ITP001 Axioms: ITP062^7.ax
%------------------------------------------------------------------------------
% File : ITP062^7 : 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 : bft.ax [Gau19]
% : HL4062^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 46 ( 6 unt; 30 typ; 0 def)
% Number of atoms : 66 ( 11 equ; 3 cnn)
% Maximal formula atoms : 11 ( 1 avg)
% Number of connectives : 468 ( 3 ~; 2 |; 12 &; 431 @)
% ( 7 <=>; 13 =>; 0 <=; 0 <~>)
% Maximal formula depth : 21 ( 11 avg; 431 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 105 ( 105 >; 0 *; 0 +; 0 <<)
% Number of symbols : 30 ( 28 usr; 1 con; 0-7 aty)
% Number of variables : 115 ( 0 ^ 84 !; 1 ?; 115 :)
% ( 30 !>; 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_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_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_2Elist_2EALL__DISTINCT,type,
c_2Elist_2EALL__DISTINCT:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > $o ) ).
thf(c_2Elist_2EAPPEND,type,
c_2Elist_2EAPPEND:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Ebft_2EBFT,type,
c_2Ebft_2EBFT:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27b > ( tyop_2Elist_2Elist @ A_27b ) ) > ( A_27b > A_27a > A_27a ) > ( tyop_2Elist_2Elist @ A_27b ) > ( tyop_2Elist_2Elist @ A_27b ) > A_27a > A_27a ) ).
thf(c_2Epred__set_2ECARD,type,
c_2Epred__set_2ECARD:
!>[A_27a: $tType] : ( ( A_27a > $o ) > tyop_2Enum_2Enum ) ).
thf(c_2Ebool_2ECOND,type,
c_2Ebool_2ECOND:
!>[A_27a: $tType] : ( $o > A_27a > A_27a > A_27a ) ).
thf(c_2Elist_2ECONS,type,
c_2Elist_2ECONS:
!>[A_27a: $tType] : ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) ) ).
thf(c_2Epred__set_2EDIFF,type,
c_2Epred__set_2EDIFF:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > 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_2EFINITE,type,
c_2Epred__set_2EFINITE:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Elist_2EFOLDR,type,
c_2Elist_2EFOLDR:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b > A_27b ) > A_27b > ( tyop_2Elist_2Elist @ A_27a ) > A_27b ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Elist_2ELENGTH,type,
c_2Elist_2ELENGTH:
!>[A_27a: $tType] : ( ( tyop_2Elist_2Elist @ A_27a ) > tyop_2Enum_2Enum ) ).
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__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_2Ebft_2ERel,type,
c_2Ebft_2ERel:
!>[A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType] : ( ( tyop_2Epair_2Eprod @ ( A_27a > ( tyop_2Elist_2Elist @ A_27b ) ) @ ( tyop_2Epair_2Eprod @ A_27c @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27d ) @ A_27e ) ) ) ) > ( tyop_2Epair_2Eprod @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum ) ) ).
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_2Ebft_2ERel__def,axiom,
! [A_27a: $tType,A_27b: $tType,A_27c: $tType,A_27d: $tType,A_27e: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27b ),V1f: A_27c,V2seen: tyop_2Elist_2Elist @ A_27a,V3fringe: tyop_2Elist_2Elist @ A_27d,V4acc: A_27e] :
( ( c_2Ebft_2ERel @ A_27a @ A_27b @ A_27c @ A_27d @ A_27e @ ( c_2Epair_2E_2C @ ( A_27a > ( tyop_2Elist_2Elist @ A_27b ) ) @ ( tyop_2Epair_2Eprod @ A_27c @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27d ) @ A_27e ) ) ) @ V0G @ ( c_2Epair_2E_2C @ A_27c @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27d ) @ A_27e ) ) @ V1f @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27a ) @ ( tyop_2Epair_2Eprod @ ( tyop_2Elist_2Elist @ A_27d ) @ A_27e ) @ V2seen @ ( c_2Epair_2E_2C @ ( tyop_2Elist_2Elist @ A_27d ) @ A_27e @ V3fringe @ V4acc ) ) ) ) )
= ( c_2Epair_2E_2C @ tyop_2Enum_2Enum @ tyop_2Enum_2Enum @ ( c_2Epred__set_2ECARD @ A_27a @ ( c_2Epred__set_2EDIFF @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27b @ V0G ) @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V2seen ) ) ) @ ( c_2Elist_2ELENGTH @ A_27d @ V3fringe ) ) ) ).
thf(thm_2Ebft_2EBFT__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0t: tyop_2Elist_2Elist @ A_27a,V1seen: tyop_2Elist_2Elist @ A_27a,V2h: A_27a,V3f: A_27a > A_27b > A_27b,V4acc: A_27b,V5G: A_27a > ( tyop_2Elist_2Elist @ A_27a )] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V5G ) )
=> ( ( ( c_2Ebft_2EBFT @ A_27b @ A_27a @ V5G @ V3f @ V1seen @ ( c_2Elist_2ENIL @ A_27a ) @ V4acc )
= V4acc )
& ( ( c_2Ebft_2EBFT @ A_27b @ A_27a @ V5G @ V3f @ V1seen @ ( c_2Elist_2ECONS @ A_27a @ V2h @ V0t ) @ V4acc )
= ( c_2Ebool_2ECOND @ A_27b @ ( c_2Ebool_2EIN @ A_27a @ V2h @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V1seen ) ) @ ( c_2Ebft_2EBFT @ A_27b @ A_27a @ V5G @ V3f @ V1seen @ V0t @ V4acc ) @ ( c_2Ebft_2EBFT @ A_27b @ A_27a @ V5G @ V3f @ ( c_2Elist_2ECONS @ A_27a @ V2h @ V1seen ) @ ( c_2Elist_2EAPPEND @ A_27a @ V0t @ ( V5G @ V2h ) ) @ ( V3f @ V2h @ V4acc ) ) ) ) ) ) ).
thf(thm_2Ebft_2EBFT__ind,axiom,
! [A_27a: $tType,A_27b: $tType,V0P: ( A_27a > ( tyop_2Elist_2Elist @ A_27a ) ) > ( A_27a > A_27b > A_27b ) > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ) > A_27b > $o] :
( ! [V1G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V2f: A_27a > A_27b > A_27b,V3seen: tyop_2Elist_2Elist @ A_27a,V4h: A_27a,V5t: tyop_2Elist_2Elist @ A_27a,V6acc: A_27b] :
( ( V0P @ V1G @ V2f @ V3seen @ ( c_2Elist_2ENIL @ A_27a ) @ V6acc )
& ( ( ( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V1G ) )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V4h @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V3seen ) ) ) )
=> ( V0P @ V1G @ V2f @ ( c_2Elist_2ECONS @ A_27a @ V4h @ V3seen ) @ ( c_2Elist_2EAPPEND @ A_27a @ V5t @ ( V1G @ V4h ) ) @ ( V2f @ V4h @ V6acc ) ) )
& ( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V1G ) )
& ( c_2Ebool_2EIN @ A_27a @ V4h @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V3seen ) ) )
=> ( V0P @ V1G @ V2f @ V3seen @ V5t @ V6acc ) ) )
=> ( V0P @ V1G @ V2f @ V3seen @ ( c_2Elist_2ECONS @ A_27a @ V4h @ V5t ) @ V6acc ) ) )
=> ! [V7v: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V8v1: A_27a > A_27b > A_27b,V9v2: tyop_2Elist_2Elist @ A_27a,V10v3: tyop_2Elist_2Elist @ A_27a,V11v4: A_27b] : ( V0P @ V7v @ V8v1 @ V9v2 @ V10v3 @ V11v4 ) ) ).
thf(thm_2Ebft_2EBFT__CONS,axiom,
! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1f: A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ),V2seen: tyop_2Elist_2Elist @ A_27a,V3fringe: tyop_2Elist_2Elist @ A_27a,V4acc: tyop_2Elist_2Elist @ A_27a,V5a: tyop_2Elist_2Elist @ A_27a,V6b: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V0G ) )
& ( V1f
= ( c_2Elist_2ECONS @ A_27a ) )
& ( V4acc
= ( c_2Elist_2EAPPEND @ A_27a @ V5a @ V6b ) ) )
=> ( ( c_2Ebft_2EBFT @ ( tyop_2Elist_2Elist @ A_27a ) @ A_27a @ V0G @ V1f @ V2seen @ V3fringe @ V4acc )
= ( c_2Elist_2EAPPEND @ A_27a @ ( c_2Ebft_2EBFT @ ( tyop_2Elist_2Elist @ A_27a ) @ A_27a @ V0G @ V1f @ V2seen @ V3fringe @ V5a ) @ V6b ) ) ) ).
thf(thm_2Ebft_2EBFT__FOLD,axiom,
! [A_27a: $tType,A_27b: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1f: A_27a > A_27b > A_27b,V2seen: tyop_2Elist_2Elist @ A_27a,V3fringe: tyop_2Elist_2Elist @ A_27a,V4acc: A_27b] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V0G ) )
=> ( ( c_2Ebft_2EBFT @ A_27b @ A_27a @ V0G @ V1f @ V2seen @ V3fringe @ V4acc )
= ( c_2Elist_2EFOLDR @ A_27a @ A_27b @ V1f @ V4acc @ ( c_2Ebft_2EBFT @ ( tyop_2Elist_2Elist @ A_27a ) @ A_27a @ V0G @ ( c_2Elist_2ECONS @ A_27a ) @ V2seen @ V3fringe @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ).
thf(thm_2Ebft_2EBFT__ALL__DISTINCT,axiom,
! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1seen: tyop_2Elist_2Elist @ A_27a,V2fringe: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V0G ) )
=> ( c_2Elist_2EALL__DISTINCT @ A_27a @ ( c_2Ebft_2EBFT @ ( tyop_2Elist_2Elist @ A_27a ) @ A_27a @ V0G @ ( c_2Elist_2ECONS @ A_27a ) @ V1seen @ V2fringe @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ).
thf(thm_2Ebft_2EBFT__REACH__1,axiom,
! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1f: A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ),V2seen: tyop_2Elist_2Elist @ A_27a,V3fringe: tyop_2Elist_2Elist @ A_27a,V4acc: tyop_2Elist_2Elist @ A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V0G ) )
& ( V1f
= ( c_2Elist_2ECONS @ A_27a ) ) )
=> ! [V5x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Ebft_2EBFT @ ( tyop_2Elist_2Elist @ A_27a ) @ A_27a @ V0G @ V1f @ V2seen @ V3fringe @ V4acc ) ) )
=> ( ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2EdirGraph_2EREACH__LIST @ A_27a @ V0G @ V3fringe ) )
| ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V4acc ) ) ) ) ) ).
thf(thm_2Ebft_2EBFT__REACH__2,axiom,
! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1f: A_27a > ( tyop_2Elist_2Elist @ A_27a ) > ( tyop_2Elist_2Elist @ A_27a ),V2seen: tyop_2Elist_2Elist @ A_27a,V3fringe: tyop_2Elist_2Elist @ A_27a,V4acc: tyop_2Elist_2Elist @ A_27a,V5x: A_27a] :
( ( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V0G ) )
& ( V1f
= ( c_2Elist_2ECONS @ A_27a ) )
& ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2EdirGraph_2EREACH__LIST @ A_27a @ ( c_2EdirGraph_2EEXCLUDE @ A_27a @ A_27a @ V0G @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V2seen ) ) @ V3fringe ) )
& ( (~) @ ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ V2seen ) ) ) )
=> ( c_2Ebool_2EIN @ A_27a @ V5x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Ebft_2EBFT @ ( tyop_2Elist_2Elist @ A_27a ) @ A_27a @ V0G @ V1f @ V2seen @ V3fringe @ V4acc ) ) ) ) ).
thf(thm_2Ebft_2EBFT__REACH__THM,axiom,
! [A_27a: $tType,V0G: A_27a > ( tyop_2Elist_2Elist @ A_27a ),V1fringe: tyop_2Elist_2Elist @ A_27a] :
( ( c_2Epred__set_2EFINITE @ A_27a @ ( c_2EdirGraph_2EParents @ A_27a @ A_27a @ V0G ) )
=> ! [V2x: A_27a] :
( ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2EdirGraph_2EREACH__LIST @ A_27a @ V0G @ V1fringe ) )
= ( c_2Ebool_2EIN @ A_27a @ V2x @ ( c_2Elist_2ELIST__TO__SET @ A_27a @ ( c_2Ebft_2EBFT @ ( tyop_2Elist_2Elist @ A_27a ) @ A_27a @ V0G @ ( c_2Elist_2ECONS @ A_27a ) @ ( c_2Elist_2ENIL @ A_27a ) @ V1fringe @ ( c_2Elist_2ENIL @ A_27a ) ) ) ) ) ) ).
%------------------------------------------------------------------------------