SET007 Axioms: SET007+722.ax


%------------------------------------------------------------------------------
% File     : SET007+722 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Preparing the Internal Approximations of Simple Closed Curves
% 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    : jordan11 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   17 (   0 unit)
%            Number of atoms       :   96 (   4 equality)
%            Maximal formula depth :   19 (   7 average)
%            Number of connectives :   84 (   5 ~  ;   1  |;  28  &)
%                                         (   3 <=>;  47 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :    9 (   0 propositional; 1-3 arity)
%            Number of functors    :   24 (   4 constant; 0-3 arity)
%            Number of variables   :   35 (   0 singleton;  35 !;   0 ?)
%            Maximal term depth    :    6 (   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(d1_jordan11,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k1_jordan11(A)
          <=> ( r1_jordan1h(A,B)
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_jordan1h(A,C)
                   => r1_xreal_0(B,C) ) ) ) ) ) ) )).

fof(t1_jordan11,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => r1_xreal_0(np__1,k1_jordan11(A)) ) )).

fof(d2_jordan11,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( B = k2_jordan11(A)
          <=> ( ~ r1_xreal_0(k1_matrix_1(k1_jordan8(A,k1_jordan11(A))),B)
              & r1_tarski(k3_goboard5(k1_jordan8(A,k1_jordan11(A)),k5_binarith(k3_jordan1h(A,k1_jordan11(A)),np__1),B),k1_jordan2c(np__2,A))
              & ! [C] :
                  ( m2_subset_1(C,k1_numbers,k5_numbers)
                 => ( r1_tarski(k3_goboard5(k1_jordan8(A,k1_jordan11(A)),k5_binarith(k3_jordan1h(A,k1_jordan11(A)),np__1),C),k1_jordan2c(np__2,A))
                   => ( r1_xreal_0(k1_matrix_1(k1_jordan8(A,k1_jordan11(A))),C)
                      | r1_xreal_0(B,C) ) ) ) ) ) ) ) )).

fof(t2_jordan11,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ r1_xreal_0(k2_jordan11(A),np__1) ) )).

fof(t3_jordan11,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ~ r1_xreal_0(k1_matrix_1(k1_jordan8(A,k1_jordan11(A))),k1_nat_1(k2_jordan11(A),np__1)) ) )).

fof(d3_jordan11,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => ! [B] :
          ( m2_subset_1(B,k1_numbers,k5_numbers)
         => ( r1_jordan1h(A,B)
           => ! [C] :
                ( m2_subset_1(C,k1_numbers,k5_numbers)
               => ( C = k3_jordan11(A,B)
                <=> ( r1_xreal_0(C,k1_matrix_1(k1_jordan8(A,B)))
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(C,D)
                            & r1_xreal_0(D,k1_nat_1(k2_nat_1(k1_card_4(np__2,k5_binarith(B,k1_jordan11(A))),k5_binarith(k2_jordan11(A),np__2)),np__2)) )
                         => r1_tarski(k3_goboard5(k1_jordan8(A,B),k5_binarith(k3_jordan1h(A,B),np__1),D),k1_jordan2c(np__2,A)) ) )
                    & ! [D] :
                        ( m2_subset_1(D,k1_numbers,k5_numbers)
                       => ( ( r1_xreal_0(D,k1_matrix_1(k1_jordan8(A,B)))
                            & ! [E] :
                                ( m2_subset_1(E,k1_numbers,k5_numbers)
                               => ( ( r1_xreal_0(D,E)
                                    & r1_xreal_0(E,k1_nat_1(k2_nat_1(k1_card_4(np__2,k5_binarith(B,k1_jordan11(A))),k5_binarith(k2_jordan11(A),np__2)),np__2)) )
                                 => r1_tarski(k3_goboard5(k1_jordan8(A,B),k5_binarith(k3_jordan1h(A,B),np__1),E),k1_jordan2c(np__2,A)) ) ) )
                         => r1_xreal_0(C,D) ) ) ) ) ) ) ) ) )).

fof(t4_jordan11,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_jordan1h(B,A)
           => k3_jordan1h(B,A) = k3_real_1(k4_real_1(k1_card_4(np__2,k5_binarith(A,k1_jordan11(B))),k5_real_1(k3_jordan1h(B,k1_jordan11(B)),np__2)),np__2) ) ) ) )).

fof(t5_jordan11,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_jordan1h(B,A)
           => r1_xreal_0(k3_jordan11(B,A),k1_nat_1(k2_nat_1(k1_card_4(np__2,k5_binarith(A,k1_jordan11(B))),k5_binarith(k2_jordan11(B),np__2)),np__2)) ) ) ) )).

fof(t6_jordan11,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_jordan1h(B,A)
           => r1_tarski(k3_goboard5(k1_jordan8(B,A),k5_binarith(k3_jordan1h(B,A),np__1),k3_jordan11(B,A)),k1_jordan2c(np__2,B)) ) ) ) )).

fof(t7_jordan11,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_jordan1h(B,A)
           => ( ~ r1_xreal_0(k3_jordan11(B,A),np__1)
              & r1_xreal_0(k3_jordan11(B,A),k1_matrix_1(k1_jordan8(B,A))) ) ) ) ) )).

fof(t8_jordan11,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_jordan1h(B,A)
           => r2_hidden(k4_tarski(k3_jordan1h(B,A),k3_jordan11(B,A)),k2_matrix_1(k1_jordan8(B,A))) ) ) ) )).

fof(t9_jordan11,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_jordan1h(B,A)
           => r2_hidden(k4_tarski(k5_binarith(k3_jordan1h(B,A),np__1),k3_jordan11(B,A)),k2_matrix_1(k1_jordan8(B,A))) ) ) ) )).

fof(t10_jordan11,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ~ ( r1_jordan1h(B,A)
              & r1_xboole_0(k3_goboard5(k1_jordan8(B,A),k5_binarith(k3_jordan1h(B,A),np__1),k5_binarith(k3_jordan11(B,A),np__1)),B) ) ) ) )).

fof(t11_jordan11,axiom,(
    ! [A] :
      ( m2_subset_1(A,k1_numbers,k5_numbers)
     => ! [B] :
          ( ( v1_topreal2(B)
            & m1_subset_1(B,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
         => ( r1_jordan1h(B,A)
           => r1_xboole_0(k3_goboard5(k1_jordan8(B,A),k5_binarith(k3_jordan1h(B,A),np__1),k3_jordan11(B,A)),B) ) ) ) )).

fof(dt_k1_jordan11,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => m2_subset_1(k1_jordan11(A),k1_numbers,k5_numbers) ) )).

fof(dt_k2_jordan11,axiom,(
    ! [A] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2)))) )
     => m2_subset_1(k2_jordan11(A),k1_numbers,k5_numbers) ) )).

fof(dt_k3_jordan11,axiom,(
    ! [A,B] :
      ( ( v1_topreal2(A)
        & m1_subset_1(A,k1_zfmisc_1(u1_struct_0(k15_euclid(np__2))))
        & m1_subset_1(B,k5_numbers) )
     => m2_subset_1(k3_jordan11(A,B),k1_numbers,k5_numbers) ) )).
%------------------------------------------------------------------------------