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 : 23 ( 0 unt; 0 def)
% Number of atoms : 119 ( 6 equ)
% Maximal formula atoms : 14 ( 5 avg)
% Number of connectives : 101 ( 5 ~; 0 |; 12 &)
% ( 5 <=>; 79 =>; 0 <=; 0 <~>)
% Maximal formula depth : 17 ( 8 avg)
% Maximal term depth : 7 ( 2 avg)
% Number of predicates : 4 ( 3 usr; 0 prp; 1-2 aty)
% Number of functors : 34 ( 34 usr; 6 con; 0-2 aty)
% Number of variables : 75 ( 74 !; 1 ?)
% SPC : FOF_SAT_RFO_SEQ
% Comments :
% Bugfixes : v7.5.0 - Fixes to the axioms.
%------------------------------------------------------------------------------
fof(mem_c_2Eveblen_2Eclosed,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eveblen_2Eclosed(A_27a),arr(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool)) ) ).
fof(mem_c_2Eveblen_2Eclub,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eveblen_2Eclub(A_27a),arr(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool)) ) ).
fof(mem_c_2Eveblen_2Econtinuous,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eveblen_2Econtinuous(A_27a,A_27b),arr(arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27b)),bool)) ) ) ).
fof(mem_c_2Eveblen_2Estrict__mono,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> mem(c_2Eveblen_2Estrict__mono(A_27a,A_27b),arr(arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27b)),bool)) ) ) ).
fof(mem_c_2Eveblen_2Eunbounded,axiom,
! [A_27a] :
( ne(A_27a)
=> mem(c_2Eveblen_2Eunbounded(A_27a),arr(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool)) ) ).
fof(conj_thm_2Eveblen_2Ebetter__induction,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ( ( p(ap(V0P,ap(c_2Eordinal_2EfromNat(A_27a),c_2Enum_2E0)))
& ! [V1a] :
( mem(V1a,ty_2Eordinal_2Eordinal(A_27a))
=> ( p(ap(V0P,V1a))
=> p(ap(V0P,ap(c_2Eordinal_2EordSUC(A_27a),V1a))) ) )
& ! [V2a] :
( mem(V2a,ty_2Eordinal_2Eordinal(A_27a))
=> ( ( p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(c_2Eordinal_2EfromNat(A_27a),c_2Enum_2E0)),V2a))
& ! [V3b] :
( 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] :
( mem(V4a,ty_2Eordinal_2Eordinal(A_27a))
=> p(ap(V0P,V4a)) ) ) ) ) ).
fof(lameq_f2516,axiom,
! [A_27a,V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ! [V3n] : ap(f2516(A_27a,V1g),V3n) = ap(ap(c_2Epair_2E_2C(ty_2Eordinal_2Eordinal(A_27a),bool),ap(V1g,V3n)),c_2Ebool_2ET) ) ).
fof(ax_thm_2Eveblen_2Eclosed__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ( p(ap(c_2Eveblen_2Eclosed(A_27a),V0A))
<=> ! [V1g] :
( mem(V1g,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eordinal_2Eordinal(A_27a)),ap(V1g,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)) ) ) ) ) ) ).
fof(ax_thm_2Eveblen_2Eunbounded__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ( p(ap(c_2Eveblen_2Eunbounded(A_27a),V0A))
<=> ! [V1a] :
( mem(V1a,ty_2Eordinal_2Eordinal(A_27a))
=> ? [V2b] :
( 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)) ) ) ) ) ) ).
fof(ax_thm_2Eveblen_2Eclub__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( 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)) ) ) ) ) ).
fof(ax_thm_2Eveblen_2Econtinuous__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( mem(V0f,arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27b)))
=> ( p(ap(c_2Eveblen_2Econtinuous(A_27a,A_27b),V0f))
<=> ! [V1A] :
( 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)) ) ) ) ) ) ) ).
fof(ax_thm_2Eveblen_2Estrict__mono__def,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( 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] :
( mem(V1x,ty_2Eordinal_2Eordinal(A_27a))
=> ! [V2y] :
( 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))) ) ) ) ) ) ) ) ).
fof(lameq_f2517,axiom,
! [A_27b,V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,A_27b))
=> ! [V1n] : ap(f2517(A_27b,V0f),V1n) = ap(ap(c_2Epair_2E_2C(A_27b,bool),ap(V0f,V1n)),c_2Ebool_2ET) ) ).
fof(conj_thm_2Eveblen_2Enrange__IN__Uinf,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [A_27b] :
( ne(A_27b)
=> ! [V0f] :
( 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)))) ) ) ) ).
fof(conj_thm_2Eveblen_2Eincreasing,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Eordinal_2Eordinal(A_27a),ty_2Eordinal_2Eordinal(A_27a)))
=> ! [V1x] :
( 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)) ) ) ) ) ).
fof(conj_thm_2Eveblen_2Eclubs__exist,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( 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))))) ) ) ) ).
fof(conj_thm_2Eveblen_2Emono__natI,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0f] :
( mem(V0f,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> ~ p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(V0f,ap(c_2Enum_2ESUC,V1n))),ap(V0f,V1n))) )
=> ! [V2m] :
( mem(V2m,ty_2Enum_2Enum)
=> ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V2m),V3n))
=> ~ p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(V0f,V3n)),ap(V0f,V2m))) ) ) ) ) ) ) ).
fof(lameq_f2518,axiom,
! [A_27a,V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ! [V7n] : ap(f2518(A_27a,V1f),V7n) = ap(ap(c_2Epair_2E_2C(ty_2Eordinal_2Eordinal(A_27a),bool),ap(V1f,V7n)),c_2Ebool_2ET) ) ).
fof(lameq_f2519,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(ty_2Enum_2Enum,arr(ty_2Eordinal_2Eordinal(A_27a),bool)))
=> ! [V8n] : ap(f2519(A_27a,V0A),V8n) = ap(ap(c_2Epair_2E_2C(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool),ap(V0A,V8n)),c_2Ebool_2ET) ) ).
fof(conj_thm_2Eveblen_2Esup__mem__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(ty_2Enum_2Enum,arr(ty_2Eordinal_2Eordinal(A_27a),bool)))
=> ! [V1f] :
( mem(V1f,arr(ty_2Enum_2Enum,ty_2Eordinal_2Eordinal(A_27a)))
=> ( ( ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(c_2Eveblen_2Eclub(A_27a),ap(V0A,V2n))) )
& ! [V3n] :
( mem(V3n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Eordinal_2Eordinal(A_27a)),ap(V0A,ap(c_2Enum_2ESUC,V3n))),ap(V0A,V3n))) )
& ! [V4n] :
( mem(V4n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Ebool_2EIN(ty_2Eordinal_2Eordinal(A_27a)),ap(V1f,V4n)),ap(V0A,V4n))) )
& ! [V5m] :
( mem(V5m,ty_2Enum_2Enum)
=> ! [V6n] :
( mem(V6n,ty_2Enum_2Enum)
=> ( p(ap(ap(c_2Earithmetic_2E_3C_3D,V5m),V6n))
=> ~ p(ap(ap(c_2Eordinal_2Eordlt(A_27a),ap(V1f,V6n)),ap(V1f,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))))) ) ) ) ) ).
fof(conj_thm_2Eveblen_2Eoleast__leq,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0P] :
( mem(V0P,arr(ty_2Eordinal_2Eordinal(A_27a),bool))
=> ! [V1a] :
( 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))) ) ) ) ) ).
fof(lameq_f2520,axiom,
! [A_27a,V0A] :
( mem(V0A,arr(ty_2Enum_2Enum,arr(ty_2Eordinal_2Eordinal(A_27a),bool)))
=> ! [V3n] : ap(f2520(A_27a,V0A),V3n) = ap(ap(c_2Epair_2E_2C(arr(ty_2Eordinal_2Eordinal(A_27a),bool),bool),ap(V0A,V3n)),c_2Ebool_2ET) ) ).
fof(conj_thm_2Eveblen_2Eclub__INTER,axiom,
! [A_27a] :
( ne(A_27a)
=> ! [V0A] :
( mem(V0A,arr(ty_2Enum_2Enum,arr(ty_2Eordinal_2Eordinal(A_27a),bool)))
=> ( ( ! [V1n] :
( mem(V1n,ty_2Enum_2Enum)
=> p(ap(c_2Eveblen_2Eclub(A_27a),ap(V0A,V1n))) )
& ! [V2n] :
( mem(V2n,ty_2Enum_2Enum)
=> p(ap(ap(c_2Epred__set_2ESUBSET(ty_2Eordinal_2Eordinal(A_27a)),ap(V0A,ap(c_2Enum_2ESUC,V2n))),ap(V0A,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))))) ) ) ) ).
%------------------------------------------------------------------------------