SET007 Axioms: SET007+131.ax


%------------------------------------------------------------------------------
% File     : SET007+131 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The Divisibility of Integers and Integer Relatively Primes
% 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    : int_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   65 (  16 unit)
%            Number of atoms       :  246 (  41 equality)
%            Maximal formula depth :   14 (   5 average)
%            Number of connectives :  201 (  20 ~  ;   5  |;  47  &)
%                                         (  12 <=>; 117 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   1 propositional; 0-3 arity)
%            Number of functors    :   20 (   8 constant; 0-2 arity)
%            Number of variables   :   98 (   0 singleton;  98 !;   0 ?)
%            Maximal term depth    :    3 (   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(t1_int_2,axiom,(
    $true )).

fof(t2_int_2,axiom,(
    $true )).

fof(t3_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_nat_1(np__0,A)
      <=> A = np__0 ) ) )).

fof(t4_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( A = np__0
              | B = np__0 )
          <=> k5_nat_1(A,B) = np__0 ) ) ) )).

fof(t5_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( A = np__0
              & B = np__0 )
          <=> k6_nat_1(A,B) = np__0 ) ) ) )).

fof(t6_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k2_nat_1(A,B) = k2_nat_1(k5_nat_1(A,B),k6_nat_1(A,B)) ) ) )).

fof(t7_int_2,axiom,(
    $true )).

fof(t8_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( m2_subset_1(k1_real_1(A),k1_numbers,k5_numbers)
      <=> A = np__0 ) ) )).

fof(t9_int_2,axiom,(
    ~ m2_subset_1(k1_real_1(np__1),k1_numbers,k5_numbers) )).

fof(t10_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( r2_int_1(np__0,A)
      <=> A = np__0 ) ) )).

fof(t11_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( r2_int_1(A,k4_xcmplx_0(A))
        & r2_int_1(k4_xcmplx_0(A),A) ) ) )).

fof(t12_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r2_int_1(A,B)
               => r2_int_1(A,k3_xcmplx_0(B,C)) ) ) ) ) )).

fof(t13_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r2_int_1(A,B)
                  & r2_int_1(B,C) )
               => r2_int_1(A,C) ) ) ) ) )).

fof(t14_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( r2_int_1(A,B)
             => r2_int_1(A,k4_xcmplx_0(B)) )
            & ( r2_int_1(A,k4_xcmplx_0(B))
             => r2_int_1(A,B) )
            & ( r2_int_1(A,B)
             => r2_int_1(k4_xcmplx_0(A),B) )
            & ( r2_int_1(k4_xcmplx_0(A),B)
             => r2_int_1(A,B) )
            & ( r2_int_1(A,B)
             => r2_int_1(k4_xcmplx_0(A),k4_xcmplx_0(B)) )
            & ( r2_int_1(k4_xcmplx_0(A),k4_xcmplx_0(B))
             => r2_int_1(A,B) )
            & ( r2_int_1(A,k4_xcmplx_0(B))
             => r2_int_1(k4_xcmplx_0(A),B) )
            & ( r2_int_1(k4_xcmplx_0(A),B)
             => r2_int_1(A,k4_xcmplx_0(B)) ) ) ) ) )).

fof(t15_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( r2_int_1(A,B)
              & r2_int_1(B,A)
              & A != B
              & A != k4_xcmplx_0(B) ) ) ) )).

fof(t16_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( r2_int_1(A,np__0)
        & r2_int_1(np__1,A)
        & r2_int_1(k1_real_1(np__1),A) ) ) )).

fof(t17_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ~ ( ( r2_int_1(A,np__1)
            | r2_int_1(A,k1_real_1(np__1)) )
          & A != np__1
          & A != k1_real_1(np__1) ) ) )).

fof(t18_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ( ( A = np__1
          | A = k1_real_1(np__1) )
       => ( r2_int_1(A,np__1)
          & r2_int_1(A,k1_real_1(np__1)) ) ) ) )).

fof(t19_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_int_1(A,B,C)
              <=> r2_int_1(C,k6_xcmplx_0(A,B)) ) ) ) ) )).

fof(t20_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => m2_subset_1(k1_int_2(A),k1_numbers,k5_numbers) ) )).

fof(t21_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r2_int_1(A,B)
          <=> r1_nat_1(k1_int_2(A),k1_int_2(B)) ) ) ) )).

fof(d1_int_2,axiom,(
    $true )).

fof(d2_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => k2_int_2(A,B) = k5_nat_1(k1_int_2(A),k1_int_2(B)) ) ) )).

fof(t22_int_2,axiom,(
    $true )).

fof(t23_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => m2_subset_1(k2_int_2(A,B),k1_numbers,k5_numbers) ) ) )).

fof(t24_int_2,axiom,(
    $true )).

fof(t25_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => r2_int_1(A,k2_int_2(A,B)) ) ) )).

fof(t26_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => r2_int_1(A,k2_int_2(B,A)) ) ) )).

fof(t27_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r2_int_1(A,C)
                  & r2_int_1(B,C) )
               => r2_int_1(k2_int_2(A,B),C) ) ) ) ) )).

fof(d3_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => k3_int_2(A,B) = k6_nat_1(k1_int_2(A),k1_int_2(B)) ) ) )).

fof(t28_int_2,axiom,(
    $true )).

fof(t29_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => m2_subset_1(k3_int_2(A,B),k1_numbers,k5_numbers) ) ) )).

fof(t30_int_2,axiom,(
    $true )).

fof(t31_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => r2_int_1(k3_int_2(A,B),A) ) ) )).

fof(t32_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => r2_int_1(k3_int_2(A,B),B) ) ) )).

fof(t33_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r2_int_1(C,A)
                  & r2_int_1(C,B) )
               => r2_int_1(C,k3_int_2(A,B)) ) ) ) ) )).

fof(t34_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( A = np__0
              | B = np__0 )
          <=> k2_int_2(A,B) = np__0 ) ) ) )).

fof(t35_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( A = np__0
              & B = np__0 )
          <=> k3_int_2(A,B) = np__0 ) ) ) )).

fof(d4_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_int_2(A,B)
          <=> k3_int_2(A,B) = np__1 ) ) ) )).

fof(t36_int_2,axiom,(
    $true )).

fof(t37_int_2,axiom,(
    $true )).

fof(t38_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( ~ ( A = np__0
                  & B = np__0 )
              & ! [C] :
                  ( v1_int_1(C)
                 => ! [D] :
                      ( v1_int_1(D)
                     => ~ ( A = k3_xcmplx_0(k3_int_2(A,B),C)
                          & B = k3_xcmplx_0(k3_int_2(A,B),D)
                          & r1_int_2(C,D) ) ) ) ) ) ) )).

fof(t39_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_int_2(A,B)
               => ( k3_int_2(k3_xcmplx_0(C,A),k3_xcmplx_0(C,B)) = k1_int_2(C)
                  & k3_int_2(k3_xcmplx_0(C,A),k3_xcmplx_0(B,C)) = k1_int_2(C)
                  & k3_int_2(k3_xcmplx_0(A,C),k3_xcmplx_0(C,B)) = k1_int_2(C)
                  & k3_int_2(k3_xcmplx_0(A,C),k3_xcmplx_0(B,C)) = k1_int_2(C) ) ) ) ) ) )).

fof(t40_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r2_int_1(A,k3_xcmplx_0(B,C))
                  & r1_int_2(B,A) )
               => r2_int_1(A,C) ) ) ) ) )).

fof(t41_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r1_int_2(A,B)
                  & r1_int_2(C,B) )
               => r1_int_2(k3_xcmplx_0(A,C),B) ) ) ) ) )).

fof(d5_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( v1_int_2(A)
      <=> ( ~ r1_xreal_0(A,np__1)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( r1_nat_1(B,A)
                  & B != np__1
                  & B != A ) ) ) ) ) )).

fof(d6_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r2_int_2(A,B)
          <=> k6_nat_1(A,B) = np__1 ) ) ) )).

fof(t42_int_2,axiom,(
    $true )).

fof(t43_int_2,axiom,(
    $true )).

fof(t44_int_2,axiom,(
    v1_int_2(np__2) )).

fof(t45_int_2,axiom,(
    $true )).

fof(t46_int_2,axiom,(
    ~ v1_int_2(np__4) )).

fof(t47_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( v1_int_2(A)
              & v1_int_2(B)
              & ~ r2_int_2(A,B)
              & A != B ) ) ) )).

fof(t48_int_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( r1_xreal_0(np__2,A)
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( v1_int_2(B)
                  & r1_nat_1(B,A) ) ) ) ) )).

fof(s1_int_2,axiom,
    ( ( p1_s1_int_2(f1_s1_int_2)
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(f1_s1_int_2,A)
              & p1_s1_int_2(A) )
           => p1_s1_int_2(k1_nat_1(A,np__1)) ) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( r1_xreal_0(f1_s1_int_2,A)
         => p1_s1_int_2(A) ) ) )).

fof(s2_int_2,axiom,
    ( ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( ( r1_xreal_0(f1_s2_int_2,A)
            & ! [B] :
                ( m2_subset_1(B,k1_numbers,k5_numbers)
               => ( r1_xreal_0(f1_s2_int_2,B)
                 => ( r1_xreal_0(A,B)
                    | p1_s2_int_2(B) ) ) ) )
         => p1_s2_int_2(A) ) )
   => ! [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
       => ( r1_xreal_0(f1_s2_int_2,A)
         => p1_s2_int_2(A) ) ) )).

fof(symmetry_r1_int_2,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => ( r1_int_2(A,B)
       => r1_int_2(B,A) ) ) )).

fof(symmetry_r2_int_2,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( r2_int_2(A,B)
       => r2_int_2(B,A) ) ) )).

fof(dt_k1_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => m2_subset_1(k1_int_2(A),k1_numbers,k5_numbers) ) )).

fof(projectivity_k1_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k1_int_2(k1_int_2(A)) = k1_int_2(A) ) )).

fof(redefinition_k1_int_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => k1_int_2(A) = k16_complex1(A) ) )).

fof(dt_k2_int_2,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => v1_int_1(k2_int_2(A,B)) ) )).

fof(commutativity_k2_int_2,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => k2_int_2(A,B) = k2_int_2(B,A) ) )).

fof(dt_k3_int_2,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => v1_int_1(k3_int_2(A,B)) ) )).

fof(commutativity_k3_int_2,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => k3_int_2(A,B) = k3_int_2(B,A) ) )).
%------------------------------------------------------------------------------