ITP001 Axioms: ITP029^7.ax


%------------------------------------------------------------------------------
% File     : ITP029^7 : 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    : gcdset.ax [Gau19]
%          : HL4029^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   38 (   8 unt;  26 typ;   0 def)
%            Number of atoms       :   34 (   4 equ;   1 cnn)
%            Maximal formula atoms :    3 (   0 avg)
%            Number of connectives :  109 (   1   ~;   1   |;   1   &;  95   @)
%                                         (   7 <=>;   4  =>;   0  <=;   0 <~>)
%            Maximal formula depth :    9 (   5 avg;  95 nst)
%            Number of types       :    3 (   2 usr)
%            Number of type conns  :   62 (  62   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   26 (  24 usr;   2 con; 0-4 aty)
%            Number of variables   :   40 (   3   ^  23   !;   1   ?;  40   :)
%                                         (  13  !>;   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_2Epair_2Eprod,type,
    tyop_2Epair_2Eprod: $tType > $tType > $tType ).

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

thf(c_2Epair_2E_2C,type,
    c_2Epair_2E_2C: 
      !>[A_27a: $tType,A_27b: $tType] : ( A_27a > A_27b > ( tyop_2Epair_2Eprod @ A_27a @ A_27b ) ) ).

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_2Earithmetic_2E_3C_3D,type,
    c_2Earithmetic_2E_3C_3D: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $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_2Epred__set_2EDELETE,type,
    c_2Epred__set_2EDELETE: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > A_27a > A_27a > $o ) ).

thf(c_2Epred__set_2EEMPTY,type,
    c_2Epred__set_2EEMPTY: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2Epred__set_2EGSPEC,type,
    c_2Epred__set_2EGSPEC: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Epair_2Eprod @ A_27a @ $o ) ) > A_27a > $o ) ).

thf(c_2Ebool_2EIN,type,
    c_2Ebool_2EIN: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).

thf(c_2Epred__set_2EINSERT,type,
    c_2Epred__set_2EINSERT: 
      !>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EINTER,type,
    c_2Epred__set_2EINTER: 
      !>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > A_27a > $o ) ).

thf(c_2Epred__set_2EMAX__SET,type,
    c_2Epred__set_2EMAX__SET: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).

thf(c_2Epred__set_2EMIN__SET,type,
    c_2Epred__set_2EMIN__SET: ( tyop_2Enum_2Enum > $o ) > tyop_2Enum_2Enum ).

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

thf(c_2Edivides_2Edivides,type,
    c_2Edivides_2Edivides: tyop_2Enum_2Enum > tyop_2Enum_2Enum > $o ).

thf(c_2Egcd_2Egcd,type,
    c_2Egcd_2Egcd: tyop_2Enum_2Enum > tyop_2Enum_2Enum > tyop_2Enum_2Enum ).

thf(c_2Egcdset_2Egcdset,type,
    c_2Egcdset_2Egcdset: ( tyop_2Enum_2Enum > $o ) > 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_2Egcdset_2Egcdset__def,axiom,
    ! [V0s: tyop_2Enum_2Enum > $o] :
      ( ( c_2Egcdset_2Egcdset @ V0s )
      = ( c_2Ebool_2ECOND @ tyop_2Enum_2Enum @ ( c_2Ebool_2E_5C_2F @ ( c_2Emin_2E_3D @ ( tyop_2Enum_2Enum > $o ) @ V0s @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) @ ( c_2Emin_2E_3D @ ( tyop_2Enum_2Enum > $o ) @ V0s @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ c_2Enum_2E0 @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) ) ) ) @ c_2Enum_2E0
        @ ( c_2Epred__set_2EMAX__SET
          @ ( c_2Epred__set_2EINTER @ tyop_2Enum_2Enum
            @ ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
              @ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V1n @ ( c_2Earithmetic_2E_3C_3D @ V1n @ ( c_2Epred__set_2EMIN__SET @ ( c_2Epred__set_2EDELETE @ tyop_2Enum_2Enum @ V0s @ c_2Enum_2E0 ) ) ) ) )
            @ ( c_2Epred__set_2EGSPEC @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum
              @ ^ [V2d: tyop_2Enum_2Enum] :
                  ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ $o @ V2d
                  @ ( c_2Ebool_2E_21 @ tyop_2Enum_2Enum
                    @ ^ [V3e: tyop_2Enum_2Enum] : ( c_2Emin_2E_3D_3D_3E @ ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V3e @ V0s ) @ ( c_2Edivides_2Edivides @ V2d @ V3e ) ) ) ) ) ) ) ) ) ).

thf(thm_2Egcdset_2Egcdset__divides,axiom,
    ! [V0s: tyop_2Enum_2Enum > $o,V1e: tyop_2Enum_2Enum] :
      ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V1e @ V0s )
     => ( c_2Edivides_2Edivides @ ( c_2Egcdset_2Egcdset @ V0s ) @ V1e ) ) ).

thf(thm_2Egcdset_2Egcdset__greatest,axiom,
    ! [V0s: tyop_2Enum_2Enum > $o,V1g: tyop_2Enum_2Enum] :
      ( ! [V2e: tyop_2Enum_2Enum] :
          ( ( c_2Ebool_2EIN @ tyop_2Enum_2Enum @ V2e @ V0s )
         => ( c_2Edivides_2Edivides @ V1g @ V2e ) )
     => ( c_2Edivides_2Edivides @ V1g @ ( c_2Egcdset_2Egcdset @ V0s ) ) ) ).

thf(thm_2Egcdset_2Egcdset__EMPTY,axiom,
    ( ( c_2Egcdset_2Egcdset @ ( c_2Epred__set_2EEMPTY @ tyop_2Enum_2Enum ) )
    = c_2Enum_2E0 ) ).

thf(thm_2Egcdset_2Egcdset__INSERT,axiom,
    ! [V0x: tyop_2Enum_2Enum,V1s: tyop_2Enum_2Enum > $o] :
      ( ( c_2Egcdset_2Egcdset @ ( c_2Epred__set_2EINSERT @ tyop_2Enum_2Enum @ V0x @ V1s ) )
      = ( c_2Egcd_2Egcd @ V0x @ ( c_2Egcdset_2Egcdset @ V1s ) ) ) ).

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