Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

### The abstract of the Mizar article:

### Basic Petri Net Concepts

**by****Pauline N. Kawamoto,****Yasushi Fuwa, and****Yatsuka Nakamura**- Received November 27, 1992
- MML identifier: PETRI

- [ Mizar article, MML identifier index ]

environ vocabulary RELAT_1, NET_1, MCART_1, ARYTM_3, BOOLE, PETRI; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, RELSET_1, MCART_1, DOMAIN_1; constructors RELSET_1, DOMAIN_1, MEMBERED, XBOOLE_0; clusters RELSET_1, PRELAMB, SUBSET_1, MEMBERED, ZFMISC_1, XBOOLE_0; requirements SUBSET, BOOLE; begin :: Redefinition of Element for Non-empty Relation definition let A, B be non empty set; let r be non empty Relation of A, B; redefine mode Element of r -> Element of [:A,B:]; end; :: Place/Transition Net Structure definition struct PT_net_Str (# Places, Transitions -> non empty set, S-T_Arcs -> non empty Relation of the Places, the Transitions, T-S_Arcs -> non empty Relation of the Transitions, the Places #); end; reserve PTN for PT_net_Str; :: Place, Transition, and Arc (s->t, t->s) Elements definition let PTN; mode place of PTN is Element of the Places of PTN; mode places of PTN is Element of the Places of PTN; mode transition of PTN is Element of the Transitions of PTN; mode transitions of PTN is Element of the Transitions of PTN; mode S-T_arc of PTN is Element of the S-T_Arcs of PTN; mode T-S_arc of PTN is Element of the T-S_Arcs of PTN; end; :: Redefinition of Relation for s->t Arcs definition let PTN; let x be S-T_arc of PTN; redefine func x`1 -> place of PTN; func x`2 -> transition of PTN; end; :: Redefinition of Relation for t->s Arcs definition let PTN; let x be T-S_arc of PTN; redefine func x`1 -> transition of PTN; func x`2 -> place of PTN; end; :: *S, S* Definitions and Theorems reserve S0 for Subset of the Places of PTN; definition let PTN, S0; func *'S0 -> Subset of the Transitions of PTN equals :: PETRI:def 1 { t where t is transition of PTN : ex f being T-S_arc of PTN, s being place of PTN st s in S0 & f = [t,s] }; func S0*' -> Subset of the Transitions of PTN equals :: PETRI:def 2 { t where t is transition of PTN : ex f being S-T_arc of PTN, s being place of PTN st s in S0 & f = [s,t] }; end; theorem :: PETRI:1 *'S0 = {f`1 where f is T-S_arc of PTN : f`2 in S0}; theorem :: PETRI:2 for x being set holds x in *'S0 iff ex f being T-S_arc of PTN, s being place of PTN st s in S0 & f = [x,s]; theorem :: PETRI:3 S0*' = {f`2 where f is S-T_arc of PTN : f`1 in S0}; theorem :: PETRI:4 for x being set holds x in S0*' iff ex f being S-T_arc of PTN, s being place of PTN st s in S0 & f = [s,x]; :: *T, T* Definitions and Theorems reserve T0 for Subset of the Transitions of PTN; definition let PTN, T0; func *'T0 -> Subset of the Places of PTN equals :: PETRI:def 3 { s where s is place of PTN : ex f being S-T_arc of PTN, t being transition of PTN st t in T0 & f = [s,t] }; func T0*' -> Subset of the Places of PTN equals :: PETRI:def 4 { s where s is place of PTN : ex f being T-S_arc of PTN, t being transition of PTN st t in T0 & f = [t,s] }; end; theorem :: PETRI:5 *'T0 = {f`1 where f is S-T_arc of PTN : f`2 in T0}; theorem :: PETRI:6 for x being set holds x in *'T0 iff ex f being S-T_arc of PTN, t being transition of PTN st t in T0 & f = [x,t]; theorem :: PETRI:7 T0*' = {f`2 where f is T-S_arc of PTN : f`1 in T0}; theorem :: PETRI:8 for x being set holds x in T0*' iff ex f being T-S_arc of PTN, t being transition of PTN st t in T0 & f = [t,x]; theorem :: PETRI:9 *'{}the Places of PTN = {}; theorem :: PETRI:10 ({}the Places of PTN)*' = {}; theorem :: PETRI:11 *'{}the Transitions of PTN = {}; theorem :: PETRI:12 ({}the Transitions of PTN)*' = {}; begin :: Deadlock-like Attribute for Place Sets definition let PTN; let IT be Subset of the Places of PTN; attr IT is Deadlock-like means :: PETRI:def 5 *'IT is Subset of IT*'; end; :: With_Deadlocks Mode for Place\Transition Nets definition let IT be PT_net_Str; attr IT is With_Deadlocks means :: PETRI:def 6 ex S being Subset of the Places of IT st S is Deadlock-like; end; definition cluster With_Deadlocks PT_net_Str; end; begin :: Trap-like Attribute for Place Sets definition let PTN; let IT be Subset of the Places of PTN; attr IT is Trap-like means :: PETRI:def 7 IT*' is Subset of *'IT; end; :: With_Traps Mode for Place\Transition Nets definition let IT be PT_net_Str; attr IT is With_Traps means :: PETRI:def 8 ex S being Subset of the Places of IT st S is Trap-like; end; definition cluster With_Traps PT_net_Str; end; definition let A, B be non empty set; let r be non empty Relation of A, B; redefine func r~ -> non empty Relation of B, A; end; begin :: Duality Definitions and Theorems for Place/Transition Nets definition let PTN; func PTN.: -> strict PT_net_Str equals :: PETRI:def 9 PT_net_Str (# the Places of PTN, the Transitions of PTN, (the T-S_Arcs of PTN)~, (the S-T_Arcs of PTN)~ #); end; theorem :: PETRI:13 PTN.:.: = the PT_net_Str of PTN; theorem :: PETRI:14 the Places of PTN = the Places of PTN.: & the Transitions of PTN = the Transitions of PTN.: & (the S-T_Arcs of PTN)~ = the T-S_Arcs of PTN.: & (the T-S_Arcs of PTN)~ = the S-T_Arcs of PTN.:; definition let PTN; let S0 be Subset of the Places of PTN; func S0.: -> Subset of the Places of PTN.: equals :: PETRI:def 10 S0; end; definition let PTN; let s be place of PTN; func s.: -> place of PTN.: equals :: PETRI:def 11 s; end; definition let PTN; let S0 be Subset of the Places of PTN.:; func .:S0 -> Subset of the Places of PTN equals :: PETRI:def 12 S0; end; definition let PTN; let s be place of PTN.:; func .:s -> place of PTN equals :: PETRI:def 13 s; end; definition let PTN; let T0 be Subset of the Transitions of PTN; func T0.: -> Subset of the Transitions of PTN.: equals :: PETRI:def 14 T0; end; definition let PTN; let t be transition of PTN; func t.: -> transition of PTN.: equals :: PETRI:def 15 t; end; definition let PTN; let T0 be Subset of the Transitions of PTN.:; func .:T0 -> Subset of the Transitions of PTN equals :: PETRI:def 16 T0; end; definition let PTN; let t be transition of PTN.:; func .:t -> transition of PTN equals :: PETRI:def 17 t; end; reserve S for Subset of the Places of PTN; theorem :: PETRI:15 S.:*' = *'S; theorem :: PETRI:16 *'(S.:) = S*'; theorem :: PETRI:17 S is Deadlock-like iff S.: is Trap-like; theorem :: PETRI:18 S is Trap-like iff S.: is Deadlock-like; theorem :: PETRI:19 for PTN being PT_net_Str, t being transition of PTN, S0 being Subset of the Places of PTN holds t in S0*' iff *'{t} meets S0; theorem :: PETRI:20 for PTN being PT_net_Str, t being transition of PTN, S0 being Subset of the Places of PTN holds t in *'S0 iff {t}*' meets S0;

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