SET007 Axioms: SET007+796.ax


%------------------------------------------------------------------------------
% File     : SET007+796 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Magnitude Relation Properties of Radix-2^k SD 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_5 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   35 (   0 unit)
%            Number of atoms       :  222 (  44 equality)
%            Maximal formula depth :   23 (   9 average)
%            Number of connectives :  199 (  12 ~  ;   4  |;  39  &)
%                                         (   4 <=>; 140 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    6 (   0 propositional; 2-3 arity)
%            Number of functors    :   29 (   7 constant; 0-4 arity)
%            Number of variables   :  112 (   0 singleton; 112 !;   0 ?)
%            Maximal term depth    :    5 (   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_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__2,A)
       => r2_hidden(k6_xcmplx_0(k1_radix_1(A),np__1),k3_radix_1(A)) ) ) )).

fof(t2_radix_5,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))
           => ( r1_xreal_0(A,np__1)
              | r2_hidden(k5_binarith(A,np__1),k2_finseq_1(B)) ) ) ) ) )).

fof(t3_radix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__2,A)
       => r1_xreal_0(np__4,k1_radix_1(A)) ) ) )).

fof(t4_radix_5,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(t5_radix_5,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)
             => ( r2_hidden(A,k2_finseq_1(C))
               => k4_radix_1(A,B,C,k10_radix_1(B,C,np__0)) = np__0 ) ) ) ) )).

fof(t6_radix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_xreal_0(np__1,A)
           => k8_radix_1(A,B,k10_radix_1(B,A,np__0)) = np__0 ) ) ) )).

fof(t7_radix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r2_hidden(np__1,k2_finseq_1(B))
              & r1_xreal_0(np__2,A) )
           => k4_radix_1(np__1,A,B,k10_radix_1(A,B,np__1)) = np__1 ) ) ) )).

fof(t8_radix_5,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)
             => ( ( r2_hidden(A,k2_finseq_1(C))
                  & r1_xreal_0(np__2,B) )
               => ( r1_xreal_0(A,np__1)
                  | k4_radix_1(A,B,C,k10_radix_1(B,C,np__1)) = np__0 ) ) ) ) ) )).

fof(t9_radix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(np__2,B) )
           => k8_radix_1(A,B,k10_radix_1(B,A,np__1)) = np__1 ) ) ) )).

fof(t10_radix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__2,A)
       => k11_radix_1(k1_radix_1(A)) = np__1 ) ) )).

fof(t11_radix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ( r1_xreal_0(np__2,A)
       => k12_radix_1(k1_radix_1(A),A) = np__0 ) ) )).

fof(t12_radix_5,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_finseq_2(C,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
               => ! [D] :
                    ( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                   => ( ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k2_finseq_1(A))
                           => k4_radix_1(E,B,A,C) = k4_radix_1(E,B,A,D) ) )
                     => k8_radix_1(A,B,C) = k8_radix_1(A,B,D) ) ) ) ) ) ) )).

fof(t13_radix_5,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_finseq_2(C,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
               => ! [D] :
                    ( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                   => ( ! [E] :
                          ( m2_subset_1(E,k1_numbers,k5_numbers)
                         => ( r2_hidden(E,k2_finseq_1(A))
                           => r1_xreal_0(k4_radix_1(E,B,A,D),k4_radix_1(E,B,A,C)) ) )
                     => r1_xreal_0(k8_radix_1(A,B,D),k8_radix_1(A,B,C)) ) ) ) ) ) ) )).

fof(t14_radix_5,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)
           => ( r1_xreal_0(np__2,B)
             => ! [C] :
                  ( m2_finseq_2(C,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                 => ! [D] :
                      ( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                     => ! [E] :
                          ( m2_finseq_2(E,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                         => ! [F] :
                              ( m2_finseq_2(F,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                             => ( ! [G] :
                                    ( m2_subset_1(G,k1_numbers,k5_numbers)
                                   => ~ ( r2_hidden(G,k2_finseq_1(A))
                                        & ~ ( k4_radix_1(G,B,A,C) = k4_radix_1(G,B,A,E)
                                            & k4_radix_1(G,B,A,D) = k4_radix_1(G,B,A,F) )
                                        & ~ ( k4_radix_1(G,B,A,D) = k4_radix_1(G,B,A,E)
                                            & k4_radix_1(G,B,A,C) = k4_radix_1(G,B,A,F) ) ) )
                               => k2_xcmplx_0(k8_radix_1(A,B,E),k8_radix_1(A,B,F)) = k2_xcmplx_0(k8_radix_1(A,B,C),k8_radix_1(A,B,D)) ) ) ) ) ) ) ) ) ) )).

fof(t15_radix_5,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_xreal_0(np__1,A)
              & r1_xreal_0(np__2,B) )
           => ! [C] :
                ( m2_finseq_2(C,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
               => ! [D] :
                    ( m2_finseq_2(D,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                   => ! [E] :
                        ( m2_finseq_2(E,k3_radix_1(B),k4_finseq_2(A,k3_radix_1(B)))
                       => ( ! [F] :
                              ( m2_subset_1(F,k1_numbers,k5_numbers)
                             => ~ ( r2_hidden(F,k2_finseq_1(A))
                                  & ~ ( k4_radix_1(F,B,A,C) = k4_radix_1(F,B,A,E)
                                      & k4_radix_1(F,B,A,D) = np__0 )
                                  & ~ ( k4_radix_1(F,B,A,D) = k4_radix_1(F,B,A,E)
                                      & k4_radix_1(F,B,A,C) = np__0 ) ) )
                         => k2_xcmplx_0(k8_radix_1(A,B,E),k8_radix_1(A,B,k10_radix_1(B,A,np__0))) = k2_xcmplx_0(k8_radix_1(A,B,C),k8_radix_1(A,B,D)) ) ) ) ) ) ) ) )).

fof(d1_radix_5,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,C)
               => ( ( r1_xreal_0(np__1,A)
                   => ( r1_xreal_0(B,A)
                      | k1_radix_5(A,B,C) = k2_xcmplx_0(k4_xcmplx_0(k1_radix_1(C)),np__1) ) )
                  & ( ~ ( r1_xreal_0(np__1,A)
                        & ~ r1_xreal_0(B,A) )
                   => k1_radix_5(A,B,C) = np__0 ) ) ) ) ) ) )).

fof(d2_radix_5,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(C),k4_finseq_2(A,k3_radix_1(C)))
                 => ( D = k2_radix_5(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(A))
                         => k4_radix_1(E,C,A,D) = k1_radix_5(E,B,C) ) ) ) ) ) ) ) )).

fof(d3_radix_5,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,C)
               => ( ( r1_xreal_0(np__1,A)
                   => ( r1_xreal_0(B,A)
                      | k3_radix_5(A,B,C) = k6_xcmplx_0(k1_radix_1(C),np__1) ) )
                  & ( ~ ( r1_xreal_0(np__1,A)
                        & ~ r1_xreal_0(B,A) )
                   => k3_radix_5(A,B,C) = np__0 ) ) ) ) ) ) )).

fof(d4_radix_5,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(C),k4_finseq_2(A,k3_radix_1(C)))
                 => ( D = k4_radix_5(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(A))
                         => k4_radix_1(E,C,A,D) = k3_radix_5(E,B,C) ) ) ) ) ) ) ) )).

fof(d5_radix_5,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,C)
               => ( ( A = B
                   => k5_radix_5(A,B,C) = np__1 )
                  & ( A != B
                   => k5_radix_5(A,B,C) = np__0 ) ) ) ) ) ) )).

fof(d6_radix_5,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(C),k4_finseq_2(A,k3_radix_1(C)))
                 => ( D = k6_radix_5(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(A))
                         => k4_radix_1(E,C,A,D) = k5_radix_5(E,B,C) ) ) ) ) ) ) ) )).

fof(d7_radix_5,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,C)
               => ( ( A = B
                   => k7_radix_5(A,B,C) = k6_xcmplx_0(k1_radix_1(C),np__1) )
                  & ( A != B
                   => k7_radix_5(A,B,C) = np__0 ) ) ) ) ) ) )).

fof(d8_radix_5,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(C),k4_finseq_2(A,k3_radix_1(C)))
                 => ( D = k8_radix_5(A,B,C)
                  <=> ! [E] :
                        ( m2_subset_1(E,k1_numbers,k5_numbers)
                       => ( r2_hidden(E,k2_finseq_1(A))
                         => k4_radix_1(E,C,A,D) = k7_radix_5(E,B,C) ) ) ) ) ) ) ) )).

fof(t16_radix_5,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,A)
                  & r1_xreal_0(np__2,C)
                  & r2_hidden(B,k2_finseq_1(A)) )
               => ! [D] :
                    ( m2_subset_1(D,k1_numbers,k5_numbers)
                   => ( r2_hidden(D,k2_finseq_1(A))
                     => k2_xcmplx_0(k4_radix_1(D,C,A,k4_radix_5(A,B,C)),k4_radix_1(D,C,A,k2_radix_5(A,B,C))) = np__0 ) ) ) ) ) ) )).

fof(t17_radix_5,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)
               => ( ( r2_hidden(B,k2_finseq_1(A))
                    & r1_xreal_0(np__2,C) )
                 => k2_xcmplx_0(k8_radix_1(A,C,k4_radix_5(A,B,C)),k8_radix_1(A,C,k2_radix_5(A,B,C))) = k8_radix_1(A,C,k10_radix_1(C,A,np__0)) ) ) ) ) ) )).

fof(t18_radix_5,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)
               => ( ( r2_hidden(B,k2_finseq_1(A))
                    & r1_xreal_0(np__2,C) )
                 => k8_radix_1(A,C,k6_radix_5(A,B,C)) = k2_xcmplx_0(k8_radix_1(A,C,k4_radix_5(A,B,C)),k8_radix_1(A,C,k10_radix_1(C,A,np__1))) ) ) ) ) ) )).

fof(t19_radix_5,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)
             => ( ( r2_hidden(B,k2_finseq_1(A))
                  & r1_xreal_0(np__2,C) )
               => k8_radix_1(k1_nat_1(A,np__1),C,k6_radix_5(k1_nat_1(A,np__1),k1_nat_1(B,np__1),C)) = k2_xcmplx_0(k8_radix_1(k1_nat_1(A,np__1),C,k6_radix_5(k1_nat_1(A,np__1),B,C)),k8_radix_1(k1_nat_1(A,np__1),C,k8_radix_5(k1_nat_1(A,np__1),B,C))) ) ) ) ) )).

fof(dt_k1_radix_5,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(k1_radix_5(A,B,C),k6_wsierp_1,k3_radix_1(C)) ) )).

fof(dt_k2_radix_5,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(k2_radix_5(A,B,C),k3_radix_1(C),k4_finseq_2(A,k3_radix_1(C))) ) )).

fof(dt_k3_radix_5,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(k3_radix_5(A,B,C),k6_wsierp_1,k3_radix_1(C)) ) )).

fof(dt_k4_radix_5,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(k4_radix_5(A,B,C),k3_radix_1(C),k4_finseq_2(A,k3_radix_1(C))) ) )).

fof(dt_k5_radix_5,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(k5_radix_5(A,B,C),k6_wsierp_1,k3_radix_1(C)) ) )).

fof(dt_k6_radix_5,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(k6_radix_5(A,B,C),k3_radix_1(C),k4_finseq_2(A,k3_radix_1(C))) ) )).

fof(dt_k7_radix_5,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(k7_radix_5(A,B,C),k6_wsierp_1,k3_radix_1(C)) ) )).

fof(dt_k8_radix_5,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(k8_radix_5(A,B,C),k3_radix_1(C),k4_finseq_2(A,k3_radix_1(C))) ) )).
%------------------------------------------------------------------------------