SET007 Axioms: SET007+800.ax


%------------------------------------------------------------------------------
% File     : SET007+800 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Correctness of Non Overwriting Programs. Part I
% 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    : prgcor_1 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   17 (   0 unit)
%            Number of atoms       :  248 (  90 equality)
%            Maximal formula depth :   48 (  13 average)
%            Number of connectives :  270 (  39 ~  ;  17  |; 119  &)
%                                         (   2 <=>;  93 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   0 propositional; 1-3 arity)
%            Number of functors    :   23 (   6 constant; 0-3 arity)
%            Number of variables   :   57 (   0 singleton;  52 !;   5 ?)
%            Maximal term depth    :    6 (   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(fc1_prgcor_1,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(B)
        & m1_relset_1(B,A,k4_numbers) )
     => ( v1_xcmplx_0(k1_funct_1(B,C))
        & v1_xreal_0(k1_funct_1(B,C))
        & v1_int_1(k1_funct_1(B,C)) ) ) )).

fof(t1_prgcor_1,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)
             => k5_binarith(k1_nat_1(A,C),k1_nat_1(B,C)) = k5_binarith(A,B) ) ) ) )).

fof(t2_prgcor_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,k4_nat_1(A,k2_nat_1(np__2,B)))
           => ( r1_xreal_0(B,np__0)
              | ( k6_xcmplx_0(k4_nat_1(A,k2_nat_1(np__2,B)),B) = k4_nat_1(A,B)
                & k1_nat_1(k4_nat_1(A,B),B) = k4_nat_1(A,k2_nat_1(np__2,B)) ) ) ) ) ) )).

fof(t3_prgcor_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(B,k4_nat_1(A,k2_nat_1(np__2,B)))
           => ( r1_xreal_0(B,np__0)
              | k3_nat_1(A,B) = k1_nat_1(k2_nat_1(k3_nat_1(A,k2_nat_1(np__2,B)),np__2),np__1) ) ) ) ) )).

fof(t4_prgcor_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(B,np__0)
              & ~ r1_xreal_0(B,k4_nat_1(A,k2_nat_1(np__2,B)))
              & k4_nat_1(A,k2_nat_1(np__2,B)) != k4_nat_1(A,B) ) ) ) )).

fof(t5_prgcor_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(B,np__0)
              & ~ r1_xreal_0(B,k4_nat_1(A,k2_nat_1(np__2,B)))
              & k3_nat_1(A,B) != k2_nat_1(k3_nat_1(A,k2_nat_1(np__2,B)),np__2) ) ) ) )).

fof(t6_prgcor_1,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)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ~ ( ! [D] :
                          ( m2_subset_1(D,k1_numbers,k5_numbers)
                         => ( ~ r1_xreal_0(C,D)
                           => r1_xreal_0(k2_nat_1(A,k3_euler_2(np__2,D)),B) ) )
                      & ~ r1_xreal_0(k2_nat_1(A,k3_euler_2(np__2,C)),B) ) ) ) ) ) )).

fof(t7_prgcor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(A,k3_finseq_1(B)) )
           => r2_hidden(A,k4_finseq_1(B)) ) ) ) )).

fof(d1_prgcor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_xreal_0(np__0,A)
           => ( r1_xreal_0(B,np__0)
              | ! [C] :
                  ( v1_int_1(C)
                 => ( C = k1_prgcor_1(A,B)
                  <=> ? [D] :
                        ( m2_finseq_1(D,k4_numbers)
                        & ? [E] :
                            ( m2_finseq_1(E,k4_numbers)
                            & ? [F] :
                                ( m2_finseq_1(F,k4_numbers)
                                & k3_finseq_1(D) = k2_xcmplx_0(A,np__1)
                                & k3_finseq_1(E) = k2_xcmplx_0(A,np__1)
                                & k3_finseq_1(F) = k2_xcmplx_0(A,np__1)
                                & ( ~ r1_xreal_0(B,A)
                                 => C = np__0 )
                                & ( r1_xreal_0(B,A)
                                 => ( k1_funct_1(D,np__1) = B
                                    & ? [G] :
                                        ( v1_int_1(G)
                                        & r1_xreal_0(np__1,G)
                                        & r1_xreal_0(G,A)
                                        & ! [H] :
                                            ( v1_int_1(H)
                                           => ( r1_xreal_0(np__1,H)
                                             => ( r1_xreal_0(G,H)
                                                | ( k1_funct_1(D,k2_xcmplx_0(H,np__1)) = k3_xcmplx_0(k1_funct_1(D,H),np__2)
                                                  & r1_xreal_0(k1_funct_1(D,k2_xcmplx_0(H,np__1)),A) ) ) ) )
                                        & k1_funct_1(D,k2_xcmplx_0(G,np__1)) = k3_xcmplx_0(k1_funct_1(D,G),np__2)
                                        & ~ r1_xreal_0(k1_funct_1(D,k2_xcmplx_0(G,np__1)),A)
                                        & k1_funct_1(F,k2_xcmplx_0(G,np__1)) = np__0
                                        & k1_funct_1(E,k2_xcmplx_0(G,np__1)) = A
                                        & ! [H] :
                                            ( v1_int_1(H)
                                           => ( ( r1_xreal_0(np__1,H)
                                                & r1_xreal_0(H,G) )
                                             => ( ( r1_xreal_0(k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)),k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))))
                                                 => ( k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)) = k6_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)))
                                                    & k1_funct_1(F,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)) = k2_xcmplx_0(k3_xcmplx_0(k1_funct_1(F,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))),np__2),np__1) ) )
                                                & ( ~ r1_xreal_0(k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)),k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))))
                                                 => ( k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)) = k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1)))
                                                    & k1_funct_1(F,k6_xcmplx_0(k2_xcmplx_0(G,np__1),H)) = k3_xcmplx_0(k1_funct_1(F,k6_xcmplx_0(k2_xcmplx_0(G,np__1),k6_xcmplx_0(H,np__1))),np__2) ) ) ) ) )
                                        & C = k1_funct_1(F,np__1) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t8_prgcor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_xreal_0(np__0,A)
           => ( r1_xreal_0(B,np__0)
              | ! [C] :
                  ( m2_finseq_1(C,k4_numbers)
                 => ! [D] :
                      ( m2_finseq_1(D,k4_numbers)
                     => ! [E] :
                          ( m2_finseq_1(E,k4_numbers)
                         => ! [F] :
                              ( v1_int_1(F)
                             => ( ( k3_finseq_1(C) = k2_xcmplx_0(A,np__1)
                                  & k3_finseq_1(D) = k2_xcmplx_0(A,np__1)
                                  & k3_finseq_1(E) = k2_xcmplx_0(A,np__1) )
                               => ( ( r1_xreal_0(B,A)
                                    & ~ ( k1_funct_1(C,np__1) = B
                                        & r1_xreal_0(np__1,F)
                                        & r1_xreal_0(F,A)
                                        & ! [G] :
                                            ( v1_int_1(G)
                                           => ( r1_xreal_0(np__1,G)
                                             => ( r1_xreal_0(F,G)
                                                | ( k1_funct_1(C,k2_xcmplx_0(G,np__1)) = k3_xcmplx_0(k1_funct_1(C,G),np__2)
                                                  & r1_xreal_0(k1_funct_1(C,k2_xcmplx_0(G,np__1)),A) ) ) ) )
                                        & k1_funct_1(C,k2_xcmplx_0(F,np__1)) = k3_xcmplx_0(k1_funct_1(C,F),np__2)
                                        & ~ r1_xreal_0(k1_funct_1(C,k2_xcmplx_0(F,np__1)),A)
                                        & k1_funct_1(E,k2_xcmplx_0(F,np__1)) = np__0
                                        & k1_funct_1(D,k2_xcmplx_0(F,np__1)) = A
                                        & ! [G] :
                                            ( v1_int_1(G)
                                           => ( ( r1_xreal_0(np__1,G)
                                                & r1_xreal_0(G,F) )
                                             => ( ( r1_xreal_0(k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))))
                                                 => ( k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k6_xcmplx_0(k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)))
                                                    & k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k2_xcmplx_0(k3_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),np__2),np__1) ) )
                                                & ( ~ r1_xreal_0(k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))))
                                                 => ( k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)))
                                                    & k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k3_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),np__2) ) ) ) ) )
                                        & k1_prgcor_1(A,B) = k1_funct_1(E,np__1) ) )
                                  | ( k3_finseq_1(C) = k2_xcmplx_0(A,np__1)
                                    & k3_finseq_1(D) = k2_xcmplx_0(A,np__1)
                                    & k3_finseq_1(E) = k2_xcmplx_0(A,np__1)
                                    & ( ~ r1_xreal_0(B,A)
                                     => k1_prgcor_1(A,B) = np__0 )
                                    & ( r1_xreal_0(B,A)
                                     => ( r2_hidden(np__1,k4_relset_1(k5_numbers,k4_numbers,C))
                                        & k1_funct_1(C,np__1) = B
                                        & r1_xreal_0(np__1,F)
                                        & r1_xreal_0(F,A)
                                        & ! [G] :
                                            ( v1_int_1(G)
                                           => ( r1_xreal_0(np__1,G)
                                             => ( r1_xreal_0(F,G)
                                                | ( r2_hidden(k2_xcmplx_0(G,np__1),k4_relset_1(k5_numbers,k4_numbers,C))
                                                  & r2_hidden(G,k4_relset_1(k5_numbers,k4_numbers,C))
                                                  & k1_funct_1(C,k2_xcmplx_0(G,np__1)) = k3_xcmplx_0(k1_funct_1(C,G),np__2)
                                                  & r1_xreal_0(k1_funct_1(C,k2_xcmplx_0(G,np__1)),A) ) ) ) )
                                        & r2_hidden(k2_xcmplx_0(F,np__1),k4_relset_1(k5_numbers,k4_numbers,C))
                                        & r2_hidden(F,k4_relset_1(k5_numbers,k4_numbers,C))
                                        & k1_funct_1(C,k2_xcmplx_0(F,np__1)) = k3_xcmplx_0(k1_funct_1(C,F),np__2)
                                        & ~ r1_xreal_0(k1_funct_1(C,k2_xcmplx_0(F,np__1)),A)
                                        & r2_hidden(k2_xcmplx_0(F,np__1),k4_relset_1(k5_numbers,k4_numbers,E))
                                        & k1_funct_1(E,k2_xcmplx_0(F,np__1)) = np__0
                                        & r2_hidden(k2_xcmplx_0(F,np__1),k4_relset_1(k5_numbers,k4_numbers,D))
                                        & k1_funct_1(D,k2_xcmplx_0(F,np__1)) = A
                                        & ! [G] :
                                            ( v1_int_1(G)
                                           => ( ( r1_xreal_0(np__1,G)
                                                & r1_xreal_0(G,F) )
                                             => ( r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)),k4_relset_1(k5_numbers,k4_numbers,D))
                                                & r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,C))
                                                & ( r1_xreal_0(k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))))
                                                 => ( r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,D))
                                                    & r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,C))
                                                    & k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k6_xcmplx_0(k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)))
                                                    & r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,E))
                                                    & r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)),k4_relset_1(k5_numbers,k4_numbers,E))
                                                    & k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k2_xcmplx_0(k3_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),np__2),np__1) ) )
                                                & ( ~ r1_xreal_0(k1_funct_1(C,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)),k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))))
                                                 => ( r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,D))
                                                    & r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)),k4_relset_1(k5_numbers,k4_numbers,D))
                                                    & k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k1_funct_1(D,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)))
                                                    & r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),G),k4_relset_1(k5_numbers,k4_numbers,E))
                                                    & r2_hidden(k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1)),k4_relset_1(k5_numbers,k4_numbers,E))
                                                    & k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),G)) = k3_xcmplx_0(k1_funct_1(E,k6_xcmplx_0(k2_xcmplx_0(F,np__1),k6_xcmplx_0(G,np__1))),np__2) ) ) ) ) )
                                        & r2_hidden(np__1,k4_relset_1(k5_numbers,k4_numbers,E))
                                        & k1_prgcor_1(A,B) = k1_funct_1(E,np__1) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t9_prgcor_1,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(B,np__0)
           => k1_prgcor_1(A,B) = k3_nat_1(A,B) ) ) ) )).

fof(t10_prgcor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ( r1_xreal_0(np__0,A)
           => ( r1_xreal_0(B,np__0)
              | k1_prgcor_1(A,B) = k5_int_1(A,B) ) ) ) ) )).

fof(t11_prgcor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( ( B = np__0
                        & C = A
                        & D = B )
                     => ( k5_int_1(A,B) = np__0
                        & k3_nat_1(C,D) = np__0 ) )
                    & ( ( r1_xreal_0(np__0,A)
                        & C = A
                        & D = B )
                     => ( r1_xreal_0(B,np__0)
                        | k5_int_1(A,B) = k3_nat_1(C,D) ) )
                    & ( ( r1_xreal_0(np__0,A)
                        & C = A
                        & D = k4_xcmplx_0(B) )
                     => ( r1_xreal_0(np__0,B)
                        | ( ( k2_nat_1(D,k3_nat_1(C,D)) = C
                           => k5_int_1(A,B) = k4_xcmplx_0(k3_nat_1(C,D)) )
                          & ( k2_nat_1(D,k3_nat_1(C,D)) != C
                           => k5_int_1(A,B) = k6_xcmplx_0(k4_xcmplx_0(k3_nat_1(C,D)),np__1) ) ) ) )
                    & ( ( C = k4_xcmplx_0(A)
                        & D = B )
                     => ( r1_xreal_0(np__0,A)
                        | r1_xreal_0(B,np__0)
                        | ( ( k2_nat_1(D,k3_nat_1(C,D)) = C
                           => k5_int_1(A,B) = k4_xcmplx_0(k3_nat_1(C,D)) )
                          & ( k2_nat_1(D,k3_nat_1(C,D)) != C
                           => k5_int_1(A,B) = k6_xcmplx_0(k4_xcmplx_0(k3_nat_1(C,D)),np__1) ) ) ) )
                    & ( ( C = k4_xcmplx_0(A)
                        & D = k4_xcmplx_0(B) )
                     => ( r1_xreal_0(np__0,A)
                        | r1_xreal_0(np__0,B)
                        | k5_int_1(A,B) = k3_nat_1(C,D) ) ) ) ) ) ) ) )).

fof(d2_prgcor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( C = k2_prgcor_1(A,B)
              <=> ? [D] :
                    ( v1_int_1(D)
                    & ( B = np__0
                     => C = np__0 )
                    & ( B != np__0
                     => ( ( r1_xreal_0(np__0,A)
                         => ( r1_xreal_0(B,np__0)
                            | C = k1_prgcor_1(A,B) ) )
                        & ( ~ ( r1_xreal_0(np__0,A)
                              & ~ r1_xreal_0(B,np__0) )
                         => ( ( r1_xreal_0(np__0,A)
                             => ( r1_xreal_0(np__0,B)
                                | ( D = k1_prgcor_1(A,k4_xcmplx_0(B))
                                  & ( k3_xcmplx_0(k4_xcmplx_0(B),D) = A
                                   => C = k4_xcmplx_0(D) )
                                  & ( k3_xcmplx_0(k4_xcmplx_0(B),D) != A
                                   => C = k6_xcmplx_0(k4_xcmplx_0(D),np__1) ) ) ) )
                            & ( ~ ( r1_xreal_0(np__0,A)
                                  & ~ r1_xreal_0(np__0,B) )
                             => ( ~ ( ~ r1_xreal_0(np__0,A)
                                    & ~ r1_xreal_0(B,np__0)
                                    & ~ ( D = k1_prgcor_1(k4_xcmplx_0(A),B)
                                        & ( k3_xcmplx_0(B,D) = k4_xcmplx_0(A)
                                         => C = k4_xcmplx_0(D) )
                                        & ( k3_xcmplx_0(B,D) != k4_xcmplx_0(A)
                                         => C = k6_xcmplx_0(k4_xcmplx_0(D),np__1) ) ) )
                                & ( ~ ( ~ r1_xreal_0(np__0,A)
                                      & ~ r1_xreal_0(B,np__0) )
                                 => C = k1_prgcor_1(k4_xcmplx_0(A),k4_xcmplx_0(B)) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t12_prgcor_1,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => k2_prgcor_1(A,B) = k5_int_1(A,B) ) ) )).

fof(dt_k1_prgcor_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => v1_int_1(k1_prgcor_1(A,B)) ) )).

fof(dt_k2_prgcor_1,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & v1_int_1(B) )
     => v1_int_1(k2_prgcor_1(A,B)) ) )).
%------------------------------------------------------------------------------