SET007 Axioms: SET007+382.ax
%------------------------------------------------------------------------------
% File : SET007+382 : TPTP v8.2.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Basic Concepts for Petri Nets with Boolean Markings
% 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 : boolmark [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 22 ( 2 unt; 0 def)
% Number of atoms : 133 ( 25 equ)
% Maximal formula atoms : 17 ( 6 avg)
% Number of connectives : 131 ( 20 ~; 0 |; 30 &)
% ( 7 <=>; 74 =>; 0 <=; 0 <~>)
% Maximal formula depth : 23 ( 9 avg)
% Maximal term depth : 7 ( 1 avg)
% Number of predicates : 21 ( 19 usr; 1 prp; 0-4 aty)
% Number of functors : 29 ( 29 usr; 8 con; 0-4 aty)
% Number of variables : 72 ( 71 !; 1 ?)
% 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_boolmark,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( m1_subset_1(E,B)
=> ( v1_funct_1(k1_funct_4(C,k2_funcop_1(D,E)))
& v1_funct_2(k1_funct_4(C,k2_funcop_1(D,E)),A,B)
& m2_relset_1(k1_funct_4(C,k2_funcop_1(D,E)),A,B) ) ) ) ) ) ) ).
fof(t2_boolmark,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( m1_subset_1(C,k1_zfmisc_1(A))
=> ! [D] :
( m1_subset_1(D,k1_zfmisc_1(A))
=> ! [E] :
( ( v1_funct_1(E)
& v1_funct_2(E,A,B)
& m2_relset_1(E,A,B) )
=> ( r1_xboole_0(k2_funct_2(A,B,E,C),k2_funct_2(A,B,E,D))
=> r1_xboole_0(C,D) ) ) ) ) ) ) ).
fof(t3_boolmark,axiom,
! [A,B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> ! [D] :
( r1_xboole_0(A,B)
=> k9_relat_1(k1_funct_4(C,k2_funcop_1(A,D)),B) = k9_relat_1(C,B) ) ) ).
fof(d1_boolmark,axiom,
! [A] :
( l1_petri(A)
=> k1_boolmark(A) = k1_fraenkel(u1_petri(A),k6_margrel1) ) ).
fof(d2_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m1_subset_1(C,u2_petri(A))
=> ( r1_boolmark(A,B,C)
<=> r1_tarski(k2_funct_2(u1_petri(A),k6_margrel1,B,k7_petri(A,k6_domain_1(u2_petri(A),C))),k6_domain_1(k6_margrel1,k8_margrel1)) ) ) ) ) ).
fof(d3_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m1_subset_1(C,u2_petri(A))
=> k2_boolmark(A,B,C) = k1_funct_4(k1_funct_4(B,k2_funcop_1(k7_petri(A,k6_domain_1(u2_petri(A),C)),k7_margrel1)),k2_funcop_1(k8_petri(A,k6_domain_1(u2_petri(A),C)),k8_margrel1)) ) ) ) ).
fof(d4_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m2_finseq_1(C,u2_petri(A))
=> ( r2_boolmark(A,B,C)
<=> ~ ( C != k1_xboole_0
& ! [D] :
( m2_finseq_1(D,k1_boolmark(A))
=> ~ ( k3_finseq_1(C) = k3_finseq_1(D)
& r1_boolmark(A,B,k4_finseq_4(k5_numbers,u2_petri(A),C,np__1))
& k4_finseq_4(k5_numbers,k1_boolmark(A),D,np__1) = k2_boolmark(A,B,k4_finseq_4(k5_numbers,u2_petri(A),C,np__1))
& ! [E] :
( m2_subset_1(E,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(k3_finseq_1(C),E)
& ~ r1_xreal_0(E,np__0)
& ~ ( r1_boolmark(A,k4_finseq_4(k5_numbers,k1_boolmark(A),D,E),k4_finseq_4(k5_numbers,u2_petri(A),C,k1_nat_1(E,np__1)))
& k4_finseq_4(k5_numbers,k1_boolmark(A),D,k1_nat_1(E,np__1)) = k2_boolmark(A,k4_finseq_4(k5_numbers,k1_boolmark(A),D,E),k4_finseq_4(k5_numbers,u2_petri(A),C,k1_nat_1(E,np__1))) ) ) ) ) ) ) ) ) ) ) ).
fof(d5_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m2_finseq_1(C,u2_petri(A))
=> ! [D] :
( m2_fraenkel(D,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ( ( C = k1_xboole_0
=> ( D = k3_boolmark(A,B,C)
<=> D = B ) )
& ( C != k1_xboole_0
=> ( D = k3_boolmark(A,B,C)
<=> ? [E] :
( m2_finseq_1(E,k1_boolmark(A))
& k3_finseq_1(C) = k3_finseq_1(E)
& D = k4_finseq_4(k5_numbers,k1_boolmark(A),E,k3_finseq_1(E))
& k4_finseq_4(k5_numbers,k1_boolmark(A),E,np__1) = k2_boolmark(A,B,k4_finseq_4(k5_numbers,u2_petri(A),C,np__1))
& ! [F] :
( m2_subset_1(F,k1_numbers,k5_numbers)
=> ~ ( ~ r1_xreal_0(k3_finseq_1(C),F)
& ~ r1_xreal_0(F,np__0)
& k4_finseq_4(k5_numbers,k1_boolmark(A),E,k1_nat_1(F,np__1)) != k2_boolmark(A,k4_finseq_4(k5_numbers,k1_boolmark(A),E,F),k4_finseq_4(k5_numbers,u2_petri(A),C,k1_nat_1(F,np__1))) ) ) ) ) ) ) ) ) ) ) ).
fof(t4_boolmark,axiom,
$true ).
fof(t5_boolmark,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C] :
( ( v1_relat_1(C)
& v1_funct_1(C) )
=> k9_relat_1(k1_funct_4(C,k2_funcop_1(A,B)),A) = k1_tarski(B) ) ) ).
fof(t6_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m1_subset_1(C,u2_petri(A))
=> ! [D] :
( m1_subset_1(D,u1_petri(A))
=> ( r2_hidden(D,k8_petri(A,k6_domain_1(u2_petri(A),C)))
=> k8_funct_2(u1_petri(A),k6_margrel1,k2_boolmark(A,B,C),D) = k8_margrel1 ) ) ) ) ) ).
fof(t7_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_petri(A))) )
=> ( v2_petri(B,A)
<=> ! [C] :
( m2_fraenkel(C,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ( k2_funct_2(u1_petri(A),k6_margrel1,C,B) = k6_domain_1(k6_margrel1,k7_margrel1)
=> ! [D] :
( m1_subset_1(D,u2_petri(A))
=> ( r1_boolmark(A,C,D)
=> k2_funct_2(u1_petri(A),k6_margrel1,k2_boolmark(A,C,D),B) = k6_domain_1(k6_margrel1,k7_margrel1) ) ) ) ) ) ) ) ).
fof(t8_boolmark,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( m2_finseq_1(B,A)
=> ! [C] :
( m2_finseq_1(C,A)
=> ! [D] :
( m2_subset_1(D,k1_numbers,k5_numbers)
=> ( ( r1_xreal_0(np__1,D)
& r1_xreal_0(D,k3_finseq_1(B)) )
=> k4_finseq_4(k5_numbers,A,k8_finseq_1(A,B,C),D) = k4_finseq_4(k5_numbers,A,B,D) ) ) ) ) ) ).
fof(t9_boolmark,axiom,
$true ).
fof(t10_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m2_finseq_1(C,u2_petri(A))
=> ! [D] :
( m2_finseq_1(D,u2_petri(A))
=> k3_boolmark(A,B,k8_finseq_1(u2_petri(A),C,D)) = k3_boolmark(A,k3_boolmark(A,B,C),D) ) ) ) ) ).
fof(t11_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m2_finseq_1(C,u2_petri(A))
=> ! [D] :
( m2_finseq_1(D,u2_petri(A))
=> ( r2_boolmark(A,B,k8_finseq_1(u2_petri(A),C,D))
=> ( r2_boolmark(A,k3_boolmark(A,B,C),D)
& r2_boolmark(A,B,C) ) ) ) ) ) ) ).
fof(t12_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m1_subset_1(C,u2_petri(A))
=> ( r1_boolmark(A,B,C)
<=> r2_boolmark(A,B,k12_finseq_1(u2_petri(A),C)) ) ) ) ) ).
fof(t13_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( m2_fraenkel(B,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ! [C] :
( m1_subset_1(C,u2_petri(A))
=> k2_boolmark(A,B,C) = k3_boolmark(A,B,k12_finseq_1(u2_petri(A),C)) ) ) ) ).
fof(t14_boolmark,axiom,
! [A] :
( l1_petri(A)
=> ! [B] :
( ( ~ v1_xboole_0(B)
& m1_subset_1(B,k1_zfmisc_1(u1_petri(A))) )
=> ( v2_petri(B,A)
<=> ! [C] :
( m2_fraenkel(C,u1_petri(A),k6_margrel1,k1_boolmark(A))
=> ( k2_funct_2(u1_petri(A),k6_margrel1,C,B) = k6_domain_1(k6_margrel1,k7_margrel1)
=> ! [D] :
( m2_finseq_1(D,u2_petri(A))
=> ( r2_boolmark(A,C,D)
=> k2_funct_2(u1_petri(A),k6_margrel1,k3_boolmark(A,C,D),B) = k6_domain_1(k6_margrel1,k7_margrel1) ) ) ) ) ) ) ) ).
fof(dt_k1_boolmark,axiom,
! [A] :
( l1_petri(A)
=> m1_fraenkel(k1_boolmark(A),u1_petri(A),k6_margrel1) ) ).
fof(dt_k2_boolmark,axiom,
! [A,B,C] :
( ( l1_petri(A)
& m1_subset_1(B,k1_boolmark(A))
& m1_subset_1(C,u2_petri(A)) )
=> m2_fraenkel(k2_boolmark(A,B,C),u1_petri(A),k6_margrel1,k1_boolmark(A)) ) ).
fof(dt_k3_boolmark,axiom,
! [A,B,C] :
( ( l1_petri(A)
& m1_subset_1(B,k1_boolmark(A))
& m1_finseq_1(C,u2_petri(A)) )
=> m2_fraenkel(k3_boolmark(A,B,C),u1_petri(A),k6_margrel1,k1_boolmark(A)) ) ).
%------------------------------------------------------------------------------