SET007 Axioms: SET007+198.ax


%------------------------------------------------------------------------------
% File     : SET007+198 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Natural Numbers
% Version  : [Urb08] axioms.
% English  :

% Refs     : [Mat90] Matuszewski (1990), Formalized Mathematics
%          : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
%          : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source   : [Urb08]
% Names    : nat_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   36 (   2 unit)
%            Number of atoms       :  172 (  28 equality)
%            Maximal formula depth :   14 (   7 average)
%            Number of connectives :  164 (  28 ~  ;   6  |;  44  &)
%                                         (   6 <=>;  80 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   22 (   1 propositional; 0-3 arity)
%            Number of functors    :   23 (   8 constant; 0-4 arity)
%            Number of variables   :   63 (   0 singleton;  59 !;   4 ?)
%            Maximal term depth    :    5 (   1 average)
% SPC      : 

% Comments : The individual reference can be found in [Mat90] by looking for
%            the name provided by [Urb08].
%          : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
%          : These set theory axioms are used in encodings of problems in
%            various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_nat_2,axiom,(
    ? [A] :
      ( m1_subset_1(A,k5_numbers)
      & ~ v1_xboole_0(A)
      & v1_ordinal1(A)
      & v2_ordinal1(A)
      & v3_ordinal1(A)
      & v4_ordinal2(A)
      & v1_xreal_0(A)
      & v2_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_xcmplx_0(A)
      & v1_int_1(A)
      & ~ v1_realset1(A) ) )).

fof(rc2_nat_2,axiom,(
    ? [A] :
      ( ~ v1_xboole_0(A)
      & v4_ordinal2(A)
      & v1_xreal_0(A)
      & v2_xreal_0(A)
      & ~ v3_xreal_0(A)
      & v1_xcmplx_0(A)
      & v1_int_1(A)
      & ~ v1_realset1(A) ) )).

fof(t1_nat_2,axiom,(
    $true )).

fof(t2_nat_2,axiom,(
    ! [A] :
      ( v1_xreal_0(A)
     => ! [B] :
          ( v1_xreal_0(B)
         => ~ ( r1_xreal_0(np__0,A)
              & ~ r1_xreal_0(B,np__0)
              & r1_xreal_0(B,k7_xcmplx_0(A,k2_xcmplx_0(k1_int_1(k7_xcmplx_0(A,B)),np__1))) ) ) ) )).

fof(t3_nat_2,axiom,(
    $true )).

fof(t4_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k3_nat_1(np__0,A) = np__0 ) )).

fof(t5_nat_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k3_nat_1(A,A) = np__1 ) )).

fof(t6_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k3_nat_1(A,np__1) = A ) )).

fof(t7_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ! [D] :
                  ( v4_ordinal2(D)
                 => ( ( r1_xreal_0(A,B)
                      & r1_xreal_0(C,B)
                      & A = k2_xcmplx_0(k5_binarith(B,C),D) )
                   => C = k2_xcmplx_0(k5_binarith(B,A),D) ) ) ) ) ) )).

fof(t8_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r2_hidden(A,k2_finseq_1(B))
           => r2_hidden(k1_nat_1(k5_binarith(B,A),np__1),k2_finseq_1(B)) ) ) ) )).

fof(t9_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ~ r1_xreal_0(A,B)
           => k1_nat_1(k5_binarith(A,k2_xcmplx_0(B,np__1)),np__1) = k5_binarith(A,B) ) ) ) )).

fof(t10_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(B,A)
           => k5_binarith(B,A) = np__0 ) ) ) )).

fof(t11_nat_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ! [B] :
          ( ( ~ v1_xboole_0(B)
            & v4_ordinal2(B) )
         => ~ r1_xreal_0(A,k5_binarith(A,B)) ) ) )).

fof(t12_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(B,A)
           => k3_power(np__2,A) = k3_xcmplx_0(k3_power(np__2,B),k3_series_1(np__2,k5_binarith(A,B))) ) ) ) )).

fof(t13_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,A)
           => r1_nat_1(k3_series_1(np__2,B),k3_series_1(np__2,A)) ) ) ) )).

fof(t14_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ~ ( ~ r1_xreal_0(B,np__0)
              & k3_nat_1(A,B) = np__0
              & r1_xreal_0(B,A) ) ) ) )).

fof(t15_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(A,B)
           => ( r1_xreal_0(A,np__0)
              | r1_xreal_0(np__1,k3_nat_1(B,A)) ) ) ) ) )).

fof(t16_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( A != np__0
           => k3_nat_1(k2_xcmplx_0(B,A),A) = k1_nat_1(k3_nat_1(B,A),np__1) ) ) ) )).

fof(t17_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( ( r1_nat_1(A,B)
                  & r1_xreal_0(np__1,B)
                  & r1_xreal_0(np__1,C)
                  & r1_xreal_0(C,A) )
               => k3_nat_1(k5_binarith(B,C),A) = k6_xcmplx_0(k3_nat_1(B,A),np__1) ) ) ) ) )).

fof(t18_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,A)
           => k3_nat_1(k3_series_1(np__2,A),k3_series_1(np__2,B)) = k3_series_1(np__2,k5_binarith(A,B)) ) ) ) )).

fof(t19_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => k4_nat_1(k3_series_1(np__2,A),np__2) = np__0 ) ) )).

fof(t20_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => ( k4_nat_1(A,np__2) = np__0
        <=> k4_nat_1(k5_binarith(A,np__1),np__2) = np__1 ) ) ) )).

fof(t21_nat_2,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & v4_ordinal2(A) )
     => ~ ( A != np__1
          & r1_xreal_0(A,np__1) ) ) )).

fof(t22_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(A,B)
           => ( r1_xreal_0(k2_xcmplx_0(A,A),B)
              | k3_nat_1(B,A) = np__1 ) ) ) ) )).

fof(t23_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_abian(A)
      <=> k4_nat_1(A,np__2) = np__0 ) ) )).

fof(t24_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ v1_abian(A)
      <=> k4_nat_1(A,np__2) = np__1 ) ) )).

fof(t25_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,C)
                  & r1_xreal_0(B,A)
                  & r1_nat_1(k2_nat_1(np__2,C),B) )
               => ( v1_abian(k3_nat_1(A,C))
                <=> v1_abian(k3_nat_1(k5_binarith(A,B),C)) ) ) ) ) ) )).

fof(t26_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(A,B)
               => r1_xreal_0(k3_nat_1(A,C),k3_nat_1(B,C)) ) ) ) ) )).

fof(t27_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,k2_nat_1(np__2,A))
           => r1_xreal_0(k3_nat_1(k1_nat_1(B,np__1),np__2),A) ) ) ) )).

fof(t28_nat_2,axiom,(
    ! [A] :
      ( ( v1_abian(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k3_nat_1(A,np__2) = k3_nat_1(k1_nat_1(A,np__1),np__2) ) )).

fof(t29_nat_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => k3_nat_1(k3_nat_1(A,B),C) = k3_nat_1(A,k2_nat_1(B,C)) ) ) ) )).

fof(d1_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( v1_realset1(A)
      <=> ( A = np__0
          | A = np__1 ) ) ) )).

fof(t30_nat_2,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ( ~ v1_realset1(A)
      <=> ( ~ v1_xboole_0(A)
          & A != np__1 ) ) ) )).

fof(t31_nat_2,axiom,(
    ! [A] :
      ( ( v4_ordinal2(A)
        & ~ v1_realset1(A) )
     => r1_xreal_0(np__2,A) ) )).

fof(s1_nat_2,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( r1_xreal_0(np__1,A)
         => ( r1_xreal_0(f3_s1_nat_2,A)
            | ! [B] :
                ( m1_subset_1(B,f1_s1_nat_2)
               => ? [C] :
                    ( m1_subset_1(C,f1_s1_nat_2)
                    & p1_s1_nat_2(A,B,C) ) ) ) ) )
   => ? [A] :
        ( m2_finseq_1(A,f1_s1_nat_2)
        & k3_finseq_1(A) = f3_s1_nat_2
        & ( k4_finseq_4(k5_numbers,f1_s1_nat_2,A,np__1) = f2_s1_nat_2
          | f3_s1_nat_2 = np__0 )
        & ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( r1_xreal_0(np__1,B)
             => ( r1_xreal_0(f3_s1_nat_2,B)
                | p1_s1_nat_2(B,k4_finseq_4(k5_numbers,f1_s1_nat_2,A,B),k4_finseq_4(k5_numbers,f1_s1_nat_2,A,k1_nat_1(B,np__1))) ) ) ) ) )).

fof(s2_nat_2,axiom,
    ( ( p1_s2_nat_2(np__2)
      & ! [A] :
          ( ( ~ v1_realset1(A)
            & m2_subset_1(A,k1_numbers,k5_numbers) )
         => ( p1_s2_nat_2(A)
           => p1_s2_nat_2(k1_nat_1(A,np__1)) ) ) )
   => ! [A] :
        ( ( ~ v1_realset1(A)
          & m2_subset_1(A,k1_numbers,k5_numbers) )
       => p1_s2_nat_2(A) ) )).
%------------------------------------------------------------------------------