SET007 Axioms: SET007+659.ax


%------------------------------------------------------------------------------
% File     : SET007+659 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Insert Sort on SCMPDS
% 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    : scpisort [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   24 (   0 unit)
%            Number of atoms       :  298 (  45 equality)
%            Maximal formula depth :   27 (  13 average)
%            Number of connectives :  286 (  12 ~  ;  16  |; 140  &)
%                                         (   1 <=>; 117 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   0 propositional; 1-3 arity)
%            Number of functors    :   64 (  29 constant; 0-4 arity)
%            Number of variables   :   92 (   0 singleton;  90 !;   2 ?)
%            Maximal term depth    :   13 (   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(d1_scpisort,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k4_numbers)
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_scpisort(A,B,C)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ( r1_xreal_0(np__1,D)
                        & r1_xreal_0(D,k3_finseq_1(A)) )
                     => k1_funct_1(A,D) = k2_scmpds_2(B,k1_scmp_gcd(k1_nat_1(C,D))) ) ) ) ) ) ) )).

fof(t1_scpisort,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k4_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(C,B)
               => r1_sfmastr3(A,B,C) ) ) ) ) )).

fof(t2_scpisort,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)
             => ? [D] :
                  ( m2_finseq_1(D,k4_numbers)
                  & k3_finseq_1(D) = B
                  & ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,E)
                          & r1_xreal_0(E,k3_finseq_1(D)) )
                       => k1_funct_1(D,E) = k2_scmpds_2(A,k1_scmp_gcd(k1_nat_1(C,E))) ) ) ) ) ) ) )).

fof(t3_scpisort,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)
             => ? [D] :
                  ( m2_finseq_1(D,k4_numbers)
                  & k3_finseq_1(D) = B
                  & r1_scpisort(D,A,C) ) ) ) ) )).

fof(t4_scpisort,axiom,(
    ! [A] :
      ( m2_finseq_1(A,k4_numbers)
     => ! [B] :
          ( m2_finseq_1(B,k4_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(np__1,D)
                      & r1_xreal_0(D,k3_finseq_1(A))
                      & r1_xreal_0(np__1,C)
                      & r1_xreal_0(C,k3_finseq_1(A))
                      & k3_finseq_1(A) = k3_finseq_1(B)
                      & k1_funct_1(A,C) = k1_funct_1(B,D)
                      & k1_funct_1(A,D) = k1_funct_1(B,C)
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( ( r1_xreal_0(np__1,E)
                              & r1_xreal_0(E,k3_finseq_1(A)) )
                           => ( E = C
                              | E = D
                              | k1_funct_1(A,E) = k1_funct_1(B,E) ) ) ) )
                   => r1_rfinseq(A,B) ) ) ) ) ) )).

fof(t5_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
     => ! [B] :
          ( m1_subset_1(B,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
         => ( ! [C] :
                ( m1_scmpds_2(C)
               => k2_scmpds_2(A,C) = k2_scmpds_2(B,C) )
           => k1_scmpds_8(A) = k1_scmpds_8(B) ) ) ) )).

fof(t6_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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)
            & v3_scmpds_5(B)
            & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
         => ! [C] :
              ( ( v4_scmpds_4(C)
                & v2_scmpds_5(C)
                & m2_subset_1(C,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)) )
             => ( ( r1_scmpds_6(B,A)
                  & r2_scmpds_6(B,A) )
               => ( r1_scmpds_6(k5_scmpds_4(B,C),A)
                  & r2_scmpds_6(k5_scmpds_4(B,C),A) ) ) ) ) ) )).

fof(t7_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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)
            & v3_scmpds_5(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)
                & v2_scmpds_4(C)
                & v3_scmpds_4(C)
                & m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) )
             => ! [D] :
                  ( m1_scmpds_2(D)
                 => ( ( r1_scmpds_6(B,A)
                      & r2_scmpds_6(B,A) )
                   => k2_scmpds_2(k9_scmpds_4(k3_scmpds_4(B,C),A),D) = k2_scmpds_2(k9_scmpds_4(C,k9_scmpds_4(B,A)),D) ) ) ) ) ) )).

fof(t8_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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)
            & v2_scmpds_4(B)
            & v3_scmpds_5(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)
                & v3_scmpds_4(C)
                & m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) )
             => ! [D] :
                  ( m1_scmpds_2(D)
                 => ( ( r1_scmpds_6(C,k9_scmpds_4(B,A))
                      & r2_scmpds_6(C,k9_scmpds_4(B,A)) )
                   => k2_scmpds_2(k9_scmpds_4(k3_scmpds_4(B,C),A),D) = k2_scmpds_2(k9_scmpds_4(C,k9_scmpds_4(B,A)),D) ) ) ) ) ) )).

fof(t9_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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)
                & v2_scmpds_4(C)
                & v3_scmpds_4(C)
                & m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) )
             => ( ( r1_scmpds_6(B,A)
                  & r2_scmpds_6(B,A) )
               => ( r1_scmpds_6(k3_scmpds_4(B,C),A)
                  & r2_scmpds_6(k3_scmpds_4(B,C),A) ) ) ) ) ) )).

fof(t10_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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)
            & v2_scmpds_4(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)
                & v3_scmpds_4(C)
                & m1_ami_1(C,k1_tarski(k4_numbers),k1_scmpds_2) )
             => ( ( r1_scmpds_6(C,k9_scmpds_4(B,A))
                  & r2_scmpds_6(C,k9_scmpds_4(B,A)) )
               => ( r1_scmpds_6(k3_scmpds_4(B,C),A)
                  & r2_scmpds_6(k3_scmpds_4(B,C),A) ) ) ) ) ) )).

fof(t11_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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] :
              ( ( v4_scmpds_4(C)
                & v2_scmpds_5(C)
                & m2_subset_1(C,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)) )
             => ( ( r1_scmpds_6(B,A)
                  & r2_scmpds_6(B,A) )
               => ( r1_scmpds_6(k5_scmpds_4(B,C),A)
                  & r2_scmpds_6(k5_scmpds_4(B,C),A) ) ) ) ) ) )).

fof(t12_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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)
            & v3_scmpds_4(B)
            & v3_scmpds_5(B)
            & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
         => ! [C] :
              ( m1_scmpds_2(C)
             => ! [D] :
                  ( m1_scmpds_2(D)
                 => ! [E] :
                      ( m1_scmpds_2(E)
                     => ! [F] :
                          ( v1_int_1(F)
                         => ! [G] :
                              ( v1_int_1(G)
                             => ! [H] :
                                  ( m2_subset_1(H,k1_numbers,k5_numbers)
                                 => ( ( r1_xreal_0(k2_xcmplx_0(k2_scmpds_2(A,E),G),k2_scmpds_2(A,D))
                                      & ! [I] :
                                          ( m1_subset_1(I,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
                                         => ( ( r1_xreal_0(k2_xcmplx_0(k2_scmpds_2(I,E),G),k2_scmpds_2(I,D))
                                              & k2_scmpds_2(I,C) = k2_scmpds_2(A,C) )
                                           => ( r1_xreal_0(k2_scmpds_2(I,k3_scmpds_2(k2_scmpds_2(A,C),F)),np__0)
                                              | ( k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),I),C) = k2_scmpds_2(I,C)
                                                & k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),I),k3_scmpds_2(k2_scmpds_2(A,C),F)) = k6_xcmplx_0(k2_scmpds_2(I,k3_scmpds_2(k2_scmpds_2(A,C),F)),H)
                                                & r1_scmpds_6(B,I)
                                                & r2_scmpds_6(B,I)
                                                & r1_xreal_0(k2_xcmplx_0(k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),I),E),G),k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),I),D)) ) ) ) ) )
                                   => ( r1_xreal_0(H,np__0)
                                      | ( r1_scmpds_6(k2_scmpds_7(C,F,H,B),A)
                                        & r2_scmpds_6(k2_scmpds_7(C,F,H,B),A) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t13_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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)
            & v3_scmpds_4(B)
            & v3_scmpds_5(B)
            & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
         => ! [C] :
              ( m1_scmpds_2(C)
             => ! [D] :
                  ( m1_scmpds_2(D)
                 => ! [E] :
                      ( m1_scmpds_2(E)
                     => ! [F] :
                          ( v1_int_1(F)
                         => ! [G] :
                              ( v1_int_1(G)
                             => ! [H] :
                                  ( m2_subset_1(H,k1_numbers,k5_numbers)
                                 => ( ( r1_xreal_0(k2_xcmplx_0(k2_scmpds_2(A,E),G),k2_scmpds_2(A,D))
                                      & ! [I] :
                                          ( m1_subset_1(I,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
                                         => ( ( r1_xreal_0(k2_xcmplx_0(k2_scmpds_2(I,E),G),k2_scmpds_2(I,D))
                                              & k2_scmpds_2(I,C) = k2_scmpds_2(A,C) )
                                           => ( r1_xreal_0(k2_scmpds_2(I,k3_scmpds_2(k2_scmpds_2(A,C),F)),np__0)
                                              | ( k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),I),C) = k2_scmpds_2(I,C)
                                                & k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),I),k3_scmpds_2(k2_scmpds_2(A,C),F)) = k6_xcmplx_0(k2_scmpds_2(I,k3_scmpds_2(k2_scmpds_2(A,C),F)),H)
                                                & r1_scmpds_6(B,I)
                                                & r2_scmpds_6(B,I)
                                                & r1_xreal_0(k2_xcmplx_0(k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),I),E),G),k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),I),D)) ) ) ) ) )
                                   => ( r1_xreal_0(H,np__0)
                                      | r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),F)),np__0)
                                      | k9_scmpds_4(k2_scmpds_7(C,F,H,B),A) = k9_scmpds_4(k2_scmpds_7(C,F,H,B),k9_scmpds_4(k5_scmpds_4(B,k12_scmpds_2(C,F,k4_xcmplx_0(H))),A)) ) ) ) ) ) ) ) ) ) ) )).

fof(t14_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_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)
            & v3_scmpds_4(B)
            & v3_scmpds_5(B)
            & m1_ami_1(B,k1_tarski(k4_numbers),k1_scmpds_2) )
         => ! [C] :
              ( m1_scmpds_2(C)
             => ! [D] :
                  ( v1_int_1(D)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ! [F] :
                            ( m1_subset_1(F,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
                           => ( k2_scmpds_2(F,C) = k2_scmpds_2(A,C)
                             => ( k2_scmpds_2(k9_scmpds_4(B,F),C) = k2_scmpds_2(F,C)
                                & k2_scmpds_2(k9_scmpds_4(B,F),k3_scmpds_2(k2_scmpds_2(A,C),D)) = k2_scmpds_2(F,k3_scmpds_2(k2_scmpds_2(A,C),D))
                                & r1_scmpds_6(B,F)
                                & r2_scmpds_6(B,F) ) ) )
                       => ( r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(A,C),D)),np__0)
                          | r1_xreal_0(E,np__0)
                          | r1_xreal_0(k4_card_1(B),np__0)
                          | C = k3_scmpds_2(k2_scmpds_2(A,C),D)
                          | ( r1_scmpds_6(k2_scmpds_7(C,D,E,B),A)
                            & r2_scmpds_6(k2_scmpds_7(C,D,E,B),A) ) ) ) ) ) ) ) ) )).

fof(d2_scpisort,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_scpisort(A,B) = k3_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k6_scmpds_2(k2_scmp_gcd,np__0),k11_scmpds_2(k2_scmp_gcd,np__1,np__0)),k11_scmpds_2(k2_scmp_gcd,np__2,k6_xcmplx_0(A,np__1))),k11_scmpds_2(k2_scmp_gcd,np__3,B)),k2_scmpds_7(k2_scmp_gcd,np__2,np__1,k3_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k12_scmpds_2(k2_scmp_gcd,np__3,np__1),k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__4,np__3)),k12_scmpds_2(k2_scmp_gcd,np__1,np__1)),k17_scmpds_2(k2_scmp_gcd,k2_scmp_gcd,np__6,np__1)),k3_scmpds_8(k2_scmp_gcd,np__6,k3_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__5,k4_xcmplx_0(np__1)),k14_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__5,np__0)),k3_scmpds_6(k2_scmp_gcd,np__5,k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__5,k4_xcmplx_0(np__1)),k17_scmpds_2(k1_scmp_gcd(np__4),k1_scmp_gcd(np__4),k4_xcmplx_0(np__1),np__0)),k17_scmpds_2(k1_scmp_gcd(np__4),k2_scmp_gcd,np__0,np__5)),k12_scmpds_2(k2_scmp_gcd,np__4,k4_xcmplx_0(np__1))),k12_scmpds_2(k2_scmp_gcd,np__6,k4_xcmplx_0(np__1))),k1_scmpds_4(k11_scmpds_2(k2_scmp_gcd,np__6,np__0)))))))) ) ) )).

fof(t15_scpisort,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k4_card_1(k1_scpisort(A,B)) = np__23 ) ) )).

fof(t16_scpisort,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__7,A)
           => v2_scmpds_4(k1_scpisort(B,A)) ) ) ) )).

fof(t17_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
     => ! [B] :
          ( m2_finseq_1(B,k4_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k4_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(k2_xcmplx_0(np__7,k2_scmpds_2(A,k1_scmp_gcd(np__6))),k2_scmpds_2(A,k1_scmp_gcd(np__4)))
                          & k2_scmpds_2(A,k2_scmp_gcd) = np__0
                          & E = k2_scmpds_2(A,k1_scmp_gcd(np__6))
                          & D = k6_xcmplx_0(k6_xcmplx_0(k2_scmpds_2(A,k1_scmp_gcd(np__4)),k2_scmpds_2(A,k1_scmp_gcd(np__6))),np__1)
                          & r1_scpisort(B,A,D)
                          & r1_scpisort(C,k9_scmpds_4(k3_scmpds_8(k2_scmp_gcd,np__6,k3_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__5,k4_xcmplx_0(np__1)),k14_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__5,np__0)),k3_scmpds_6(k2_scmp_gcd,np__5,k5_scmpds_4(k5_scmpds_4(k5_scmpds_4(k6_scmpds_4(k17_scmpds_2(k2_scmp_gcd,k1_scmp_gcd(np__4),np__5,k4_xcmplx_0(np__1)),k17_scmpds_2(k1_scmp_gcd(np__4),k1_scmp_gcd(np__4),k4_xcmplx_0(np__1),np__0)),k17_scmpds_2(k1_scmp_gcd(np__4),k2_scmp_gcd,np__0,np__5)),k12_scmpds_2(k2_scmp_gcd,np__4,k4_xcmplx_0(np__1))),k12_scmpds_2(k2_scmp_gcd,np__6,k4_xcmplx_0(np__1))),k1_scmpds_4(k11_scmpds_2(k2_scmp_gcd,np__6,np__0))))),A),D)
                          & k3_finseq_1(B) = k3_finseq_1(C)
                          & r1_sfmastr3(B,np__1,E) )
                       => ( r1_xreal_0(k3_finseq_1(B),E)
                          | ( r1_rfinseq(B,C)
                            & r1_sfmastr3(C,np__1,k1_nat_1(E,np__1))
                            & ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ( r1_xreal_0(F,k3_finseq_1(B))
                                 => ( r1_xreal_0(F,k1_nat_1(E,np__1))
                                    | k1_funct_1(B,F) = k1_funct_1(C,F) ) ) )
                            & ! [F] :
                                ( m2_subset_1(F,k1_numbers,k5_numbers)
                               => ~ ( r1_xreal_0(np__1,F)
                                    & r1_xreal_0(F,k1_nat_1(E,np__1))
                                    & ! [G] :
                                        ( m2_subset_1(G,k1_numbers,k5_numbers)
                                       => ~ ( r1_xreal_0(np__1,G)
                                            & r1_xreal_0(G,k1_nat_1(E,np__1))
                                            & k1_funct_1(C,F) = k1_funct_1(B,G) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t18_scpisort,axiom,(
    ! [A] :
      ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
     => ! [B] :
          ( m2_finseq_1(B,k4_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k4_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__6,D)
                          & k3_finseq_1(B) = E
                          & k3_finseq_1(C) = E
                          & r1_scpisort(B,A,D)
                          & r1_scpisort(C,k9_scmpds_4(k1_scpisort(E,k1_nat_1(D,np__1)),A),D) )
                       => ( r1_rfinseq(B,C)
                          & r1_sfmastr3(C,np__1,E) ) ) ) ) ) ) ) )).

fof(s1_scpisort,axiom,
    ( ( ~ r1_xreal_0(f5_s1_scpisort,np__0)
      & p1_s1_scpisort(k1_scmpds_8(f1_s1_scpisort))
      & ! [A] :
          ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
         => ( ( p1_s1_scpisort(k1_scmpds_8(A))
              & k2_scmpds_2(A,f3_s1_scpisort) = k2_scmpds_2(f1_s1_scpisort,f3_s1_scpisort) )
           => ( r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f1_s1_scpisort,f3_s1_scpisort),f4_s1_scpisort)),np__0)
              | ( k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(f2_s1_scpisort,k12_scmpds_2(f3_s1_scpisort,f4_s1_scpisort,k4_xcmplx_0(f5_s1_scpisort))),A),f3_s1_scpisort) = k2_scmpds_2(A,f3_s1_scpisort)
                & k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(f2_s1_scpisort,k12_scmpds_2(f3_s1_scpisort,f4_s1_scpisort,k4_xcmplx_0(f5_s1_scpisort))),A),k3_scmpds_2(k2_scmpds_2(f1_s1_scpisort,f3_s1_scpisort),f4_s1_scpisort)) = k6_xcmplx_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f1_s1_scpisort,f3_s1_scpisort),f4_s1_scpisort)),f5_s1_scpisort)
                & r1_scmpds_6(f2_s1_scpisort,A)
                & r2_scmpds_6(f2_s1_scpisort,A)
                & p1_s1_scpisort(k1_scmpds_8(k9_scmpds_4(k5_scmpds_4(f2_s1_scpisort,k12_scmpds_2(f3_s1_scpisort,f4_s1_scpisort,k4_xcmplx_0(f5_s1_scpisort))),A))) ) ) ) ) )
   => ( ~ ( ~ p1_s1_scpisort(f1_s1_scpisort)
          & p1_s1_scpisort(f1_s1_scpisort) )
      & r1_scmpds_6(k2_scmpds_7(f3_s1_scpisort,f4_s1_scpisort,f5_s1_scpisort,f2_s1_scpisort),f1_s1_scpisort)
      & r2_scmpds_6(k2_scmpds_7(f3_s1_scpisort,f4_s1_scpisort,f5_s1_scpisort,f2_s1_scpisort),f1_s1_scpisort) ) )).

fof(s2_scpisort,axiom,
    ( ( ~ r1_xreal_0(f5_s2_scpisort,np__0)
      & ~ r1_xreal_0(k2_scmpds_2(f1_s2_scpisort,k3_scmpds_2(k2_scmpds_2(f1_s2_scpisort,f3_s2_scpisort),f4_s2_scpisort)),np__0)
      & p1_s2_scpisort(k1_scmpds_8(f1_s2_scpisort))
      & ! [A] :
          ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
         => ( ( p1_s2_scpisort(k1_scmpds_8(A))
              & k2_scmpds_2(A,f3_s2_scpisort) = k2_scmpds_2(f1_s2_scpisort,f3_s2_scpisort) )
           => ( r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f1_s2_scpisort,f3_s2_scpisort),f4_s2_scpisort)),np__0)
              | ( k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(f2_s2_scpisort,k12_scmpds_2(f3_s2_scpisort,f4_s2_scpisort,k4_xcmplx_0(f5_s2_scpisort))),A),f3_s2_scpisort) = k2_scmpds_2(A,f3_s2_scpisort)
                & k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(f2_s2_scpisort,k12_scmpds_2(f3_s2_scpisort,f4_s2_scpisort,k4_xcmplx_0(f5_s2_scpisort))),A),k3_scmpds_2(k2_scmpds_2(f1_s2_scpisort,f3_s2_scpisort),f4_s2_scpisort)) = k6_xcmplx_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f1_s2_scpisort,f3_s2_scpisort),f4_s2_scpisort)),f5_s2_scpisort)
                & r1_scmpds_6(f2_s2_scpisort,A)
                & r2_scmpds_6(f2_s2_scpisort,A)
                & p1_s2_scpisort(k1_scmpds_8(k9_scmpds_4(k5_scmpds_4(f2_s2_scpisort,k12_scmpds_2(f3_s2_scpisort,f4_s2_scpisort,k4_xcmplx_0(f5_s2_scpisort))),A))) ) ) ) ) )
   => ( ~ ( ~ p1_s2_scpisort(f1_s2_scpisort)
          & p1_s2_scpisort(f1_s2_scpisort) )
      & k9_scmpds_4(k2_scmpds_7(f3_s2_scpisort,f4_s2_scpisort,f5_s2_scpisort,f2_s2_scpisort),f1_s2_scpisort) = k9_scmpds_4(k2_scmpds_7(f3_s2_scpisort,f4_s2_scpisort,f5_s2_scpisort,f2_s2_scpisort),k9_scmpds_4(k5_scmpds_4(f2_s2_scpisort,k12_scmpds_2(f3_s2_scpisort,f4_s2_scpisort,k4_xcmplx_0(f5_s2_scpisort))),f1_s2_scpisort)) ) )).

fof(s3_scpisort,axiom,
    ( ( ~ r1_xreal_0(f5_s3_scpisort,np__0)
      & p1_s3_scpisort(k1_scmpds_8(f1_s3_scpisort))
      & ! [A] :
          ( m1_subset_1(A,k4_card_3(u5_ami_1(k1_tarski(k4_numbers),k1_scmpds_2)))
         => ( ( p1_s3_scpisort(k1_scmpds_8(A))
              & k2_scmpds_2(A,f3_s3_scpisort) = k2_scmpds_2(f1_s3_scpisort,f3_s3_scpisort) )
           => ( r1_xreal_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f1_s3_scpisort,f3_s3_scpisort),f4_s3_scpisort)),np__0)
              | ( k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(f2_s3_scpisort,k12_scmpds_2(f3_s3_scpisort,f4_s3_scpisort,k4_xcmplx_0(f5_s3_scpisort))),A),f3_s3_scpisort) = k2_scmpds_2(A,f3_s3_scpisort)
                & k2_scmpds_2(k9_scmpds_4(k5_scmpds_4(f2_s3_scpisort,k12_scmpds_2(f3_s3_scpisort,f4_s3_scpisort,k4_xcmplx_0(f5_s3_scpisort))),A),k3_scmpds_2(k2_scmpds_2(f1_s3_scpisort,f3_s3_scpisort),f4_s3_scpisort)) = k6_xcmplx_0(k2_scmpds_2(A,k3_scmpds_2(k2_scmpds_2(f1_s3_scpisort,f3_s3_scpisort),f4_s3_scpisort)),f5_s3_scpisort)
                & r1_scmpds_6(f2_s3_scpisort,A)
                & r2_scmpds_6(f2_s3_scpisort,A)
                & p1_s3_scpisort(k1_scmpds_8(k9_scmpds_4(k5_scmpds_4(f2_s3_scpisort,k12_scmpds_2(f3_s3_scpisort,f4_s3_scpisort,k4_xcmplx_0(f5_s3_scpisort))),A))) ) ) ) ) )
   => ( ~ ( ~ p1_s3_scpisort(f1_s3_scpisort)
          & p1_s3_scpisort(f1_s3_scpisort) )
      & r1_xreal_0(k2_scmpds_2(k9_scmpds_4(k2_scmpds_7(f3_s3_scpisort,f4_s3_scpisort,f5_s3_scpisort,f2_s3_scpisort),f1_s3_scpisort),k3_scmpds_2(k2_scmpds_2(f1_s3_scpisort,f3_s3_scpisort),f4_s3_scpisort)),np__0)
      & p1_s3_scpisort(k1_scmpds_8(k9_scmpds_4(k2_scmpds_7(f3_s3_scpisort,f4_s3_scpisort,f5_s3_scpisort,f2_s3_scpisort),f1_s3_scpisort))) ) )).

fof(dt_k1_scpisort,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( v1_ami_3(k1_scpisort(A,B),k1_tarski(k4_numbers),k1_scmpds_2)
        & v1_scmpds_3(k1_scpisort(A,B))
        & m1_ami_1(k1_scpisort(A,B),k1_tarski(k4_numbers),k1_scmpds_2) ) ) )).
%------------------------------------------------------------------------------