SET007 Axioms: SET007+754.ax


%------------------------------------------------------------------------------
% File     : SET007+754 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Improvement of Radix-2^k Signed-Digit Number for High Speed Circuit
% 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_3 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   54 (   4 unit)
%            Number of atoms       :  233 (  29 equality)
%            Maximal formula depth :   15 (   7 average)
%            Number of connectives :  188 (   9 ~  ;   1  |;  50  &)
%                                         (   4 <=>; 124 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   1 propositional; 0-3 arity)
%            Number of functors    :   40 (   6 constant; 0-4 arity)
%            Number of variables   :  127 (   0 singleton; 125 !;   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(fc1_radix_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ~ v1_xboole_0(k1_radix_3(A)) ) )).

fof(fc2_radix_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ~ v1_xboole_0(k2_radix_3(A)) ) )).

fof(t1_radix_3,axiom,(
    $true )).

fof(t2_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ( r2_hidden(B,k2_radix_3(A))
           => ( r1_xreal_0(k6_xcmplx_0(k4_xcmplx_0(k1_radix_1(k5_binarith(A,np__1))),np__1),B)
              & r1_xreal_0(B,k1_radix_1(k5_binarith(A,np__1))) ) ) ) ) )).

fof(t3_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k1_radix_3(A),k2_radix_3(A)) ) )).

fof(t4_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k1_radix_3(A),k1_radix_3(k1_nat_1(A,np__1))) ) )).

fof(t5_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__2,A)
       => r1_tarski(k2_radix_3(A),k3_radix_1(A)) ) ) )).

fof(t6_radix_3,axiom,(
    r2_hidden(np__0,k1_radix_3(np__0)) )).

fof(t7_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r2_hidden(np__0,k1_radix_3(A)) ) )).

fof(t8_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r2_hidden(np__0,k2_radix_3(A)) ) )).

fof(t9_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( r2_hidden(B,k2_radix_3(A))
         => v1_int_1(B) ) ) )).

fof(t10_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k2_radix_3(A),k6_wsierp_1) ) )).

fof(t11_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => r1_tarski(k1_radix_3(A),k6_wsierp_1) ) )).

fof(t12_radix_3,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)))
                 => ( r2_hidden(A,k2_finseq_1(B))
                   => m2_subset_1(k1_funct_1(D,A),k6_wsierp_1,k4_radix_3(C)) ) ) ) ) ) )).

fof(d3_radix_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(k1_radix_1(k5_binarith(B,np__1)),A)
             => k5_radix_3(A,B) = np__1 )
            & ( ~ r1_xreal_0(k4_xcmplx_0(k1_radix_1(k5_binarith(B,np__1))),A)
             => k5_radix_3(A,B) = k4_xcmplx_0(np__1) )
            & ( r1_xreal_0(k4_xcmplx_0(k1_radix_1(k5_binarith(B,np__1))),A)
             => ( r1_xreal_0(k1_radix_1(k5_binarith(B,np__1)),A)
                | k5_radix_3(A,B) = np__0 ) ) ) ) ) )).

fof(d4_radix_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => k6_radix_3(A,B) = k6_xcmplx_0(A,k3_xcmplx_0(k1_radix_1(B),k5_radix_3(A,B))) ) ) )).

fof(t13_radix_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__2,B)
           => ( r1_xreal_0(k4_xcmplx_0(np__1),k5_radix_3(A,B))
              & r1_xreal_0(k5_radix_3(A,B),np__1) ) ) ) ) )).

fof(t14_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ( ( r1_xreal_0(np__2,A)
              & r2_hidden(B,k3_radix_1(A)) )
           => ( r1_xreal_0(k4_xcmplx_0(k1_radix_1(k5_binarith(A,np__1))),k6_radix_3(B,A))
              & r1_xreal_0(k6_radix_3(B,A),k6_xcmplx_0(k1_radix_1(k5_binarith(A,np__1)),np__1)) ) ) ) ) )).

fof(t15_radix_3,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__2,B)
           => r2_hidden(k5_radix_3(A,B),k3_radix_3(B)) ) ) ) )).

fof(t16_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r1_xreal_0(np__2,A)
                  & r2_hidden(B,k3_radix_1(A))
                  & r2_hidden(C,k3_radix_1(A)) )
               => r2_hidden(k2_xcmplx_0(k6_radix_3(B,A),k5_radix_3(C,A)),k4_radix_3(A)) ) ) ) ) )).

fof(t17_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__2,A)
       => k5_radix_3(np__0,A) = np__0 ) ) )).

fof(d5_radix_3,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(B),k4_finseq_2(C,k4_radix_3(B)))
                 => ( ( r2_hidden(A,k2_finseq_1(C))
                     => k7_radix_3(A,B,C,D) = k1_funct_1(D,A) )
                    & ( A = np__0
                     => k7_radix_3(A,B,C,D) = np__0 ) ) ) ) ) ) )).

fof(d6_radix_3,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,k3_radix_1(B),k4_finseq_2(C,k3_radix_1(B)))
                 => ( ( r2_hidden(A,k2_finseq_1(C))
                     => k8_radix_3(A,B,C,D) = k2_xcmplx_0(k6_radix_3(k4_radix_1(A,B,C,D),B),k5_radix_3(k4_radix_1(k5_binarith(A,np__1),B,C,D),B)) )
                    & ( A = k1_nat_1(C,np__1)
                     => k8_radix_3(A,B,C,D) = k5_radix_3(k4_radix_1(k5_binarith(A,np__1),B,C,D),B) )
                    & ~ ( ~ r2_hidden(A,k2_finseq_1(C))
                        & A != k1_nat_1(C,np__1)
                        & k8_radix_3(A,B,C,D) != np__0 ) ) ) ) ) ) )).

fof(t18_radix_3,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,k3_radix_1(A),k4_finseq_2(C,k3_radix_1(A)))
                 => ( ( r1_xreal_0(np__2,A)
                      & r2_hidden(B,k2_finseq_1(k1_nat_1(C,np__1))) )
                   => m2_subset_1(k8_radix_3(B,A,C,D),k6_wsierp_1,k4_radix_3(A)) ) ) ) ) ) )).

fof(d7_radix_3,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,k3_radix_1(B),k4_finseq_2(C,k3_radix_1(B)))
                 => ( ( r1_xreal_0(np__2,B)
                      & r2_hidden(A,k2_finseq_1(k1_nat_1(C,np__1))) )
                   => k9_radix_3(A,B,C,D) = k8_radix_3(A,B,C,D) ) ) ) ) ) )).

fof(d8_radix_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,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
             => ! [D] :
                  ( m2_finseq_2(D,k4_radix_3(B),k4_finseq_2(k1_nat_1(A,np__1),k4_radix_3(B)))
                 => ( D = k10_radix_3(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(k1_nat_1(A,np__1)))
                         => k7_radix_3(E,B,k1_nat_1(A,np__1),D) = k9_radix_3(E,B,A,C) ) ) ) ) ) ) ) )).

fof(t19_radix_3,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)))
                 => ( r2_hidden(A,k2_finseq_1(B))
                   => m2_subset_1(k7_radix_3(A,C,B,D),k6_wsierp_1,k4_radix_3(C)) ) ) ) ) ) )).

fof(t20_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( v1_int_1(C)
             => ( ( r1_xreal_0(np__2,A)
                  & r2_hidden(B,k3_radix_1(A))
                  & r2_hidden(C,k4_radix_3(A)) )
               => r2_hidden(k6_radix_3(k2_xcmplx_0(B,C),A),k3_radix_3(A)) ) ) ) ) )).

fof(d9_radix_3,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(B),k4_finseq_2(C,k4_radix_3(B)))
                 => k11_radix_3(A,B,C,D) = k7_radix_3(A,B,C,D) ) ) ) ) )).

fof(d10_radix_3,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(B),k4_finseq_2(C,k4_radix_3(B)))
                 => k12_radix_3(A,B,C,D) = k3_xcmplx_0(k2_wsierp_1(k1_radix_1(B),k5_binarith(A,np__1)),k11_radix_3(A,B,C,D)) ) ) ) ) )).

fof(d11_radix_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,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
             => ! [D] :
                  ( m2_finseq_2(D,k6_wsierp_1,k4_finseq_2(A,k6_wsierp_1))
                 => ( D = k13_radix_3(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_wsierp_1,D,E) = k12_radix_3(E,B,A,C) ) ) ) ) ) ) ) )).

fof(d12_radix_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,k4_radix_3(B),k4_finseq_2(A,k4_radix_3(B)))
             => k14_radix_3(A,B,C) = k7_wsierp_1(k13_radix_3(A,B,C)) ) ) ) )).

fof(t21_radix_3,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)
                 => ( ( r2_hidden(D,k2_finseq_1(A))
                      & r1_xreal_0(np__2,B) )
                   => k7_radix_3(D,B,k1_nat_1(k1_nat_1(A,np__1),np__1),k10_radix_3(k1_nat_1(A,np__1),B,k10_radix_1(B,k1_nat_1(A,np__1),C))) = k7_radix_3(D,B,k1_nat_1(A,np__1),k10_radix_3(A,B,k10_radix_1(B,A,k4_nat_1(C,k2_wsierp_1(k1_radix_1(B),A))))) ) ) ) ) ) )).

fof(t22_radix_3,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)
               => ( ( r1_xreal_0(np__2,B)
                    & r1_radix_1(A,C,B) )
                 => C = k14_radix_3(k1_nat_1(A,np__1),B,k10_radix_3(A,B,k10_radix_1(B,A,C))) ) ) ) ) ) )).

fof(dt_k1_radix_3,axiom,(
    $true )).

fof(dt_k2_radix_3,axiom,(
    $true )).

fof(dt_k3_radix_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v1_xboole_0(k3_radix_3(A))
        & m1_subset_1(k3_radix_3(A),k1_zfmisc_1(k6_wsierp_1)) ) ) )).

fof(redefinition_k3_radix_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => k3_radix_3(A) = k1_radix_3(A) ) )).

fof(dt_k4_radix_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => ( ~ v1_xboole_0(k4_radix_3(A))
        & m1_subset_1(k4_radix_3(A),k1_zfmisc_1(k6_wsierp_1)) ) ) )).

fof(redefinition_k4_radix_3,axiom,(
    ! [A] :
      ( m1_subset_1(A,k5_numbers)
     => k4_radix_3(A) = k2_radix_3(A) ) )).

fof(dt_k5_radix_3,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & m1_subset_1(B,k5_numbers) )
     => v1_int_1(k5_radix_3(A,B)) ) )).

fof(dt_k6_radix_3,axiom,(
    ! [A,B] :
      ( ( v1_int_1(A)
        & m1_subset_1(B,k5_numbers) )
     => v1_int_1(k6_radix_3(A,B)) ) )).

fof(dt_k7_radix_3,axiom,(
    ! [A,B,C,D] :
      ( ( 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(C,k4_radix_3(B))) )
     => v1_int_1(k7_radix_3(A,B,C,D)) ) )).

fof(dt_k8_radix_3,axiom,(
    ! [A,B,C,D] :
      ( ( 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(C,k3_radix_1(B))) )
     => v1_int_1(k8_radix_3(A,B,C,D)) ) )).

fof(dt_k9_radix_3,axiom,(
    ! [A,B,C,D] :
      ( ( 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(C,k3_radix_1(B))) )
     => m2_subset_1(k9_radix_3(A,B,C,D),k6_wsierp_1,k4_radix_3(B)) ) )).

fof(dt_k10_radix_3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k4_finseq_2(A,k3_radix_1(B))) )
     => m2_finseq_2(k10_radix_3(A,B,C),k4_radix_3(B),k4_finseq_2(k1_nat_1(A,np__1),k4_radix_3(B))) ) )).

fof(dt_k11_radix_3,axiom,(
    ! [A,B,C,D] :
      ( ( 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(C,k4_radix_3(B))) )
     => m2_subset_1(k11_radix_3(A,B,C,D),k1_numbers,k6_wsierp_1) ) )).

fof(dt_k12_radix_3,axiom,(
    ! [A,B,C,D] :
      ( ( 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(C,k4_radix_3(B))) )
     => m2_subset_1(k12_radix_3(A,B,C,D),k1_numbers,k6_wsierp_1) ) )).

fof(dt_k13_radix_3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k4_finseq_2(A,k4_radix_3(B))) )
     => m2_finseq_2(k13_radix_3(A,B,C),k6_wsierp_1,k4_finseq_2(A,k6_wsierp_1)) ) )).

fof(dt_k14_radix_3,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k4_finseq_2(A,k4_radix_3(B))) )
     => v1_int_1(k14_radix_3(A,B,C)) ) )).

fof(d1_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k1_radix_3(A) = a_1_0_radix_3(A) ) )).

fof(d2_radix_3,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k2_radix_3(A) = a_1_1_radix_3(A) ) )).

fof(fraenkel_a_1_0_radix_3,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_0_radix_3(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k6_wsierp_1)
            & A = C
            & r1_xreal_0(k4_xcmplx_0(k1_radix_1(k5_binarith(B,np__1))),C)
            & r1_xreal_0(C,k6_xcmplx_0(k1_radix_1(k5_binarith(B,np__1)),np__1)) ) ) ) )).

fof(fraenkel_a_1_1_radix_3,axiom,(
    ! [A,B] :
      ( m2_subset_1(B,k1_numbers,k5_numbers)
     => ( r2_hidden(A,a_1_1_radix_3(B))
      <=> ? [C] :
            ( m2_subset_1(C,k1_numbers,k6_wsierp_1)
            & A = C
            & r1_xreal_0(k6_xcmplx_0(k4_xcmplx_0(k1_radix_1(k5_binarith(B,np__1))),np__1),C)
            & r1_xreal_0(C,k1_radix_1(k5_binarith(B,np__1))) ) ) ) )).
%------------------------------------------------------------------------------