SET007 Axioms: SET007+304.ax


%------------------------------------------------------------------------------
% File     : SET007+304 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Fix Point Theorem for Compact Spaces
% 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    : ali2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :    5 (   1 unit)
%            Number of atoms       :   47 (   3 equality)
%            Maximal formula depth :   14 (   9 average)
%            Number of connectives :   50 (   8 ~  ;   0  |;  29  &)
%                                         (   1 <=>;  12 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   15 (   1 propositional; 0-3 arity)
%            Number of functors    :    8 (   3 constant; 0-4 arity)
%            Number of variables   :   13 (   0 singleton;  11 !;   2 ?)
%            Maximal term depth    :    4 (   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_ali2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) )
         => ( m1_ali2(B,A)
          <=> ? [C] :
                ( m1_subset_1(C,k1_numbers)
                & ~ r1_xreal_0(C,np__0)
                & ~ r1_xreal_0(np__1,C)
                & ! [D] :
                    ( m1_subset_1(D,u1_struct_0(A))
                   => ! [E] :
                        ( m1_subset_1(E,u1_struct_0(A))
                       => r1_xreal_0(k4_metric_1(A,k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D),k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,E)),k4_real_1(C,k4_metric_1(A,D,E))) ) ) ) ) ) ) )).

fof(t1_ali2,axiom,(
    $true )).

fof(t2_ali2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_ali2(B,A)
         => ~ ( v2_compts_1(k5_pcomps_1(A))
              & ! [C] :
                  ( m1_subset_1(C,u1_struct_0(A))
                 => ~ ( k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,C) = C
                      & ! [D] :
                          ( m1_subset_1(D,u1_struct_0(A))
                         => ( k8_funct_2(u1_struct_0(A),u1_struct_0(A),B,D) = D
                           => D = C ) ) ) ) ) ) ) )).

fof(dt_m1_ali2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ! [B] :
          ( m1_ali2(B,A)
         => ( v1_funct_1(B)
            & v1_funct_2(B,u1_struct_0(A),u1_struct_0(A))
            & m2_relset_1(B,u1_struct_0(A),u1_struct_0(A)) ) ) ) )).

fof(existence_m1_ali2,axiom,(
    ! [A] :
      ( ( ~ v3_struct_0(A)
        & v6_metric_1(A)
        & v7_metric_1(A)
        & v8_metric_1(A)
        & v9_metric_1(A)
        & l1_metric_1(A) )
     => ? [B] : m1_ali2(B,A) ) )).
%------------------------------------------------------------------------------