SET007 Axioms: SET007+760.ax


%------------------------------------------------------------------------------
% File     : SET007+760 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Integers by Binary Arithmetics and Addition of Integers
% 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    : binari_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   42 (   0 unit)
%            Number of atoms       :  243 (  43 equality)
%            Maximal formula depth :   14 (   8 average)
%            Number of connectives :  245 (  44 ~  ;   9  |;  64  &)
%                                         (   1 <=>; 127 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   10 (   0 propositional; 1-3 arity)
%            Number of functors    :   34 (   8 constant; 0-4 arity)
%            Number of variables   :  100 (   0 singleton;  98 !;   2 ?)
%            Maximal term depth    :    7 (   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_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( ~ r1_xreal_0(A,np__0)
       => r1_xreal_0(k1_nat_1(A,np__1),k2_nat_1(A,np__2)) ) ) )).

fof(t2_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_xreal_0(A,k3_series_1(np__2,A)) ) )).

fof(t3_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k7_euclid(A,k5_euclid(A),k5_euclid(A)) = k5_euclid(A) ) )).

fof(t4_binari_4,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)
             => ~ ( r1_xreal_0(C,A)
                  & r1_xreal_0(A,B)
                  & C != A
                  & ~ ( r1_xreal_0(k1_nat_1(C,np__1),A)
                      & r1_xreal_0(A,B) ) ) ) ) ) )).

fof(t5_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => ! [C] :
              ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
             => ( ( B = k5_euclid(A)
                  & C = k5_euclid(A) )
               => k7_binarith(A,B,C) = k5_euclid(A) ) ) ) ) )).

fof(t6_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => ! [C] :
              ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
             => ( ( B = k5_euclid(A)
                  & C = k5_euclid(A) )
               => k10_binarith(A,B,C) = k5_euclid(A) ) ) ) ) )).

fof(t7_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => ( B = k5_euclid(A)
           => k4_binari_2(A,B) = np__0 ) ) ) )).

fof(t8_binari_4,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)
             => ( r1_xreal_0(k1_nat_1(A,B),k5_real_1(C,np__1))
               => ( ~ r1_xreal_0(C,A)
                  & ~ r1_xreal_0(C,B) ) ) ) ) ) )).

fof(t9_binari_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_xreal_0(A,k2_xcmplx_0(B,C))
               => ( r1_xreal_0(np__0,B)
                  | r1_xreal_0(np__0,C)
                  | ( ~ r1_xreal_0(B,A)
                    & ~ r1_xreal_0(C,A) ) ) ) ) ) ) )).

fof(t10_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)
             => ( r1_xreal_0(k1_nat_1(B,C),k5_real_1(k3_series_1(np__2,A),np__1))
               => k11_binarith(A,k1_binari_3(A,B),k1_binari_3(A,C)) = k7_margrel1 ) ) ) ) )).

fof(t11_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)
             => ( r1_xreal_0(k1_nat_1(B,C),k5_real_1(k3_series_1(np__2,A),np__1))
               => k9_binarith(A,k10_binarith(A,k1_binari_3(A,B),k1_binari_3(A,C))) = k1_nat_1(B,C) ) ) ) ) )).

fof(t12_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => ( k4_finseq_4(k5_numbers,k6_margrel1,B,A) = k8_margrel1
           => r1_xreal_0(k3_series_1(np__2,k5_binarith(A,np__1)),k9_binarith(A,B)) ) ) ) )).

fof(t13_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)
             => ( r1_xreal_0(k1_nat_1(B,C),k5_real_1(k3_series_1(np__2,k5_binarith(A,np__1)),np__1))
               => k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,k1_binari_3(A,B),k1_binari_3(A,C)),A) = k7_margrel1 ) ) ) ) )).

fof(t14_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( r1_xreal_0(k1_nat_1(A,B),k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1))
               => k4_binari_2(C,k10_binarith(C,k1_binari_3(C,A),k1_binari_3(C,B))) = k1_nat_1(A,B) ) ) ) ) )).

fof(t15_binari_4,axiom,(
    ! [A] :
      ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
     => ( A = k13_binarith(k6_margrel1,k8_margrel1)
       => k4_binari_2(np__1,A) = k1_real_1(np__1) ) ) )).

fof(t16_binari_4,axiom,(
    ! [A] :
      ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
     => ( A = k13_binarith(k6_margrel1,k7_margrel1)
       => k4_binari_2(np__1,A) = np__0 ) ) )).

fof(t17_binari_4,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k1_binarith(k8_margrel1,A) = k8_margrel1 ) )).

fof(t18_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ( r1_xreal_0(np__0,k5_real_1(k3_series_1(np__2,k5_binarith(A,np__1)),np__1))
        & r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(A,np__1))),np__0) ) ) )).

fof(t19_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => ! [C] :
              ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
             => ( ( B = k5_euclid(A)
                  & C = k5_euclid(A) )
               => r1_binarith(A,B,C) ) ) ) ) )).

fof(t20_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_int_1(B)
         => k6_int_1(k3_xcmplx_0(B,A),A) = np__0 ) ) )).

fof(d1_binari_4,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)
             => ( C = k1_binari_4(A,B)
              <=> ( r1_xreal_0(B,k3_series_1(np__2,C))
                  & r1_xreal_0(A,C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(B,k3_series_1(np__2,D))
                          & r1_xreal_0(A,D) )
                       => r1_xreal_0(C,D) ) ) ) ) ) ) ) )).

fof(t21_binari_4,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)
             => ( r1_xreal_0(B,A)
               => r1_xreal_0(k1_binari_4(C,B),k1_binari_4(C,A)) ) ) ) ) )).

fof(t22_binari_4,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)
             => ( r1_xreal_0(B,A)
               => r1_xreal_0(k1_binari_4(B,C),k1_binari_4(A,C)) ) ) ) ) )).

fof(t23_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => k1_binari_4(A,np__1) = A ) ) )).

fof(t24_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(A,k3_series_1(np__2,B))
           => k1_binari_4(B,A) = B ) ) ) )).

fof(t25_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( ~ r1_xreal_0(A,k3_series_1(np__2,B))
              & r1_xreal_0(k1_binari_4(B,A),B) ) ) ) )).

fof(d2_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( ~ r1_xreal_0(np__0,B)
             => k2_binari_4(A,B) = k1_binari_3(A,k1_prepower(k2_xcmplx_0(k3_series_1(np__2,k1_binari_4(A,k1_prepower(B))),B))) )
            & ( r1_xreal_0(np__0,B)
             => k2_binari_4(A,B) = k1_binari_3(A,k1_prepower(B)) ) ) ) ) )).

fof(t26_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_binari_4(A,np__0) = k5_euclid(A) ) )).

fof(t27_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_int_1(B)
         => ( ( r1_xreal_0(B,k5_real_1(k3_series_1(np__2,k5_binarith(A,np__1)),np__1))
              & r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(A,np__1))),B) )
           => k4_binari_2(A,k2_binari_4(A,B)) = B ) ) ) )).

fof(t28_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( k6_int_1(B,k3_series_1(np__2,A)) = k6_int_1(C,k3_series_1(np__2,A))
               => ( ( ~ ( r1_xreal_0(np__0,B)
                        & r1_xreal_0(np__0,C) )
                    & ~ ( ~ r1_xreal_0(np__0,B)
                        & ~ r1_xreal_0(np__0,C) ) )
                  | k2_binari_4(A,B) = k2_binari_4(A,C) ) ) ) ) ) )).

fof(t29_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( r1_int_1(B,C,k3_series_1(np__2,A))
               => ( ( ~ ( r1_xreal_0(np__0,B)
                        & r1_xreal_0(np__0,C) )
                    & ~ ( ~ r1_xreal_0(np__0,B)
                        & ~ r1_xreal_0(np__0,C) ) )
                  | k2_binari_4(A,B) = k2_binari_4(A,C) ) ) ) ) ) )).

fof(t30_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)
             => ( k4_nat_1(B,k3_series_1(np__2,A)) = k4_nat_1(C,k3_series_1(np__2,A))
               => k1_binari_3(A,B) = k1_binari_3(A,C) ) ) ) ) )).

fof(t31_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(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)
             => ( r1_int_1(B,C,k3_series_1(np__2,A))
               => k1_binari_3(A,B) = k1_binari_3(A,C) ) ) ) ) )).

fof(t32_binari_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,C)
                  & r1_xreal_0(C,A) )
               => k4_finseq_4(k5_numbers,k6_margrel1,k2_binari_4(k1_nat_1(A,np__1),B),C) = k4_finseq_4(k5_numbers,k6_margrel1,k2_binari_4(A,B),C) ) ) ) ) )).

fof(t33_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ? [C] :
              ( m1_subset_1(C,k6_margrel1)
              & k2_binari_4(k1_nat_1(A,np__1),B) = k8_finseq_1(k6_margrel1,k2_binari_4(A,B),k13_binarith(k6_margrel1,C)) ) ) ) )).

fof(t34_binari_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ? [C] :
              ( m1_subset_1(C,k6_margrel1)
              & k1_binari_3(k1_nat_1(A,np__1),B) = k8_finseq_1(k6_margrel1,k1_binari_3(A,B),k13_binarith(k6_margrel1,C)) ) ) ) )).

fof(t35_binari_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( ( r1_xreal_0(k1_real_1(k3_series_1(np__2,C)),k2_xcmplx_0(A,B))
                  & r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),A)
                  & r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),B) )
               => ( r1_xreal_0(np__0,A)
                  | r1_xreal_0(np__0,B)
                  | k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(k1_nat_1(C,np__1),k2_binari_4(k1_nat_1(C,np__1),A),k2_binari_4(k1_nat_1(C,np__1),B)),k1_nat_1(C,np__1)) = k8_margrel1 ) ) ) ) ) )).

fof(t36_binari_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( ( r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),k2_xcmplx_0(A,B))
                  & r1_xreal_0(k2_xcmplx_0(A,B),k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1))
                  & r1_xreal_0(np__0,A)
                  & r1_xreal_0(np__0,B) )
               => k4_binari_2(C,k10_binarith(C,k2_binari_4(C,A),k2_binari_4(C,B))) = k2_xcmplx_0(A,B) ) ) ) ) )).

fof(t37_binari_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( ( r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(k1_nat_1(C,np__1),np__1))),k2_xcmplx_0(A,B))
                  & r1_xreal_0(k2_xcmplx_0(A,B),k5_real_1(k3_series_1(np__2,k5_binarith(k1_nat_1(C,np__1),np__1)),np__1))
                  & r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),A)
                  & r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),B) )
               => ( r1_xreal_0(np__0,A)
                  | r1_xreal_0(np__0,B)
                  | k4_binari_2(k1_nat_1(C,np__1),k10_binarith(k1_nat_1(C,np__1),k2_binari_4(k1_nat_1(C,np__1),A),k2_binari_4(k1_nat_1(C,np__1),B))) = k2_xcmplx_0(A,B) ) ) ) ) ) )).

fof(t38_binari_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( ( ~ v1_xboole_0(C)
                & m2_subset_1(C,k1_numbers,k5_numbers) )
             => ( ( r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),A)
                  & r1_xreal_0(A,k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1))
                  & r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),B)
                  & r1_xreal_0(B,k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1))
                  & r1_xreal_0(k1_real_1(k3_series_1(np__2,k5_binarith(C,np__1))),k2_xcmplx_0(A,B))
                  & r1_xreal_0(k2_xcmplx_0(A,B),k5_real_1(k3_series_1(np__2,k5_binarith(C,np__1)),np__1)) )
               => ( ( ~ ( r1_xreal_0(np__0,A)
                        & ~ r1_xreal_0(np__0,B) )
                    & ~ ( ~ r1_xreal_0(np__0,A)
                        & r1_xreal_0(np__0,B) ) )
                  | k4_binari_2(C,k10_binarith(C,k2_binari_4(C,A),k2_binari_4(C,B))) = k2_xcmplx_0(A,B) ) ) ) ) ) )).

fof(dt_k1_binari_4,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k1_binari_4(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k2_binari_4,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & v1_int_1(B) )
     => m2_finseq_2(k2_binari_4(A,B),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) )).
%------------------------------------------------------------------------------