SET007 Axioms: SET007+107.ax


%------------------------------------------------------------------------------
% File     : SET007+107 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Real Function Uniform Continuity
% 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    : fcont_2 [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   22 (   2 unit)
%            Number of atoms       :  153 (  10 equality)
%            Maximal formula depth :   23 (   9 average)
%            Number of connectives :  150 (  19 ~  ;   0  |;  65  &)
%                                         (   1 <=>;  65 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   1 propositional; 0-4 arity)
%            Number of functors    :   21 (   3 constant; 0-4 arity)
%            Number of variables   :   66 (   0 singleton;  66 !;   0 ?)
%            Maximal term depth    :    5 (   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_fcont_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( r1_fcont_2(A,B)
        <=> ( r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
            & ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ~ ( ~ r1_xreal_0(C,np__0)
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,np__0)
                            & ! [E] :
                                ( m1_subset_1(E,k1_numbers)
                               => ! [F] :
                                    ( m1_subset_1(F,k1_numbers)
                                   => ~ ( r2_hidden(E,B)
                                        & r2_hidden(F,B)
                                        & ~ r1_xreal_0(D,k18_complex1(k5_real_1(E,F)))
                                        & r1_xreal_0(C,k18_complex1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,A,E),k2_seq_1(k1_numbers,k1_numbers,A,F)))) ) ) ) ) ) ) ) ) ) ) )).

fof(t1_fcont_2,axiom,(
    $true )).

fof(t2_fcont_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ( ( r1_fcont_2(C,A)
          & r1_tarski(B,A) )
       => r1_fcont_2(C,B) ) ) )).

fof(t3_fcont_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( ( r1_fcont_2(C,A)
              & r1_fcont_2(D,B) )
           => r1_fcont_2(k6_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) )).

fof(t4_fcont_2,axiom,(
    ! [A,B,C] :
      ( ( v1_funct_1(C)
        & m2_relset_1(C,k1_numbers,k1_numbers) )
     => ! [D] :
          ( ( v1_funct_1(D)
            & m2_relset_1(D,k1_numbers,k1_numbers) )
         => ( ( r1_fcont_2(C,A)
              & r1_fcont_2(D,B) )
           => r1_fcont_2(k7_seq_1(k1_numbers,k1_numbers,C,D),k3_xboole_0(A,B)) ) ) ) )).

fof(t5_fcont_2,axiom,(
    ! [A,B] :
      ( m1_subset_1(B,k1_numbers)
     => ! [C] :
          ( ( v1_funct_1(C)
            & m2_relset_1(C,k1_numbers,k1_numbers) )
         => ( r1_fcont_2(C,A)
           => r1_fcont_2(k13_seq_1(k1_numbers,k1_numbers,C,B),A) ) ) ) )).

fof(t6_fcont_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_fcont_2(B,A)
       => r1_fcont_2(k16_seq_1(k1_numbers,k1_numbers,B),A) ) ) )).

fof(t7_fcont_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_fcont_2(B,A)
       => r1_fcont_2(k21_seq_1(k1_numbers,k1_numbers,B),A) ) ) )).

fof(t8_fcont_2,axiom,(
    ! [A,B,C,D,E] :
      ( ( v1_funct_1(E)
        & m2_relset_1(E,k1_numbers,k1_numbers) )
     => ! [F] :
          ( ( v1_funct_1(F)
            & m2_relset_1(F,k1_numbers,k1_numbers) )
         => ( ( r1_fcont_2(E,A)
              & r1_fcont_2(F,B)
              & r3_rfunct_1(E,C)
              & r3_rfunct_1(F,D) )
           => r1_fcont_2(k8_seq_1(k1_numbers,k1_numbers,E,F),k3_xboole_0(k3_xboole_0(k3_xboole_0(A,C),B),D)) ) ) ) )).

fof(t9_fcont_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r1_fcont_2(B,A)
       => r2_fcont_1(B,A) ) ) )).

fof(t10_fcont_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( r3_fcont_1(B,A)
       => r1_fcont_2(B,A) ) ) )).

fof(t11_fcont_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ( ( v1_rcomp_1(B)
              & r2_fcont_1(A,B) )
           => r1_fcont_2(A,B) ) ) ) )).

fof(t12_fcont_2,axiom,(
    $true )).

fof(t13_fcont_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_zfmisc_1(k1_numbers))
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
              & v1_rcomp_1(A)
              & r1_fcont_2(B,A) )
           => v1_rcomp_1(k10_relset_1(k1_numbers,k1_numbers,B,A)) ) ) ) )).

fof(t14_fcont_2,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_zfmisc_1(k1_numbers))
         => ~ ( B != k1_xboole_0
              & r1_tarski(B,k4_relset_1(k1_numbers,k1_numbers,A))
              & v1_rcomp_1(B)
              & r1_fcont_2(A,B)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( r2_hidden(C,B)
                          & r2_hidden(D,B)
                          & k2_seq_1(k1_numbers,k1_numbers,A,C) = k4_seq_4(k10_relset_1(k1_numbers,k1_numbers,A,B))
                          & k2_seq_1(k1_numbers,k1_numbers,A,D) = k5_seq_4(k10_relset_1(k1_numbers,k1_numbers,A,B)) ) ) ) ) ) ) )).

fof(t15_fcont_2,axiom,(
    ! [A,B] :
      ( ( v1_funct_1(B)
        & m2_relset_1(B,k1_numbers,k1_numbers) )
     => ( ( r1_tarski(A,k4_relset_1(k1_numbers,k1_numbers,B))
          & r1_partfun2(k1_numbers,k1_numbers,B,A) )
       => r1_fcont_2(B,A) ) ) )).

fof(t16_fcont_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_xreal_0(A,B)
                  & r2_fcont_1(C,k1_rcomp_1(A,B)) )
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( r2_hidden(D,k4_subset_1(k1_numbers,k1_rcomp_1(k2_seq_1(k1_numbers,k1_numbers,C,A),k2_seq_1(k1_numbers,k1_numbers,C,B)),k1_rcomp_1(k2_seq_1(k1_numbers,k1_numbers,C,B),k2_seq_1(k1_numbers,k1_numbers,C,A))))
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ~ ( r2_hidden(E,k1_rcomp_1(A,B))
                                & D = k2_seq_1(k1_numbers,k1_numbers,C,E) ) ) ) ) ) ) ) ) )).

fof(t17_fcont_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_xreal_0(A,B)
                  & r2_fcont_1(C,k1_rcomp_1(A,B)) )
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( r2_hidden(D,k1_rcomp_1(k5_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))),k4_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B)))))
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ~ ( r2_hidden(E,k1_rcomp_1(A,B))
                                & D = k2_seq_1(k1_numbers,k1_numbers,C,E) ) ) ) ) ) ) ) ) )).

fof(t18_fcont_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ~ ( v2_funct_1(C)
                  & r1_xreal_0(A,B)
                  & r2_fcont_1(C,k1_rcomp_1(A,B))
                  & ~ r1_rfunct_2(C,k1_rcomp_1(A,B))
                  & ~ r2_rfunct_2(C,k1_rcomp_1(A,B)) ) ) ) ) )).

fof(t19_fcont_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ~ ( v2_funct_1(C)
                  & r1_xreal_0(A,B)
                  & r2_fcont_1(C,k1_rcomp_1(A,B))
                  & ~ ( k5_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))) = k2_seq_1(k1_numbers,k1_numbers,C,A)
                      & k4_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))) = k2_seq_1(k1_numbers,k1_numbers,C,B) )
                  & ~ ( k5_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))) = k2_seq_1(k1_numbers,k1_numbers,C,B)
                      & k4_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))) = k2_seq_1(k1_numbers,k1_numbers,C,A) ) ) ) ) ) )).

fof(t20_fcont_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_xreal_0(A,B)
                  & r2_fcont_1(C,k1_rcomp_1(A,B)) )
               => k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B)) = k1_rcomp_1(k5_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))),k4_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B)))) ) ) ) ) )).

fof(t21_fcont_2,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( ( v1_funct_1(C)
                & v2_funct_1(C)
                & m2_relset_1(C,k1_numbers,k1_numbers) )
             => ( ( r1_xreal_0(A,B)
                  & r2_fcont_1(C,k1_rcomp_1(A,B)) )
               => r2_fcont_1(k2_partfun2(k1_numbers,k1_numbers,C),k1_rcomp_1(k5_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))),k4_seq_4(k10_relset_1(k1_numbers,k1_numbers,C,k1_rcomp_1(A,B))))) ) ) ) ) )).
%------------------------------------------------------------------------------