SET007 Axioms: SET007+173.ax


%------------------------------------------------------------------------------
% File     : SET007+173 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Binary Arithmetics
% 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    : binarith [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   87 (  13 unit)
%            Number of atoms       :  313 (  69 equality)
%            Maximal formula depth :   15 (   6 average)
%            Number of connectives :  251 (  25 ~  ;   2  |;  53  &)
%                                         (   5 <=>; 166 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   12 (   1 propositional; 0-3 arity)
%            Number of functors    :   39 (   9 constant; 0-5 arity)
%            Number of variables   :  180 (   0 singleton; 180 !;   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(fc1_binarith,axiom,(
    ! [A,B] :
      ( ( v2_margrel1(A)
        & v2_margrel1(B) )
     => v2_margrel1(k1_binarith(A,B)) ) )).

fof(fc2_binarith,axiom,(
    ! [A,B] :
      ( ( v2_margrel1(A)
        & v2_margrel1(B) )
     => v2_margrel1(k2_binarith(A,B)) ) )).

fof(t1_binarith,axiom,(
    $true )).

fof(t2_binarith,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( ~ v1_xboole_0(C)
             => ! [D] :
                  ( m1_subset_1(D,C)
                 => ! [E] :
                      ( m2_finseq_2(E,C,k4_finseq_2(B,C))
                     => ( r2_hidden(A,k2_finseq_1(B))
                       => k4_finseq_4(k5_numbers,C,k8_finseq_1(C,E,k12_finseq_1(C,D)),A) = k4_finseq_4(k5_numbers,C,E,A) ) ) ) ) ) ) )).

fof(t3_binarith,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ~ v1_xboole_0(B)
         => ! [C] :
              ( m1_subset_1(C,B)
             => ! [D] :
                  ( m2_finseq_2(D,B,k4_finseq_2(A,B))
                 => k4_finseq_4(k5_numbers,B,k8_finseq_1(B,D,k12_finseq_1(B,C)),k23_binop_2(A,np__1)) = C ) ) ) ) )).

fof(t4_binarith,axiom,(
    $true )).

fof(t5_binarith,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ~ ( r2_hidden(A,k2_finseq_1(B))
              & v1_xboole_0(A) ) ) ) )).

fof(d1_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k1_binarith(A,B) = k9_margrel1(k10_margrel1(k9_margrel1(A),k9_margrel1(B))) ) ) )).

fof(d2_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k2_binarith(A,B) = k1_binarith(k10_margrel1(k9_margrel1(A),B),k10_margrel1(A,k9_margrel1(B))) ) ) )).

fof(t6_binarith,axiom,(
    $true )).

fof(t7_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k1_binarith(A,k7_margrel1) = A ) )).

fof(t8_binarith,axiom,(
    $true )).

fof(t9_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k9_margrel1(k10_margrel1(A,B)) = k1_binarith(k9_margrel1(A),k9_margrel1(B)) ) ) )).

fof(t10_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k9_margrel1(k1_binarith(A,B)) = k10_margrel1(k9_margrel1(A),k9_margrel1(B)) ) ) )).

fof(t11_binarith,axiom,(
    $true )).

fof(t12_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k10_margrel1(A,B) = k9_margrel1(k1_binarith(k9_margrel1(A),k9_margrel1(B))) ) ) )).

fof(t13_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k2_binarith(k8_margrel1,A) = k9_margrel1(A) ) )).

fof(t14_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k2_binarith(k7_margrel1,A) = A ) )).

fof(t15_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k2_binarith(A,A) = k7_margrel1 ) )).

fof(t16_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k10_margrel1(A,A) = A ) )).

fof(t17_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k2_binarith(A,k9_margrel1(A)) = k8_margrel1 ) )).

fof(t18_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k1_binarith(A,k9_margrel1(A)) = k8_margrel1 ) )).

fof(t19_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k1_binarith(A,k8_margrel1) = k8_margrel1 ) )).

fof(t20_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => ! [C] :
              ( v2_margrel1(C)
             => k1_binarith(k1_binarith(A,B),C) = k1_binarith(A,k1_binarith(B,C)) ) ) ) )).

fof(t21_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => k1_binarith(A,A) = A ) )).

fof(t22_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => ! [C] :
              ( v2_margrel1(C)
             => k10_margrel1(A,k1_binarith(B,C)) = k1_binarith(k10_margrel1(A,B),k10_margrel1(A,C)) ) ) ) )).

fof(t23_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => ! [C] :
              ( v2_margrel1(C)
             => k1_binarith(A,k10_margrel1(B,C)) = k10_margrel1(k1_binarith(A,B),k1_binarith(A,C)) ) ) ) )).

fof(t24_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k1_binarith(A,k10_margrel1(A,B)) = A ) ) )).

fof(t25_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k10_margrel1(A,k1_binarith(A,B)) = A ) ) )).

fof(t26_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k1_binarith(A,k10_margrel1(k9_margrel1(A),B)) = k1_binarith(A,B) ) ) )).

fof(t27_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k10_margrel1(A,k1_binarith(k9_margrel1(A),B)) = k10_margrel1(A,B) ) ) )).

fof(t28_binarith,axiom,(
    $true )).

fof(t29_binarith,axiom,(
    $true )).

fof(t30_binarith,axiom,(
    $true )).

fof(t31_binarith,axiom,(
    $true )).

fof(t32_binarith,axiom,(
    $true )).

fof(t33_binarith,axiom,(
    k4_binarith(k8_margrel1,k7_margrel1) = k8_margrel1 )).

fof(t34_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => ! [C] :
              ( v2_margrel1(C)
             => k2_binarith(k2_binarith(A,B),C) = k2_binarith(A,k2_binarith(B,C)) ) ) ) )).

fof(t35_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k2_binarith(A,k10_margrel1(k9_margrel1(A),B)) = k1_binarith(A,B) ) ) )).

fof(t36_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k1_binarith(A,k2_binarith(A,B)) = k1_binarith(A,B) ) ) )).

fof(t37_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => k1_binarith(A,k2_binarith(k9_margrel1(A),B)) = k1_binarith(A,k9_margrel1(B)) ) ) )).

fof(t38_binarith,axiom,(
    ! [A] :
      ( v2_margrel1(A)
     => ! [B] :
          ( v2_margrel1(B)
         => ! [C] :
              ( v2_margrel1(C)
             => k10_margrel1(A,k2_binarith(B,C)) = k2_binarith(k10_margrel1(A,B),k10_margrel1(A,C)) ) ) ) )).

fof(d3_binarith,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( ( r1_xreal_0(np__0,k6_xcmplx_0(A,B))
             => k5_binarith(A,B) = k6_xcmplx_0(A,B) )
            & ( ~ r1_xreal_0(np__0,k6_xcmplx_0(A,B))
             => k5_binarith(A,B) = np__0 ) ) ) ) )).

fof(t39_binarith,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => k5_binarith(k2_xcmplx_0(A,B),B) = A ) ) )).

fof(d4_binarith,axiom,(
    ! [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 = k6_binarith(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) = k11_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,D)) ) ) ) ) ) ) )).

fof(d5_binarith,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))
             => ! [D] :
                  ( m2_finseq_2(D,k6_margrel1,k4_finseq_2(A,k6_margrel1))
                 => ( D = k7_binarith(A,B,C)
                  <=> ( k4_finseq_4(k5_numbers,k6_margrel1,D,np__1) = k7_margrel1
                      & ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r1_xreal_0(np__1,E)
                           => ( r1_xreal_0(A,E)
                              | k4_finseq_4(k5_numbers,k6_margrel1,D,k23_binop_2(E,np__1)) = k3_binarith(k3_binarith(k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,E),k4_finseq_4(k5_numbers,k6_margrel1,C,E)),k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,E),k4_finseq_4(k5_numbers,k6_margrel1,D,E))),k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,C,E),k4_finseq_4(k5_numbers,k6_margrel1,D,E))) ) ) ) ) ) ) ) ) ) )).

fof(d6_binarith,axiom,(
    ! [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,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ( C = k8_binarith(A,B)
              <=> ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r2_hidden(D,k2_finseq_1(A))
                     => k4_finseq_4(k5_numbers,k5_numbers,C,D) = k2_cqc_lang(k5_numbers,k4_finseq_4(k5_numbers,k6_margrel1,B,D),k7_margrel1,np__0,k3_series_1(np__2,k5_binarith(D,np__1))) ) ) ) ) ) ) )).

fof(d7_binarith,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k6_margrel1,k4_finseq_2(A,k6_margrel1))
         => k9_binarith(A,B) = k2_finsop_1(k5_numbers,k8_binarith(A,B),k47_binop_2) ) ) )).

fof(d8_binarith,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))
             => ! [D] :
                  ( m2_finseq_2(D,k6_margrel1,k4_finseq_2(A,k6_margrel1))
                 => ( D = k10_binarith(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(A))
                         => k4_finseq_4(k5_numbers,k6_margrel1,D,E) = k4_binarith(k4_binarith(k4_finseq_4(k5_numbers,k6_margrel1,B,E),k4_finseq_4(k5_numbers,k6_margrel1,C,E)),k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,C),E)) ) ) ) ) ) ) ) )).

fof(d9_binarith,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))
             => k11_binarith(A,B,C) = k3_binarith(k3_binarith(k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,A),k4_finseq_4(k5_numbers,k6_margrel1,C,A)),k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,B,A),k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,C),A))),k12_margrel1(k4_finseq_4(k5_numbers,k6_margrel1,C,A),k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,C),A))) ) ) ) )).

fof(d10_binarith,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))
             => ( r1_binarith(A,B,C)
              <=> k11_binarith(A,B,C) = k7_margrel1 ) ) ) ) )).

fof(t40_binarith,axiom,(
    ! [A] :
      ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
     => ( A = k12_finseq_1(k6_margrel1,k7_margrel1)
        | A = k12_finseq_1(k6_margrel1,k8_margrel1) ) ) )).

fof(t41_binarith,axiom,(
    ! [A] :
      ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
     => ( A = k12_finseq_1(k6_margrel1,k7_margrel1)
       => k9_binarith(np__1,A) = np__0 ) ) )).

fof(t42_binarith,axiom,(
    ! [A] :
      ( m2_finseq_2(A,k6_margrel1,k4_finseq_2(np__1,k6_margrel1))
     => ( A = k12_finseq_1(k6_margrel1,k8_margrel1)
       => k9_binarith(np__1,A) = np__1 ) ) )).

fof(t43_binarith,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))
             => ! [D] :
                  ( m1_subset_1(D,k6_margrel1)
                 => ! [E] :
                      ( m1_subset_1(E,k6_margrel1)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ( r2_hidden(F,k2_finseq_1(A))
                           => k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(k23_binop_2(A,np__1),k12_binarith(A,np__1,k6_margrel1,B,k13_binarith(k6_margrel1,D)),k12_binarith(A,np__1,k6_margrel1,C,k13_binarith(k6_margrel1,E))),F) = k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(A,B,C),F) ) ) ) ) ) ) ) )).

fof(t44_binarith,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))
             => ! [D] :
                  ( m1_subset_1(D,k6_margrel1)
                 => ! [E] :
                      ( m1_subset_1(E,k6_margrel1)
                     => k11_binarith(A,B,C) = k4_finseq_4(k5_numbers,k6_margrel1,k7_binarith(k23_binop_2(A,np__1),k12_binarith(A,np__1,k6_margrel1,B,k13_binarith(k6_margrel1,D)),k12_binarith(A,np__1,k6_margrel1,C,k13_binarith(k6_margrel1,E))),k23_binop_2(A,np__1)) ) ) ) ) ) )).

fof(t45_binarith,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))
             => ! [D] :
                  ( m1_subset_1(D,k6_margrel1)
                 => ! [E] :
                      ( m1_subset_1(E,k6_margrel1)
                     => k10_binarith(k23_binop_2(A,np__1),k12_binarith(A,np__1,k6_margrel1,B,k13_binarith(k6_margrel1,D)),k12_binarith(A,np__1,k6_margrel1,C,k13_binarith(k6_margrel1,E))) = k12_binarith(A,np__1,k6_margrel1,k10_binarith(A,B,C),k13_binarith(k6_margrel1,k4_binarith(k4_binarith(D,E),k11_binarith(A,B,C)))) ) ) ) ) ) )).

fof(t46_binarith,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] :
              ( m1_subset_1(C,k6_margrel1)
             => k9_binarith(k23_binop_2(A,np__1),k12_binarith(A,np__1,k6_margrel1,B,k13_binarith(k6_margrel1,C))) = k23_binop_2(k9_binarith(A,B),k2_cqc_lang(k5_numbers,C,k7_margrel1,np__0,k3_series_1(np__2,A))) ) ) ) )).

fof(t47_binarith,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))
             => k23_binop_2(k9_binarith(A,k10_binarith(A,B,C)),k2_cqc_lang(k5_numbers,k11_binarith(A,B,C),k7_margrel1,np__0,k3_series_1(np__2,A))) = k23_binop_2(k9_binarith(A,B),k9_binarith(A,C)) ) ) ) )).

fof(t48_binarith,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))
             => ( r1_binarith(A,B,C)
               => k9_binarith(A,k10_binarith(A,B,C)) = k23_binop_2(k9_binarith(A,B),k9_binarith(A,C)) ) ) ) ) )).

fof(t49_binarith,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ! [C] :
              ( v4_ordinal2(C)
             => ( ( r1_xreal_0(A,B)
                  & r1_xreal_0(A,C)
                  & k5_binarith(B,A) = k5_binarith(C,A) )
               => B = C ) ) ) ) )).

fof(t50_binarith,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(A,B)
           => k5_binarith(B,A) = k6_xcmplx_0(B,A) ) ) ) )).

fof(t51_binarith,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => k5_binarith(A,A) = np__0 ) )).

fof(t52_binarith,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => r1_xreal_0(k5_binarith(A,B),A) ) ) )).

fof(t53_binarith,axiom,(
    ! [A] :
      ( v4_ordinal2(A)
     => ! [B] :
          ( v4_ordinal2(B)
         => ( r1_xreal_0(B,A)
           => k2_xcmplx_0(k5_binarith(A,B),B) = A ) ) ) )).

fof(s1_binarith,axiom,
    ( ( p1_s1_binarith(np__1)
      & ! [A] :
          ( ( ~ v1_xboole_0(A)
            & m2_subset_1(A,k1_numbers,k5_numbers) )
         => ( p1_s1_binarith(A)
           => p1_s1_binarith(k23_binop_2(A,np__1)) ) ) )
   => ! [A] :
        ( ( ~ v1_xboole_0(A)
          & m2_subset_1(A,k1_numbers,k5_numbers) )
       => p1_s1_binarith(A) ) )).

fof(dt_k1_binarith,axiom,(
    $true )).

fof(commutativity_k1_binarith,axiom,(
    ! [A,B] :
      ( ( v2_margrel1(A)
        & v2_margrel1(B) )
     => k1_binarith(A,B) = k1_binarith(B,A) ) )).

fof(dt_k2_binarith,axiom,(
    $true )).

fof(commutativity_k2_binarith,axiom,(
    ! [A,B] :
      ( ( v2_margrel1(A)
        & v2_margrel1(B) )
     => k2_binarith(A,B) = k2_binarith(B,A) ) )).

fof(dt_k3_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k6_margrel1)
        & m1_subset_1(B,k6_margrel1) )
     => m1_subset_1(k3_binarith(A,B),k6_margrel1) ) )).

fof(commutativity_k3_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k6_margrel1)
        & m1_subset_1(B,k6_margrel1) )
     => k3_binarith(A,B) = k3_binarith(B,A) ) )).

fof(redefinition_k3_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k6_margrel1)
        & m1_subset_1(B,k6_margrel1) )
     => k3_binarith(A,B) = k1_binarith(A,B) ) )).

fof(dt_k4_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k6_margrel1)
        & m1_subset_1(B,k6_margrel1) )
     => m1_subset_1(k4_binarith(A,B),k6_margrel1) ) )).

fof(commutativity_k4_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k6_margrel1)
        & m1_subset_1(B,k6_margrel1) )
     => k4_binarith(A,B) = k4_binarith(B,A) ) )).

fof(redefinition_k4_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k6_margrel1)
        & m1_subset_1(B,k6_margrel1) )
     => k4_binarith(A,B) = k2_binarith(A,B) ) )).

fof(dt_k5_binarith,axiom,(
    ! [A,B] :
      ( ( v4_ordinal2(A)
        & v4_ordinal2(B) )
     => m2_subset_1(k5_binarith(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k6_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k6_margrel1)) )
     => m2_finseq_2(k6_binarith(A,B),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) )).

fof(dt_k7_binarith,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k6_margrel1))
        & m1_subset_1(C,k4_finseq_2(A,k6_margrel1)) )
     => m2_finseq_2(k7_binarith(A,B,C),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) )).

fof(dt_k8_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k6_margrel1)) )
     => m2_finseq_2(k8_binarith(A,B),k5_numbers,k4_finseq_2(A,k5_numbers)) ) )).

fof(dt_k9_binarith,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k6_margrel1)) )
     => m2_subset_1(k9_binarith(A,B),k1_numbers,k5_numbers) ) )).

fof(dt_k10_binarith,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k6_margrel1))
        & m1_subset_1(C,k4_finseq_2(A,k6_margrel1)) )
     => m2_finseq_2(k10_binarith(A,B,C),k6_margrel1,k4_finseq_2(A,k6_margrel1)) ) )).

fof(dt_k11_binarith,axiom,(
    ! [A,B,C] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k4_finseq_2(A,k6_margrel1))
        & m1_subset_1(C,k4_finseq_2(A,k6_margrel1)) )
     => m1_subset_1(k11_binarith(A,B,C),k6_margrel1) ) )).

fof(dt_k12_binarith,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k4_finseq_2(A,C))
        & m1_subset_1(E,k4_finseq_2(B,C)) )
     => m2_finseq_2(k12_binarith(A,B,C,D,E),C,k4_finseq_2(k23_binop_2(A,B),C)) ) )).

fof(redefinition_k12_binarith,axiom,(
    ! [A,B,C,D,E] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & ~ v1_xboole_0(C)
        & m1_subset_1(D,k4_finseq_2(A,C))
        & m1_subset_1(E,k4_finseq_2(B,C)) )
     => k12_binarith(A,B,C,D,E) = k7_finseq_1(D,E) ) )).

fof(dt_k13_binarith,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => m2_finseq_2(k13_binarith(A,B),A,k4_finseq_2(np__1,A)) ) )).

fof(redefinition_k13_binarith,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(A)
        & m1_subset_1(B,A) )
     => k13_binarith(A,B) = k5_finseq_1(B) ) )).
%------------------------------------------------------------------------------