enum,V2x: A_27a] :
          ( ( c_2EbasicSize_2Esum__size @ A_27a @ A_27b @ V0f @ V1g @ ( c_2Esum_2EINL @ A_27a @ A_27b @ V2x ) )
          = ( V0f @ V2x ) )
      & ! [V3f: A_27a > tyop_2Enum_2Enum,V4g: A_27b > tyop_2Enum_2Enum,V5y: A_27b] :
          ( ( c_2EbasicSize_2Esum__size @ A_27a @ A_27b @ V3f @ V4g @ ( c_2Esum_2EINR @ A_27a @ A_27b @ V5y ) )
          = ( V4g @ V5y ) ) ) ).

thf(thm_2EbasicSize_2Eoption__size__def,axiom,
    ! [A_27a: $tType] :
      ( ! [V0f: A_27a > tyop_2Enum_2Enum] :
          ( ( c_2EbasicSize_2Eoption__size @ A_27a @ V0f @ ( c_2Eoption_2ENONE @ A_27a ) )
          = c_2Enum_2E0 )
      & ! [V1f: A_27a > tyop_2Enum_2Enum,V2x: A_27a] :
          ( ( c_2EbasicSize_2Eoption__size @ A_27a @ V1f @ ( c_2Eoption_2ESOME @ A_27a @ V2x ) )
          = ( c_2Enum_2ESUC @ ( V1f @ V2x ) ) ) ) ).

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