:: Basic Concepts for Petri Nets with Boolean Markings.Boolean Markings and the Firability/Firing of Transitions
:: by Pauline N. Kawamoto, Yasushi Fuwa and Yatsuka Nakamura
::
:: Received October 8, 1993
:: Copyright (c) 1993 Association of Mizar Users

begin

theorem Th1: :: BOOLMARK:1

for A, B being non empty set
for f being Function of A,B
for C being Subset of
for v being Element of B holds f +* (C --> v) is Function of A,B

proof end;

theorem Th2: :: BOOLMARK:2

for X, Y being non empty set
for A, B being Subset of
for f being Function of X,Y st f .: A misses f .: B holds
A misses B

proof end;

theorem Th3: :: BOOLMARK:3

for A, B being set
for f being Function
for x being set st A misses B holds
(f +* (A --> x)) .: B = f .: B

proof end;

begin

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 ;
correctness
coherence
Funcs the Places of PTN,BOOLEAN is FUNCTION_DOMAIN of the Places of PTN, BOOLEAN ;
;

end;

:: deftheorem defines Bool_marks_of BOOLMARK:def 1 :

for PTN being PT_net_Str holds Bool_marks_of PTN = Funcs the Places of PTN,BOOLEAN ;

definition

let PTN be PT_net_Str ;
mode Boolean_marking of PTN is Element of Bool_marks_of PTN;

end;

definition

let PTN be PT_net_Str ;
let M0 be Boolean_marking of PTN;
let t be transition of ;
pred t is_firable_on M0 means :Def2: :: BOOLMARK:def 2
M0 .: (*' {t}) c= {TRUE };

end;

:: deftheorem Def2 defines is_firable_on BOOLMARK:def 2 :

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for t being transition of holds
( t is_firable_on M0 iff M0 .: (*' {t}) c= {TRUE } );

notation

let PTN be PT_net_Str ;
let M0 be Boolean_marking of PTN;
let t be transition of ;
antonym t is_not_firable_on M0 for t is_firable_on M0;

end;

definition

let PTN be PT_net_Str ;
let M0 be Boolean_marking of PTN;
let t be transition of ;
func Firing t,M0 -> Boolean_marking of PTN equals :: BOOLMARK:def 3
(M0 +* ((*' {t}) --> FALSE )) +* (({t} *' ) --> TRUE );
coherence
(M0 +* ((*' {t}) --> FALSE )) +* (({t} *' ) --> TRUE ) is Boolean_marking of PTN

proof end;

correctness
;

end;

:: deftheorem defines Firing BOOLMARK:def 3 :

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for t being transition of holds Firing t,M0 = (M0 +* ((*' {t}) --> FALSE )) +* (({t} *' ) --> TRUE );

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 :Def4: :: 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 Element of 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) ) ) ) );

end;

:: deftheorem Def4 defines is_firable_on BOOLMARK:def 4 :

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for Q being FinSequence of the Transitions of PTN holds
( Q is_firable_on M0 iff ( 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 Element of 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) ) ) ) ) );

notation

let PTN be PT_net_Str ;
let M0 be Boolean_marking of PTN;
let Q be FinSequence of the Transitions of PTN;
antonym Q is_not_firable_on M0 for Q is_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 :Def5: :: 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 Element of NAT st i < len Q & i > 0 holds
M /. (i + 1) = Firing (Q /. (i + 1)),(M /. i) ) );
existence
( ( Q = {} implies ex b₁ being Boolean_marking of PTN st b₁ = M0 ) & ( not Q = {} implies ex b₁ being Boolean_marking of PTN ex M being FinSequence of Bool_marks_of PTN st
( len Q = len M & b₁ = M /. (len M) & M /. 1 = Firing (Q /. 1),M0 & ( for i being Element of NAT st i < len Q & i > 0 holds
M /. (i + 1) = Firing (Q /. (i + 1)),(M /. i) ) ) ) )

proof end;

uniqueness
for b₁, b₂ being Boolean_marking of PTN holds
( ( Q = {} & b₁ = M0 & b₂ = M0 implies b₁ = b₂ ) & ( not Q = {} & ex M being FinSequence of Bool_marks_of PTN st
( len Q = len M & b₁ = M /. (len M) & M /. 1 = Firing (Q /. 1),M0 & ( for i being Element of NAT st i < len Q & i > 0 holds
M /. (i + 1) = Firing (Q /. (i + 1)),(M /. i) ) ) & ex M being FinSequence of Bool_marks_of PTN st
( len Q = len M & b₂ = M /. (len M) & M /. 1 = Firing (Q /. 1),M0 & ( for i being Element of NAT st i < len Q & i > 0 holds
M /. (i + 1) = Firing (Q /. (i + 1)),(M /. i) ) ) implies b₁ = b₂ ) )

proof end;

correctness
consistency
for b₁ being Boolean_marking of PTN holds verum;
;

end;

:: deftheorem Def5 defines Firing BOOLMARK:def 5 :

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for Q being FinSequence of the Transitions of PTN
for b₄ being Boolean_marking of PTN holds
( ( Q = {} implies ( b₄ = Firing Q,M0 iff b₄ = M0 ) ) & ( not Q = {} implies ( b₄ = Firing Q,M0 iff ex M being FinSequence of Bool_marks_of PTN st
( len Q = len M & b₄ = M /. (len M) & M /. 1 = Firing (Q /. 1),M0 & ( for i being Element of NAT st i < len Q & i > 0 holds
M /. (i + 1) = Firing (Q /. (i + 1)),(M /. i) ) ) ) ) );

theorem :: BOOLMARK:4

canceled;

theorem Th5: :: BOOLMARK:5

for A being non empty set
for y being set
for f being Function holds (f +* (A --> y)) .: A = {y}

proof end;

theorem Th6: :: BOOLMARK:6

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for t being transition of
for s being place of st s in {t} *' holds
(Firing t,M0) . s = TRUE

proof end;

Lm1: now

let PTN be PT_net_Str ; :: thesis: for Sd being non empty Subset of st ( for M0 being Boolean_marking of PTN st M0 .: Sd = {FALSE } holds
for t being transition of st t is_firable_on M0 holds
(Firing t,M0) .: Sd = {FALSE } ) holds
Sd is Deadlock-like
let Sd be non empty Subset of ; :: thesis: ( ( for M0 being Boolean_marking of PTN st M0 .: Sd = {FALSE } holds
for t being transition of st t is_firable_on M0 holds
(Firing t,M0) .: Sd = {FALSE } ) implies Sd is Deadlock-like )
set M0 = (the Places of PTN --> TRUE ) +* (Sd --> FALSE );
A1: [#] the Places of PTN = the Places of PTN ;
( dom (the Places of PTN --> TRUE ) = the Places of PTN & dom (Sd --> FALSE ) = Sd ) by FUNCOP_1:19;
then A2: dom ((the Places of PTN --> TRUE ) +* (Sd --> FALSE )) = the Places of PTN \/ Sd by FUNCT_4:def 1
.= the Places of PTN by A1, SUBSET_1:28 ;
A3: rng ((the Places of PTN --> TRUE ) +* (Sd --> FALSE )) c= (rng (the Places of PTN --> TRUE )) \/ (rng (Sd --> FALSE )) by FUNCT_4:18;
rng (Sd --> FALSE ) c= {FALSE } by FUNCOP_1:19;
then A4: rng (Sd --> FALSE ) c= BOOLEAN by XBOOLE_1:1;
rng (the Places of PTN --> TRUE ) c= {TRUE } by FUNCOP_1:19;
then rng (the Places of PTN --> TRUE ) c= BOOLEAN by XBOOLE_1:1;
then (rng (the Places of PTN --> TRUE )) \/ (rng (Sd --> FALSE )) c= BOOLEAN by A4, XBOOLE_1:8;
then rng ((the Places of PTN --> TRUE ) +* (Sd --> FALSE )) c= BOOLEAN by A3, XBOOLE_1:1;
then reconsider M0 = (the Places of PTN --> TRUE ) +* (Sd --> FALSE ) as Boolean_marking of PTN by A2, FUNCT_2:def 2;
assume A5: for M0 being Boolean_marking of PTN st M0 .: Sd = {FALSE } holds
for t being transition of st t is_firable_on M0 holds
(Firing t,M0) .: Sd = {FALSE } ; :: thesis: Sd is Deadlock-like
assume not Sd is Deadlock-like ; :: thesis: contradiction
then not *' Sd c= Sd *' by PETRI:def 5;
then consider t being transition of such that
A6: t in *' Sd and
A7: not t in Sd *' by SUBSET_1:7;
{t} *' meets Sd by A6, PETRI:20;
then consider s being set such that
A8: s in ({t} *' ) /\ Sd by XBOOLE_0:4;
s in {t} *' by A8, XBOOLE_0:def 4;
then A9: (Firing t,M0) . s = TRUE by Th6;
s in Sd by A8, XBOOLE_0:def 4;
then TRUE in (Firing t,M0) .: Sd by A9, FUNCT_2:43;
then A10: (Firing t,M0) .: Sd <> {FALSE } by TARSKI:def 1;
Sd misses *' {t} by A7, PETRI:19;
then A11: (the Places of PTN --> TRUE ) .: (*' {t}) = M0 .: (*' {t}) by Th3;
( rng (the Places of PTN --> TRUE ) c= {TRUE } & (the Places of PTN --> TRUE ) .: (*' {t}) c= rng (the Places of PTN --> TRUE ) ) by FUNCOP_1:19, RELAT_1:144;
then M0 .: (*' {t}) c= {TRUE } by A11, XBOOLE_1:1;
then A12: t is_firable_on M0 by Def2;
M0 .: Sd = {FALSE } by Th5;
hence contradiction by A5, A12, A10; :: thesis: verum

end;

theorem :: BOOLMARK:7

for PTN being PT_net_Str
for Sd being non empty Subset of holds
( Sd is Deadlock-like iff for M0 being Boolean_marking of PTN st M0 .: Sd = {FALSE } holds
for t being transition of st t is_firable_on M0 holds
(Firing t,M0) .: Sd = {FALSE } )

proof end;

theorem Th8: :: BOOLMARK:8

for D being non empty set
for Q0, Q1 being FinSequence of D
for i being Element of NAT st 1 <= i & i <= len Q0 holds
(Q0 ^ Q1) /. i = Q0 /. i

proof end;

theorem :: BOOLMARK:9

canceled;

theorem Th10: :: BOOLMARK:10

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for Q0, Q1 being FinSequence of the Transitions of PTN holds Firing (Q0 ^ Q1),M0 = Firing Q1,(Firing Q0,M0)

proof end;

theorem Th11: :: BOOLMARK:11

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for 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 )

proof end;

theorem Th12: :: BOOLMARK:12

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for t being transition of holds
( t is_firable_on M0 iff <*t*> is_firable_on M0 )

proof end;

theorem Th13: :: BOOLMARK:13

for PTN being PT_net_Str
for M0 being Boolean_marking of PTN
for t being transition of holds Firing t,M0 = Firing <*t*>,M0

proof end;

theorem :: BOOLMARK:14

for PTN being PT_net_Str
for Sd being non empty Subset of holds
( Sd is Deadlock-like iff for M0 being Boolean_marking of PTN st M0 .: Sd = {FALSE } holds
for Q being FinSequence of the Transitions of PTN st Q is_firable_on M0 holds
(Firing Q,M0) .: Sd = {FALSE } )

proof end;