SET007 Axioms: SET007+107.ax
%------------------------------------------------------------------------------
% File : SET007+107 : TPTP v8.2.0. 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 unt; 0 def)
% Number of atoms : 153 ( 10 equ)
% Maximal formula atoms : 14 ( 6 avg)
% Number of connectives : 150 ( 19 ~; 0 |; 65 &)
% ( 1 <=>; 65 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 17 ( 15 usr; 1 prp; 0-4 aty)
% Number of functors : 21 ( 21 usr; 3 con; 0-4 aty)
% Number of variables : 66 ( 66 !; 0 ?)
% 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))))) ) ) ) ) ).
%------------------------------------------------------------------------------