SET007 Axioms: SET007+642.ax
%------------------------------------------------------------------------------
% File : SET007+642 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Integrability of Bounded Total Functions
% 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 : integra4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 31 ( 0 unt; 0 def)
% Number of atoms : 297 ( 21 equ)
% Maximal formula atoms : 24 ( 9 avg)
% Number of connectives : 274 ( 8 ~; 3 |; 141 &)
% ( 3 <=>; 119 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 10 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-3 aty)
% Number of functors : 36 ( 36 usr; 4 con; 0-4 aty)
% Number of variables : 99 ( 97 !; 2 ?)
% 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_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ( k3_integra1(A) = np__0
=> k3_finseq_1(B) = np__1 ) ) ) ).
fof(t2_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ( r3_integra1(A,k5_rfunct_1(A,A))
& k13_integra1(A,k5_rfunct_1(A,A)) = k3_integra1(A) ) ) ).
fof(t3_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( ( v1_partfun1(B,A,k1_numbers)
& k5_relset_1(A,k1_numbers,B) = k1_seq_4(C) )
<=> B = k13_seq_1(A,k1_numbers,k5_rfunct_1(A,A),C) ) ) ) ) ).
fof(t4_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m1_subset_1(C,k1_numbers)
=> ( k5_relset_1(A,k1_numbers,B) = k1_seq_4(C)
=> ( r3_integra1(A,B)
& k13_integra1(A,B) = k4_real_1(C,k3_integra1(A)) ) ) ) ) ) ).
fof(t5_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m1_subset_1(B,k1_numbers)
=> ? [C] :
( v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m2_relset_1(C,A,k1_numbers)
& k5_relset_1(A,k1_numbers,C) = k1_seq_4(B)
& r3_rfunct_1(C,A) ) ) ) ).
fof(t6_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( m3_integra1(C,A,k8_integra1(A))
=> ( k3_integra1(A) = np__0
=> ( r3_integra1(A,B)
& k13_integra1(A,B) = np__0 ) ) ) ) ) ).
fof(t7_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ~ ( r3_rfunct_1(B,A)
& r3_integra1(A,B)
& ! [C] :
( m1_subset_1(C,k1_numbers)
=> ~ ( r1_xreal_0(k4_pscomp_1(k5_relset_1(A,k1_numbers,B)),C)
& r1_xreal_0(C,k3_pscomp_1(k5_relset_1(A,k1_numbers,B)))
& k13_integra1(A,B) = k4_real_1(C,k3_integra1(A)) ) ) ) ) ) ).
fof(t8_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m2_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ( ( r3_rfunct_1(B,A)
& v4_seq_2(k2_integra2(A,C))
& k2_seq_2(k2_integra2(A,C)) = np__0 )
=> ( v4_seq_2(k4_integra2(A,B,C))
& k2_seq_2(k4_integra2(A,B,C)) = k12_integra1(A,B) ) ) ) ) ) ).
fof(t9_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m2_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ( ( r3_rfunct_1(B,A)
& v4_seq_2(k2_integra2(A,C))
& k2_seq_2(k2_integra2(A,C)) = np__0 )
=> ( v4_seq_2(k3_integra2(A,B,C))
& k2_seq_2(k3_integra2(A,B,C)) = k11_integra1(A,B) ) ) ) ) ) ).
fof(t10_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( r3_rfunct_1(B,A)
=> ( r1_integra1(A,B)
& r2_integra1(A,B) ) ) ) ) ).
fof(d1_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( m3_integra1(B,A,k8_integra1(A))
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r1_integra4(A,B,C)
<=> ( k3_finseq_1(B) = C
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> k1_goboard1(B,D) = k3_real_1(k4_pscomp_1(A),k4_real_1(k6_real_1(k3_integra1(A),k3_finseq_1(B)),D)) ) ) ) ) ) ) ) ).
fof(t11_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k8_integra1(A))
& m2_relset_1(B,k5_numbers,k8_integra1(A))
& v4_seq_2(k2_integra2(A,B))
& k2_seq_2(k2_integra2(A,B)) = np__0 ) ) ).
fof(t12_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( r3_rfunct_1(B,A)
=> ( r3_integra1(A,B)
<=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,k8_integra1(A))
& m2_relset_1(C,k5_numbers,k8_integra1(A)) )
=> ( ( v4_seq_2(k2_integra2(A,C))
& k2_seq_2(k2_integra2(A,C)) = np__0 )
=> k5_real_1(k2_seq_2(k3_integra2(A,B,C)),k2_seq_2(k4_integra2(A,B,C))) = np__0 ) ) ) ) ) ) ).
fof(t13_integra4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( v1_partfun1(k19_rfunct_3(A,B),A,k1_numbers)
& v1_partfun1(k20_rfunct_3(A,B),A,k1_numbers) ) ) ) ).
fof(t14_integra4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( r1_rfunct_1(C,B)
=> r1_rfunct_1(k19_rfunct_3(A,C),B) ) ) ) ).
fof(t15_integra4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> r2_rfunct_1(k19_rfunct_3(A,C),B) ) ) ).
fof(t16_integra4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( r2_rfunct_1(C,B)
=> r1_rfunct_1(k20_rfunct_3(A,C),B) ) ) ) ).
fof(t17_integra4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> r2_rfunct_1(k20_rfunct_3(A,C),B) ) ) ).
fof(t18_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( r1_rfunct_1(C,A)
=> v1_seq_4(k5_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,C,B))) ) ) ) ).
fof(t19_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B,C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k1_numbers) )
=> ( r2_rfunct_1(C,A)
=> v2_seq_4(k5_relset_1(A,k1_numbers,k2_partfun1(A,k1_numbers,C,B))) ) ) ) ).
fof(t20_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( ( r3_rfunct_1(B,A)
& r3_integra1(A,B) )
=> r3_integra1(A,k19_rfunct_3(A,B)) ) ) ) ).
fof(t21_integra4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( v1_funct_1(B)
& m2_relset_1(B,A,k1_numbers) )
=> k20_rfunct_3(A,B) = k19_rfunct_3(A,k16_seq_1(A,k1_numbers,B)) ) ) ).
fof(t22_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( ( r3_rfunct_1(B,A)
& r3_integra1(A,B) )
=> r3_integra1(A,k20_rfunct_3(A,B)) ) ) ) ).
fof(t23_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( ( r3_rfunct_1(B,A)
& r3_integra1(A,B) )
=> ( r3_integra1(A,k21_seq_1(A,k1_numbers,B))
& r1_xreal_0(k18_complex1(k13_integra1(A,B)),k13_integra1(A,k21_seq_1(A,k1_numbers,B))) ) ) ) ) ).
fof(t24_integra4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ( ! [D] :
( m1_subset_1(D,k1_numbers)
=> ! [E] :
( m1_subset_1(E,k1_numbers)
=> ( ( r2_hidden(D,B)
& r2_hidden(E,B) )
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,C,D),k2_seq_1(B,k1_numbers,C,E))),A) ) ) )
=> r1_xreal_0(k5_real_1(k3_pscomp_1(k5_relset_1(B,k1_numbers,C)),k4_pscomp_1(k5_relset_1(B,k1_numbers,C))),A) ) ) ) ) ).
fof(t25_integra4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,k1_numbers)
& m2_relset_1(D,B,k1_numbers) )
=> ( ( r3_rfunct_1(C,B)
& r1_xreal_0(np__0,A)
& ! [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(k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,D,E),k2_seq_1(B,k1_numbers,D,F))),k4_real_1(A,k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,C,E),k2_seq_1(B,k1_numbers,C,F))))) ) ) ) )
=> r1_xreal_0(k5_real_1(k3_pscomp_1(k5_relset_1(B,k1_numbers,D)),k4_pscomp_1(k5_relset_1(B,k1_numbers,D))),k4_real_1(A,k5_real_1(k3_pscomp_1(k5_relset_1(B,k1_numbers,C)),k4_pscomp_1(k5_relset_1(B,k1_numbers,C))))) ) ) ) ) ) ).
fof(t26_integra4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,k1_numbers)
& m2_relset_1(D,B,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,k1_numbers)
& m2_relset_1(E,B,k1_numbers) )
=> ( ( r3_rfunct_1(C,B)
& r3_rfunct_1(D,B)
& r1_xreal_0(np__0,A)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( r2_hidden(F,B)
& r2_hidden(G,B) )
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,E,F),k2_seq_1(B,k1_numbers,E,G))),k4_real_1(A,k3_real_1(k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,C,F),k2_seq_1(B,k1_numbers,C,G))),k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,D,F),k2_seq_1(B,k1_numbers,D,G)))))) ) ) ) )
=> r1_xreal_0(k5_real_1(k3_pscomp_1(k5_relset_1(B,k1_numbers,E)),k4_pscomp_1(k5_relset_1(B,k1_numbers,E))),k4_real_1(A,k3_real_1(k5_real_1(k3_pscomp_1(k5_relset_1(B,k1_numbers,C)),k4_pscomp_1(k5_relset_1(B,k1_numbers,C))),k5_real_1(k3_pscomp_1(k5_relset_1(B,k1_numbers,D)),k4_pscomp_1(k5_relset_1(B,k1_numbers,D)))))) ) ) ) ) ) ) ).
fof(t27_integra4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,k1_numbers)
& m2_relset_1(D,B,k1_numbers) )
=> ( ( r3_rfunct_1(C,B)
& r3_integra1(B,C)
& r3_rfunct_1(D,B)
& ! [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(k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,D,E),k2_seq_1(B,k1_numbers,D,F))),k4_real_1(A,k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,C,E),k2_seq_1(B,k1_numbers,C,F))))) ) ) ) )
=> ( r1_xreal_0(A,np__0)
| r3_integra1(B,D) ) ) ) ) ) ) ).
fof(t28_integra4,axiom,
! [A] :
( m1_subset_1(A,k1_numbers)
=> ! [B] :
( ( v1_integra1(B)
& m1_subset_1(B,k1_zfmisc_1(k1_numbers)) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,B,k1_numbers)
& m2_relset_1(C,B,k1_numbers) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,B,k1_numbers)
& m2_relset_1(D,B,k1_numbers) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,B,k1_numbers)
& m2_relset_1(E,B,k1_numbers) )
=> ( ( r3_rfunct_1(C,B)
& r3_integra1(B,C)
& r3_rfunct_1(D,B)
& r3_integra1(B,D)
& r3_rfunct_1(E,B)
& ! [F] :
( m1_subset_1(F,k1_numbers)
=> ! [G] :
( m1_subset_1(G,k1_numbers)
=> ( ( r2_hidden(F,B)
& r2_hidden(G,B) )
=> r1_xreal_0(k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,E,F),k2_seq_1(B,k1_numbers,E,G))),k4_real_1(A,k3_real_1(k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,C,F),k2_seq_1(B,k1_numbers,C,G))),k18_complex1(k5_real_1(k2_seq_1(B,k1_numbers,D,F),k2_seq_1(B,k1_numbers,D,G)))))) ) ) ) )
=> ( r1_xreal_0(A,np__0)
| r3_integra1(B,E) ) ) ) ) ) ) ) ).
fof(t29_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,k1_numbers)
& m2_relset_1(C,A,k1_numbers) )
=> ( ( r3_rfunct_1(B,A)
& r3_integra1(A,B)
& r3_rfunct_1(C,A)
& r3_integra1(A,C) )
=> r3_integra1(A,k8_seq_1(A,k1_numbers,B,C)) ) ) ) ) ).
fof(t30_integra4,axiom,
! [A] :
( ( v1_integra1(A)
& m1_subset_1(A,k1_zfmisc_1(k1_numbers)) )
=> ! [B] :
( ( v1_funct_1(B)
& v1_funct_2(B,A,k1_numbers)
& m2_relset_1(B,A,k1_numbers) )
=> ( ( r3_rfunct_1(B,A)
& r3_integra1(A,B)
& r3_rfunct_1(k4_rfunct_1(A,k1_numbers,B),A) )
=> ( r2_hidden(np__0,k5_relset_1(A,k1_numbers,B))
| r3_integra1(A,k4_rfunct_1(A,k1_numbers,B)) ) ) ) ) ).
%------------------------------------------------------------------------------