Volume 5, 1993

University of Bialystok

Copyright (c) 1993 Association of Mizar Users

### The abstract of the Mizar article:

### Basic Concepts for Petri Nets with Boolean Markings

**by****Pauline N. Kawamoto,****Yasushi Fuwa, and****Yatsuka Nakamura**- Received October 8, 1993
- MML identifier: BOOLMARK

- [ Mizar article, MML identifier index ]

environ vocabulary FUNCT_1, FUNCT_4, FUNCOP_1, RELAT_1, BOOLE, SUBSET_1, FINSEQ_1, PETRI, FRAENKEL, NET_1, MARGREL1, FUNCT_2, ARYTM_3, ARYTM_1, BOOLMARK, FINSEQ_4; notation TARSKI, XBOOLE_0, SUBSET_1, XCMPLX_0, XREAL_0, NAT_1, RELAT_1, FUNCT_1, FUNCT_2, DOMAIN_1, FRAENKEL, FUNCOP_1, FINSEQ_1, FINSEQ_4, FUNCT_4, MARGREL1, PETRI; constructors NAT_1, DOMAIN_1, FINSEQ_4, FUNCT_4, MARGREL1, PETRI, REAL_1, XREAL_0, MEMBERED, XBOOLE_0; clusters SUBSET_1, FINSEQ_1, RELSET_1, MARGREL1, FUNCOP_1, XREAL_0, ARYTM_3, ZFMISC_1, XBOOLE_0; requirements REAL, NUMERALS, BOOLE, SUBSET, ARITHM; begin theorem :: BOOLMARK:1 for A, B being non empty set, f being Function of A,B, C being Subset of A, v being Element of B holds f +* (C-->v) is Function of A,B; theorem :: BOOLMARK:2 for X, Y being non empty set, A, B being Subset of X, f being Function of X,Y st f.:A misses f.:B holds A misses B; theorem :: BOOLMARK:3 for A, B being set, f being Function, x being set holds A misses B implies (f +* (A --> x)).:B = f.:B; :: Main definitions, theorems block begin :: Boolean Markings of Place/Transition Net definition let PTN be PT_net_Str; func Bool_marks_of PTN -> FUNCTION_DOMAIN of the Places of PTN, BOOLEAN equals :: BOOLMARK:def 1 Funcs(the Places of PTN, BOOLEAN); end; definition let PTN be PT_net_Str; mode Boolean_marking of PTN is Element of Bool_marks_of PTN; end; :: Firable and Firing Conditions for Transitions definition let PTN be PT_net_Str; let M0 be Boolean_marking of PTN; let t be transition of PTN; pred t is_firable_on M0 means :: BOOLMARK:def 2 M0.:*'{t} c= {TRUE}; antonym t is_not_firable_on M0; end; definition let PTN be PT_net_Str; let M0 be Boolean_marking of PTN; let t be transition of PTN; func Firing(t,M0) -> Boolean_marking of PTN equals :: BOOLMARK:def 3 M0 +* (*'{t}-->FALSE) +* ({t}*'-->TRUE); end; :: Firable and Firing Conditions for Transition Sequences definition let PTN be PT_net_Str; let M0 be Boolean_marking of PTN; let Q be FinSequence of the Transitions of PTN; pred Q is_firable_on M0 means :: BOOLMARK:def 4 Q = {} or ex M being FinSequence of Bool_marks_of PTN st len Q = len M & Q/.1 is_firable_on M0 & M/.1 = Firing(Q/.1,M0) & for i being Nat st i < len Q & i > 0 holds Q/.(i+1) is_firable_on M/.i & M/.(i+1) = Firing(Q/.(i+1),M/.i); antonym Q is_not_firable_on M0; end; definition let PTN be PT_net_Str; let M0 be Boolean_marking of PTN; let Q be FinSequence of the Transitions of PTN; func Firing(Q,M0) -> Boolean_marking of PTN means :: BOOLMARK:def 5 it = M0 if Q = {} otherwise ex M being FinSequence of Bool_marks_of PTN st len Q = len M & it = M/.len M & M/.1 = Firing(Q/.1,M0) & for i being Nat st i < len Q & i > 0 holds M/.(i+1) = Firing(Q/.(i+1),M/.i); end; canceled; theorem :: BOOLMARK:5 for A being non empty set, y being set, f being Function holds (f+*(A --> y)).:A = {y}; theorem :: BOOLMARK:6 for PTN being PT_net_Str, M0 being Boolean_marking of PTN, t being transition of PTN, s being place of PTN st s in {t}*' holds Firing(t,M0).s = TRUE; theorem :: BOOLMARK:7 for PTN being PT_net_Str, Sd being non empty Subset of the Places of PTN holds Sd is Deadlock-like iff for M0 being Boolean_marking of PTN st M0.:Sd = {FALSE} for t being transition of PTN st t is_firable_on M0 holds Firing(t,M0).:Sd = {FALSE}; theorem :: BOOLMARK:8 for D being non empty set for Q0,Q1 being FinSequence of D, i being Nat st 1<=i & i<=len Q0 holds (Q0^Q1)/.i=Q0/.i; canceled; theorem :: BOOLMARK:10 for PTN being PT_net_Str, M0 being Boolean_marking of PTN, Q0, Q1 being FinSequence of the Transitions of PTN holds Firing(Q0^Q1,M0) = Firing(Q1,Firing(Q0,M0)); theorem :: BOOLMARK:11 for PTN being PT_net_Str, M0 being Boolean_marking of PTN, Q0, Q1 being FinSequence of the Transitions of PTN st Q0^Q1 is_firable_on M0 holds Q1 is_firable_on Firing(Q0,M0) & Q0 is_firable_on M0; theorem :: BOOLMARK:12 for PTN being PT_net_Str, M0 being Boolean_marking of PTN, t being transition of PTN holds t is_firable_on M0 iff <*t*> is_firable_on M0; theorem :: BOOLMARK:13 for PTN being PT_net_Str, M0 being Boolean_marking of PTN, t being transition of PTN holds Firing(t,M0) = Firing(<*t*>,M0); theorem :: BOOLMARK:14 for PTN being PT_net_Str, Sd being non empty Subset of the Places of PTN holds Sd is Deadlock-like iff for M0 being Boolean_marking of PTN st M0.:Sd = {FALSE} for Q being FinSequence of the Transitions of PTN st Q is_firable_on M0 holds Firing(Q,M0).:Sd = {FALSE};

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