SET007 Axioms: SET007+160.ax
%------------------------------------------------------------------------------
% File : SET007+160 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Properties of Caratheodor's Measure
% 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 : measure4 [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 44 ( 6 unt; 0 def)
% Number of atoms : 244 ( 29 equ)
% Maximal formula atoms : 18 ( 5 avg)
% Number of connectives : 219 ( 19 ~; 0 |; 103 &)
% ( 6 <=>; 91 =>; 0 <=; 0 <~>)
% Maximal formula depth : 18 ( 8 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 23 ( 21 usr; 1 prp; 0-3 aty)
% Number of functors : 33 ( 33 usr; 9 con; 0-5 aty)
% Number of variables : 133 ( 128 !; 5 ?)
% 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(fc1_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ( ~ v1_xboole_0(k3_measure4(A,B))
& v1_prob_1(k3_measure4(A,B),A)
& v1_finsub_1(k3_measure4(A,B))
& v2_finsub_1(k3_measure4(A,B))
& v1_measure1(k3_measure4(A,B),A) ) ) ).
fof(t1_measure4,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ( ( r1_supinf_1(k1_supinf_2,A)
& r1_supinf_1(k1_supinf_2,B)
& r1_supinf_1(k1_supinf_2,C) )
=> k2_supinf_2(k2_supinf_2(A,B),C) = k2_supinf_2(A,k2_supinf_2(B,C)) ) ) ) ) ).
fof(t2_measure4,axiom,
! [A] :
( m1_subset_1(A,k3_supinf_1)
=> ! [B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ~ ( A != k5_supinf_1
& A != k4_supinf_1
& ~ ( r1_supinf_1(k2_supinf_2(B,A),C)
<=> r1_supinf_1(B,k4_supinf_2(C,A)) ) ) ) ) ) ).
fof(t3_measure4,axiom,
$true ).
fof(t4_measure4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m2_relset_1(C,k5_numbers,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ! [E] :
( m2_subset_1(E,k1_zfmisc_1(A),B)
=> ( ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,F) = k3_xboole_0(E,k8_funct_2(k5_numbers,B,C,F)) )
=> k3_tarski(k2_relat_1(D)) = k3_xboole_0(E,k3_tarski(k2_relat_1(C))) ) ) ) ) ) ).
fof(t5_measure4,axiom,
! [A,B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k5_numbers,B)
& m2_relset_1(C,k5_numbers,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,k5_numbers,B)
& m2_relset_1(D,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,D,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,D,k1_nat_1(E,np__1)) = k2_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(E,np__1)),k8_funct_2(k5_numbers,B,D,E)) ) )
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B) )
=> ( ( k8_funct_2(k5_numbers,B,E,np__0) = k8_funct_2(k5_numbers,B,C,np__0)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,E,k1_nat_1(F,np__1)) = k4_xboole_0(k8_funct_2(k5_numbers,B,C,k1_nat_1(F,np__1)),k8_funct_2(k5_numbers,B,D,F)) ) )
=> k3_tarski(k2_relat_1(C)) = k3_tarski(k2_relat_1(E)) ) ) ) ) ) ) ).
fof(t6_measure4,axiom,
! [A] :
( ~ v1_xboole_0(k1_zfmisc_1(A))
& v1_prob_1(k1_zfmisc_1(A),A)
& v1_measure1(k1_zfmisc_1(A),A)
& m1_subset_1(k1_zfmisc_1(A),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(t7_measure4,axiom,
! [A,B] :
( m1_subset_1(B,k3_supinf_1)
=> ! [C] :
( m1_subset_1(C,k3_supinf_1)
=> ? [D] :
( v1_funct_1(D)
& v1_funct_2(D,k1_zfmisc_1(A),k3_supinf_1)
& m2_relset_1(D,k1_zfmisc_1(A),k3_supinf_1)
& ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( ( E = k1_xboole_0
=> k1_measure1(k1_zfmisc_1(A),D,E) = B )
& ( E != k1_xboole_0
=> k1_measure1(k1_zfmisc_1(A),D,E) = C ) ) ) ) ) ) ).
fof(t8_measure4,axiom,
! [A] :
? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k3_supinf_1)
& m2_relset_1(B,k1_zfmisc_1(A),k3_supinf_1)
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> k1_measure1(k1_zfmisc_1(A),B,C) = k1_supinf_2 ) ) ).
fof(t9_measure4,axiom,
$true ).
fof(t10_measure4,axiom,
$true ).
fof(t11_measure4,axiom,
! [A] :
? [B] :
( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k3_supinf_1)
& m2_relset_1(B,k1_zfmisc_1(A),k3_supinf_1)
& v6_supinf_2(B,k1_zfmisc_1(A))
& k1_funct_1(B,k1_xboole_0) = k1_supinf_2
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r1_tarski(C,D)
=> ( r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,C),k1_measure1(k1_zfmisc_1(A),B,D))
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(E,k5_numbers,k1_zfmisc_1(A)) )
=> r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,k5_setfam_1(A,k1_measure4(A,E))),k19_supinf_2(k2_measure4(k5_numbers,k1_zfmisc_1(A),k3_supinf_1,E,B))) ) ) ) ) ) ) ).
fof(d1_measure4,axiom,
$true ).
fof(d2_measure4,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k3_supinf_1)
& m2_relset_1(B,k1_zfmisc_1(A),k3_supinf_1) )
=> ( m1_measure4(B,A)
<=> ( v6_supinf_2(B,k1_zfmisc_1(A))
& k1_funct_1(B,k1_xboole_0) = k1_supinf_2
& ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r1_tarski(C,D)
=> ( r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,C),k1_measure1(k1_zfmisc_1(A),B,D))
& ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,k1_zfmisc_1(A))
& m2_relset_1(E,k5_numbers,k1_zfmisc_1(A)) )
=> r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,k5_setfam_1(A,k1_measure4(A,E))),k19_supinf_2(k2_measure4(k5_numbers,k1_zfmisc_1(A),k3_supinf_1,E,B))) ) ) ) ) ) ) ) ) ).
fof(d3_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ! [C] :
( ( ~ v1_xboole_0(C)
& m1_subset_1(C,k1_zfmisc_1(k1_zfmisc_1(A))) )
=> ( C = k3_measure4(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,C)
<=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ! [F] :
( m1_subset_1(F,k1_zfmisc_1(A))
=> ( ( r1_tarski(E,D)
& r1_tarski(F,k4_xboole_0(A,D)) )
=> r1_supinf_1(k2_supinf_2(k1_measure1(k1_zfmisc_1(A),B,E),k1_measure1(k1_zfmisc_1(A),B,F)),k1_measure1(k1_zfmisc_1(A),B,k4_subset_1(A,E,F))) ) ) ) ) ) ) ) ) ).
fof(t12_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> r1_supinf_1(k1_measure1(k1_zfmisc_1(A),B,k4_subset_1(A,C,D)),k2_supinf_2(k1_measure1(k1_zfmisc_1(A),B,C),k1_measure1(k1_zfmisc_1(A),B,D))) ) ) ) ).
fof(t13_measure4,axiom,
$true ).
fof(t14_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( r2_hidden(C,k3_measure4(A,B))
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,k1_zfmisc_1(A))
=> ( ( r1_tarski(D,C)
& r1_tarski(E,k4_xboole_0(A,C)) )
=> k2_supinf_2(k1_measure1(k1_zfmisc_1(A),B,D),k1_measure1(k1_zfmisc_1(A),B,E)) = k1_measure1(k1_zfmisc_1(A),B,k4_subset_1(A,D,E)) ) ) ) ) ) ) ).
fof(t15_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( ( r2_hidden(C,k3_measure4(A,B))
& r2_hidden(D,k3_measure4(A,B))
& r1_xboole_0(D,C) )
=> k1_measure1(k1_zfmisc_1(A),B,k4_subset_1(A,C,D)) = k2_supinf_2(k1_measure1(k1_zfmisc_1(A),B,C),k1_measure1(k1_zfmisc_1(A),B,D)) ) ) ) ) ).
fof(t16_measure4,axiom,
! [A,B,C] :
( m1_measure4(C,B)
=> ( r2_hidden(A,k3_measure4(B,C))
=> r2_hidden(k4_xboole_0(B,A),k3_measure4(B,C)) ) ) ).
fof(t17_measure4,axiom,
! [A,B,C,D] :
( m1_measure4(D,A)
=> ( ( r2_hidden(B,k3_measure4(A,D))
& r2_hidden(C,k3_measure4(A,D)) )
=> r2_hidden(k2_xboole_0(B,C),k3_measure4(A,D)) ) ) ).
fof(t18_measure4,axiom,
! [A,B,C,D] :
( m1_measure4(D,A)
=> ( ( r2_hidden(B,k3_measure4(A,D))
& r2_hidden(C,k3_measure4(A,D)) )
=> r2_hidden(k3_xboole_0(B,C),k3_measure4(A,D)) ) ) ).
fof(t19_measure4,axiom,
! [A,B,C,D] :
( m1_measure4(D,A)
=> ( ( r2_hidden(B,k3_measure4(A,D))
& r2_hidden(C,k3_measure4(A,D)) )
=> r2_hidden(k4_xboole_0(B,C),k3_measure4(A,D)) ) ) ).
fof(t20_measure4,axiom,
! [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)
& v1_funct_2(C,k5_numbers,B)
& m2_relset_1(C,k5_numbers,B) )
=> ! [D] :
( m2_subset_1(D,k1_zfmisc_1(A),B)
=> ? [E] :
( v1_funct_1(E)
& v1_funct_2(E,k5_numbers,B)
& m2_relset_1(E,k5_numbers,B)
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> k8_funct_2(k5_numbers,B,E,F) = k3_measure1(A,B,D,k8_funct_2(k5_numbers,B,C,F)) ) ) ) ) ) ).
fof(t21_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ( ~ v1_xboole_0(k3_measure4(A,B))
& v1_prob_1(k3_measure4(A,B),A)
& v1_measure1(k3_measure4(A,B),A)
& m1_subset_1(k3_measure4(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(d4_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,k3_measure4(A,B),k3_supinf_1)
& m2_relset_1(C,k3_measure4(A,B),k3_supinf_1) )
=> ( C = k5_measure4(A,B)
<=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ( r2_hidden(D,k3_measure4(A,B))
=> k1_funct_1(C,D) = k1_measure1(k1_zfmisc_1(A),B,D) ) ) ) ) ) ).
fof(t22_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> m1_measure1(k5_measure4(A,B),A,k3_measure4(A,B)) ) ).
fof(t23_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> m3_measure1(k5_measure4(A,B),A,k3_measure4(A,B)) ) ).
fof(t24_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ( k1_measure1(k1_zfmisc_1(A),B,C) = k1_supinf_2
=> r2_hidden(C,k3_measure4(A,B)) ) ) ) ).
fof(t25_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> r1_measure3(A,k3_measure4(A,B),k7_measure4(A,B)) ) ).
fof(dt_m1_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ( v1_funct_1(B)
& v1_funct_2(B,k1_zfmisc_1(A),k3_supinf_1)
& m2_relset_1(B,k1_zfmisc_1(A),k3_supinf_1) ) ) ).
fof(existence_m1_measure4,axiom,
! [A] :
? [B] : m1_measure4(B,A) ).
fof(dt_k1_measure4,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> m1_subset_1(k1_measure4(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ).
fof(redefinition_k1_measure4,axiom,
! [A,B] :
( ( v1_funct_1(B)
& v1_funct_2(B,k5_numbers,k1_zfmisc_1(A))
& m1_relset_1(B,k5_numbers,k1_zfmisc_1(A)) )
=> k1_measure4(A,B) = k2_relat_1(B) ) ).
fof(dt_k2_measure4,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,B)
& m1_relset_1(D,A,B)
& v1_funct_1(E)
& v1_funct_2(E,B,C)
& m1_relset_1(E,B,C) )
=> ( v1_funct_1(k2_measure4(A,B,C,D,E))
& v1_funct_2(k2_measure4(A,B,C,D,E),A,C)
& m2_relset_1(k2_measure4(A,B,C,D,E),A,C) ) ) ).
fof(redefinition_k2_measure4,axiom,
! [A,B,C,D,E] :
( ( ~ v1_xboole_0(B)
& ~ v1_xboole_0(C)
& v1_funct_1(D)
& v1_funct_2(D,A,B)
& m1_relset_1(D,A,B)
& v1_funct_1(E)
& v1_funct_2(E,B,C)
& m1_relset_1(E,B,C) )
=> k2_measure4(A,B,C,D,E) = k5_relat_1(D,E) ) ).
fof(dt_k3_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ( ~ v1_xboole_0(k3_measure4(A,B))
& m1_subset_1(k3_measure4(A,B),k1_zfmisc_1(k1_zfmisc_1(A))) ) ) ).
fof(dt_k4_measure4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_measure2(C,A,B) )
=> m2_subset_1(k4_measure4(A,B,C),k1_zfmisc_1(A),B) ) ).
fof(redefinition_k4_measure4,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& v1_prob_1(B,A)
& v1_measure1(B,A)
& m1_subset_1(B,k1_zfmisc_1(k1_zfmisc_1(A)))
& m1_measure2(C,A,B) )
=> k4_measure4(A,B,C) = k3_tarski(C) ) ).
fof(dt_k5_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> ( v1_funct_1(k5_measure4(A,B))
& v1_funct_2(k5_measure4(A,B),k3_measure4(A,B),k3_supinf_1)
& m2_relset_1(k5_measure4(A,B),k3_measure4(A,B),k3_supinf_1) ) ) ).
fof(dt_k6_measure4,axiom,
! [A,B,C] :
( ( m1_measure4(B,A)
& m1_subset_1(C,k3_measure4(A,B)) )
=> m1_subset_1(k6_measure4(A,B,C),k3_supinf_1) ) ).
fof(redefinition_k6_measure4,axiom,
! [A,B,C] :
( ( m1_measure4(B,A)
& m1_subset_1(C,k3_measure4(A,B)) )
=> k6_measure4(A,B,C) = k1_funct_1(B,C) ) ).
fof(dt_k7_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> m3_measure1(k7_measure4(A,B),A,k3_measure4(A,B)) ) ).
fof(redefinition_k7_measure4,axiom,
! [A,B] :
( m1_measure4(B,A)
=> k7_measure4(A,B) = k5_measure4(A,B) ) ).
%------------------------------------------------------------------------------