ITP001 Axioms: ITP046^7.ax


%------------------------------------------------------------------------------
% File     : ITP046^7 : 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    : dirGraph.ax [Gau19]
%          : HL4046^7.ax [TPAP]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   39 (  10 unt;  24 typ;   0 def)
%            Number of atoms       :   51 (   7 equ;   3 cnn)
%            Maximal formula atoms :    6 (   1 avg)
%            Number of connectives :  198 (   3   ~;   1   |;   2   &; 180   @)
%                                         (   8 <=>;   4  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   15 (   7 avg; 180 nst)
%            Number of types       :    2 (   1 usr)
%            Number of type conns  :   71 (  71   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :   25 (  23 usr;   1 con; 0-5 aty)
%            Number of variables   :   73 (   5   ^  46   !;   2   ?;  73   :)
%                                         (  20  !>;   0  ?*;   0  @-;   0  @+)
% SPC      : TH1_SAT_EQU_NAR

% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tyop_2Elist_2Elist,type,
    tyop_2Elist_2Elist: $tType > $tType ).

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

thf(tyop_2Emin_2Efun,type,
    tyop_2Emin_2Efun: $tType > $tType > $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_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_2EEMPTY,type,
    c_2Epred__set_2EEMPTY: 
      !>[A_27a: $tType] : ( A_27a > $o ) ).

thf(c_2EdirGraph_2EEXCLUDE,type,
    c_2EdirGraph_2EEXCLUDE: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Elist_2Elist @ A_27a ) ) > ( A_27b > $o ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) ) ).

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_2Elist_2ELIST__TO__SET,type,
    c_2Elist_2ELIST__TO__SET: 
      !>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).

thf(c_2Elist_2ENIL,type,
    c_2Elist_2ENIL: 
      !>[A_27a: $tType] : ( tyop_2Elist_2Elist @ A_27a ) ).

thf(c_2EdirGraph_2EParents,type,
    c_2EdirGraph_2EParents: 
      !>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > ( tyop_2Elist_2Elist @ A_27b ) ) > A_27a > $o ) ).

thf(c_2EdirGraph_2EREACH,type,
    c_2EdirGraph_2EREACH: 
      !>[A_27a: $tType] : ( ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) ) > A_27a > A_27a > $o ) ).

thf(c_2EdirGraph_2EREACH__LIST,type,
    c_2EdirGraph_2EREACH__LIST: 
      !>[A_27a: $tType] : ( ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) ) > ( tyop_2Elist_2Elist @ A_27a ) > A_27a > $o ) ).

thf(c_2Erelation_2ERTC,type,
    c_2Erelation_2ERTC: 
      !>[A_27a: $tType] : ( ( A_27a > A_27a > $o ) > A_27a > A_27a > $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_2EdirGraph_2EParents__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27b )] :
      ( ( c_2EdirGraph_2EParents @ A_27a @ A_27b @ V0G )
      = ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a
        @ ^ [V1x: A_27a] : ( c_2Epair_2E_2C @ A_27a @ $o @ V1x @ ( c_2Ebool_2E_7E @ ( c_2Emin_2E_3D @ ( tyop_2Elist_2Elist @ A_27b ) @ ( V0G @ V1x ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) ) ).

thf(thm_2EdirGraph_2EREACH__def,axiom,
    ! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a )] :
      ( ( c_2EdirGraph_2EREACH @ A_27a @ V0G )
      = ( c_2Erelation_2ERTC @ A_27a
        @ ^ [V1x: A_27a,V2y: A_27a] : ( c_2Ebool_2EIN @ A_27a @ V2y @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( V0G @ V1x ) ) ) ) ) ).

thf(thm_2EdirGraph_2EREACH__LIST__def,axiom,
    ! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1nodes: tyop_2Elist_2Elist @ A_27a,V2y: A_27a] :
      ( ( c_2EdirGraph_2EREACH__LIST @ A_27a @ V0G @ V1nodes @ V2y )
    <=> ? [V3x: A_27a] :
          ( ( c_2Ebool_2EIN @ A_27a @ V3x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1nodes ) )
          & ( c_2Ebool_2EIN @ A_27a @ V2y @ ( c_2EdirGraph_2EREACH @ A_27a @ V0G @ V3x ) ) ) ) ).

thf(thm_2EdirGraph_2EEXCLUDE__def,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0G: A_27b > ( tyop_2Elist_2Elist @ A_27a ),V1ex: A_27b > $o,V2node: A_27b] :
      ( ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27b @ V0G @ V1ex @ V2node )
      = ( c_2Ebool_2ECOND @ ( tyop_2Elist_2Elist @ A_27a ) @ ( c_2Ebool_2EIN @ A_27b @ V2node @ V1ex ) @ ( c_2Elist_2ENIL @ A_27a ) @ ( V0G @ V2node ) ) ) ).

thf(thm_2EdirGraph_2EEXCLUDE__LEM,axiom,
    ! [A_27a: $tType,A_27b: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27b ),V1x: A_27a,V2l: A_27a > $o] :
      ( ( c_2EdirGraph_2EEXCLUDE @ A_27b @ A_27a @ V0G @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ V2l ) )
      = ( c_2EdirGraph_2EEXCLUDE @ A_27b @ A_27a @ ( c_2EdirGraph_2EEXCLUDE @ A_27b @ A_27a @ V0G @ V2l ) @ ( c_2Epred__set_2EINSERT @ A_27a @ V1x @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ).

thf(thm_2EdirGraph_2EREACH__EXCLUDE,axiom,
    ! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1x: A_27a > $o] :
      ( ( c_2EdirGraph_2EREACH @ A_27a @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a @ V0G @ V1x ) )
      = ( c_2Erelation_2ERTC @ A_27a
        @ ^ [V2x_27: A_27a,V3y: A_27a] : ( c_2Ebool_2E_2F_5C @ ( c_2Ebool_2E_7E @ ( c_2Ebool_2EIN @ A_27a @ V2x_27 @ V1x ) ) @ ( c_2Ebool_2EIN @ A_27a @ V3y @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( V0G @ V2x_27 ) ) ) ) ) ) ).

thf(thm_2EdirGraph_2EREACH__LEM1,axiom,
    ! [A_27a: $tType,V0p: A_27a,V1G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V2seen: A_27a > $o] :
      ( ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V0p @ V2seen ) )
     => ( ( c_2EdirGraph_2EREACH @ A_27a @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a @ V1G @ V2seen ) @ V0p )
        = ( c_2Epred__set_2EINSERT @ A_27a @ V0p @ ( c_2EdirGraph_2EREACH__LIST @ A_27a @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a @ V1G @ ( c_2Epred__set_2EINSERT @ A_27a @ V0p @ V2seen ) ) @ ( V1G @ V0p ) ) ) ) ) ).

thf(thm_2EdirGraph_2EREACH__LEM2,axiom,
    ! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1x: A_27a,V2y: A_27a] :
      ( ( c_2EdirGraph_2EREACH @ A_27a @ V0G @ V1x @ V2y )
     => ! [V3z: A_27a] :
          ( ( (~) @ ( c_2EdirGraph_2EREACH @ A_27a @ V0G @ V3z @ V2y ) )
         => ( c_2EdirGraph_2EREACH @ A_27a @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a @ V0G @ ( c_2Epred__set_2EINSERT @ A_27a @ V3z @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) @ V1x @ V2y ) ) ) ).

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