SET007 Axioms: SET007+755.ax


%------------------------------------------------------------------------------
% File     : SET007+755 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : High Speed Adder Algorithm with Radix-2^k Sub Signed-Digit Number
% 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    : radix_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   13 (   0 unit)
%            Number of atoms       :   82 (  11 equality)
%            Maximal formula depth :   15 (  10 average)
%            Number of connectives :   70 (   1 ~  ;   0  |;  18  &)
%                                         (   1 <=>;  50 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   0 propositional; 1-3 arity)
%            Number of functors    :   27 (   7 constant; 0-5 arity)
%            Number of variables   :   46 (   0 singleton;  46 !;   0 ?)
%            Maximal term depth    :    8 (   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_radix_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ ( r1_xreal_0(np__2,A)
          & r1_xreal_0(k1_radix_1(A),np__2) ) ) )).

fof(t2_radix_4,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( r1_xreal_0(np__3,C)
               => k5_radix_3(k2_xcmplx_0(k5_radix_3(A,C),k5_radix_3(B,C)),C) = np__0 ) ) ) ) )).

fof(t3_radix_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(np__2,A)
               => k7_radix_3(k1_nat_1(C,np__1),A,k1_nat_1(C,np__1),k10_radix_3(C,A,k10_radix_1(A,C,B))) = k5_radix_3(k4_radix_1(C,A,C,k10_radix_1(A,C,B)),A) ) ) ) ) )).

fof(t4_radix_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__2,A)
              & r1_radix_1(np__1,B,A) )
           => k7_radix_3(k1_nat_1(np__1,np__1),A,k1_nat_1(np__1,np__1),k10_radix_3(np__1,A,k10_radix_1(A,np__1,B))) = k5_radix_3(B,A) ) ) ) )).

fof(t5_radix_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(np__1,C)
                  & r1_xreal_0(np__3,A)
                  & r1_radix_1(k1_nat_1(C,np__1),B,A) )
               => k7_radix_3(k1_nat_1(C,np__1),A,k1_nat_1(C,np__1),k10_radix_3(C,A,k10_radix_1(A,C,k4_nat_1(B,k2_wsierp_1(k1_radix_1(A),C))))) = k5_radix_3(k4_radix_1(C,A,C,k10_radix_1(A,C,B)),A) ) ) ) ) )).

fof(t6_radix_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__2,A)
              & r1_radix_1(np__1,B,A) )
           => k7_radix_3(np__1,A,k1_nat_1(np__1,np__1),k10_radix_3(np__1,A,k10_radix_1(A,np__1,B))) = k6_xcmplx_0(B,k3_xcmplx_0(k5_radix_3(B,A),k1_radix_1(A))) ) ) ) )).

fof(t7_radix_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(np__1,C)
                  & r1_xreal_0(np__2,A)
                  & r1_radix_1(k1_nat_1(C,np__1),B,A) )
               => k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),C),k7_radix_3(k1_nat_1(C,np__1),A,k1_nat_1(k1_nat_1(C,np__1),np__1),k10_radix_3(k1_nat_1(C,np__1),A,k10_radix_1(A,k1_nat_1(C,np__1),B)))) = k2_xcmplx_0(k6_xcmplx_0(k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),C),k4_radix_1(k1_nat_1(C,np__1),A,k1_nat_1(C,np__1),k10_radix_1(A,k1_nat_1(C,np__1),B))),k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),k1_nat_1(C,np__1)),k5_radix_3(k4_radix_1(k1_nat_1(C,np__1),A,k1_nat_1(C,np__1),k10_radix_1(A,k1_nat_1(C,np__1),B)),A))),k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),C),k5_radix_3(k4_radix_1(C,A,k1_nat_1(C,np__1),k10_radix_1(A,k1_nat_1(C,np__1),B)),A))) ) ) ) ) )).

fof(d1_radix_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)
             => ! [D] :
                  ( m2_finseq_2(D,k4_radix_3(C),k4_finseq_2(B,k4_radix_3(C)))
                 => ! [E] :
                      ( m2_finseq_2(E,k4_radix_3(C),k4_finseq_2(B,k4_radix_3(C)))
                     => ( ( r2_hidden(A,k2_finseq_1(B))
                          & r1_xreal_0(np__2,C) )
                       => k1_radix_4(A,B,C,D,E) = k2_xcmplx_0(k6_radix_3(k2_xcmplx_0(k7_radix_3(A,C,B,D),k7_radix_3(A,C,B,E)),C),k5_radix_3(k2_xcmplx_0(k7_radix_3(k5_binarith(A,np__1),C,B,D),k7_radix_3(k5_binarith(A,np__1),C,B,E)),C)) ) ) ) ) ) ) )).

fof(d2_radix_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
             => ! [D] :
                  ( m2_finseq_2(D,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
                 => ! [E] :
                      ( m2_finseq_2(E,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
                     => ( E = k2_radix_4(A,B,C,D)
                      <=> ! [F] :
                            ( m2_subset_1(F,k1_numbers,k5_numbers)
                           => ( r2_hidden(F,k2_finseq_1(A))
                             => k7_radix_3(F,B,A,E) = k1_radix_4(F,A,B,C,D) ) ) ) ) ) ) ) ) )).

fof(t8_radix_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)
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ( ( r2_hidden(E,k2_finseq_1(A))
                          & r1_xreal_0(np__2,B) )
                       => k1_radix_4(E,k1_nat_1(k1_nat_1(A,np__1),np__1),B,k10_radix_3(k1_nat_1(A,np__1),B,k10_radix_1(B,k1_nat_1(A,np__1),C)),k10_radix_3(k1_nat_1(A,np__1),B,k10_radix_1(B,k1_nat_1(A,np__1),D))) = k1_radix_4(E,k1_nat_1(A,np__1),B,k10_radix_3(A,B,k10_radix_1(B,A,k4_nat_1(C,k2_wsierp_1(k1_radix_1(B),A)))),k10_radix_3(A,B,k10_radix_1(B,A,k4_nat_1(D,k2_wsierp_1(k1_radix_1(B),A))))) ) ) ) ) ) ) )).

fof(t9_radix_4,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( m2_subset_1(B,k1_numbers,k5_numbers)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( ( r1_xreal_0(np__3,B)
                        & r1_radix_1(A,C,B)
                        & r1_radix_1(A,D,B) )
                     => k1_nat_1(C,D) = k14_radix_3(k1_nat_1(A,np__1),B,k2_radix_4(k1_nat_1(A,np__1),B,k10_radix_3(A,B,k10_radix_1(B,A,C)),k10_radix_3(A,B,k10_radix_1(B,A,D)))) ) ) ) ) ) ) )).

fof(dt_k1_radix_4,axiom,(
    ! [A,B,C,D,E] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k4_finseq_2(B,k4_radix_3(C)))
        & m1_subset_1(E,k4_finseq_2(B,k4_radix_3(C))) )
     => m2_subset_1(k1_radix_4(A,B,C,D,E),k6_wsierp_1,k4_radix_3(C)) ) )).

fof(dt_k2_radix_4,axiom,(
    ! [A,B,C,D] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k4_finseq_2(A,k4_radix_3(B)))
        & m1_subset_1(D,k4_finseq_2(A,k4_radix_3(B))) )
     => m2_finseq_2(k2_radix_4(A,B,C,D),k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B))) ) )).
%------------------------------------------------------------------------------