SET007 Axioms: SET007+109.ax
%------------------------------------------------------------------------------
% File : SET007+109 : TPTP v8.2.0. 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 unt; 0 def)
% Number of atoms : 143 ( 12 equ)
% Maximal formula atoms : 18 ( 11 avg)
% Number of connectives : 164 ( 33 ~; 0 |; 55 &)
% ( 0 <=>; 76 =>; 0 <=; 0 <~>)
% Maximal formula depth : 22 ( 16 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-4 aty)
% Number of functors : 12 ( 12 usr; 3 con; 0-4 aty)
% Number of variables : 54 ( 53 !; 1 ?)
% 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)) ) ) ) ) ) ).
%------------------------------------------------------------------------------