SET007 Axioms: SET007+379.ax


%------------------------------------------------------------------------------
% File     : SET007+379 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Euclid's Algorithm
% 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    : ami_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   18 (   4 unit)
%            Number of atoms       :  103 (  40 equality)
%            Maximal formula depth :   16 (   7 average)
%            Number of connectives :   97 (  12 ~  ;   2  |;  38  &)
%                                         (   3 <=>;  42 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   14 (   1 propositional; 0-4 arity)
%            Number of functors    :   47 (  13 constant; 0-5 arity)
%            Number of variables   :   30 (   0 singleton;  25 !;   5 ?)
%            Maximal term depth    :    8 (   2 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_ami_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( r1_xreal_0(np__0,A)
              & r1_xreal_0(np__0,B) )
           => r1_xreal_0(np__0,k5_int_1(A,B)) ) ) ) )).

fof(t2_ami_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( r1_xreal_0(np__0,A)
              & r1_xreal_0(np__0,B) )
           => ( k4_nat_1(k1_int_2(A),k1_int_2(B)) = k6_int_1(A,B)
              & k3_nat_1(k1_int_2(A),k1_int_2(B)) = k5_int_1(A,B) ) ) ) ) )).

fof(d1_ami_4,axiom,(
    k1_ami_4 = k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__0),k3_ami_3(k15_ami_3(np__2),k15_ami_3(np__1))),k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__1),k7_ami_3(k15_ami_3(np__0),k15_ami_3(np__1))),k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__2),k3_ami_3(k15_ami_3(np__0),k15_ami_3(np__2))),k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__3),k10_ami_3(k16_ami_3(np__0),k15_ami_3(np__1))),k14_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__4),k5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))))) )).

fof(t3_ami_4,axiom,(
    $true )).

fof(t4_ami_4,axiom,(
    k1_relat_1(k1_ami_4) = k3_enumset1(k16_ami_3(np__0),k16_ami_3(np__1),k16_ami_3(np__2),k16_ami_3(np__3),k16_ami_3(np__4)) )).

fof(t5_ami_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
     => ( r1_tarski(k1_ami_4,A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__0)
             => ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__1)
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__0)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__0))
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__1)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1))
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__2)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)) ) ) ) ) ) )).

fof(t6_ami_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
     => ( r1_tarski(k1_ami_4,A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__1)
             => ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__2)
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__0)) = k5_int_1(k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__0)),k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)))
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__1)) = k6_int_1(k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__0)),k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)))
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__2)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__2)) ) ) ) ) ) )).

fof(t7_ami_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
     => ( r1_tarski(k1_ami_4,A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__2)
             => ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__3)
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__0)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__2))
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__1)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1))
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__2)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__2)) ) ) ) ) ) )).

fof(t8_ami_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
     => ( r1_tarski(k1_ami_4,A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__3)
             => ( ( ~ r1_xreal_0(k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)),np__0)
                 => k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__0) )
                & ( r1_xreal_0(k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)),np__0)
                 => k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1))) = k16_ami_3(np__4) )
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__0)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__0))
                & k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,np__1)),k15_ami_3(np__1)) = k2_ami_3(k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B),k15_ami_3(np__1)) ) ) ) ) ) )).

fof(t9_ami_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
     => ( r1_tarski(k1_ami_4,A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( k6_ami_1(k1_tarski(k4_numbers),k1_ami_3,k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B)) = k16_ami_3(np__4)
                 => k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k1_nat_1(B,C)) = k11_ami_1(k1_tarski(k4_numbers),k1_ami_3,k10_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),B) ) ) ) ) ) )).

fof(t10_ami_4,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_ami_3)))
     => ( ( r1_ami_3(k1_tarski(k4_numbers),k1_ami_3,A,k16_ami_3(np__0))
          & r1_tarski(k1_ami_4,A) )
       => ! [B] :
            ( v1_int_1(B)
           => ! [C] :
                ( v1_int_1(C)
               => ( ( k2_ami_3(A,k15_ami_3(np__0)) = B
                    & k2_ami_3(A,k15_ami_3(np__1)) = C )
                 => ( r1_xreal_0(B,np__0)
                    | r1_xreal_0(C,np__0)
                    | k2_ami_3(k12_ami_1(k1_tarski(k4_numbers),k1_ami_3,A),k15_ami_3(np__0)) = k3_int_2(B,C) ) ) ) ) ) ) )).

fof(d2_ami_4,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k14_ami_1(k1_tarski(k4_numbers),k1_ami_3),k14_ami_1(k1_tarski(k4_numbers),k1_ami_3)) )
     => ( A = k2_ami_4
      <=> ! [B] :
            ( m1_ami_1(B,k1_tarski(k4_numbers),k1_ami_3)
           => ! [C] :
                ( m1_ami_1(C,k1_tarski(k4_numbers),k1_ami_3)
               => ( r2_hidden(k4_tarski(B,C),A)
                <=> ? [D] :
                      ( v1_int_1(D)
                      & ? [E] :
                          ( v1_int_1(E)
                          & ~ r1_xreal_0(D,np__0)
                          & ~ r1_xreal_0(E,np__0)
                          & B = k18_ami_3(k15_ami_3(np__0),k15_ami_3(np__1),D,E)
                          & C = k17_ami_3(k15_ami_3(np__0),k3_int_2(D,E)) ) ) ) ) ) ) ) )).

fof(t11_ami_4,axiom,(
    ! [A] :
      ( r2_hidden(A,k1_relat_1(k2_ami_4))
    <=> ? [B] :
          ( v1_int_1(B)
          & ? [C] :
              ( v1_int_1(C)
              & ~ r1_xreal_0(B,np__0)
              & ~ r1_xreal_0(C,np__0)
              & A = k18_ami_3(k15_ami_3(np__0),k15_ami_3(np__1),B,C) ) ) ) )).

fof(t12_ami_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ~ ( ~ r1_xreal_0(A,np__0)
              & ~ r1_xreal_0(B,np__0)
              & k1_funct_1(k2_ami_4,k18_ami_3(k15_ami_3(np__0),k15_ami_3(np__1),A,B)) != k17_ami_3(k15_ami_3(np__0),k3_int_2(A,B)) ) ) ) )).

fof(t13_ami_4,axiom,(
    r1_ami_1(k1_tarski(k4_numbers),k1_ami_3,k17_ami_1(k1_tarski(k4_numbers),k1_ami_3,k12_ami_3(k1_tarski(k4_numbers),k1_ami_3,k16_ami_3(np__0)),k1_ami_4),k2_ami_4) )).

fof(s1_ami_4,axiom,
    ( ( ~ r1_xreal_0(f4_s1_ami_4,np__0)
      & ~ r1_xreal_0(f3_s1_ami_4,f4_s1_ami_4)
      & f1_s1_ami_4(np__0) = f3_s1_ami_4
      & f2_s1_ami_4(np__0) = f4_s1_ami_4
      & ! [A] :
          ( m2_subset_1(A,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(f2_s1_ami_4(A),np__0)
           => ( f1_s1_ami_4(k1_nat_1(A,np__1)) = f2_s1_ami_4(A)
              & f2_s1_ami_4(k1_nat_1(A,np__1)) = k4_nat_1(f1_s1_ami_4(A),f2_s1_ami_4(A)) ) ) ) )
   => ? [A] :
        ( m2_subset_1(A,k1_numbers,k5_numbers)
        & f1_s1_ami_4(A) = k6_nat_1(f3_s1_ami_4,f4_s1_ami_4)
        & f2_s1_ami_4(A) = np__0 ) )).

fof(dt_k1_ami_4,axiom,
    ( v1_ami_3(k1_ami_4,k1_tarski(k4_numbers),k1_ami_3)
    & m1_ami_1(k1_ami_4,k1_tarski(k4_numbers),k1_ami_3) )).

fof(dt_k2_ami_4,axiom,
    ( v1_funct_1(k2_ami_4)
    & m2_relset_1(k2_ami_4,k14_ami_1(k1_tarski(k4_numbers),k1_ami_3),k14_ami_1(k1_tarski(k4_numbers),k1_ami_3)) )).
%------------------------------------------------------------------------------