SET007 Axioms: SET007+869.ax
%------------------------------------------------------------------------------
% File : SET007+869 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : 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 : mesfunc3 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 21 ( 0 unt; 0 def)
% Number of atoms : 267 ( 53 equ)
% Maximal formula atoms : 30 ( 12 avg)
% Number of connectives : 292 ( 46 ~; 4 |; 115 &)
% ( 3 <=>; 124 =>; 0 <=; 0 <~>)
% Maximal formula depth : 29 ( 15 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of predicates : 20 ( 19 usr; 0 prp; 1-5 aty)
% Number of functors : 29 ( 29 usr; 10 con; 0-4 aty)
% Number of variables : 113 ( 106 !; 7 ?)
% 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_mesfunc3,axiom,
! [A] :
( m2_subset_1(A,k1_numbers,k5_numbers)
=> ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k6_supinf_1)
& m2_relset_1(C,k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k6_supinf_1) )
=> ! [D] :
( m2_finseq_1(D,k6_supinf_1)
=> ! [E] :
( m2_finseq_1(E,k6_supinf_1)
=> ( ( k5_finsop_1(D) = k2_finseq_1(A)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k5_finsop_1(D))
& ! [G] :
( m2_finseq_1(G,k6_supinf_1)
=> ~ ( k5_finsop_1(G) = k2_finseq_1(B)
& k2_seq_1(k5_numbers,k6_supinf_1,D,F) = k15_rvsum_1(G)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k5_finsop_1(G))
=> k2_seq_1(k5_numbers,k6_supinf_1,G,H) = k2_seq_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k6_supinf_1,C,k4_tarski(F,H)) ) ) ) ) ) )
& k5_finsop_1(E) = k2_finseq_1(B)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(F,k5_finsop_1(E))
& ! [G] :
( m2_finseq_1(G,k6_supinf_1)
=> ~ ( k5_finsop_1(G) = k2_finseq_1(A)
& k2_seq_1(k5_numbers,k6_supinf_1,E,F) = k15_rvsum_1(G)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k5_finsop_1(G))
=> k2_seq_1(k5_numbers,k6_supinf_1,G,H) = k2_seq_1(k2_zfmisc_1(k2_finseq_1(A),k2_finseq_1(B)),k6_supinf_1,C,k4_tarski(H,F)) ) ) ) ) ) ) )
=> k15_rvsum_1(D) = k15_rvsum_1(E) ) ) ) ) ) ) ).
fof(t2_mesfunc3,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ! [B] :
( m2_finseq_1(B,k6_supinf_1)
=> ( A = B
=> k5_convfun1(A) = k15_rvsum_1(B) ) ) ) ).
fof(t3_mesfunc3,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) )
=> ~ ( r2_mesfunc2(A,B,C)
& ! [D] :
( ( v1_prob_2(D)
& m2_finseq_1(D,B) )
=> ! [E] :
( m2_finseq_1(E,k3_supinf_1)
=> ~ ( k1_relat_1(C) = k3_tarski(k2_relat_1(D))
& k5_finsop_1(D) = k5_finsop_1(E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k5_finsop_1(D))
=> ! [G] :
( r2_hidden(G,k1_funct_1(D,F))
=> k4_mesfunc1(A,C,G) = k4_mesfunc1(k5_numbers,E,F) ) ) )
& ! [F] :
~ ( r2_hidden(F,k1_relat_1(C))
& ! [G] :
( m2_finseq_1(G,k3_supinf_1)
=> ~ ( k5_finsop_1(G) = k5_finsop_1(E)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k5_finsop_1(G))
=> k4_mesfunc1(k5_numbers,G,H) = k2_extreal1(k4_mesfunc1(k5_numbers,E,H),k4_mesfunc1(A,k3_mesfunc2(k1_funct_1(D,H),A),F)) ) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_mesfunc3,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ( v1_prob_2(B)
<=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r2_hidden(C,k5_finsop_1(B))
& r2_hidden(D,k5_finsop_1(B)) )
=> ( C = D
| r1_xboole_0(k1_funct_1(B,C),k1_funct_1(B,D)) ) ) ) ) ) ) ).
fof(t5_mesfunc3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( ~ v1_xboole_0(C)
& v1_prob_1(C,A)
& v1_measure1(C,A)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [D] :
( ( v1_prob_2(D)
& m2_finseq_1(D,C) )
=> ! [E] :
( m2_finseq_1(E,C)
=> ( ( k5_finsop_1(E) = k5_finsop_1(D)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k5_finsop_1(E))
=> k1_funct_1(E,F) = k3_xboole_0(B,k1_funct_1(D,F)) ) ) )
=> ( v1_prob_2(E)
& m2_finseq_1(E,C) ) ) ) ) ) ) ).
fof(t6_mesfunc3,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m2_finseq_1(D,A)
=> ( ( k5_finsop_1(D) = k5_finsop_1(C)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k5_finsop_1(D))
=> k1_funct_1(D,E) = k3_xboole_0(B,k1_funct_1(C,E)) ) ) )
=> k3_tarski(k2_relat_1(D)) = k3_xboole_0(B,k3_tarski(k2_relat_1(C))) ) ) ) ) ).
fof(t7_mesfunc3,axiom,
! [A,B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_subset_1(C,k1_numbers,k5_numbers)
=> ( r2_hidden(C,k5_finsop_1(B))
=> ( r1_tarski(k1_funct_1(B,C),k3_tarski(k2_relat_1(B)))
& k3_xboole_0(k1_funct_1(B,C),k3_tarski(k2_relat_1(B))) = k1_funct_1(B,C) ) ) ) ) ).
fof(t8_mesfunc3,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_prob_2(D)
& m2_finseq_1(D,B) )
=> k5_finsop_1(D) = k5_finsop_1(k5_finseqop(B,k3_supinf_1,D,C)) ) ) ) ) ).
fof(t9_mesfunc3,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_prob_2(D)
& m2_finseq_1(D,B) )
=> k4_mesfunc1(B,C,k3_tarski(k2_relat_1(D))) = k5_convfun1(k5_finseqop(B,k3_supinf_1,D,C)) ) ) ) ) ).
fof(t10_mesfunc3,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ! [B] :
( m2_finseq_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( k5_finsop_1(A) = k5_finsop_1(B)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k5_finsop_1(B))
=> k4_mesfunc1(k5_numbers,B,D) = k2_extreal1(C,k4_mesfunc1(k5_numbers,A,D)) ) ) )
=> ( ( ~ ( C != k4_supinf_1
& C != k5_supinf_1 )
& ? [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
& r2_hidden(D,k5_finsop_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,k5_finsop_1(A))
& r1_supinf_1(k4_mesfunc1(k5_numbers,A,D),k1_supinf_2) ) )
| k5_convfun1(B) = k2_extreal1(C,k5_convfun1(A)) ) ) ) ) ) ).
fof(t11_mesfunc3,axiom,
! [A] :
( m2_finseq_1(A,k6_supinf_1)
=> m2_finseq_1(A,k3_supinf_1) ) ).
fof(d1_mesfunc3,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] :
( ( v1_prob_2(D)
& m2_finseq_1(D,B) )
=> ! [E] :
( m2_finseq_1(E,k3_supinf_1)
=> ( r1_mesfunc3(A,B,C,D,E)
<=> ( k1_relat_1(C) = k3_tarski(k2_relat_1(D))
& k5_finsop_1(D) = k5_finsop_1(E)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( r2_hidden(F,k5_finsop_1(D))
=> ! [G] :
( r2_hidden(G,k1_funct_1(D,F))
=> k4_mesfunc1(A,C,G) = k4_mesfunc1(k5_numbers,E,F) ) ) ) ) ) ) ) ) ) ) ).
fof(t12_mesfunc3,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) )
=> ~ ( r2_mesfunc2(A,B,C)
& ! [D] :
( ( v1_prob_2(D)
& m2_finseq_1(D,B) )
=> ! [E] :
( m2_finseq_1(E,k3_supinf_1)
=> ~ r1_mesfunc3(A,B,C,D,E) ) ) ) ) ) ) ).
fof(t13_mesfunc3,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_prob_2(C)
& m2_finseq_1(C,B) )
=> ? [D] :
( v1_prob_2(D)
& m2_finseq_1(D,B)
& k3_tarski(k2_relat_1(C)) = k3_tarski(k2_relat_1(D))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ( r2_hidden(E,k5_finsop_1(D))
=> ( k1_funct_1(D,E) != k1_xboole_0
& ? [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
& r2_hidden(F,k5_finsop_1(C))
& k1_funct_1(C,F) = k1_funct_1(D,E) ) ) ) ) ) ) ) ) ).
fof(t14_mesfunc3,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) )
=> ~ ( r2_mesfunc2(A,B,C)
& ! [D] :
( r2_hidden(D,k1_relat_1(C))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,C,D)) )
& ! [D] :
( ( v1_prob_2(D)
& m2_finseq_1(D,B) )
=> ! [E] :
( m2_finseq_1(E,k3_supinf_1)
=> ~ ( r1_mesfunc3(A,B,C,D,E)
& k4_mesfunc1(k5_numbers,E,np__1) = k1_supinf_2
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,F)
& r2_hidden(F,k5_finsop_1(E)) )
=> ( ~ r1_supinf_1(k4_mesfunc1(k5_numbers,E,F),k1_supinf_2)
& ~ r1_supinf_1(k4_supinf_1,k4_mesfunc1(k5_numbers,E,F)) ) ) ) ) ) ) ) ) ) ) ).
fof(t15_mesfunc3,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] :
( ( v1_prob_2(D)
& m2_finseq_1(D,B) )
=> ! [E] :
( m2_finseq_1(E,k3_supinf_1)
=> ! [F] :
( m1_subset_1(F,A)
=> ~ ( r1_mesfunc3(A,B,C,D,E)
& r2_hidden(F,k1_relat_1(C))
& ! [G] :
( m2_finseq_1(G,k3_supinf_1)
=> ~ ( k5_finsop_1(G) = k5_finsop_1(E)
& ! [H] :
( m2_subset_1(H,k1_numbers,k5_numbers)
=> ( r2_hidden(H,k5_finsop_1(G))
=> k4_mesfunc1(k5_numbers,G,H) = k2_extreal1(k4_mesfunc1(k5_numbers,E,H),k4_mesfunc1(A,k3_mesfunc2(k1_funct_1(D,H),A),F)) ) )
& k4_mesfunc1(A,C,F) = k5_convfun1(G) ) ) ) ) ) ) ) ) ) ).
fof(t16_mesfunc3,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ! [B] :
( m2_finseq_1(B,k6_supinf_1)
=> ( A = B
=> k5_convfun1(A) = k15_rvsum_1(B) ) ) ) ).
fof(t17_mesfunc3,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ( r2_hidden(B,k5_finsop_1(A))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(k5_numbers,A,B)) ) )
=> ( ! [B] :
( m2_subset_1(B,k1_numbers,k5_numbers)
=> ~ ( r2_hidden(B,k5_finsop_1(A))
& k4_mesfunc1(k5_numbers,A,B) = k4_supinf_1 ) )
| k5_convfun1(A) = k4_supinf_1 ) ) ) ).
fof(d2_mesfunc3,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) )
=> ( ( r2_mesfunc2(A,B,D)
& ! [E] :
( r2_hidden(E,k1_relat_1(D))
=> r1_supinf_1(k1_supinf_2,k4_mesfunc1(A,D,E)) ) )
=> ( k1_relat_1(D) = k1_xboole_0
| ! [E] :
( m1_subset_1(E,k3_supinf_1)
=> ( E = k1_mesfunc3(A,B,C,D)
<=> ? [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)
& r1_mesfunc3(A,B,D,F,G)
& k4_mesfunc1(k5_numbers,G,np__1) = k1_supinf_2
& ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__2,I)
& r2_hidden(I,k5_finsop_1(G)) )
=> ( ~ r1_supinf_1(k4_mesfunc1(k5_numbers,G,I),k1_supinf_2)
& ~ r1_supinf_1(k4_supinf_1,k4_mesfunc1(k5_numbers,G,I)) ) ) )
& k5_finsop_1(H) = k5_finsop_1(F)
& ! [I] :
( m2_subset_1(I,k1_numbers,k5_numbers)
=> ( r2_hidden(I,k5_finsop_1(H))
=> k4_mesfunc1(k5_numbers,H,I) = k2_extreal1(k4_mesfunc1(k5_numbers,G,I),k4_mesfunc1(k5_numbers,k5_finseqop(B,k3_supinf_1,F,C),I)) ) )
& E = k5_convfun1(H) ) ) ) ) ) ) ) ) ) ) ) ).
fof(t18_mesfunc3,axiom,
! [A] :
( m2_finseq_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( C = k3_finseq_1(A)
& ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( r2_hidden(D,k5_finsop_1(A))
=> k4_mesfunc1(k5_numbers,A,D) = B ) ) )
=> k5_convfun1(A) = k2_extreal1(C,B) ) ) ) ) ).
fof(dt_k1_mesfunc3,axiom,
! [A,B,C,D] :
( ( ~ v1_xboole_0(A)
& ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m3_measure1(C,A,B)
& v1_funct_1(D)
& m1_relset_1(D,A,k3_supinf_1) )
=> m1_subset_1(k1_mesfunc3(A,B,C,D),k3_supinf_1) ) ).
%------------------------------------------------------------------------------