SET007 Axioms: SET007+916.ax
%------------------------------------------------------------------------------
% File : SET007+916 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Linearity of Lebesgue Integral of Simple Valued Function
% 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 : mesfunc4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 6 ( 0 unt; 0 def)
% Number of atoms : 124 ( 24 equ)
% Maximal formula atoms : 24 ( 20 avg)
% Number of connectives : 132 ( 14 ~; 5 |; 54 &)
% ( 0 <=>; 59 =>; 0 <=; 0 <~>)
% Maximal formula depth : 26 ( 21 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-5 aty)
% Number of functors : 17 ( 17 usr; 6 con; 0-6 aty)
% Number of variables : 49 ( 49 !; 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(t1_mesfunc4,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ! [B] :
( m2_finseq_1(B,k3_supinf_1)
=> ! [C] :
( m2_finseq_1(C,k3_supinf_1)
=> ( ( ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(A))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(k5_numbers,A,D)) ) )
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k4_finseq_1(B))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(k5_numbers,B,D)) ) )
& k4_finseq_1(A) = k4_finseq_1(B)
& C = k13_mesfunc1(k5_numbers,A,B) )
=> k5_convfun1(C) = k2_supinf_2(k5_convfun1(A),k5_convfun1(B)) ) ) ) ) ).
fof(t2_mesfunc4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,k3_supinf_1) )
=> ! [F] :
( ( v1_prob_2(F)
& m2_finseq_1(F,B) )
=> ! [G] :
( m2_finseq_1(G,k3_supinf_1)
=> ! [H] :
( m2_finseq_1(H,k3_supinf_1)
=> ( ( r2_mesfunc2(A,B,E)
& ! [I] :
( r2_hidden(I,k1_relat_1(E))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,E,I)) )
& r1_mesfunc3(A,B,E,F,G)
& k4_finseq_1(H) = k4_finseq_1(F)
& ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ( r2_hidden(I,k4_finseq_1(H))
=> k4_mesfunc1(k5_numbers,H,I) = k2_extreal1(k4_mesfunc1(k5_numbers,G,I),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,F,C),I)) ) )
& k3_finseq_1(F) = D )
=> ( k1_relat_1(E) = k1_xboole_0
| k1_mesfunc3(A,B,C,E) = k5_convfun1(H) ) ) ) ) ) ) ) ) ) ) ).
fof(t3_mesfunc4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k3_supinf_1) )
=> ! [D] :
( m3_measure1(D,A,B)
=> ! [E] :
( ( v1_prob_2(E)
& m2_finseq_1(E,B) )
=> ! [F] :
( m2_finseq_1(F,k3_supinf_1)
=> ! [G] :
( m2_finseq_1(G,k3_supinf_1)
=> ( ( r2_mesfunc2(A,B,C)
& ! [H] :
( r2_hidden(H,k1_relat_1(C))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,C,H)) )
& r1_mesfunc3(A,B,C,E,F)
& k4_finseq_1(G) = k4_finseq_1(E)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k4_finseq_1(G))
=> k4_mesfunc1(k5_numbers,G,H) = k2_extreal1(k4_mesfunc1(k5_numbers,F,H),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,E,D),H)) ) ) )
=> ( k1_relat_1(C) = k1_xboole_0
| k1_mesfunc3(A,B,D,C) = k5_convfun1(G) ) ) ) ) ) ) ) ) ) ).
fof(t4_mesfunc4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& m2_relset_1(C,A,k3_supinf_1) )
=> ! [D] :
( m3_measure1(D,A,B)
=> ~ ( r2_mesfunc2(A,B,C)
& k1_relat_1(C) != k1_xboole_0
& ! [E] :
( r2_hidden(E,k1_relat_1(C))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,C,E)) )
& ! [E] :
( ( v1_prob_2(E)
& m2_finseq_1(E,B) )
=> ! [F] :
( m2_finseq_1(F,k3_supinf_1)
=> ! [G] :
( m2_finseq_1(G,k3_supinf_1)
=> ~ ( r1_mesfunc3(A,B,C,E,F)
& k4_finseq_1(G) = k4_finseq_1(E)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k4_finseq_1(G))
=> k4_mesfunc1(k5_numbers,G,H) = k2_extreal1(k4_mesfunc1(k5_numbers,F,H),k4_mesfunc1(k5_numbers,k1_partfun1(k5_numbers,B,B,k3_supinf_1,E,D),H)) ) )
& k1_mesfunc3(A,B,D,C) = k5_convfun1(G) ) ) ) ) ) ) ) ) ) ).
fof(t5_mesfunc4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k3_supinf_1) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,k3_supinf_1) )
=> ( ( r2_mesfunc2(A,B,D)
& ! [F] :
( r2_hidden(F,k1_relat_1(D))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,D,F)) )
& r2_mesfunc2(A,B,E)
& k1_relat_1(E) = k1_relat_1(D)
& ! [F] :
( r2_hidden(F,k1_relat_1(E))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,E,F)) ) )
=> ( k1_relat_1(D) = k1_xboole_0
| ( r2_mesfunc2(A,B,k13_mesfunc1(A,D,E))
& k1_relat_1(k13_mesfunc1(A,D,E)) != k1_xboole_0
& ! [F] :
( r2_hidden(F,k1_relat_1(k13_mesfunc1(A,D,E)))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,k13_mesfunc1(A,D,E),F)) )
& k1_mesfunc3(A,B,C,k13_mesfunc1(A,D,E)) = k2_supinf_2(k1_mesfunc3(A,B,C,D),k1_mesfunc3(A,B,C,E)) ) ) ) ) ) ) ) ) ).
fof(t6_mesfunc4,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( m3_measure1(C,A,B)
=> ! [D] :
( ( v1_funct_1(D)
& m2_relset_1(D,A,k3_supinf_1) )
=> ! [E] :
( ( v1_funct_1(E)
& m2_relset_1(E,A,k3_supinf_1) )
=> ! [F] :
( m1_subset_1(F,k3_supinf_1)
=> ( ( r2_mesfunc2(A,B,D)
& ! [G] :
( r2_hidden(G,k1_relat_1(D))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,D,G)) )
& r1_supinf_1(k1_supinf_2,F)
& k1_relat_1(E) = k1_relat_1(D)
& ! [G] :
( r2_hidden(G,k1_relat_1(E))
=> k4_mesfunc1(A,E,G) = k2_extreal1(F,k4_mesfunc1(A,D,G)) ) )
=> ( k1_relat_1(D) = k1_xboole_0
| r1_supinf_1(k4_supinf_1,F)
| k1_mesfunc3(A,B,C,E) = k2_extreal1(F,k1_mesfunc3(A,B,C,D)) ) ) ) ) ) ) ) ) ).
%------------------------------------------------------------------------------