SET007 Axioms: SET007+475.ax


%------------------------------------------------------------------------------
% File     : SET007+475 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The First Part of Jordan's Theorem for Special Polygons
% 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    : gobrd12 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   12 (   1 unit)
%            Number of atoms       :  158 (  10 equality)
%            Maximal formula depth :   19 (  11 average)
%            Number of connectives :  169 (  23 ~  ;   2  |; 102  &)
%                                         (   2 <=>;  40 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   1 propositional; 0-4 arity)
%            Number of functors    :   26 (   4 constant; 0-4 arity)
%            Number of variables   :   36 (   0 singleton;  32 !;   4 ?)
%            Maximal term depth    :    7 (   2 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_gobrd12,axiom,(
    $true )).

fof(t2_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
                  & r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) )
               => r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) ) ) ) ) )).

fof(t3_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
                  & r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) )
               => k6_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))),k4_connsp_3(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))) = k5_subset_1(u1_struct_0(k15_euclid(np__2)),k3_goboard5(k3_goboard2(A),B,C),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) ) ) ) ) )).

fof(t4_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
                  & r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) )
               => ( v2_connsp_1(k6_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))),k4_connsp_3(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))),k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))))
                  & k4_connsp_3(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) = k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)) ) ) ) ) ) )).

fof(t6_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ( m1_connsp_3(k4_subset_1(u1_struct_0(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))),k4_connsp_3(k15_euclid(np__2),k2_goboard9(A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))),k4_connsp_3(k15_euclid(np__2),k3_goboard9(A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)))),k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))))
        & k4_connsp_3(k15_euclid(np__2),k2_goboard9(A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) = k2_goboard9(A)
        & k4_connsp_3(k15_euclid(np__2),k3_goboard9(A),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A))) = k3_goboard9(A) ) ) )).

fof(t7_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [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_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
                          & r1_xreal_0(C,k1_matrix_1(k3_goboard2(A)))
                          & r1_xreal_0(D,k3_finseq_1(k3_goboard2(A)))
                          & r1_xreal_0(E,k1_matrix_1(k3_goboard2(A)))
                          & r2_gobrd10(B,C,D,E) )
                       => ( r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A)))
                        <=> r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),D,E)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) ) ) ) ) ) ) )).

fof(t8_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( m2_finseq_1(B,k5_numbers)
         => ! [C] :
              ( m2_finseq_1(C,k5_numbers)
             => ( ( k3_finseq_1(B) = k3_finseq_1(C)
                  & ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r1_xreal_0(np__1,D)
                       => ( r1_xreal_0(k3_finseq_1(B),D)
                          | r2_gobrd10(k4_finseq_4(k5_numbers,k5_numbers,B,D),k4_finseq_4(k5_numbers,k5_numbers,C,D),k4_finseq_4(k5_numbers,k5_numbers,B,k1_nat_1(D,np__1)),k4_finseq_4(k5_numbers,k5_numbers,C,k1_nat_1(D,np__1))) ) ) )
                  & ! [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)
                             => ( ( r2_hidden(D,k4_finseq_1(B))
                                  & E = k1_funct_1(B,D)
                                  & F = k1_funct_1(C,D) )
                               => ( r1_xreal_0(E,k3_finseq_1(k3_goboard2(A)))
                                  & r1_xreal_0(F,k1_matrix_1(k3_goboard2(A))) ) ) ) ) ) )
               => ( ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ~ ( r2_hidden(D,k4_finseq_1(B))
                          & r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),k4_finseq_4(k5_numbers,k5_numbers,B,D),k4_finseq_4(k5_numbers,k5_numbers,C,D))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) )
                  | ! [D] :
                      ( m2_subset_1(D,k1_numbers,k5_numbers)
                     => ( r2_hidden(D,k4_finseq_1(B))
                       => r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),k4_finseq_4(k5_numbers,k5_numbers,B,D),k4_finseq_4(k5_numbers,k5_numbers,C,D))),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) ) ) ) ) ) ) )).

fof(t9_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ? [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
          & ? [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
              & r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
              & r1_xreal_0(C,k1_matrix_1(k3_goboard2(A)))
              & r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) ) ) )).

fof(t10_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(B,k3_finseq_1(k3_goboard2(A)))
                  & r1_xreal_0(C,k1_matrix_1(k3_goboard2(A))) )
               => r1_tarski(k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(A),B,C)),k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A))) ) ) ) ) )).

fof(t11_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)) = k4_subset_1(u1_struct_0(k15_euclid(np__2)),k2_goboard9(A),k3_goboard9(A)) ) )).

fof(t5_gobrd12,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & ~ v5_seqm_3(A)
        & v1_topreal1(A)
        & v2_topreal1(A)
        & v1_finseq_6(A,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(A)
        & v2_goboard5(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,A)) = k3_tarski(a_1_0_gobrd12(A)) ) )).

fof(fraenkel_a_1_0_gobrd12,axiom,(
    ! [A,B] :
      ( ( ~ v1_xboole_0(B)
        & ~ v5_seqm_3(B)
        & v1_topreal1(B)
        & v2_topreal1(B)
        & v1_finseq_6(B,u1_struct_0(k15_euclid(np__2)))
        & v1_goboard5(B)
        & v2_goboard5(B)
        & m2_finseq_1(B,u1_struct_0(k15_euclid(np__2))) )
     => ( r2_hidden(A,a_1_0_gobrd12(B))
      <=> ? [C,D] :
            ( m2_subset_1(C,k1_numbers,k5_numbers)
            & m2_subset_1(D,k1_numbers,k5_numbers)
            & A = k6_pre_topc(k3_pre_topc(k15_euclid(np__2),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,B))),k4_connsp_3(k15_euclid(np__2),k1_tops_1(k15_euclid(np__2),k3_goboard5(k3_goboard2(B),C,D)),k3_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,B))))
            & r1_xreal_0(C,k3_finseq_1(k3_goboard2(B)))
            & r1_xreal_0(D,k1_matrix_1(k3_goboard2(B))) ) ) ) )).
%------------------------------------------------------------------------------