SET007 Axioms: SET007+467.ax


%------------------------------------------------------------------------------
% File     : SET007+467 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Adjacency Concept for Pairs of Natural Numbers
% 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    : gobrd10 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   16 (   1 unit)
%            Number of atoms       :  157 (  26 equality)
%            Maximal formula depth :   34 (  12 average)
%            Number of connectives :  153 (  12 ~  ;   5  |;  54  &)
%                                         (   4 <=>;  78 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   11 (   1 propositional; 0-4 arity)
%            Number of functors    :   14 (   3 constant; 0-4 arity)
%            Number of variables   :   69 (   0 singleton;  69 !;   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(d1_gobrd10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_gobrd10(A,B)
          <=> ( B = k1_nat_1(A,np__1)
              | A = k1_nat_1(B,np__1) ) ) ) ) )).

fof(t1_gobrd10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_gobrd10(A,B)
           => r1_gobrd10(k1_nat_1(A,np__1),k1_nat_1(B,np__1)) ) ) ) )).

fof(t2_gobrd10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( ( r1_gobrd10(A,B)
              & r1_xreal_0(np__1,A)
              & r1_xreal_0(np__1,B) )
           => r1_gobrd10(k5_binarith(A,np__1),k5_binarith(B,np__1)) ) ) ) )).

fof(d2_gobrd10,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_gobrd10(A,B,C,D)
                  <=> ( ( r1_gobrd10(A,C)
                        & B = D )
                      | ( A = C
                        & r1_gobrd10(B,D) ) ) ) ) ) ) ) )).

fof(t3_gobrd10,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_gobrd10(A,C,B,D)
                   => r2_gobrd10(k1_nat_1(A,np__1),k1_nat_1(C,np__1),k1_nat_1(B,np__1),k1_nat_1(D,np__1)) ) ) ) ) ) )).

fof(t4_gobrd10,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_gobrd10(A,C,B,D)
                      & r1_xreal_0(np__1,A)
                      & r1_xreal_0(np__1,B)
                      & r1_xreal_0(np__1,C)
                      & r1_xreal_0(np__1,D) )
                   => r2_gobrd10(k5_binarith(A,np__1),k5_binarith(C,np__1),k5_binarith(B,np__1),k5_binarith(D,np__1)) ) ) ) ) ) )).

fof(d3_gobrd10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k5_numbers)
             => ( C = k1_gobrd10(A,B)
              <=> ( k3_finseq_1(C) = A
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( ( r1_xreal_0(np__1,D)
                          & r1_xreal_0(D,A) )
                       => k1_funct_1(C,D) = B ) ) ) ) ) ) ) )).

fof(t5_gobrd10,axiom,(
    $true )).

fof(t6_gobrd10,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(B,A)
                  & r1_xreal_0(C,A)
                  & ! [D] :
                      ( m2_finseq_1(D,k5_numbers)
                     => ~ ( k1_funct_1(D,np__1) = B
                          & k1_funct_1(D,k3_finseq_1(D)) = C
                          & k3_finseq_1(D) = k1_nat_1(k1_nat_1(k5_binarith(B,C),k5_binarith(C,B)),np__1)
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ( ( r1_xreal_0(np__1,E)
                                      & r1_xreal_0(E,k3_finseq_1(D))
                                      & F = k1_funct_1(D,E) )
                                   => r1_xreal_0(F,A) ) ) )
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ~ ( r1_xreal_0(np__1,E)
                                  & ~ r1_xreal_0(k3_finseq_1(D),E)
                                  & k1_funct_1(D,k1_nat_1(E,np__1)) != k1_nat_1(k4_finseq_4(k5_numbers,k5_numbers,D,E),np__1)
                                  & k1_funct_1(D,E) != k1_nat_1(k4_finseq_4(k5_numbers,k5_numbers,D,k1_nat_1(E,np__1)),np__1) ) ) ) ) ) ) ) ) )).

fof(t7_gobrd10,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(B,A)
                  & r1_xreal_0(C,A)
                  & ! [D] :
                      ( m2_finseq_1(D,k5_numbers)
                     => ~ ( k1_funct_1(D,np__1) = B
                          & k1_funct_1(D,k3_finseq_1(D)) = C
                          & k3_finseq_1(D) = k1_nat_1(k1_nat_1(k5_binarith(B,C),k5_binarith(C,B)),np__1)
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ! [F] :
                                  ( m2_subset_1(F,k1_numbers,k5_numbers)
                                 => ( ( r1_xreal_0(np__1,E)
                                      & r1_xreal_0(E,k3_finseq_1(D))
                                      & F = k1_funct_1(D,E) )
                                   => r1_xreal_0(F,A) ) ) )
                          & ! [E] :
                              ( m2_subset_1(E,k1_numbers,k5_numbers)
                             => ( r1_xreal_0(np__1,E)
                               => ( r1_xreal_0(k3_finseq_1(D),E)
                                  | r1_gobrd10(k4_finseq_4(k5_numbers,k5_numbers,D,E),k4_finseq_4(k5_numbers,k5_numbers,D,k1_nat_1(E,np__1))) ) ) ) ) ) ) ) ) ) )).

fof(t8_gobrd10,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)
                     => ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ~ ( r1_xreal_0(C,A)
                              & r1_xreal_0(D,B)
                              & r1_xreal_0(E,A)
                              & r1_xreal_0(F,B)
                              & ! [G] :
                                  ( m2_finseq_1(G,k5_numbers)
                                 => ! [H] :
                                      ( m2_finseq_1(H,k5_numbers)
                                     => ~ ( ! [I] :
                                              ( m2_subset_1(I,k1_numbers,k5_numbers)
                                             => ! [J] :
                                                  ( m2_subset_1(J,k1_numbers,k5_numbers)
                                                 => ! [K] :
                                                      ( m2_subset_1(K,k1_numbers,k5_numbers)
                                                     => ( ( r2_hidden(I,k4_finseq_1(G))
                                                          & J = k1_funct_1(G,I)
                                                          & K = k1_funct_1(H,I) )
                                                       => ( r1_xreal_0(J,A)
                                                          & r1_xreal_0(K,B) ) ) ) ) )
                                          & k1_funct_1(G,np__1) = C
                                          & k1_funct_1(G,k3_finseq_1(G)) = E
                                          & k1_funct_1(H,np__1) = D
                                          & k1_funct_1(H,k3_finseq_1(H)) = F
                                          & k3_finseq_1(G) = k3_finseq_1(H)
                                          & k3_finseq_1(G) = k1_nat_1(k1_nat_1(k1_nat_1(k1_nat_1(k5_binarith(C,E),k5_binarith(E,C)),k5_binarith(D,F)),k5_binarith(F,D)),np__1)
                                          & ! [I] :
                                              ( m2_subset_1(I,k1_numbers,k5_numbers)
                                             => ( r1_xreal_0(np__1,I)
                                               => ( r1_xreal_0(k3_finseq_1(G),I)
                                                  | r2_gobrd10(k4_finseq_4(k5_numbers,k5_numbers,G,I),k4_finseq_4(k5_numbers,k5_numbers,H,I),k4_finseq_4(k5_numbers,k5_numbers,G,k1_nat_1(I,np__1)),k4_finseq_4(k5_numbers,k5_numbers,H,k1_nat_1(I,np__1))) ) ) ) ) ) ) ) ) ) ) ) ) ) )).

fof(t9_gobrd10,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C,D] :
              ( m1_subset_1(D,k1_zfmisc_1(C))
             => ! [E] :
                  ( m1_matrix_1(E,k1_zfmisc_1(C),A,B)
                 => ( ! [F] :
                        ( m2_subset_1(F,k1_numbers,k5_numbers)
                       => ! [G] :
                            ( m2_subset_1(G,k1_numbers,k5_numbers)
                           => ! [H] :
                                ( m2_subset_1(H,k1_numbers,k5_numbers)
                               => ! [I] :
                                    ( m2_subset_1(I,k1_numbers,k5_numbers)
                                   => ( ( r2_hidden(F,k2_finseq_1(A))
                                        & r2_hidden(H,k2_finseq_1(A))
                                        & r2_hidden(G,k2_finseq_1(B))
                                        & r2_hidden(I,k2_finseq_1(B))
                                        & r2_gobrd10(F,G,H,I) )
                                     => ( r1_tarski(k3_matrix_1(k1_zfmisc_1(C),E,F,G),D)
                                      <=> r1_tarski(k3_matrix_1(k1_zfmisc_1(C),E,H,I),D) ) ) ) ) ) )
                   => ( ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ! [G] :
                              ( m2_subset_1(G,k1_numbers,k5_numbers)
                             => ~ ( r2_hidden(F,k2_finseq_1(A))
                                  & r2_hidden(G,k2_finseq_1(B))
                                  & r1_tarski(k3_matrix_1(k1_zfmisc_1(C),E,F,G),D) ) ) )
                      | ! [F] :
                          ( m2_subset_1(F,k1_numbers,k5_numbers)
                         => ! [G] :
                              ( m2_subset_1(G,k1_numbers,k5_numbers)
                             => ( ( r2_hidden(F,k2_finseq_1(A))
                                  & r2_hidden(G,k2_finseq_1(B)) )
                               => r1_tarski(k3_matrix_1(k1_zfmisc_1(C),E,F,G),D) ) ) ) ) ) ) ) ) ) )).

fof(symmetry_r1_gobrd10,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ( r1_gobrd10(A,B)
       => r1_gobrd10(B,A) ) ) )).

fof(irreflexivity_r1_gobrd10,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => ~ r1_gobrd10(A,A) ) )).

fof(dt_k1_gobrd10,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => m2_finseq_1(k1_gobrd10(A,B),k5_numbers) ) )).

fof(redefinition_k1_gobrd10,axiom,(
    ! [A,B] :
      ( ( m1_subset_1(A,k5_numbers)
        & m1_subset_1(B,k5_numbers) )
     => k1_gobrd10(A,B) = k2_finseq_2(A,B) ) )).
%------------------------------------------------------------------------------