ITP001 Axioms: ITP046^7.ax
%------------------------------------------------------------------------------
% File : ITP046^7 : TPTP v8.2.0. 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 ) ) ) ).
%------------------------------------------------------------------------------