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    :   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))))) ) ) ) ).

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