ITP001 Axioms: ITP029^7.ax
%------------------------------------------------------------------------------
% File : ITP029^7 : 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 : 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 ) ) ) ).
%------------------------------------------------------------------------------