ITP001 Axioms: ITP046^5.ax
%------------------------------------------------------------------------------
% File : ITP046^5 : TPTP v8.2.0. Bugfixed v7.5.0.
% Domain : Interactive Theorem Proving
% Axioms : HOL4 set theory export, chainy mode
% Version : [BG+19] axioms.
% English :
% Refs : [BG+19] Brown et al. (2019), GRUNGE: A Grand Unified ATP Chall
% : [Gau20] Gauthier (2020), Email to Geoff Sutcliffe
% Source : [BG+19]
% Names : dirGraph^2.ax [Gau20]
% : HL4046^5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 16 ( 0 unt; 4 typ; 0 def)
% Number of atoms : 209 ( 6 equ; 0 cnn)
% Maximal formula atoms : 24 ( 13 avg)
% Number of connectives : 396 ( 2 ~; 0 |; 2 &; 368 @)
% ( 1 <=>; 23 =>; 0 <=; 0 <~>)
% Maximal formula depth : 24 ( 11 avg; 368 nst)
% Number of types : 1 ( 0 usr)
% Number of type conns : 6 ( 6 >; 0 *; 0 +; 0 <<)
% Number of symbols : 25 ( 24 usr; 20 con; 0-2 aty)
% Number of variables : 43 ( 5 ^ 37 !; 1 ?; 43 :)
% SPC : TH0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
thf(tp_c_2EdirGraph_2EEXCLUDE,type,
c_2EdirGraph_2EEXCLUDE: del > del > $i ).
thf(mem_c_2EdirGraph_2EEXCLUDE,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( arr @ ( arr @ A_27b @ bool ) @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) ) ) ) ).
thf(tp_c_2EdirGraph_2EParents,type,
c_2EdirGraph_2EParents: del > del > $i ).
thf(mem_c_2EdirGraph_2EParents,axiom,
! [A_27a: del,A_27b: del] : ( mem @ ( c_2EdirGraph_2EParents @ A_27a @ A_27b ) @ ( arr @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( arr @ A_27a @ bool ) ) ) ).
thf(tp_c_2EdirGraph_2EREACH,type,
c_2EdirGraph_2EREACH: del > $i ).
thf(mem_c_2EdirGraph_2EREACH,axiom,
! [A_27a: del] : ( mem @ ( c_2EdirGraph_2EREACH @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( arr @ A_27a @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(tp_c_2EdirGraph_2EREACH__LIST,type,
c_2EdirGraph_2EREACH__LIST: del > $i ).
thf(mem_c_2EdirGraph_2EREACH__LIST,axiom,
! [A_27a: del] : ( mem @ ( c_2EdirGraph_2EREACH__LIST @ A_27a ) @ ( arr @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( arr @ ( ty_2Elist_2Elist @ A_27a ) @ ( arr @ A_27a @ bool ) ) ) ) ).
thf(ax_thm_2EdirGraph_2EParents__def,axiom,
! [A_27a: del,A_27b: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ( ( ap @ ( c_2EdirGraph_2EParents @ A_27a @ A_27b ) @ V0G )
= ( ap @ ( c_2Epred__set_2EGSPEC @ A_27a @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] : ( ap @ ( ap @ ( c_2Epair_2E_2C @ A_27a @ bool ) @ V1x ) @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Emin_2E_3D @ ( ty_2Elist_2Elist @ A_27b ) ) @ ( ap @ V0G @ V1x ) ) @ ( c_2Elist_2ENIL @ A_27b ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EdirGraph_2EREACH__def,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ( ( ap @ ( c_2EdirGraph_2EREACH @ A_27a ) @ V0G )
= ( ap @ ( c_2Erelation_2ERTC @ A_27a )
@ ( lam @ A_27a
@ ^ [V1x: $i] :
( lam @ A_27a
@ ^ [V2y: $i] : ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ V0G @ V1x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EdirGraph_2EREACH__LIST__def,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ! [V1nodes: $i] :
( ( mem @ V1nodes @ ( ty_2Elist_2Elist @ A_27a ) )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EdirGraph_2EREACH__LIST @ A_27a ) @ V0G ) @ V1nodes ) @ V2y ) )
<=> ? [V3x: $i] :
( ( mem @ V3x @ A_27a )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3x ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ V1nodes ) ) )
& ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2y ) @ ( ap @ ( ap @ ( c_2EdirGraph_2EREACH @ A_27a ) @ V0G ) @ V3x ) ) ) ) ) ) ) ) ).
thf(ax_thm_2EdirGraph_2EEXCLUDE__def,axiom,
! [A_27a: del,A_27b: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27b @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ! [V1ex: $i] :
( ( mem @ V1ex @ ( arr @ A_27b @ bool ) )
=> ! [V2node: $i] :
( ( mem @ V2node @ A_27b )
=> ( ( ap @ ( ap @ ( ap @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27b ) @ V0G ) @ V1ex ) @ V2node )
= ( ap @ ( ap @ ( ap @ ( c_2Ebool_2ECOND @ ( ty_2Elist_2Elist @ A_27a ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27b ) @ V2node ) @ V1ex ) ) @ ( c_2Elist_2ENIL @ A_27a ) ) @ ( ap @ V0G @ V2node ) ) ) ) ) ) ).
thf(conj_thm_2EdirGraph_2EEXCLUDE__LEM,axiom,
! [A_27a: del,A_27b: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27b ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2l: $i] :
( ( mem @ V2l @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( ap @ ( c_2EdirGraph_2EEXCLUDE @ A_27b @ A_27a ) @ V0G ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ V2l ) )
= ( ap @ ( ap @ ( c_2EdirGraph_2EEXCLUDE @ A_27b @ A_27a ) @ ( ap @ ( ap @ ( c_2EdirGraph_2EEXCLUDE @ A_27b @ A_27a ) @ V0G ) @ V2l ) ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V1x ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) ) ) ) ).
thf(conj_thm_2EdirGraph_2EREACH__EXCLUDE,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ ( arr @ A_27a @ bool ) )
=> ( ( ap @ ( c_2EdirGraph_2EREACH @ A_27a ) @ ( ap @ ( ap @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a ) @ V0G ) @ V1x ) )
= ( ap @ ( c_2Erelation_2ERTC @ A_27a )
@ ( lam @ A_27a
@ ^ [V2x_27: $i] :
( lam @ A_27a
@ ^ [V3y: $i] : ( ap @ ( ap @ c_2Ebool_2E_2F_5C @ ( ap @ c_2Ebool_2E_7E @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V2x_27 ) @ V1x ) ) ) @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V3y ) @ ( ap @ ( c_2Elist_2ELIST__TO__SET @ A_27a ) @ ( ap @ V0G @ V2x_27 ) ) ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EdirGraph_2EREACH__LEM1,axiom,
! [A_27a: del,V0p: $i] :
( ( mem @ V0p @ A_27a )
=> ! [V1G: $i] :
( ( mem @ V1G @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ! [V2seen: $i] :
( ( mem @ V2seen @ ( arr @ A_27a @ bool ) )
=> ( ~ ( p @ ( ap @ ( ap @ ( c_2Ebool_2EIN @ A_27a ) @ V0p ) @ V2seen ) )
=> ( ( ap @ ( ap @ ( c_2EdirGraph_2EREACH @ A_27a ) @ ( ap @ ( ap @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a ) @ V1G ) @ V2seen ) ) @ V0p )
= ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0p ) @ ( ap @ ( ap @ ( c_2EdirGraph_2EREACH__LIST @ A_27a ) @ ( ap @ ( ap @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a ) @ V1G ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V0p ) @ V2seen ) ) ) @ ( ap @ V1G @ V0p ) ) ) ) ) ) ) ) ).
thf(conj_thm_2EdirGraph_2EREACH__LEM2,axiom,
! [A_27a: del,V0G: $i] :
( ( mem @ V0G @ ( arr @ A_27a @ ( ty_2Elist_2Elist @ A_27a ) ) )
=> ! [V1x: $i] :
( ( mem @ V1x @ A_27a )
=> ! [V2y: $i] :
( ( mem @ V2y @ A_27a )
=> ( ( p @ ( ap @ ( ap @ ( ap @ ( c_2EdirGraph_2EREACH @ A_27a ) @ V0G ) @ V1x ) @ V2y ) )
=> ! [V3z: $i] :
( ( mem @ V3z @ A_27a )
=> ( ~ ( p @ ( ap @ ( ap @ ( ap @ ( c_2EdirGraph_2EREACH @ A_27a ) @ V0G ) @ V3z ) @ V2y ) )
=> ( p @ ( ap @ ( ap @ ( ap @ ( c_2EdirGraph_2EREACH @ A_27a ) @ ( ap @ ( ap @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a ) @ V0G ) @ ( ap @ ( ap @ ( c_2Epred__set_2EINSERT @ A_27a ) @ V3z ) @ ( c_2Epred__set_2EEMPTY @ A_27a ) ) ) ) @ V1x ) @ V2y ) ) ) ) ) ) ) ) ).
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