SET007 Axioms: SET007+610.ax


%------------------------------------------------------------------------------
% File     : SET007+610 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Recursive Euclide 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    : scmp_gcd [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   27 (   7 unit)
%            Number of atoms       :  133 (  56 equality)
%            Maximal formula depth :   20 (   7 average)
%            Number of connectives :  117 (  11 ~  ;   3  |;  54  &)
%                                         (   1 <=>;  48 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   0 propositional; 1-3 arity)
%            Number of functors    :   74 (  25 constant; 0-4 arity)
%            Number of variables   :   38 (   0 singleton;  38 !;   0 ?)
%            Maximal term depth    :   16 (   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_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(A,np__0)
           => k6_nat_1(B,A) = k6_nat_1(A,k4_nat_1(B,A)) ) ) ) )).

fof(t2_scmp_gcd,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_xreal_0(np__0,A)
           => ( r1_xreal_0(B,np__0)
              | k3_int_2(A,B) = k3_int_2(B,k6_int_1(A,B)) ) ) ) ) )).

fof(t3_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ( k2_scmpds_3(A) = B
           => k2_scmpds_3(k1_nat_1(A,np__2)) = k1_nat_1(k2_nat_1(np__2,k3_nat_1(k1_int_2(B),np__2)),np__4) ) ) ) )).

fof(d1_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_scmp_gcd(A) = k15_ami_3(A) ) )).

fof(t4_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( A != B
              & k1_scmp_gcd(A) = k1_scmp_gcd(B) ) ) ) )).

fof(t5_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k3_scmpds_2(A,B) = k1_scmp_gcd(k1_nat_1(A,B)) ) ) )).

fof(t6_scmp_gcd,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
     => ! [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_scmpds_2,A) = k2_scmpds_3(k1_nat_1(B,C))
               => k19_scmpds_2(A,k4_xcmplx_0(C)) = k2_scmpds_3(B) ) ) ) ) )).

fof(d2_scmp_gcd,axiom,(
    k2_scmp_gcd = k1_scmp_gcd(np__0) )).

fof(d3_scmp_gcd,axiom,(
    k3_scmp_gcd = k1_scmp_gcd(np__1) )).

fof(t7_scmp_gcd,axiom,(
    k2_scmp_gcd != k3_scmp_gcd )).

fof(t8_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
     => ! [B] :
          ( ( v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2)
            & v1_scmpds_3(B)
            & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
         => k4_card_1(k5_scmpds_4(B,A)) = k1_nat_1(k4_card_1(B),np__1) ) ) )).

fof(t9_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
     => ! [B] :
          ( m2_subset_1(B,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
         => k4_card_1(k6_scmpds_4(A,B)) = np__2 ) ) )).

fof(t10_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
     => ! [B] :
          ( ( v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2)
            & v1_scmpds_3(B)
            & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
         => ( k1_funct_1(k5_scmpds_4(B,A),k2_scmpds_3(k4_card_1(B))) = A
            & r2_hidden(k2_scmpds_3(k4_card_1(B)),k1_relat_1(k5_scmpds_4(B,A))) ) ) ) )).

fof(t11_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k2_zfmisc_1(u3_ami_1(k1_tarski(k4_numbers),k1_scmpds_2),k13_finseq_1(k2_xboole_0(k3_tarski(k1_tarski(k4_numbers)),u1_struct_0(k1_scmpds_2)))),u4_ami_1(k1_tarski(k4_numbers),k1_scmpds_2))
     => ! [B] :
          ( ( v1_ami_3(B,k1_tarski(k4_numbers),k1_scmpds_2)
            & v1_scmpds_3(B)
            & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
         => ! [C] :
              ( ( v1_ami_3(C,k1_tarski(k4_numbers),k1_scmpds_2)
                & v1_scmpds_3(C)
                & m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) )
             => k1_funct_1(k3_scmpds_4(k5_scmpds_4(B,A),C),k2_scmpds_3(k4_card_1(B))) = A ) ) ) )).

fof(d4_scmp_gcd,axiom,(
    k4_scmp_gcd = k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k6_scmpds_2(k2_scmp_gcd,np__0),k6_scmpds_2(k3_scmp_gcd,np__7)),k7_scmpds_2(k3_scmp_gcd,k27_scmpds_1)),k4_scmpds_2(np__2)),k5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)),k9_scmpds_2(k3_scmp_gcd,np__3,np__9)),k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__6,np__3)),k16_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__2,np__3)),k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__7,np__3)),k17_scmpds_2(k3_scmp_gcd,k2_scmp_gcd,k1_nat_1(np__4,k26_scmpds_1),np__1)),k12_scmpds_2(k2_scmp_gcd,np__1,np__4)),k7_scmpds_2(k3_scmp_gcd,k27_scmpds_1)),k4_scmpds_2(k4_xcmplx_0(np__7))),k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__2,np__6)),k5_scmpds_2(k3_scmp_gcd)) )).

fof(t12_scmp_gcd,axiom,(
    k4_card_1(k4_scmp_gcd) = np__15 )).

fof(t13_scmp_gcd,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(np__15,A)
      <=> r2_hidden(k2_scmpds_3(A),k1_relat_1(k4_scmp_gcd)) ) ) )).

fof(t14_scmp_gcd,axiom,
    ( k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__0)) = k6_scmpds_2(k2_scmp_gcd,np__0)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__1)) = k6_scmpds_2(k3_scmp_gcd,np__7)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__2)) = k7_scmpds_2(k3_scmp_gcd,k27_scmpds_1)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__3)) = k4_scmpds_2(np__2)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__4)) = k5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__5)) = k9_scmpds_2(k3_scmp_gcd,np__3,np__9)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__6)) = k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__6,np__3)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__7)) = k16_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__2,np__3)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__8)) = k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__7,np__3)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__9)) = k17_scmpds_2(k3_scmp_gcd,k2_scmp_gcd,k1_nat_1(np__4,k26_scmpds_1),np__1)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__10)) = k12_scmpds_2(k2_scmp_gcd,np__1,np__4)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__11)) = k7_scmpds_2(k3_scmp_gcd,k27_scmpds_1)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__12)) = k4_scmpds_2(k4_xcmplx_0(np__7))
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__13)) = k17_scmpds_2(k3_scmp_gcd,k3_scmp_gcd,np__2,np__6)
    & k1_funct_1(k4_scmp_gcd,k2_scmpds_3(np__14)) = k5_scmpds_2(k3_scmp_gcd) )).

fof(t15_scmp_gcd,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
     => ( r1_tarski(k2_scmpds_4(k4_scmp_gcd),A)
       => ( k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4)) = k2_scmpds_3(np__5)
          & k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k2_scmp_gcd) = np__0
          & k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k3_scmp_gcd) = np__7
          & k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k1_scmp_gcd(k1_nat_1(np__7,k27_scmpds_1))) = k2_scmpds_3(np__2)
          & k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k1_scmp_gcd(np__9)) = k2_scmpds_2(A,k1_scmp_gcd(np__9))
          & k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),np__4),k1_scmp_gcd(np__10)) = k2_scmpds_2(A,k1_scmp_gcd(np__10)) ) ) ) )).

fof(t16_scmp_gcd,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
     => ~ ( r1_tarski(k4_scmp_gcd,A)
          & k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A) = k2_scmpds_3(np__5)
          & ~ r1_xreal_0(k2_scmpds_2(A,k3_scmp_gcd),np__0)
          & k2_scmpds_2(A,k2_scmp_gcd) = np__0
          & r1_xreal_0(np__0,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)))
          & r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)),k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)))
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( k8_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B)) = k5_scmpds_2(k3_scmp_gcd)
                  & k2_scmpds_2(A,k3_scmp_gcd) = k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k3_scmp_gcd)
                  & k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)) = k3_int_2(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)),k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)))
                  & ! [C] :
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(C,k2_xcmplx_0(k2_scmpds_2(A,k3_scmp_gcd),np__1))
                       => ( r1_xreal_0(C,np__1)
                          | k2_scmpds_2(A,k1_scmp_gcd(C)) = k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k1_scmp_gcd(C)) ) ) ) ) ) ) ) )).

fof(t17_scmp_gcd,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
     => ~ ( r1_tarski(k4_scmp_gcd,A)
          & k6_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A) = k2_scmpds_3(np__5)
          & ~ r1_xreal_0(k2_scmpds_2(A,k3_scmp_gcd),np__0)
          & k2_scmpds_2(A,k2_scmp_gcd) = np__0
          & r1_xreal_0(np__0,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)))
          & r1_xreal_0(np__0,k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)))
          & ! [B] :
              ( m2_subset_1(B,k1_numbers,k5_numbers)
             => ~ ( k8_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B)) = k5_scmpds_2(k3_scmp_gcd)
                  & k2_scmpds_2(A,k3_scmp_gcd) = k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k3_scmp_gcd)
                  & k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)) = k3_int_2(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__2)),k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,k3_scmp_gcd),np__3)))
                  & ! [C] :
                      ( m2_subset_1(C,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(C,k2_xcmplx_0(k2_scmpds_2(A,k3_scmp_gcd),np__1))
                       => ( r1_xreal_0(C,np__1)
                          | k2_scmpds_2(A,k1_scmp_gcd(C)) = k2_scmpds_2(k11_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k10_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),B),k1_scmp_gcd(C)) ) ) ) ) ) ) ) )).

fof(t18_scmp_gcd,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
     => ( r1_tarski(k2_scmpds_4(k4_scmp_gcd),A)
       => ! [B] :
            ( v1_int_1(B)
           => ! [C] :
                ( v1_int_1(C)
               => ( ( k2_scmpds_2(A,k1_scmp_gcd(np__9)) = B
                    & k2_scmpds_2(A,k1_scmp_gcd(np__10)) = C
                    & r1_xreal_0(np__0,B)
                    & r1_xreal_0(np__0,C) )
                 => k2_scmpds_2(k12_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,A),k1_scmp_gcd(np__9)) = k3_int_2(B,C) ) ) ) ) ) )).

fof(t19_scmp_gcd,axiom,(
    ! [A] :
      ( m1_ami_1(A,k1_tarski(k4_numbers),k1_scmpds_2)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r1_xreal_0(np__0,C)
                  & r1_xreal_0(C,B)
                  & A = k4_funct_4(k1_scmp_gcd(np__9),k1_scmp_gcd(np__10),B,C) )
               => v11_ami_1(k17_ami_1(k1_tarski(k4_numbers),k1_scmpds_2,k2_scmpds_4(k4_scmp_gcd),A),k1_tarski(k4_numbers),k1_scmpds_2) ) ) ) ) )).

fof(dt_k1_scmp_gcd,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => m1_scmpds_2(k1_scmp_gcd(A)) ) )).

fof(dt_k2_scmp_gcd,axiom,(
    m1_scmpds_2(k2_scmp_gcd) )).

fof(dt_k3_scmp_gcd,axiom,(
    m1_scmpds_2(k3_scmp_gcd) )).

fof(dt_k4_scmp_gcd,axiom,
    ( v1_ami_3(k4_scmp_gcd,k1_tarski(k4_numbers),k1_scmpds_2)
    & v1_scmpds_3(k4_scmp_gcd)
    & m1_ami_1(k4_scmp_gcd,k1_tarski(k4_numbers),k1_scmpds_2) )).
%------------------------------------------------------------------------------