SET007 Axioms: SET007+109.ax


%------------------------------------------------------------------------------
% File     : SET007+109 : TPTP v7.5.0. Released v3.4.0.
% Domain   : Set Theory
% Axioms   : Average Value Theorems for Real Functions of One Variable
% 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    : rolle [Urb08]

% Status   : Satisfiable
% Syntax   : Number of formulae    :   12 (   0 unit)
%            Number of atoms       :  143 (  12 equality)
%            Maximal formula depth :   22 (  16 average)
%            Number of connectives :  164 (  33 ~  ;   0  |;  55  &)
%                                         (   0 <=>;  76 =>;   0 <=)
%                                         (   0 <~>;   0 ~|;   0 ~&)
%            Number of predicates  :   13 (   0 propositional; 1-4 arity)
%            Number of functors    :   12 (   3 constant; 0-4 arity)
%            Number of variables   :   54 (   0 singleton;  53 !;   1 ?)
%            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_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ~ ( r2_fcont_1(C,k1_rcomp_1(A,B))
                    & k2_seq_1(k1_numbers,k1_numbers,C,A) = k2_seq_1(k1_numbers,k1_numbers,C,B)
                    & r2_fdiff_1(C,k2_rcomp_1(A,B))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( r2_hidden(D,k2_rcomp_1(A,B))
                            & k1_fdiff_1(C,D) = np__0 ) ) ) ) ) ) ) )).

fof(t2_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,np__0)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ~ ( r2_fcont_1(C,k1_rcomp_1(A,k3_real_1(A,B)))
                    & k2_seq_1(k1_numbers,k1_numbers,C,A) = k2_seq_1(k1_numbers,k1_numbers,C,k3_real_1(A,B))
                    & r2_fdiff_1(C,k2_rcomp_1(A,k3_real_1(A,B)))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,np__0)
                            & ~ r1_xreal_0(np__1,D)
                            & k1_fdiff_1(C,k3_real_1(A,k4_real_1(D,B))) = np__0 ) ) ) ) ) ) ) )).

fof(t3_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ~ ( r2_fcont_1(C,k1_rcomp_1(A,B))
                    & r2_fdiff_1(C,k2_rcomp_1(A,B))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( r2_hidden(D,k2_rcomp_1(A,B))
                            & k1_fdiff_1(C,D) = k6_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,B),k2_seq_1(k1_numbers,k1_numbers,C,A)),k5_real_1(B,A)) ) ) ) ) ) ) ) )).

fof(t4_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,np__0)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ~ ( r2_fcont_1(C,k1_rcomp_1(A,k3_real_1(A,B)))
                    & r2_fdiff_1(C,k2_rcomp_1(A,k3_real_1(A,B)))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( ~ r1_xreal_0(D,np__0)
                            & ~ r1_xreal_0(np__1,D)
                            & k2_seq_1(k1_numbers,k1_numbers,C,k3_real_1(A,B)) = k3_real_1(k2_seq_1(k1_numbers,k1_numbers,C,A),k4_real_1(B,k1_fdiff_1(C,k3_real_1(A,k4_real_1(D,B))))) ) ) ) ) ) ) ) )).

fof(t5_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [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) )
                   => ~ ( r2_fcont_1(C,k1_rcomp_1(A,B))
                        & r2_fdiff_1(C,k2_rcomp_1(A,B))
                        & r2_fcont_1(D,k1_rcomp_1(A,B))
                        & r2_fdiff_1(D,k2_rcomp_1(A,B))
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ~ ( r2_hidden(E,k2_rcomp_1(A,B))
                                & k4_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,B),k2_seq_1(k1_numbers,k1_numbers,C,A)),k1_fdiff_1(D,E)) = k4_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,D,B),k2_seq_1(k1_numbers,k1_numbers,D,A)),k1_fdiff_1(C,E)) ) ) ) ) ) ) ) ) )).

fof(t6_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,np__0)
           => ! [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) )
                   => ~ ( r2_fcont_1(C,k1_rcomp_1(A,k3_real_1(A,B)))
                        & r2_fdiff_1(C,k2_rcomp_1(A,k3_real_1(A,B)))
                        & r2_fcont_1(D,k1_rcomp_1(A,k3_real_1(A,B)))
                        & r2_fdiff_1(D,k2_rcomp_1(A,k3_real_1(A,B)))
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ~ ( r2_hidden(E,k2_rcomp_1(A,k3_real_1(A,B)))
                                & k1_fdiff_1(D,E) = np__0 ) )
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ~ ( ~ r1_xreal_0(E,np__0)
                                & ~ r1_xreal_0(np__1,E)
                                & k6_real_1(k5_real_1(k2_seq_1(k1_numbers,k1_numbers,C,k3_real_1(A,B)),k2_seq_1(k1_numbers,k1_numbers,C,A)),k5_real_1(k2_seq_1(k1_numbers,k1_numbers,D,k3_real_1(A,B)),k2_seq_1(k1_numbers,k1_numbers,D,A))) = k6_real_1(k1_fdiff_1(C,k3_real_1(A,k4_real_1(E,B))),k1_fdiff_1(D,k3_real_1(A,k4_real_1(E,B)))) ) ) ) ) ) ) ) ) )).

fof(t7_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ( ( r2_fdiff_1(C,k2_rcomp_1(A,B))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ( r2_hidden(D,k2_rcomp_1(A,B))
                         => k1_fdiff_1(C,D) = np__0 ) ) )
                 => r1_partfun2(k1_numbers,k1_numbers,C,k2_rcomp_1(A,B)) ) ) ) ) ) )).

fof(t8_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [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) )
                   => ( ( r2_fdiff_1(C,k2_rcomp_1(A,B))
                        & r2_fdiff_1(D,k2_rcomp_1(A,B))
                        & ! [E] :
                            ( m1_subset_1(E,k1_numbers)
                           => ( r2_hidden(E,k2_rcomp_1(A,B))
                             => k1_fdiff_1(C,E) = k1_fdiff_1(D,E) ) ) )
                     => ( r1_partfun2(k1_numbers,k1_numbers,k7_seq_1(k1_numbers,k1_numbers,C,D),k2_rcomp_1(A,B))
                        & ? [E] :
                            ( m1_subset_1(E,k1_numbers)
                            & ! [F] :
                                ( m1_subset_1(F,k1_numbers)
                               => ( r2_hidden(F,k2_rcomp_1(A,B))
                                 => k2_seq_1(k1_numbers,k1_numbers,C,F) = k3_real_1(k2_seq_1(k1_numbers,k1_numbers,D,F),E) ) ) ) ) ) ) ) ) ) ) )).

fof(t9_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ( ( r2_fdiff_1(C,k2_rcomp_1(A,B))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( r2_hidden(D,k2_rcomp_1(A,B))
                            & r1_xreal_0(k1_fdiff_1(C,D),np__0) ) ) )
                 => r1_rfunct_2(C,k2_rcomp_1(A,B)) ) ) ) ) ) )).

fof(t10_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ( ( r2_fdiff_1(C,k2_rcomp_1(A,B))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ~ ( r2_hidden(D,k2_rcomp_1(A,B))
                            & r1_xreal_0(np__0,k1_fdiff_1(C,D)) ) ) )
                 => r2_rfunct_2(C,k2_rcomp_1(A,B)) ) ) ) ) ) )).

fof(t11_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ( ( r2_fdiff_1(C,k2_rcomp_1(A,B))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ( r2_hidden(D,k2_rcomp_1(A,B))
                         => r1_xreal_0(np__0,k1_fdiff_1(C,D)) ) ) )
                 => r3_rfunct_2(C,k2_rcomp_1(A,B)) ) ) ) ) ) )).

fof(t12_rolle,axiom,(
    ! [A] :
      ( m1_subset_1(A,k1_numbers)
     => ! [B] :
          ( m1_subset_1(B,k1_numbers)
         => ( ~ r1_xreal_0(B,A)
           => ! [C] :
                ( ( v1_funct_1(C)
                  & m2_relset_1(C,k1_numbers,k1_numbers) )
               => ( ( r2_fdiff_1(C,k2_rcomp_1(A,B))
                    & ! [D] :
                        ( m1_subset_1(D,k1_numbers)
                       => ( r2_hidden(D,k2_rcomp_1(A,B))
                         => r1_xreal_0(k1_fdiff_1(C,D),np__0) ) ) )
                 => r4_rfunct_2(C,k2_rcomp_1(A,B)) ) ) ) ) ) )).
%------------------------------------------------------------------------------