ITP001 Axioms: ITP105_5.ax
%------------------------------------------------------------------------------
% File : ITP105_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 : veblen_2.ax [Gau20]
% : HL4105_5.ax [TPAP]
% Status : Satisfiable
% Syntax : Number of formulae : 33 ( 5 unt; 10 typ; 0 def)
% Number of atoms : 459 ( 6 equ)
% Maximal formula atoms : 13 ( 13 avg)
% Number of connectives : 67 ( 5 ~; 0 |; 12 &)
% ( 5 <=>; 45 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 7 avg)
% Maximal term depth : 2 ( 1 avg)
% Number of FOOLs : 374 ( 374 fml; 0 var)
% Number of types : 1 ( 0 usr)
% Number of type conns : 17 ( 10 >; 7 *; 0 +; 0 <<)
% Number of predicates : 31 ( 30 usr; 8 prp; 0-2 aty)
% Number of functors : 10 ( 10 usr; 0 con; 1-2 aty)
% Number of variables : 75 ( 74 !; 1 ?; 75 :)
% SPC : TF0_SAT_EQU_NAR
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
tff(tp_c_2Eveblen_2Eclosed,type,
c_2Eveblen_2Eclosed: del > $i ).
tff(mem_c_2Eveblen_2Eclosed,axiom,
! [A_27a: del] : mem(c_2Eveblen_2Eclosed(A_27a),arr(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool)) ).
tff(tp_c_2Eveblen_2Eclub,type,
c_2Eveblen_2Eclub: del > $i ).
tff(mem_c_2Eveblen_2Eclub,axiom,
! [A_27a: del] : mem(c_2Eveblen_2Eclub(A_27a),arr(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool)) ).
tff(tp_c_2Eveblen_2Econtinuous,type,
c_2Eveblen_2Econtinuous: ( del * del ) > $i ).
tff(mem_c_2Eveblen_2Econtinuous,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eveblen_2Econtinuous(A_27a,A_27b),arr(arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27b)),bool)) ).
tff(tp_c_2Eveblen_2Estrict__mono,type,
c_2Eveblen_2Estrict__mono: ( del * del ) > $i ).
tff(mem_c_2Eveblen_2Estrict__mono,axiom,
! [A_27a: del,A_27b: del] : mem(c_2Eveblen_2Estrict__mono(A_27a,A_27b),arr(arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27b)),bool)) ).
tff(tp_c_2Eveblen_2Eunbounded,type,
c_2Eveblen_2Eunbounded: del > $i ).
tff(mem_c_2Eveblen_2Eunbounded,axiom,
! [A_27a: del] : mem(c_2Eveblen_2Eunbounded(A_27a),arr(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool)) ).
tff(conj_thm_2Eveblen_2Ebetter__induction,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ( ( p(ap(V0P,ap(c_2Eordinal_2EfromNat(A_27a),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))))
& ! [V1a: $i] :
( mem(V1a,ty_2Eordinal_2Eordinal(A_27a))
=> ( p(ap(V0P,V1a))
=> p(ap(V0P,ap(c_2Eordinal_2EordSUC(A_27a),V1a))) ) )
& ! [V2a: $i] :
( mem(V2a,ty_2Eordinal_2Eordinal(A_27a))
=> ( ( p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(c_2Eordinal_2EfromNat(A_27a),inj__ty_2Enum_2Enum(fo__c_2Enum_2E0))),V2a))
& ! [V3b: $i] :
( mem(V3b,ty_2Eordinal_2Eordinal(A_27a))
=> ( p(ap(ap(c_2Eordinal_2Eordlt(A_27a),V3b),V2a))
=> p(ap(V0P,V3b)) ) ) )
=> p(ap(V0P,ap(c_2Eordinal_2Esup(A_27a),ap(c_2Eordinal_2Epreds(A_27a),V2a)))) ) ) )
=> ! [V4a: $i] :
( mem(V4a,ty_2Eordinal_2Eordinal(A_27a))
=> p(ap(V0P,V4a)) ) ) ) ).
tff(lamtp_f2516,type,
f2516: ( del * $i ) > $i ).
tff(lameq_f2516,axiom,
! [A_27a: del,V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ! [V3n: tp__ty_2Enum_2Enum] : ( ap(f2516(A_27a,V1g),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Epair_2E_2C(ty_2Eordinal_2Eordinal(A_27a),bool),ap(V1g,inj__ty_2Enum_2Enum(V3n))),inj__o(fo__c_2Ebool_2ET)) ) ) ).
tff(ax_thm_2Eveblen_2Eclosed__def,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ( p(ap(c_2Eveblen_2Eclosed(A_27a),V0A))
<=> ! [V1g: $i] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ( ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(ty_2Eordinal_2Eordinal(A_27a)),ap(V1g,inj__ty_2Enum_2Enum(V2n))),V0A))
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eordinal_2Eordinal(A_27a)),ap(c_2Eordinal_2Esup(A_27a),ap(c_2Epred__set_2EGSPEC(ty_2Eordinal_2Eordinal(A_27a),ty_2Enum_2Enum),f2516(A_27a,V1g)))),V0A)) ) ) ) ) ).
tff(ax_thm_2Eveblen_2Eunbounded__def,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ( p(ap(c_2Eveblen_2Eunbounded(A_27a),V0A))
<=> ! [V1a: $i] :
( mem(V1a,ty_2Eordinal_2Eordinal(A_27a))
=> ? [V2b: $i] :
( mem(V2b,ty_2Eordinal_2Eordinal(A_27a))
& p(ap(ap(c_2Ebool_2EIN(ty_2Eordinal_2Eordinal(A_27a)),V2b),V0A))
& p(ap(ap(c_2Eordinal_2Eordlt(A_27a),V1a),V2b)) ) ) ) ) ).
tff(ax_thm_2Eveblen_2Eclub__def,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ( p(ap(c_2Eveblen_2Eclub(A_27a),V0A))
<=> ( p(ap(c_2Eveblen_2Eclosed(A_27a),V0A))
& p(ap(c_2Eveblen_2Eunbounded(A_27a),V0A)) ) ) ) ).
tff(ax_thm_2Eveblen_2Econtinuous__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27b)))
=> ( p(ap(c_2Eveblen_2Econtinuous(A_27a,A_27b),V0f))
<=> ! [V1A: $i] :
( mem(V1A,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ( p(ap(ap(c_2Ecardinal_2Ecardleq(ty_2Eordinal_2Eordinal(A_27a),ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),V1A),c_2Epred__set_2EUNIV(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a))))
=> ( ap(V0f,ap(c_2Eordinal_2Esup(A_27a),V1A)) = ap(c_2Eordinal_2Esup(A_27b),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27b)),V0f),V1A)) ) ) ) ) ) ).
tff(ax_thm_2Eveblen_2Estrict__mono__def,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27b)))
=> ( p(ap(c_2Eveblen_2Estrict__mono(A_27a,A_27b),V0f))
<=> ! [V1x: $i] :
( mem(V1x,ty_2Eordinal_2Eordinal(A_27a))
=> ! [V2y: $i] :
( mem(V2y,ty_2Eordinal_2Eordinal(A_27a))
=> ( p(ap(ap(c_2Eordinal_2Eordlt(A_27a),V1x),V2y))
=> p(ap(ap(c_2Eordinal_2Eordlt(A_27b),ap(V0f,V1x)),ap(V0f,V2y))) ) ) ) ) ) ).
tff(lamtp_f2517,type,
f2517: ( del * $i ) > $i ).
tff(lameq_f2517,axiom,
! [A_27b: del,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27b))
=> ! [V1n: tp__ty_2Enum_2Enum] : ( ap(f2517(A_27b,V0f),inj__ty_2Enum_2Enum(V1n)) = ap(ap(c_2Epair_2E_2C(A_27b,bool),ap(V0f,inj__ty_2Enum_2Enum(V1n))),inj__o(fo__c_2Ebool_2ET)) ) ) ).
tff(conj_thm_2Eveblen_2Enrange__IN__Uinf,axiom,
! [A_27a: del,A_27b: del,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27b))
=> p(ap(ap(c_2Ecardinal_2Ecardleq(A_27b,ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)),ap(c_2Epred__set_2EGSPEC(A_27b,ty_2Enum_2Enum),f2517(A_27b,V0f))),c_2Epred__set_2EUNIV(ty_2Esum_2Esum(ty_2Enum_2Enum,A_27a)))) ) ).
tff(conj_thm_2Eveblen_2Eincreasing,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27a)))
=> ! [V1x: $i] :
( mem(V1x,ty_2Eordinal_2Eordinal(A_27a))
=> ( ( p(ap(c_2Eveblen_2Estrict__mono(A_27a,A_27a),V0f))
& p(ap(c_2Eveblen_2Econtinuous(A_27a,A_27a),V0f)) )
=> ~ p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(V0f,V1x)),V1x)) ) ) ) ).
tff(conj_thm_2Eveblen_2Eclubs__exist,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27a)))
=> ( ( p(ap(c_2Eveblen_2Estrict__mono(A_27a,A_27a),V0f))
& p(ap(c_2Eveblen_2Econtinuous(A_27a,A_27a),V0f)) )
=> p(ap(c_2Eveblen_2Eclub(A_27a),ap(ap(c_2Epred__set_2EIMAGE(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27a)),V0f),c_2Epred__set_2EUNIV(ty_2Eordinal_2Eordinal(A_27a))))) ) ) ).
tff(conj_thm_2Eveblen_2Emono__natI,axiom,
! [A_27a: del,V0f: $i] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ( ! [V1n: tp__ty_2Enum_2Enum] : ~ p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(V0f,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V1n)))),ap(V0f,inj__ty_2Enum_2Enum(V1n))))
=> ! [V2m: tp__ty_2Enum_2Enum,V3n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V2m)),inj__ty_2Enum_2Enum(V3n)))
=> ~ p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(V0f,inj__ty_2Enum_2Enum(V3n))),ap(V0f,inj__ty_2Enum_2Enum(V2m)))) ) ) ) ).
tff(lamtp_f2518,type,
f2518: ( del * $i ) > $i ).
tff(lameq_f2518,axiom,
! [A_27a: del,V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ! [V7n: tp__ty_2Enum_2Enum] : ( ap(f2518(A_27a,V1f),inj__ty_2Enum_2Enum(V7n)) = ap(ap(c_2Epair_2E_2C(ty_2Eordinal_2Eordinal(A_27a),bool),ap(V1f,inj__ty_2Enum_2Enum(V7n))),inj__o(fo__c_2Ebool_2ET)) ) ) ).
tff(lamtp_f2519,type,
f2519: ( del * $i ) > $i ).
tff(lameq_f2519,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(ty_2Enum_2Enum,arr(ty_2Eordinal_2Eordinal(A_27a),bool)))
=> ! [V8n: tp__ty_2Enum_2Enum] : ( ap(f2519(A_27a,V0A),inj__ty_2Enum_2Enum(V8n)) = ap(ap(c_2Epair_2E_2C(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool),ap(V0A,inj__ty_2Enum_2Enum(V8n))),inj__o(fo__c_2Ebool_2ET)) ) ) ).
tff(conj_thm_2Eveblen_2Esup__mem__INTER,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(ty_2Enum_2Enum,arr(ty_2Eordinal_2Eordinal(A_27a),bool)))
=> ! [V1f: $i] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ( ( ! [V2n: tp__ty_2Enum_2Enum] : p(ap(c_2Eveblen_2Eclub(A_27a),ap(V0A,inj__ty_2Enum_2Enum(V2n))))
& ! [V3n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Eordinal_2Eordinal(A_27a)),ap(V0A,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V3n)))),ap(V0A,inj__ty_2Enum_2Enum(V3n))))
& ! [V4n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Ebool_2EIN(ty_2Eordinal_2Eordinal(A_27a)),ap(V1f,inj__ty_2Enum_2Enum(V4n))),ap(V0A,inj__ty_2Enum_2Enum(V4n))))
& ! [V5m: tp__ty_2Enum_2Enum,V6n: tp__ty_2Enum_2Enum] :
( p(ap(ap(c_2Earithmetic_2E_3C_3D,inj__ty_2Enum_2Enum(V5m)),inj__ty_2Enum_2Enum(V6n)))
=> ~ p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(V1f,inj__ty_2Enum_2Enum(V6n))),ap(V1f,inj__ty_2Enum_2Enum(V5m)))) ) )
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eordinal_2Eordinal(A_27a)),ap(c_2Eordinal_2Esup(A_27a),ap(c_2Epred__set_2EGSPEC(ty_2Eordinal_2Eordinal(A_27a),ty_2Enum_2Enum),f2518(A_27a,V1f)))),ap(c_2Epred__set_2EBIGINTER(ty_2Eordinal_2Eordinal(A_27a)),ap(c_2Epred__set_2EGSPEC(arr(ty_2Eordinal_2Eordinal(A_27a),bool),ty_2Enum_2Enum),f2519(A_27a,V0A))))) ) ) ) ).
tff(conj_thm_2Eveblen_2Eoleast__leq,axiom,
! [A_27a: del,V0P: $i] :
( mem(V0P,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ! [V1a: $i] :
( mem(V1a,ty_2Eordinal_2Eordinal(A_27a))
=> ( p(ap(V0P,V1a))
=> ~ p(ap(ap(c_2Eordinal_2Eordlt(A_27a),V1a),ap(c_2Eordinal_2Eoleast(A_27a),V0P))) ) ) ) ).
tff(lamtp_f2520,type,
f2520: ( del * $i ) > $i ).
tff(lameq_f2520,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(ty_2Enum_2Enum,arr(ty_2Eordinal_2Eordinal(A_27a),bool)))
=> ! [V3n: tp__ty_2Enum_2Enum] : ( ap(f2520(A_27a,V0A),inj__ty_2Enum_2Enum(V3n)) = ap(ap(c_2Epair_2E_2C(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool),ap(V0A,inj__ty_2Enum_2Enum(V3n))),inj__o(fo__c_2Ebool_2ET)) ) ) ).
tff(conj_thm_2Eveblen_2Eclub__INTER,axiom,
! [A_27a: del,V0A: $i] :
( mem(V0A,arr(ty_2Enum_2Enum,arr(ty_2Eordinal_2Eordinal(A_27a),bool)))
=> ( ( ! [V1n: tp__ty_2Enum_2Enum] : p(ap(c_2Eveblen_2Eclub(A_27a),ap(V0A,inj__ty_2Enum_2Enum(V1n))))
& ! [V2n: tp__ty_2Enum_2Enum] : p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Eordinal_2Eordinal(A_27a)),ap(V0A,ap(c_2Enum_2ESUC,inj__ty_2Enum_2Enum(V2n)))),ap(V0A,inj__ty_2Enum_2Enum(V2n)))) )
=> p(ap(c_2Eveblen_2Eclub(A_27a),ap(c_2Epred__set_2EBIGINTER(ty_2Eordinal_2Eordinal(A_27a)),ap(c_2Epred__set_2EGSPEC(arr(ty_2Eordinal_2Eordinal(A_27a),bool),ty_2Enum_2Enum),f2520(A_27a,V0A))))) ) ) ).
%------------------------------------------------------------------------------