ITP001 Axioms: ITP104^7.ax
%------------------------------------------------------------------------------
% File : ITP104^7 : 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 : ucord.ax [Gau19]
% : HL4104^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 36 ( 7 unt; 22 typ; 0 def)
% Number of atoms : 38 ( 3 equ; 4 cnn)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 191 ( 4 ~; 1 |; 1 &; 177 @)
% ( 7 <=>; 1 =>; 0 <=; 0 <~>)
% Maximal formula depth : 13 ( 7 avg; 177 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 69 ( 69 >; 0 *; 0 +; 0 <<)
% Number of symbols : 22 ( 20 usr; 1 con; 0-4 aty)
% Number of variables : 41 ( 2 ^ 23 !; 1 ?; 41 :)
% ( 15 !>; 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_2Eordinal_2Eordinal,type,
tyop_2Eordinal_2Eordinal: $tType > $tType ).
thf(tyop_2Epair_2Eprod,type,
tyop_2Epair_2Eprod: $tType > $tType > $tType ).
thf(tyop_2Esum_2Esum,type,
tyop_2Esum_2Esum: $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_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_2Epred__set_2EUNIV,type,
c_2Epred__set_2EUNIV:
!>[A_27a: $tType] : ( A_27a > $o ) ).
thf(c_2Ebool_2E_5C_2F,type,
c_2Ebool_2E_5C_2F: $o > $o > $o ).
thf(c_2Ecardinal_2Ecardleq,type,
c_2Ecardinal_2Ecardleq:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > $o ) > ( A_27b > $o ) > $o ) ).
thf(c_2Epred__set_2Ecountable,type,
c_2Epred__set_2Ecountable:
!>[A_27a: $tType] : ( ( A_27a > $o ) > $o ) ).
thf(c_2Eucord_2Eomega1,type,
c_2Eucord_2Eomega1:
!>[A_27a: $tType] : ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ).
thf(c_2Eordinal_2Eordlt,type,
c_2Eordinal_2Eordlt:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) ).
thf(c_2Eordinal_2Epreds,type,
c_2Eordinal_2Epreds:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) ).
thf(c_2Eordinal_2Esup,type,
c_2Eordinal_2Esup:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
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_2Eucord_2Eomega1__def,axiom,
! [A_27a: $tType] :
( ( c_2Eucord_2Eomega1 @ A_27a )
= ( c_2Eordinal_2Esup @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) )
@ ^ [V0a: tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) )] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ $o @ V0a @ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Eordinal_2Epreds @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ V0a ) ) ) ) ) ) ).
thf(thm_2Eucord_2Eucinf__uncountable,axiom,
! [A_27a: $tType] : ( (~) @ ( c_2Epred__set_2Ecountable @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ) ) ) ).
thf(thm_2Eucord_2EUnum__cardlt__ucinf,axiom,
! [A_27a: $tType] : ( (~) @ ( c_2Ecardinal_2Ecardleq @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ tyop_2Enum_2Enum @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) ) ) ).
thf(thm_2Eucord_2EUnum__cardle__ucinf,axiom,
! [A_27a: $tType] : ( c_2Ecardinal_2Ecardleq @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Epred__set_2EUNIV @ tyop_2Enum_2Enum ) @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ) ) ).
thf(thm_2Eucord_2Eucord__sup__exists__lemma,axiom,
! [A_27a: $tType] :
( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) )
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) )
@ ^ [V0a: tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) )] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ $o @ V0a @ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Eordinal_2Epreds @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ V0a ) ) ) )
@ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) ) ) ).
thf(thm_2Eucord_2Ex__lt__omega1__countable,axiom,
! [A_27a: $tType,V0x: tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) )] :
( ( c_2Eordinal_2Eordlt @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ V0x @ ( c_2Eucord_2Eomega1 @ A_27a ) )
= ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Eordinal_2Epreds @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ V0x ) ) ) ).
thf(thm_2Eucord_2Eomega1__not__countable,axiom,
! [A_27a: $tType] : ( (~) @ ( c_2Epred__set_2Ecountable @ ( tyop_2Eordinal_2Eordinal @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) ) @ ( c_2Eordinal_2Epreds @ ( tyop_2Esum_2Esum @ A_27a @ ( tyop_2Enum_2Enum > $o ) ) @ ( c_2Eucord_2Eomega1 @ A_27a ) ) ) ) ).
%------------------------------------------------------------------------------