SET007 Axioms: SET007+539.ax


%------------------------------------------------------------------------------
% File     : SET007+539 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Binary Arithmetics. Binary Sequences
% 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   39 (   5 unit)
%            Number of atoms       :  179 (  72 equality)
%            Maximal formula depth :   13 (   6 average)
%            Number of connectives :  170 (  30 ~  ;   4  |;  39  &)
%                                         (   4 <=>;  93 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   10 (   0 propositional; 1-3 arity)
%            Number of functors    :   36 (   8 constant; 0-5 arity)
%            Number of variables   :   68 (   0 singleton;  68 !;   0 ?)
%            Maximal term depth    :    6 (   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_3,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))
         => ~ r1_xreal_0(k3_series_1(np__2,A),k9_binarith(A,B)) ) ) )).

fof(t2_binari_3,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))
             => ( k9_binarith(A,B) = k9_binarith(A,C)
               => B = C ) ) ) ) )).

fof(t3_binari_3,axiom,(
    ! [A] :
      ( ( v1_relat_1(A)
        & v1_funct_1(A)
        & v1_finseq_1(A) )
     => ! [B] :
          ( ( v1_relat_1(B)
            & v1_funct_1(B)
            & v1_finseq_1(B) )
         => ( k3_finseq_5(A) = k3_finseq_5(B)
           => A = B ) ) ) )).

fof(t4_binari_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k5_euclid(k1_nat_1(A,np__1)) = k8_finseq_1(k1_numbers,k5_euclid(A),k13_binarith(k1_numbers,np__0)) ) )).

fof(t5_binari_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r2_hidden(k5_euclid(A),k13_finseq_1(k6_margrel1)) ) )).

fof(t6_binari_3,axiom,(
    ! [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)
           => k6_binarith(A,B) = k4_finseqop(k1_numbers,A,np__1) ) ) ) )).

fof(t7_binari_3,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)
           => k9_binarith(A,B) = np__0 ) ) ) )).

fof(t8_binari_3,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)
           => k9_binarith(A,k6_binarith(A,B)) = k5_real_1(k3_series_1(np__2,A),np__1) ) ) ) )).

fof(t9_binari_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k4_finseq_5(k1_numbers,k5_euclid(A)) = k5_euclid(A) ) )).

fof(t10_binari_3,axiom,(
    ! [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_finseq_5(k6_margrel1,k6_binarith(A,B)) = k6_binarith(A,B) ) ) ) )).

fof(t11_binari_3,axiom,(
    k2_binari_2(np__1) = k13_binarith(k6_margrel1,k8_margrel1) )).

fof(t12_binari_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => k9_binarith(A,k2_binari_2(A)) = np__1 ) )).

fof(t13_binari_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k6_margrel1)
     => ! [B] :
          ( m1_subset_1(B,k6_margrel1)
         => ( ~ ( k3_binarith(A,B) = k8_margrel1
                & A != k8_margrel1
                & B != k8_margrel1 )
            & ( ( A = k8_margrel1
                | B = k8_margrel1 )
             => k3_binarith(A,B) = k8_margrel1 )
            & ( k3_binarith(A,B) = k7_margrel1
             => ( A = k7_margrel1
                & B = k7_margrel1 ) )
            & ( ( A = k7_margrel1
                & B = k7_margrel1 )
             => k3_binarith(A,B) = k7_margrel1 ) ) ) ) )).

fof(t14_binari_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k6_margrel1)
     => ! [B] :
          ( m1_subset_1(B,k6_margrel1)
         => ( k11_binarith(np__1,k13_binarith(k6_margrel1,A),k13_binarith(k6_margrel1,B)) = k8_margrel1
          <=> ( A = k8_margrel1
              & B = k8_margrel1 ) ) ) ) )).

fof(t15_binari_3,axiom,(
    k6_binarith(np__1,k13_binarith(k6_margrel1,k7_margrel1)) = k13_binarith(k6_margrel1,k8_margrel1) )).

fof(t16_binari_3,axiom,(
    k6_binarith(np__1,k13_binarith(k6_margrel1,k8_margrel1)) = k13_binarith(k6_margrel1,k7_margrel1) )).

fof(t17_binari_3,axiom,(
    k10_binarith(np__1,k13_binarith(k6_margrel1,k7_margrel1),k13_binarith(k6_margrel1,k7_margrel1)) = k13_binarith(k6_margrel1,k7_margrel1) )).

fof(t18_binari_3,axiom,
    ( k10_binarith(np__1,k13_binarith(k6_margrel1,k7_margrel1),k13_binarith(k6_margrel1,k8_margrel1)) = k13_binarith(k6_margrel1,k8_margrel1)
    & k10_binarith(np__1,k13_binarith(k6_margrel1,k8_margrel1),k13_binarith(k6_margrel1,k7_margrel1)) = k13_binarith(k6_margrel1,k8_margrel1) )).

fof(t19_binari_3,axiom,(
    k10_binarith(np__1,k13_binarith(k6_margrel1,k8_margrel1),k13_binarith(k6_margrel1,k8_margrel1)) = k13_binarith(k6_margrel1,k7_margrel1) )).

fof(t20_binari_3,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))
             => ( ( k4_finseq_4(k5_numbers,k6_margrel1,B,A) = k8_margrel1
                  & k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,k2_binari_2(A)),A) = k8_margrel1 )
               => ! [D] :
                    ( ( ~ v1_xboole_0(D)
                      & m2_subset_1(D,k1_numbers,k5_numbers) )
                   => ( r1_xreal_0(D,A)
                     => ( D = np__1
                        | ( k4_finseq_4(k5_numbers,k6_margrel1,B,D) = k8_margrel1
                          & k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,k2_binari_2(A)),D) = k8_margrel1 ) ) ) ) ) ) ) ) )).

fof(t21_binari_3,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
              & k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,k2_binari_2(A)),A) = k8_margrel1 )
           => k7_binarith(A,B,k2_binari_2(A)) = k6_binarith(A,k2_binari_2(A)) ) ) ) )).

fof(t22_binari_3,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))
             => ( ( C = k5_euclid(A)
                  & k4_finseq_4(k5_numbers,k6_margrel1,B,A) = k8_margrel1
                  & k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,k2_binari_2(A)),A) = k8_margrel1 )
               => B = k6_binarith(A,C) ) ) ) ) )).

fof(t23_binari_3,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)
           => k7_binarith(A,k6_binarith(A,B),k2_binari_2(A)) = k6_binarith(A,k2_binari_2(A)) ) ) ) )).

fof(t24_binari_3,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))
             => ( C = k5_euclid(A)
               => ( k11_binarith(A,B,k2_binari_2(A)) = k8_margrel1
                <=> B = k6_binarith(A,C) ) ) ) ) ) )).

fof(t25_binari_3,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)
           => k10_binarith(A,k6_binarith(A,B),k2_binari_2(A)) = B ) ) ) )).

fof(d1_binari_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
             => ( C = k1_binari_3(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r2_hidden(D,k2_finseq_1(A))
                     => k4_finseq_4(k5_numbers,k6_margrel1,C,D) = k2_cqc_lang(k6_margrel1,k4_nat_1(k3_nat_1(B,k3_series_1(np__2,k5_binarith(D,np__1))),np__2),np__0,k7_margrel1,k8_margrel1) ) ) ) ) ) ) )).

fof(t26_binari_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_binari_3(A,np__0) = k5_euclid(A) ) )).

fof(t27_binari_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_series_1(np__2,A),B)
           => k1_funct_1(k1_binari_3(k1_nat_1(A,np__1),B),k1_nat_1(A,np__1)) = k7_margrel1 ) ) ) )).

fof(t28_binari_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_series_1(np__2,A),B)
           => k1_binari_3(k1_nat_1(A,np__1),B) = k12_binarith(A,np__1,k6_margrel1,k1_binari_3(A,B),k13_binarith(k6_margrel1,k7_margrel1)) ) ) ) )).

fof(t29_binari_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_binari_3(k1_nat_1(A,np__1),k3_series_1(np__2,A)) = k8_finseq_1(k1_numbers,k5_euclid(A),k13_binarith(k1_numbers,np__1)) ) )).

fof(t30_binari_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(k3_series_1(np__2,A),B)
           => ( r1_xreal_0(k3_series_1(np__2,k1_nat_1(A,np__1)),B)
              | k1_funct_1(k1_binari_3(k1_nat_1(A,np__1),B),k1_nat_1(A,np__1)) = k8_margrel1 ) ) ) ) )).

fof(t31_binari_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(k3_series_1(np__2,A),B)
           => ( r1_xreal_0(k3_series_1(np__2,k1_nat_1(A,np__1)),B)
              | k1_binari_3(k1_nat_1(A,np__1),B) = k12_binarith(A,np__1,k6_margrel1,k1_binari_3(A,k5_binarith(B,k3_series_1(np__2,A))),k13_binarith(k6_margrel1,k8_margrel1)) ) ) ) ) )).

fof(t32_binari_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_series_1(np__2,A),B)
           => ! [C] :
                ( m2_finseq_2(C,k6_margrel1,k4_finseq_2(A,k6_margrel1))
               => ( C = k5_euclid(A)
                 => ( k1_binari_3(A,B) = k6_binarith(A,C)
                  <=> B = k5_real_1(k3_series_1(np__2,A),np__1) ) ) ) ) ) ) )).

fof(t33_binari_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_series_1(np__2,A),k1_nat_1(B,np__1))
           => k11_binarith(A,k1_binari_3(A,B),k2_binari_2(A)) = k7_margrel1 ) ) ) )).

fof(t34_binari_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_series_1(np__2,A),k1_nat_1(B,np__1))
           => k1_binari_3(A,k1_nat_1(B,np__1)) = k10_binarith(A,k1_binari_3(A,B),k2_binari_2(A)) ) ) ) )).

fof(t35_binari_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k1_binari_3(k1_nat_1(A,np__1),B) = k7_finseq_1(k13_binarith(k1_numbers,k4_nat_1(B,np__2)),k1_binari_3(A,k3_nat_1(B,np__2))) ) ) )).

fof(t36_binari_3,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_subset_1(A,k1_numbers,k5_numbers) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ~ r1_xreal_0(k3_series_1(np__2,A),B)
           => k9_binarith(A,k1_binari_3(A,B)) = B ) ) ) )).

fof(t37_binari_3,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))
         => k1_binari_3(A,k9_binarith(A,B)) = B ) ) )).

fof(dt_k1_binari_3,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_finseq_2(k1_binari_3(A,B),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) )).
%------------------------------------------------------------------------------