ITP001 Axioms: ITP105^7.ax
%------------------------------------------------------------------------------
% File : ITP105^7 : 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 : veblen.ax [Gau19]
% : HL4105^7.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 56 ( 5 unt; 36 typ; 0 def)
% Number of atoms : 69 ( 2 equ; 6 cnn)
% Maximal formula atoms : 12 ( 1 avg)
% Number of connectives : 354 ( 6 ~; 1 |; 12 &; 307 @)
% ( 12 <=>; 16 =>; 0 <=; 0 <~>)
% Maximal formula depth : 16 ( 10 avg; 307 nst)
% Number of types : 3 ( 2 usr)
% Number of type conns : 93 ( 93 >; 0 *; 0 +; 0 <<)
% Number of symbols : 36 ( 34 usr; 3 con; 0-5 aty)
% Number of variables : 102 ( 5 ^ 67 !; 2 ?; 102 :)
% ( 28 !>; 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_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_2Epred__set_2EBIGINTER,type,
c_2Epred__set_2EBIGINTER:
!>[A_27a: $tType] : ( ( ( A_27a > $o ) > $o ) > 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_2Epred__set_2EIMAGE,type,
c_2Epred__set_2EIMAGE:
!>[A_27a: $tType,A_27b: $tType] : ( ( A_27a > A_27b ) > ( A_27a > $o ) > A_27b > $o ) ).
thf(c_2Ebool_2EIN,type,
c_2Ebool_2EIN:
!>[A_27a: $tType] : ( A_27a > ( A_27a > $o ) > $o ) ).
thf(c_2Epred__set_2ESUBSET,type,
c_2Epred__set_2ESUBSET:
!>[A_27a: $tType] : ( ( A_27a > $o ) > ( A_27a > $o ) > $o ) ).
thf(c_2Enum_2ESUC,type,
c_2Enum_2ESUC: tyop_2Enum_2Enum > tyop_2Enum_2Enum ).
thf(c_2Ebool_2ET,type,
c_2Ebool_2ET: $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_2Eveblen_2Eclosed,type,
c_2Eveblen_2Eclosed:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > $o ) ).
thf(c_2Eveblen_2Eclub,type,
c_2Eveblen_2Eclub:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > $o ) ).
thf(c_2Eveblen_2Econtinuous,type,
c_2Eveblen_2Econtinuous:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27b ) ) > $o ) ).
thf(c_2Eordinal_2EfromNat,type,
c_2Eordinal_2EfromNat:
!>[A_27a: $tType] : ( tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2Eoleast,type,
c_2Eordinal_2Eoleast:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
thf(c_2Eordinal_2EordSUC,type,
c_2Eordinal_2EordSUC:
!>[A_27a: $tType] : ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ).
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_2Eveblen_2Estrict__mono,type,
c_2Eveblen_2Estrict__mono:
!>[A_27a: $tType,A_27b: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27b ) ) > $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_2Eveblen_2Eunbounded,type,
c_2Eveblen_2Eunbounded:
!>[A_27a: $tType] : ( ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $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_2Eveblen_2Eclosed__def,axiom,
! [A_27a: $tType,V0A: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eveblen_2Eclosed @ A_27a @ V0A )
<=> ! [V1g: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a )] :
( ! [V2n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( V1g @ V2n ) @ V0A )
=> ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ( c_2Eordinal_2Esup @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ $o @ ( V1g @ V3n ) @ c_2Ebool_2ET ) ) )
@ V0A ) ) ) ).
thf(thm_2Eveblen_2Eunbounded__def,axiom,
! [A_27a: $tType,V0A: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eveblen_2Eunbounded @ A_27a @ V0A )
<=> ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
? [V2b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V2b @ V0A )
& ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ V2b ) ) ) ).
thf(thm_2Eveblen_2Eclub__def,axiom,
! [A_27a: $tType,V0A: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Eveblen_2Eclub @ A_27a @ V0A )
<=> ( ( c_2Eveblen_2Eclosed @ A_27a @ V0A )
& ( c_2Eveblen_2Eunbounded @ A_27a @ V0A ) ) ) ).
thf(thm_2Eveblen_2Econtinuous__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27b )] :
( ( c_2Eveblen_2Econtinuous @ A_27a @ A_27b @ V0f )
<=> ! [V1A: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( c_2Ecardinal_2Ecardleq @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) @ V1A @ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) )
=> ( ( V0f @ ( c_2Eordinal_2Esup @ A_27a @ V1A ) )
= ( c_2Eordinal_2Esup @ A_27b @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27b ) @ V0f @ V1A ) ) ) ) ) ).
thf(thm_2Eveblen_2Estrict__mono__def,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27b )] :
( ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27b @ V0f )
<=> ! [V1x: tyop_2Eordinal_2Eordinal @ A_27a,V2y: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V1x @ V2y )
=> ( c_2Eordinal_2Eordlt @ A_27b @ ( V0f @ V1x ) @ ( V0f @ V2y ) ) ) ) ).
thf(thm_2Eveblen_2Ebetter__induction,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ( V0P @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) )
& ! [V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( V0P @ V1a )
=> ( V0P @ ( c_2Eordinal_2EordSUC @ A_27a @ V1a ) ) )
& ! [V2a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eordinal_2Eordlt @ A_27a @ ( c_2Eordinal_2EfromNat @ A_27a @ c_2Enum_2E0 ) @ V2a )
& ! [V3b: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( c_2Eordinal_2Eordlt @ A_27a @ V3b @ V2a )
=> ( V0P @ V3b ) ) )
=> ( V0P @ ( c_2Eordinal_2Esup @ A_27a @ ( c_2Eordinal_2Epreds @ A_27a @ V2a ) ) ) ) )
=> ! [V4a: tyop_2Eordinal_2Eordinal @ A_27a] : ( V0P @ V4a ) ) ).
thf(thm_2Eveblen_2Enrange__IN__Uinf,axiom,
! [A_27a: $tType,A_27b: $tType,V0f: tyop_2Enum_2Enum > A_27b] :
( c_2Ecardinal_2Ecardleq @ A_27b @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a )
@ ( c_2Epred__set_2EGSPEC @ A_27b @ tyop_2Enum_2Enum
@ ^ [V1n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ A_27b @ $o @ ( V0f @ V1n ) @ c_2Ebool_2ET ) )
@ ( c_2Epred__set_2EUNIV @ ( tyop_2Esum_2Esum @ tyop_2Enum_2Enum @ A_27a ) ) ) ).
thf(thm_2Eveblen_2Eincreasing,axiom,
! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a ),V1x: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27a @ V0f )
& ( c_2Eveblen_2Econtinuous @ A_27a @ A_27a @ V0f ) )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V1x ) @ V1x ) ) ) ).
thf(thm_2Eveblen_2Eclubs__exist,axiom,
! [A_27a: $tType,V0f: ( tyop_2Eordinal_2Eordinal @ A_27a ) > ( tyop_2Eordinal_2Eordinal @ A_27a )] :
( ( ( c_2Eveblen_2Estrict__mono @ A_27a @ A_27a @ V0f )
& ( c_2Eveblen_2Econtinuous @ A_27a @ A_27a @ V0f ) )
=> ( c_2Eveblen_2Eclub @ A_27a @ ( c_2Epred__set_2EIMAGE @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ V0f @ ( c_2Epred__set_2EUNIV @ ( tyop_2Eordinal_2Eordinal @ A_27a ) ) ) ) ) ).
thf(thm_2Eveblen_2Emono__natI,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a )] :
( ! [V1n: tyop_2Enum_2Enum] : ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ ( c_2Enum_2ESUC @ V1n ) ) @ ( V0f @ V1n ) ) )
=> ! [V2m: tyop_2Enum_2Enum,V3n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V2m @ V3n )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V3n ) @ ( V0f @ V2m ) ) ) ) ) ).
thf(thm_2Eveblen_2Esup__mem__INTER,axiom,
! [A_27a: $tType,V0f: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ),V1A: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ! [V2n: tyop_2Enum_2Enum] : ( c_2Eveblen_2Eclub @ A_27a @ ( V1A @ V2n ) )
& ! [V3n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( V1A @ ( c_2Enum_2ESUC @ V3n ) ) @ ( V1A @ V3n ) )
& ! [V4n: tyop_2Enum_2Enum] : ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( V0f @ V4n ) @ ( V1A @ V4n ) )
& ! [V5m: tyop_2Enum_2Enum,V6n: tyop_2Enum_2Enum] :
( ( c_2Earithmetic_2E_3C_3D @ V5m @ V6n )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ ( V0f @ V6n ) @ ( V0f @ V5m ) ) ) ) )
=> ( c_2Ebool_2EIN @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ( c_2Eordinal_2Esup @ A_27a
@ ( c_2Epred__set_2EGSPEC @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ tyop_2Enum_2Enum
@ ^ [V7n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ $o @ ( V0f @ V7n ) @ c_2Ebool_2ET ) ) )
@ ( c_2Epred__set_2EBIGINTER @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ( c_2Epred__set_2EGSPEC @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ tyop_2Enum_2Enum
@ ^ [V8n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ $o @ ( V1A @ V8n ) @ c_2Ebool_2ET ) ) ) ) ) ).
thf(thm_2Eveblen_2Eoleast__leq,axiom,
! [A_27a: $tType,V0P: ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o,V1a: tyop_2Eordinal_2Eordinal @ A_27a] :
( ( V0P @ V1a )
=> ( (~) @ ( c_2Eordinal_2Eordlt @ A_27a @ V1a @ ( c_2Eordinal_2Eoleast @ A_27a @ V0P ) ) ) ) ).
thf(thm_2Eveblen_2Eclub__INTER,axiom,
! [A_27a: $tType,V0A: tyop_2Enum_2Enum > ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o] :
( ( ! [V1n: tyop_2Enum_2Enum] : ( c_2Eveblen_2Eclub @ A_27a @ ( V0A @ V1n ) )
& ! [V2n: tyop_2Enum_2Enum] : ( c_2Epred__set_2ESUBSET @ ( tyop_2Eordinal_2Eordinal @ A_27a ) @ ( V0A @ ( c_2Enum_2ESUC @ V2n ) ) @ ( V0A @ V2n ) ) )
=> ( c_2Eveblen_2Eclub @ A_27a
@ ( c_2Epred__set_2EBIGINTER @ ( tyop_2Eordinal_2Eordinal @ A_27a )
@ ( c_2Epred__set_2EGSPEC @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ tyop_2Enum_2Enum
@ ^ [V3n: tyop_2Enum_2Enum] : ( c_2Epair_2E_2C @ ( ( tyop_2Eordinal_2Eordinal @ A_27a ) > $o ) @ $o @ ( V0A @ V3n ) @ c_2Ebool_2ET ) ) ) ) ) ).
%------------------------------------------------------------------------------