SET007 Axioms: SET007+137.ax


%------------------------------------------------------------------------------
% File     : SET007+137 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : The de l'Hospital Theorem
% 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    : l_hospit [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   10 (   0 unit)
%            Number of atoms       :  205 (  28 equality)
%            Maximal formula depth :   25 (  19 average)
%            Number of connectives :  235 (  40 ~  ;   0  |; 131  &)
%                                         (   0 <=>;  64 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   17 (   0 propositional; 1-3 arity)
%            Number of functors    :   24 (   3 constant; 0-4 arity)
%            Number of variables   :   58 (   0 singleton;  48 !;  10 ?)
%            Maximal term depth    :    6 (   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(t1_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ( r1_fcont_1(A,B)
              & ! [C] :
                  ( m1_subset_1(C,k1_numbers)
                 => ! [D] :
                      ( m1_subset_1(D,k1_numbers)
                     => ~ ( ~ r1_xreal_0(B,C)
                          & ~ r1_xreal_0(D,B)
                          & ! [E] :
                              ( m1_subset_1(E,k1_numbers)
                             => ! [F] :
                                  ( m1_subset_1(F,k1_numbers)
                                 => ~ ( ~ r1_xreal_0(E,C)
                                      & ~ r1_xreal_0(B,E)
                                      & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
                                      & ~ r1_xreal_0(D,F)
                                      & ~ r1_xreal_0(F,B)
                                      & r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) ) ) ) )
           => r1_limfunc3(A,B) ) ) ) )).

fof(t2_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r4_limfunc2(A,B)
                    & k2_limfunc2(A,B) = C )
                 => ( ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,B)
                            & ! [E] :
                                ( m1_subset_1(E,k1_numbers)
                               => ~ ( ~ r1_xreal_0(D,E)
                                    & ~ r1_xreal_0(E,B)
                                    & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
                    & ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,k5_numbers,k1_numbers)
                          & m2_relset_1(D,k5_numbers,k1_numbers) )
                       => ( ( v4_seq_2(D)
                            & k2_seq_2(D) = B
                            & r1_tarski(k1_rfunct_2(D),k3_xboole_0(k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
                         => ( v4_seq_2(k2_rfunct_2(A,D))
                            & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) )
                & ( ( ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,B)
                            & ! [E] :
                                ( m1_subset_1(E,k1_numbers)
                               => ~ ( ~ r1_xreal_0(D,E)
                                    & ~ r1_xreal_0(E,B)
                                    & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
                    & ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,k5_numbers,k1_numbers)
                          & m2_relset_1(D,k5_numbers,k1_numbers) )
                       => ( ( v4_seq_2(D)
                            & k2_seq_2(D) = B
                            & r1_tarski(k1_rfunct_2(D),k3_xboole_0(k4_relset_1(k1_numbers,k1_numbers,A),k4_limfunc1(B))) )
                         => ( v4_seq_2(k2_rfunct_2(A,D))
                            & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) )
                 => ( r4_limfunc2(A,B)
                    & k2_limfunc2(A,B) = C ) ) ) ) ) ) )).

fof(t3_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ( ( r1_limfunc2(A,B)
                    & k1_limfunc2(A,B) = C )
                 => ( ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(B,D)
                            & ! [E] :
                                ( m1_subset_1(E,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,D)
                                    & ~ r1_xreal_0(B,E)
                                    & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
                    & ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,k5_numbers,k1_numbers)
                          & m2_relset_1(D,k5_numbers,k1_numbers) )
                       => ( ( v4_seq_2(D)
                            & k2_seq_2(D) = B
                            & r1_tarski(k1_rfunct_2(D),k3_xboole_0(k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
                         => ( v4_seq_2(k2_rfunct_2(A,D))
                            & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) ) )
                & ( ( ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(B,D)
                            & ! [E] :
                                ( m1_subset_1(E,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,D)
                                    & ~ r1_xreal_0(B,E)
                                    & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) )
                    & ! [D] :
                        ( ( v1_funct_1(D)
                          & v1_funct_2(D,k5_numbers,k1_numbers)
                          & m2_relset_1(D,k5_numbers,k1_numbers) )
                       => ( ( v4_seq_2(D)
                            & k2_seq_2(D) = B
                            & r1_tarski(k1_rfunct_2(D),k3_xboole_0(k4_relset_1(k1_numbers,k1_numbers,A),k12_prob_1(B))) )
                         => ( v4_seq_2(k2_rfunct_2(A,D))
                            & k2_seq_2(k2_rfunct_2(A,D)) = C ) ) ) )
                 => ( r1_limfunc2(A,B)
                    & k1_limfunc2(A,B) = C ) ) ) ) ) ) )).

fof(t4_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ? [C] :
                ( m1_rcomp_1(C,B)
                & r1_tarski(k4_xboole_0(C,k1_tarski(B)),k4_relset_1(k1_numbers,k1_numbers,A)) )
           => ! [C] :
                ( m1_subset_1(C,k1_numbers)
               => ! [D] :
                    ( m1_subset_1(D,k1_numbers)
                   => ~ ( ~ r1_xreal_0(B,C)
                        & ~ r1_xreal_0(D,B)
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ~ ( ~ r1_xreal_0(E,C)
                                    & ~ r1_xreal_0(B,E)
                                    & r2_hidden(E,k4_relset_1(k1_numbers,k1_numbers,A))
                                    & ~ r1_xreal_0(D,F)
                                    & ~ r1_xreal_0(F,B)
                                    & r2_hidden(F,k4_relset_1(k1_numbers,k1_numbers,A)) ) ) ) ) ) ) ) ) ) )).

fof(t5_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ? [D] :
                    ( m1_rcomp_1(D,C)
                    & r2_fdiff_1(A,D)
                    & r2_fdiff_1(B,D)
                    & r1_tarski(k4_xboole_0(D,k1_tarski(C)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
                    & r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D))))
                    & k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
                    & k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
                    & r2_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) )
               => r2_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) ) ) ) ) )).

fof(t6_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ? [D] :
                    ( m1_rcomp_1(D,C)
                    & r2_fdiff_1(A,D)
                    & r2_fdiff_1(B,D)
                    & r1_tarski(k4_xboole_0(D,k1_tarski(C)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
                    & r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D))))
                    & k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
                    & k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
                    & r3_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) )
               => r3_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) ) ) ) ) )).

fof(t7_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ? [D] :
                    ( m1_subset_1(D,k1_numbers)
                    & ~ r1_xreal_0(D,np__0)
                    & r2_fdiff_1(A,k2_rcomp_1(C,k3_real_1(C,D)))
                    & r2_fdiff_1(B,k2_rcomp_1(C,k3_real_1(C,D)))
                    & r1_tarski(k2_rcomp_1(C,k3_real_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
                    & r1_tarski(k1_rcomp_1(C,k3_real_1(C,D)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(C,k3_real_1(C,D))),k2_fdiff_1(B,k2_rcomp_1(C,k3_real_1(C,D))))))
                    & k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
                    & k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
                    & r1_fcont_1(A,C)
                    & r1_fcont_1(B,C)
                    & r4_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(C,k3_real_1(C,D))),k2_fdiff_1(B,k2_rcomp_1(C,k3_real_1(C,D)))),C) )
               => ( r4_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
                  & ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ~ r1_xreal_0(D,np__0)
                      & k2_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) = k2_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(C,k3_real_1(C,D))),k2_fdiff_1(B,k2_rcomp_1(C,k3_real_1(C,D)))),C) ) ) ) ) ) ) )).

fof(t8_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ? [D] :
                    ( m1_subset_1(D,k1_numbers)
                    & ~ r1_xreal_0(D,np__0)
                    & r2_fdiff_1(A,k2_rcomp_1(k5_real_1(C,D),C))
                    & r2_fdiff_1(B,k2_rcomp_1(k5_real_1(C,D),C))
                    & r1_tarski(k2_rcomp_1(k5_real_1(C,D),C),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
                    & r1_tarski(k1_rcomp_1(k5_real_1(C,D),C),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(k5_real_1(C,D),C)),k2_fdiff_1(B,k2_rcomp_1(k5_real_1(C,D),C)))))
                    & k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
                    & k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
                    & r1_fcont_1(A,C)
                    & r1_fcont_1(B,C)
                    & r1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(k5_real_1(C,D),C)),k2_fdiff_1(B,k2_rcomp_1(k5_real_1(C,D),C))),C) )
               => ( r1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
                  & ? [D] :
                      ( m1_subset_1(D,k1_numbers)
                      & ~ r1_xreal_0(D,np__0)
                      & k1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) = k1_limfunc2(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,k2_rcomp_1(k5_real_1(C,D),C)),k2_fdiff_1(B,k2_rcomp_1(k5_real_1(C,D),C))),C) ) ) ) ) ) ) )).

fof(t9_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ? [D] :
                    ( m1_rcomp_1(D,C)
                    & r2_fdiff_1(A,D)
                    & r2_fdiff_1(B,D)
                    & r1_tarski(k4_xboole_0(D,k1_tarski(C)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
                    & r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D))))
                    & k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
                    & k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
                    & r1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) )
               => ( r1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
                  & ? [D] :
                      ( m1_rcomp_1(D,C)
                      & k1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) = k1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) ) ) ) ) ) ) )).

fof(t10_l_hospit,axiom,(
    ! [A] :
      ( ( v1_funct_1(A)
        & m2_relset_1(A,k1_numbers,k1_numbers) )
     => ! [B] :
          ( ( v1_funct_1(B)
            & m2_relset_1(B,k1_numbers,k1_numbers) )
         => ! [C] :
              ( m1_subset_1(C,k1_numbers)
             => ( ? [D] :
                    ( m1_rcomp_1(D,C)
                    & r2_fdiff_1(A,D)
                    & r2_fdiff_1(B,D)
                    & r1_tarski(k4_xboole_0(D,k1_tarski(C)),k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,A,B)))
                    & r1_tarski(D,k4_relset_1(k1_numbers,k1_numbers,k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D))))
                    & k2_seq_1(k1_numbers,k1_numbers,A,C) = np__0
                    & k2_seq_1(k1_numbers,k1_numbers,B,C) = np__0
                    & r1_fcont_1(k2_rfunct_1(k1_numbers,k1_numbers,k2_fdiff_1(A,D),k2_fdiff_1(B,D)),C) )
               => ( r1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C)
                  & k1_limfunc3(k2_rfunct_1(k1_numbers,k1_numbers,A,B),C) = k6_real_1(k1_fdiff_1(A,C),k1_fdiff_1(B,C)) ) ) ) ) ) )).
%------------------------------------------------------------------------------