SET007 Axioms: SET007+643.ax


%------------------------------------------------------------------------------
% File     : SET007+643 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : High-Speed Algorithms for RSA Cryptograms
% 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_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   0 unit)
%            Number of atoms       :  219 (  40 equality)
%            Maximal formula depth :   28 (  11 average)
%            Number of connectives :  191 (   7 ~  ;   3  |;  38  &)
%                                         (   4 <=>; 139 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    8 (   0 propositional; 1-3 arity)
%            Number of functors    :   44 (   6 constant; 0-6 arity)
%            Number of variables   :  143 (   0 singleton; 143 !;   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_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => k4_nat_1(A,np__1) = np__0 ) )).

fof(t2_radix_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ~ r1_xreal_0(C,np__0)
               => k6_int_1(k2_xcmplx_0(k6_int_1(A,C),k6_int_1(B,C)),C) = k6_int_1(k2_xcmplx_0(A,k6_int_1(B,C)),C) ) ) ) ) )).

fof(t3_radix_2,axiom,(
    ! [A] :
      ( v1_int_1(A)
     => ! [B] :
          ( v1_int_1(B)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ~ r1_xreal_0(C,np__0)
               => k6_int_1(k3_xcmplx_0(A,B),C) = k6_int_1(k3_xcmplx_0(A,k6_int_1(B,C)),C) ) ) ) ) )).

fof(t4_radix_2,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(B,np__0)
                  | k3_nat_1(k4_nat_1(A,k2_wsierp_1(B,C)),k2_wsierp_1(B,k5_binarith(C,np__1))) = k4_nat_1(k3_nat_1(A,k2_wsierp_1(B,k5_binarith(C,np__1))),B) ) ) ) ) ) )).

fof(t5_radix_2,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))
           => r2_hidden(k1_nat_1(A,np__1),k2_finseq_1(k1_nat_1(B,np__1))) ) ) ) )).

fof(t6_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ~ r1_xreal_0(k1_radix_1(A),np__0) ) )).

fof(t7_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_finseq_2(B,k3_radix_1(A),k4_finseq_2(np__1,k3_radix_1(A)))
         => k8_radix_1(np__1,A,B) = k4_radix_1(np__1,A,np__1,B) ) ) )).

fof(t8_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( v1_int_1(B)
         => k2_xcmplx_0(k12_radix_1(B,A),k3_xcmplx_0(k11_radix_1(B),k1_radix_1(A))) = B ) ) )).

fof(t9_radix_2,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(A),k4_finseq_2(k1_nat_1(B,np__1),k3_radix_1(A)))
             => ! [D] :
                  ( m2_finseq_2(D,k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A)))
                 => ( ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(B))
                         => k1_funct_1(C,E) = k1_funct_1(D,E) ) )
                   => k7_wsierp_1(k7_radix_1(k1_nat_1(B,np__1),A,C)) = k7_wsierp_1(k4_wsierp_1(k1_numbers,k6_wsierp_1,k7_radix_1(B,A,D),k13_binarith(k6_wsierp_1,k6_radix_1(k1_nat_1(B,np__1),A,k1_nat_1(B,np__1),C)))) ) ) ) ) ) )).

fof(t10_radix_2,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(A),k4_finseq_2(k1_nat_1(B,np__1),k3_radix_1(A)))
             => ! [D] :
                  ( m2_finseq_2(D,k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A)))
                 => ( ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(B))
                         => k1_funct_1(C,E) = k1_funct_1(D,E) ) )
                   => k2_xcmplx_0(k8_radix_1(B,A,D),k3_xcmplx_0(k2_wsierp_1(k1_radix_1(A),B),k4_radix_1(k1_nat_1(B,np__1),A,k1_nat_1(B,np__1),C))) = k8_radix_1(k1_nat_1(B,np__1),A,C) ) ) ) ) ) )).

fof(t11_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,B)
           => ! [C] :
                ( m2_finseq_2(C,k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A)))
               => ! [D] :
                    ( m2_finseq_2(D,k3_radix_1(A),k4_finseq_2(B,k3_radix_1(A)))
                   => ( r1_xreal_0(np__2,A)
                     => k2_xcmplx_0(k8_radix_1(B,A,k14_radix_1(B,A,C,D)),k3_xcmplx_0(k11_radix_1(k2_xcmplx_0(k4_radix_1(B,A,B,C),k4_radix_1(B,A,B,D))),k2_wsierp_1(k1_radix_1(A),B))) = k2_xcmplx_0(k8_radix_1(B,A,C),k8_radix_1(B,A,D)) ) ) ) ) ) ) )).

fof(d1_radix_2,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,k5_numbers,k4_finseq_2(C,k5_numbers))
                 => k1_radix_2(A,B,C,D) = k2_nat_1(k2_wsierp_1(k1_radix_1(B),k5_binarith(A,np__1)),k3_wsierp_1(D,A)) ) ) ) ) )).

fof(d2_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k5_numbers,k4_finseq_2(A,k5_numbers))
                 => ( D = k2_radix_2(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(A))
                         => k4_finseq_4(k5_numbers,k5_numbers,D,E) = k1_radix_2(E,B,A,C) ) ) ) ) ) ) ) )).

fof(d3_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => k3_radix_2(A,B,C) = k9_wsierp_1(k2_radix_2(A,B,C)) ) ) ) )).

fof(d4_radix_2,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)
             => k4_radix_2(A,B,C) = k3_nat_1(k4_nat_1(C,k2_wsierp_1(k1_radix_1(B),A)),k2_wsierp_1(k1_radix_1(B),k5_binarith(A,np__1))) ) ) ) )).

fof(d5_radix_2,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,k5_numbers,k4_finseq_2(B,k5_numbers))
                 => ( D = k5_radix_2(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(B))
                         => k3_wsierp_1(D,E) = k4_radix_2(E,A,C) ) ) ) ) ) ) ) )).

fof(t12_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                 => ( C = D
                   => k2_radix_2(A,B,C) = k7_radix_1(A,B,D) ) ) ) ) ) )).

fof(t13_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_2(C,k5_numbers,k4_finseq_2(A,k5_numbers))
             => ! [D] :
                  ( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                 => ( C = D
                   => k3_radix_2(A,B,C) = k8_radix_1(A,B,D) ) ) ) ) ) )).

fof(t14_radix_2,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)
             => k5_radix_2(C,B,A) = k10_radix_1(C,B,A) ) ) ) )).

fof(t15_radix_2,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_radix_1(A,B,C)
                 => B = k3_radix_2(A,C,k5_radix_2(C,A,B)) ) ) ) ) ) )).

fof(d6_radix_2,axiom,(
    ! [A] :
      ( v1_int_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)
                 => ! [E] :
                      ( m2_subset_1(E,k1_numbers,k5_numbers)
                     => ! [F] :
                          ( m2_finseq_2(F,k3_radix_1(D),k4_finseq_2(E,k3_radix_1(D)))
                         => k6_radix_2(A,B,C,D,E,F) = k6_int_1(k3_xcmplx_0(A,k4_radix_1(C,D,E,F)),B) ) ) ) ) ) ) )).

fof(d7_radix_2,axiom,(
    ! [A] :
      ( v1_int_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)
                 => ! [E] :
                      ( m2_finseq_2(E,k3_radix_1(B),k4_finseq_2(D,k3_radix_1(B)))
                     => ( r1_xreal_0(np__1,D)
                       => ! [F] :
                            ( m2_finseq_2(F,k6_wsierp_1,k4_finseq_2(D,k6_wsierp_1))
                           => ( F = k7_radix_2(A,B,C,D,E)
                            <=> ( k1_funct_1(F,np__1) = k6_radix_2(A,C,D,B,D,E)
                                & ! [G] :
                                    ( m2_subset_1(G,k1_numbers,k5_numbers)
                                   => ~ ( r1_xreal_0(np__1,G)
                                        & r1_xreal_0(G,k6_xcmplx_0(D,np__1))
                                        & ! [H] :
                                            ( v1_int_1(H)
                                           => ! [I] :
                                                ( v1_int_1(I)
                                               => ~ ( H = k1_funct_1(F,G)
                                                    & I = k1_funct_1(F,k1_nat_1(G,np__1))
                                                    & I = k6_int_1(k2_xcmplx_0(k3_xcmplx_0(k1_radix_1(B),H),k6_radix_2(A,C,k5_binarith(D,G),B,D,E)),C) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t16_radix_2,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__1,A)
       => ! [B] :
            ( v1_int_1(B)
           => ! [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)
                       => ( r1_radix_1(A,C,E)
                         => ( r1_xreal_0(D,np__0)
                            | ! [F] :
                                ( m2_finseq_2(F,k3_radix_1(E),k4_finseq_2(A,k3_radix_1(E)))
                               => ( F = k10_radix_1(E,A,C)
                                 => k1_funct_1(k7_radix_2(B,E,D,A,F),A) = k6_int_1(k3_xcmplx_0(B,C),D) ) ) ) ) ) ) ) ) ) ) )).

fof(d8_radix_2,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_finseq_2(E,k5_numbers,k4_finseq_2(A,k5_numbers))
                     => k8_radix_2(A,B,C,D,E) = k4_nat_1(k2_wsierp_1(D,k4_finseq_4(k5_numbers,k5_numbers,E,C)),B) ) ) ) ) ) )).

fof(d9_radix_2,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_finseq_2(E,k5_numbers,k4_finseq_2(D,k5_numbers))
                     => ( r1_xreal_0(np__1,D)
                       => ! [F] :
                            ( m2_finseq_2(F,k5_numbers,k4_finseq_2(D,k5_numbers))
                           => ( F = k9_radix_2(A,B,C,D,E)
                            <=> ( k3_wsierp_1(F,np__1) = k8_radix_2(D,B,D,C,E)
                                & ! [G] :
                                    ( m2_subset_1(G,k1_numbers,k5_numbers)
                                   => ~ ( r1_xreal_0(np__1,G)
                                        & r1_xreal_0(G,k6_xcmplx_0(D,np__1))
                                        & ! [H] :
                                            ( m2_subset_1(H,k1_numbers,k5_numbers)
                                           => ! [I] :
                                                ( m2_subset_1(I,k1_numbers,k5_numbers)
                                               => ~ ( H = k3_wsierp_1(F,G)
                                                    & I = k3_wsierp_1(F,k1_nat_1(G,np__1))
                                                    & I = k4_nat_1(k2_nat_1(k4_nat_1(k2_wsierp_1(H,k1_radix_1(A)),B),k8_radix_2(D,B,k5_binarith(D,G),C,E)),B) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t17_radix_2,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)
                   => ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r1_radix_1(A,E,C)
                         => ( r1_xreal_0(D,np__0)
                            | ! [F] :
                                ( m2_finseq_2(F,k5_numbers,k4_finseq_2(A,k5_numbers))
                               => ( F = k5_radix_2(C,A,E)
                                 => k3_wsierp_1(k9_radix_2(C,D,B,A,F),A) = k4_nat_1(k2_wsierp_1(B,E),D) ) ) ) ) ) ) ) ) ) ) )).

fof(dt_k1_radix_2,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,k5_numbers)) )
     => m2_subset_1(k1_radix_2(A,B,C,D),k1_numbers,k5_numbers) ) )).

fof(dt_k2_radix_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k4_finseq_2(A,k5_numbers)) )
     => m2_finseq_2(k2_radix_2(A,B,C),k5_numbers,k4_finseq_2(A,k5_numbers)) ) )).

fof(dt_k3_radix_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k4_finseq_2(A,k5_numbers)) )
     => m2_subset_1(k3_radix_2(A,B,C),k1_numbers,k5_numbers) ) )).

fof(dt_k4_radix_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers) )
     => m2_subset_1(k4_radix_2(A,B,C),k1_numbers,k5_numbers) ) )).

fof(dt_k5_radix_2,axiom,(
    ! [A,B,C] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers) )
     => m2_finseq_2(k5_radix_2(A,B,C),k5_numbers,k4_finseq_2(B,k5_numbers)) ) )).

fof(dt_k6_radix_2,axiom,(
    ! [A,B,C,D,E,F] :
      ( ( v1_int_1(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k5_numbers)
        & m1_subset_1(E,k5_numbers)
        & m1_subset_1(F,k4_finseq_2(E,k3_radix_1(D))) )
     => v1_int_1(k6_radix_2(A,B,C,D,E,F)) ) )).

fof(dt_k7_radix_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_int_1(A)
        & m1_subset_1(B,k5_numbers)
        & m1_subset_1(C,k5_numbers)
        & m1_subset_1(D,k5_numbers)
        & m1_subset_1(E,k4_finseq_2(D,k3_radix_1(B))) )
     => m2_finseq_2(k7_radix_2(A,B,C,D,E),k6_wsierp_1,k4_finseq_2(D,k6_wsierp_1)) ) )).

fof(dt_k8_radix_2,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,k5_numbers)
        & m1_subset_1(E,k4_finseq_2(A,k5_numbers)) )
     => m2_subset_1(k8_radix_2(A,B,C,D,E),k1_numbers,k5_numbers) ) )).

fof(dt_k9_radix_2,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,k5_numbers)
        & m1_subset_1(E,k4_finseq_2(D,k5_numbers)) )
     => m2_finseq_2(k9_radix_2(A,B,C,D,E),k5_numbers,k4_finseq_2(D,k5_numbers)) ) )).
%------------------------------------------------------------------------------