SET007 Axioms: SET007+583.ax


%------------------------------------------------------------------------------
% File     : SET007+583 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : On the Components of the Complement of a Special Polygonal Curve
% 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    : sprect_4 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    8 (   1 unit)
%            Number of atoms       :   55 (   5 equality)
%            Maximal formula depth :   12 (   8 average)
%            Number of connectives :   60 (  13 ~  ;   2  |;  25  &)
%                                         (   0 <=>;  20 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   23 (   1 propositional; 0-5 arity)
%            Number of functors    :   23 (   5 constant; 0-4 arity)
%            Number of variables   :   16 (   0 singleton;  16 !;   0 ?)
%            Maximal term depth    :    8 (   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(cc1_sprect_4,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v2_pre_topc(A)
        & v3_compts_1(A)
        & l1_pre_topc(A) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(u1_struct_0(A)))
         => ( v6_compts_1(B,A)
           => v4_pre_topc(B,A) ) ) ) )).

fof(t1_sprect_4,axiom,(
    ! [A] :
      ( ( v4_topreal1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( ( v4_pre_topc(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ~ ( ~ r1_xboole_0(k5_topreal1(np__2,A),B)
              & ~ r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),B)
              & k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k4_jordan3(A,k1_jordan5c(k5_topreal1(np__2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))),B) != k1_struct_0(k15_euclid(np__2),k1_jordan5c(k5_topreal1(np__2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)))) ) ) ) )).

fof(t2_sprect_4,axiom,(
    ! [A] :
      ( ( ~ v1_xboole_0(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ( ( v4_topreal1(A)
              & B = k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)) )
           => k5_topreal1(np__2,k3_jordan3(A,B)) = k1_xboole_0 ) ) ) )).

fof(t3_sprect_4,axiom,(
    $true )).

fof(t4_sprect_4,axiom,(
    ! [A] :
      ( ( v4_topreal1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m2_subset_1(C,k1_numbers,k5_numbers)
             => ( ( r1_xreal_0(np__1,C)
                  & r2_hidden(B,k5_topreal1(np__2,k1_jordan3(u1_struct_0(k15_euclid(np__2)),A,C,k3_finseq_1(A)))) )
               => ( r1_xreal_0(k3_finseq_1(A),C)
                  | r1_jordan5c(k5_topreal1(np__2,A),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,C),B) ) ) ) ) ) )).

fof(t5_sprect_4,axiom,(
    ! [A] :
      ( ( v4_topreal1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( m1_subset_1(B,u1_struct_0(k15_euclid(np__2)))
         => ! [C] :
              ( m1_subset_1(C,u1_struct_0(k15_euclid(np__2)))
             => ! [D] :
                  ( m2_subset_1(D,k1_numbers,k5_numbers)
                 => ( ( r1_xreal_0(np__1,D)
                      & r2_hidden(B,k4_topreal1(np__2,A,D))
                      & r2_hidden(C,k3_topreal1(np__2,B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k1_nat_1(D,np__1)))) )
                   => ( r1_xreal_0(k3_finseq_1(A),D)
                      | r1_jordan5c(k5_topreal1(np__2,A),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),B,C) ) ) ) ) ) ) )).

fof(t6_sprect_4,axiom,(
    ! [A] :
      ( ( v4_topreal1(A)
        & m2_finseq_1(A,u1_struct_0(k15_euclid(np__2))) )
     => ! [B] :
          ( ( v4_pre_topc(B,k15_euclid(np__2))
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ~ ( ~ r1_xboole_0(k5_topreal1(np__2,A),B)
              & ~ r2_hidden(k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)),B)
              & k5_subset_1(u1_struct_0(k15_euclid(np__2)),k5_topreal1(np__2,k3_jordan3(A,k2_jordan5c(k5_topreal1(np__2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A))))),B) != k1_struct_0(k15_euclid(np__2),k2_jordan5c(k5_topreal1(np__2,A),B,k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,np__1),k4_finseq_4(k5_numbers,u1_struct_0(k15_euclid(np__2)),A,k3_finseq_1(A)))) ) ) ) )).

fof(t7_sprect_4,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))) )
     => k2_goboard9(A) != k3_goboard9(A) ) )).
%------------------------------------------------------------------------------