ITP001 Axioms: ITP105_5.ax


%------------------------------------------------------------------------------
% File     : ITP105_5 : TPTP v9.0.0. 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))))) ) ) ).

%------------------------------------------------------------------------------