:: PNPROC_1 semantic presentation

theorem Th1: :: PNPROC_1:1

for i being Nat
for x being set st i > 0 holds
{[i,x]} is FinSubsequence

proof end;

Lm1: for p being FinSubsequence st Seq p = {} holds
p = {}

proof end;

theorem Th2: :: PNPROC_1:2

for q being FinSubsequence holds
( q = {} iff Seq q = {} )

proof end;

theorem Th3: :: PNPROC_1:3

for i being Nat
for x being set
for q being FinSubsequence st q = {[i,x]} holds
Seq q = <*x*>

proof end;

registration

cluster -> finite set ;
coherence

proof end;

end;

theorem Th4: :: PNPROC_1:4

for x being set
for q being FinSubsequence st Seq q = <*x*> holds
ex i being Nat st q = {[i,x]}

proof end;

theorem Th5: :: PNPROC_1:5

for x1, y1, x2, y2 being set holds
( not {[x1,y1],[x2,y2]} is FinSequence or ( x1 = 1 & x2 = 1 & y1 = y2 ) or ( x1 = 1 & x2 = 2 ) or ( x1 = 2 & x2 = 1 ) )

proof end;

theorem Th6: :: PNPROC_1:6

for x1, x2 being set holds <*x1,x2*> = {[1,x1],[2,x2]}

proof end;

Lm2: for j, k, l being Nat st ( ( 1 <= j & j <= l ) or ( l + 1 <= j & j <= l + k ) ) holds
( 1 <= j & j <= l + k )

proof end;

theorem Th7: :: PNPROC_1:7

for p being FinSubsequence holds Card p = len (Seq p)

proof end;

Lm3: for X, Y being set holds
( ( for x being set st x in X holds
not x in Y ) iff X misses Y )

proof end;

theorem Th8: :: PNPROC_1:8

for P, R being Relation st dom P misses dom R holds
P misses R

proof end;

theorem Th9: :: PNPROC_1:9

for X, Y being set
for P, R being Relation st X misses Y holds
P | X misses R | Y

proof end;

Lm4: for q being FinSubsequence holds dom (Seq q) = dom (Sgm (dom q))

proof end;

Lm5: for q being FinSubsequence holds rng q = rng (Seq q)

proof end;

theorem Th10: :: PNPROC_1:10

for f, g, h being Function st f c= h & g c= h & f misses g holds
dom f misses dom g

proof end;

theorem :: PNPROC_1:11

for Y being set
for R being Relation holds Y | R c= R | (R " Y)

proof end;

theorem Th12: :: PNPROC_1:12

for Y being set
for f being Function holds Y | f = f | (f " Y)

proof end;

definition

let P be set ;
mode marking of P -> Function means :Def1: :: PNPROC_1:def 1
( dom it = P & rng it c= NAT );
existence

proof end;

end;

:: deftheorem Def1 defines marking PNPROC_1:def 1 :

notation

let P be set ;
let m be marking of P;
let p be set ;
synonym m multitude_of p for P . m;

end;

definition

let P be set ;
let m be marking of P;
let p be set ;
:: original: multitude_of
redefine func m multitude_of p -> Nat;
coherence

proof end;

:: original: multitude_of
redefine func m . p -> Nat;
coherence

proof end;

end;

scheme :: PNPROC_1:sch 1
MarkingLambda{ F₁() -> set , F₂( set ) -> Nat } :

ex m being marking of F₁() st
for p being set st p in F₁() holds
m multitude_of p = F₂(p)

proof end;

definition

let P be set ;
let m1, m2 be marking of P;
:: original: =
redefine pred m1 = m2 means :: PNPROC_1:def 2
for p being set st p in P holds
m1 multitude_of p = m2 multitude_of p;
compatibility

proof end;

end;

:: deftheorem defines = PNPROC_1:def 2 :

definition

let P be set ;
func {$} P -> marking of P equals :: PNPROC_1:def 3
P --> 0;
coherence

proof end;

end;

:: deftheorem defines {$} PNPROC_1:def 3 :

definition

let P be set ;
let m1, m2 be marking of P;
pred m1 c= m2 means :Def4: :: PNPROC_1:def 4
for p being set st p in P holds
m1 multitude_of p <= m2 multitude_of p;
reflexivity ;

end;

:: deftheorem Def4 defines c= PNPROC_1:def 4 :

Lm6: for p, P being set st p in P holds
({$} P) multitude_of p = 0

proof end;

theorem Th13: :: PNPROC_1:13

for P being set
for m being marking of P holds {$} P c= m

proof end;

theorem Th14: :: PNPROC_1:14

for P being set
for m1, m2, m3 being marking of P st m1 c= m2 & m2 c= m3 holds
m1 c= m3

proof end;

definition

let P be set ;
let m1, m2 be marking of P;
func m1 + m2 -> marking of P means :Def5: :: PNPROC_1:def 5
for p being set st p in P holds
it multitude_of p = (m1 multitude_of p) + (m2 multitude_of p);
existence

proof end;

uniqueness

proof end;

commutativity ;

end;

:: deftheorem Def5 defines + PNPROC_1:def 5 :

theorem :: PNPROC_1:15

for P being set
for m being marking of P holds m + ({$} P) = m

proof end;

definition

let P be set ;
let m1, m2 be marking of P;
assume A1: m2 c= m1 ;
func m1 - m2 -> marking of P means :Def6: :: PNPROC_1:def 6
for p being set st p in P holds
it multitude_of p = (m1 multitude_of p) - (m2 multitude_of p);
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def6 defines - PNPROC_1:def 6 :

theorem Th16: :: PNPROC_1:16

for P being set
for m1, m2 being marking of P holds m1 c= m1 + m2

proof end;

theorem :: PNPROC_1:17

for P being set
for m being marking of P holds m - ({$} P) = m

proof end;

theorem :: PNPROC_1:18

for P being set
for m1, m2, m3 being marking of P st m1 c= m2 & m2 c= m3 holds
m3 - m2 c= m3 - m1

proof end;

theorem Th19: :: PNPROC_1:19

for P being set
for m1, m2 being marking of P holds (m1 + m2) - m2 = m1

proof end;

theorem Th20: :: PNPROC_1:20

for P being set
for m, m1, m2 being marking of P st m c= m1 & m1 c= m2 holds
m1 - m c= m2 - m

proof end;

theorem Th21: :: PNPROC_1:21

for P being set
for m1, m2, m3 being marking of P st m1 c= m2 holds
(m2 + m3) - m1 = (m2 - m1) + m3

proof end;

theorem :: PNPROC_1:22

for P being set
for m1, m2 being marking of P st m1 c= m2 & m2 c= m1 holds
m1 = m2

proof end;

theorem Th23: :: PNPROC_1:23

for P being set
for m1, m2, m3 being marking of P holds (m1 + m2) + m3 = m1 + (m2 + m3)

proof end;

theorem :: PNPROC_1:24

for P being set
for m1, m2, m3, m4 being marking of P st m1 c= m2 & m3 c= m4 holds
m1 + m3 c= m2 + m4

proof end;

theorem :: PNPROC_1:25

for P being set
for m1, m2 being marking of P st m1 c= m2 holds
m2 - m1 c= m2

proof end;

theorem Th26: :: PNPROC_1:26

for P being set
for m1, m2, m3, m4 being marking of P st m1 c= m2 & m3 c= m4 & m4 c= m1 holds
m1 - m4 c= m2 - m3

proof end;

theorem Th27: :: PNPROC_1:27

for P being set
for m1, m2 being marking of P st m1 c= m2 holds
m2 = (m2 - m1) + m1

proof end;

theorem Th28: :: PNPROC_1:28

for P being set
for m1, m2 being marking of P holds (m1 + m2) - m1 = m2

proof end;

theorem Th29: :: PNPROC_1:29

for P being set
for m2, m3, m1 being marking of P st m2 + m3 c= m1 holds
(m1 - m2) - m3 = m1 - (m2 + m3)

proof end;

theorem :: PNPROC_1:30

for P being set
for m3, m2, m1 being marking of P st m3 c= m2 & m2 c= m1 holds
m1 - (m2 - m3) = (m1 - m2) + m3

proof end;

theorem Th31: :: PNPROC_1:31

for P being set
for m being marking of P holds m in Funcs P,NAT

proof end;

theorem Th32: :: PNPROC_1:32

for x, P being set st x in Funcs P,NAT holds
x is marking of P

proof end;

definition

let P be set ;
mode transition of P -> set means :Def7: :: PNPROC_1:def 7
ex m1, m2 being marking of P st it = [m1,m2];
existence

proof end;

end;

:: deftheorem Def7 defines transition PNPROC_1:def 7 :

notation

let P be set ;
let t be transition of P;
synonym Pre t for P `1 ;
synonym Post t for P `2 ;

end;

definition

let P be set ;
let t be transition of P;
:: original: Pre
redefine func Pre t -> marking of P;
coherence

proof end;

:: original: Post
redefine func Post t -> marking of P;
coherence

proof end;

end;

definition

let P be set ;
let m be marking of P;
let t be transition of P;
func fire t,m -> marking of P equals :Def8: :: PNPROC_1:def 8
(m - (Pre t)) + (Post t) if Pre t c= m
otherwise m;
coherence ;
consistency ;

end;

:: deftheorem Def8 defines fire PNPROC_1:def 8 :

theorem :: PNPROC_1:33

for P being set
for m being marking of P
for t1, t2 being transition of P st (Pre t1) + (Pre t2) c= m holds
fire t2,(fire t1,m) = (((m - (Pre t1)) - (Pre t2)) + (Post t1)) + (Post t2)

proof end;

definition

let P be set ;
let t be transition of P;
func fire t -> Function means :Def9: :: PNPROC_1:def 9
( dom it = Funcs P,NAT & ( for m being marking of P holds it . m = fire t,m ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def9 defines fire PNPROC_1:def 9 :

theorem Th34: :: PNPROC_1:34

for P being set
for t being transition of P holds rng (fire t) c= Funcs P,NAT

proof end;

theorem :: PNPROC_1:35

for P being set
for m being marking of P
for t2, t1 being transition of P holds fire t2,(fire t1,m) = ((fire t2) * (fire t1)) . m

proof end;

definition

let P be set ;
mode Petri_net of P -> non empty set means :Def10: :: PNPROC_1:def 10
for x being set st x in it holds
x is transition of P;
existence

proof end;

end;

:: deftheorem Def10 defines Petri_net PNPROC_1:def 10 :

definition

let P be set ;
let N be Petri_net of P;
:: original: Element
redefine mode Element of N -> transition of P;
coherence by Def10;

end;

definition

let P be set ;
let N be Petri_net of P;
mode firing-sequence of N is Element of N * ;

end;

definition

let P be set ;
let N be Petri_net of P;
let C be firing-sequence of N;
func fire C -> Function means :Def11: :: PNPROC_1:def 11
ex F being Function-yielding FinSequence st
( it = compose F,(Funcs P,NAT ) & len F = len C & ( for i being Nat st i in dom C holds
F . i = fire (C /. i) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def11 defines fire PNPROC_1:def 11 :

theorem :: PNPROC_1:36

for P being set
for N being Petri_net of P holds fire (<*> N) = id (Funcs P,NAT )

proof end;

theorem Th37: :: PNPROC_1:37

for P being set
for N being Petri_net of P
for e being Element of N holds fire <*e*> = fire e

proof end;

theorem Th38: :: PNPROC_1:38

for P being set
for N being Petri_net of P
for e being Element of N holds (fire e) * (id (Funcs P,NAT )) = fire e

proof end;

theorem :: PNPROC_1:39

for P being set
for N being Petri_net of P
for e1, e2 being Element of N holds fire <*e1,e2*> = (fire e2) * (fire e1)

proof end;

theorem Th40: :: PNPROC_1:40

for P being set
for N being Petri_net of P
for C being firing-sequence of N holds
( dom (fire C) = Funcs P,NAT & rng (fire C) c= Funcs P,NAT )

proof end;

theorem Th41: :: PNPROC_1:41

for P being set
for N being Petri_net of P
for C1, C2 being firing-sequence of N holds fire (C1 ^ C2) = (fire C2) * (fire C1)

proof end;

theorem :: PNPROC_1:42

for P being set
for N being Petri_net of P
for e being Element of N
for C being firing-sequence of N holds fire (C ^ <*e*>) = (fire e) * (fire C)

proof end;

definition

let P be set ;
let N be Petri_net of P;
let C be firing-sequence of N;
let m be marking of P;
func fire C,m -> marking of P equals :: PNPROC_1:def 12
(fire C) . m;
coherence

proof end;

end;

:: deftheorem defines fire PNPROC_1:def 12 :

definition

let P be set ;
let N be Petri_net of P;
mode process of N is Subset of (N * );

end;

registration

cluster FinSequence-like -> finite FinSubsequence-like set ;
coherence by FINSEQ_1:68;

end;

definition

let P be set ;
let N be Petri_net of P;
let R1, R2 be process of N;
func R1 before R2 -> process of N equals :: PNPROC_1:def 13
{ (C1 ^ C2) where C1, C2 is firing-sequence of N : ( C1 in R1 & C2 in R2 ) } ;
coherence

proof end;

end;

:: deftheorem defines before PNPROC_1:def 13 :

registration

let P be set ;
let N be Petri_net of P;
let R1, R2 be non empty process of N;
cluster R1 before R2 -> non empty ;
coherence

proof end;

end;

theorem Th43: :: PNPROC_1:43

for P being set
for N being Petri_net of P
for R1, R2, R being process of N holds (R1 \/ R2) before R = (R1 before R) \/ (R2 before R)

proof end;

theorem Th44: :: PNPROC_1:44

for P being set
for N being Petri_net of P
for R, R1, R2 being process of N holds R before (R1 \/ R2) = (R before R1) \/ (R before R2)

proof end;

theorem Th45: :: PNPROC_1:45

for P being set
for N being Petri_net of P
for C1, C2 being firing-sequence of N holds {C1} before {C2} = {(C1 ^ C2)}

proof end;

theorem :: PNPROC_1:46

for P being set
for N being Petri_net of P
for C1, C2, C being firing-sequence of N holds {C1,C2} before {C} = {(C1 ^ C),(C2 ^ C)}

proof end;

theorem :: PNPROC_1:47

for P being set
for N being Petri_net of P
for C, C1, C2 being firing-sequence of N holds {C} before {C1,C2} = {(C ^ C1),(C ^ C2)}

proof end;

definition

let P be set ;
let N be Petri_net of P;
let R1, R2 be process of N;
func R1 concur R2 -> process of N equals :: PNPROC_1:def 14
{ C where C is firing-sequence of N : ex q1, q2 being FinSubsequence st
( C = q1 \/ q2 & q1 misses q2 & Seq q1 in R1 & Seq q2 in R2 ) } ;
coherence

proof end;

commutativity

proof end;

end;

:: deftheorem defines concur PNPROC_1:def 14 :

theorem Th48: :: PNPROC_1:48

for P being set
for N being Petri_net of P
for R1, R2, R being process of N holds (R1 \/ R2) concur R = (R1 concur R) \/ (R2 concur R)

proof end;

theorem Th49: :: PNPROC_1:49

for P being set
for N being Petri_net of P
for e1, e2 being Element of N holds {<*e1*>} concur {<*e2*>} = {<*e1,e2*>,<*e2,e1*>}

proof end;

theorem :: PNPROC_1:50

for P being set
for N being Petri_net of P
for e1, e2, e being Element of N holds {<*e1*>,<*e2*>} concur {<*e*>} = {<*e1,e*>,<*e2,e*>,<*e,e1*>,<*e,e2*>}

proof end;

theorem :: PNPROC_1:51

for P being set
for N being Petri_net of P
for R1, R2, R3 being process of N holds (R1 before R2) before R3 = R1 before (R2 before R3)

proof end;

definition

let p be FinSubsequence;
let i be Nat;
set X = { (i + k) where k is Nat : k in dom p } ;
func i Shift p -> FinSubsequence means :Def15: :: PNPROC_1:def 15
( dom it = { (i + k) where k is Nat : k in dom p } & ( for j being Nat st j in dom p holds
it . (i + j) = p . j ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def15 defines Shift PNPROC_1:def 15 :

theorem Th52: :: PNPROC_1:52

for q being FinSubsequence holds 0 Shift q = q

proof end;

theorem :: PNPROC_1:53

for i, j being Nat
for q being FinSubsequence holds (i + j) Shift q = i Shift (j Shift q)

proof end;

theorem Th54: :: PNPROC_1:54

for i being Nat
for p being FinSequence st p <> {} holds
dom (i Shift p) = { j1 where j1 is Nat : ( i + 1 <= j1 & j1 <= i + (len p) ) }

proof end;

theorem Th55: :: PNPROC_1:55

for i being Nat
for q being FinSubsequence holds
( q = {} iff i Shift q = {} )

proof end;

theorem Th56: :: PNPROC_1:56

for i being Nat
for q being FinSubsequence ex ss being FinSubsequence st
( dom ss = dom q & rng ss = dom (i Shift q) & ( for k being Nat st k in dom q holds
ss . k = i + k ) & ss is one-to-one )

proof end;

Lm7: for i being Nat
for p being FinSequence ex fs being FinSequence st
( dom fs = dom p & rng fs = dom (i Shift p) & ( for k being Nat st k in dom p holds
fs . k = i + k ) & fs is one-to-one )

proof end;

theorem Th57: :: PNPROC_1:57

for i being Nat
for q being FinSubsequence holds Card q = Card (i Shift q)

proof end;

theorem Th58: :: PNPROC_1:58

for i being Nat
for p being FinSequence holds dom p = dom (Seq (i Shift p))

proof end;

theorem Th59: :: PNPROC_1:59

for k, i being Nat
for p being FinSequence st k in dom p holds
(Sgm (dom (i Shift p))) . k = i + k

proof end;

theorem Th60: :: PNPROC_1:60

for k, i being Nat
for p being FinSequence st k in dom p holds
(Seq (i Shift p)) . k = p . k

proof end;

theorem Th61: :: PNPROC_1:61

for i being Nat
for p being FinSequence holds Seq (i Shift p) = p

proof end;

theorem Th62: :: PNPROC_1:62

for p1, p2 being FinSequence holds dom (p1 \/ ((len p1) Shift p2)) = Seg ((len p1) + (len p2))

proof end;

theorem Th63: :: PNPROC_1:63

for i being Nat
for p1 being FinSequence
for p2 being FinSubsequence st len p1 <= i holds
dom p1 misses dom (i Shift p2)

proof end;

theorem Th64: :: PNPROC_1:64

for p1, p2 being FinSequence holds p1 ^ p2 = p1 \/ ((len p1) Shift p2)

proof end;

theorem Th65: :: PNPROC_1:65

for i being Nat
for p1 being FinSequence
for p2 being FinSubsequence st i >= len p1 holds
p1 misses i Shift p2

proof end;

theorem :: PNPROC_1:66

for P being set
for N being Petri_net of P
for R1, R2, R3 being process of N holds (R1 concur R2) concur R3 = R1 concur (R2 concur R3)

proof end;

theorem :: PNPROC_1:67

for P being set
for N being Petri_net of P
for R1, R2 being process of N holds R1 before R2 c= R1 concur R2

proof end;

theorem :: PNPROC_1:68

for P being set
for N being Petri_net of P
for R1, P1, R2, P2 being process of N st R1 c= P1 & R2 c= P2 holds
R1 before R2 c= P1 before P2

proof end;

theorem :: PNPROC_1:69

for P being set
for N being Petri_net of P
for R1, P1, R2, P2 being process of N st R1 c= P1 & R2 c= P2 holds
R1 concur R2 c= P1 concur P2

proof end;

Lm8: for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 & q2 c= p2 holds
dom (q1 \/ ((len p1) Shift q2)) c= dom (p1 ^ p2)

proof end;

Lm9: for p1 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 holds
q1 misses (len p1) Shift q2

proof end;

Lm10: for i being Nat
for p, q being FinSubsequence st q c= p holds
dom (i Shift q) c= dom (i Shift p)

proof end;

theorem Th70: :: PNPROC_1:70

for i being Nat
for p, q being FinSubsequence st q c= p holds
i Shift q c= i Shift p

proof end;

theorem Th71: :: PNPROC_1:71

for p1, p2 being FinSequence holds (len p1) Shift p2 c= p1 ^ p2

proof end;

theorem Th72: :: PNPROC_1:72

for i being Nat
for q1, q2 being FinSubsequence st dom q1 misses dom q2 holds
dom (i Shift q1) misses dom (i Shift q2)

proof end;

theorem Th73: :: PNPROC_1:73

for i being Nat
for q, q1, q2 being FinSubsequence st q = q1 \/ q2 & q1 misses q2 holds
(i Shift q1) \/ (i Shift q2) = i Shift q

proof end;

theorem Th74: :: PNPROC_1:74

for i being Nat
for q being FinSubsequence holds dom (Seq q) = dom (Seq (i Shift q))

proof end;

theorem Th75: :: PNPROC_1:75

for k, i being Nat
for q being FinSubsequence st k in dom (Seq q) holds
ex j being Nat st
( j = (Sgm (dom q)) . k & (Sgm (dom (i Shift q))) . k = i + j )

proof end;

theorem Th76: :: PNPROC_1:76

for k, i being Nat
for q being FinSubsequence st k in dom (Seq q) holds
(Seq (i Shift q)) . k = (Seq q) . k

proof end;

theorem Th77: :: PNPROC_1:77

for i being Nat
for q being FinSubsequence holds Seq q = Seq (i Shift q)

proof end;

theorem Th78: :: PNPROC_1:78

for k, i being Nat
for q being FinSubsequence st dom q c= Seg k holds
dom (i Shift q) c= Seg (i + k)

proof end;

theorem Th79: :: PNPROC_1:79

for p being FinSequence
for q1, q2 being FinSubsequence st q1 c= p holds
ex ss being FinSubsequence st ss = q1 \/ ((len p) Shift q2)

proof end;

Lm11: for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q2 <> {} & q1 c= p1 & q2 c= p2 holds
Sgm ((dom q1) \/ (dom ((len p1) Shift q2))) = (Sgm (dom q1)) ^ (Sgm (dom ((len p1) Shift q2)))

proof end;

theorem Th80: :: PNPROC_1:80

for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 & q2 c= p2 holds
ex ss being FinSubsequence st
( ss = q1 \/ ((len p1) Shift q2) & dom (Seq ss) = Seg ((len (Seq q1)) + (len (Seq q2))) )

proof end;

theorem Th81: :: PNPROC_1:81

proof end;

theorem Th82: :: PNPROC_1:82

for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 & q2 c= p2 holds
ex ss being FinSubsequence st
( ss = q1 \/ ((len p1) Shift q2) & (Seq q1) ^ (Seq q2) = Seq ss )

proof end;

theorem :: PNPROC_1:83

for P being set
for N being Petri_net of P
for R1, R2, P1, P2 being process of N holds (R1 concur R2) before (P1 concur P2) c= (R1 before P1) concur (R2 before P2)

proof end;

registration

let P be set ;
let N be Petri_net of P;
let R1, R2 be non empty process of N;
cluster R1 concur R2 -> non empty ;
coherence

proof end;

end;

definition

let P be set ;
let N be Petri_net of P;
func NeutralProcess N -> non empty process of N equals :: PNPROC_1:def 16
{(<*> N)};
coherence ;

end;

:: deftheorem defines NeutralProcess PNPROC_1:def 16 :

definition

let P be set ;
let N be Petri_net of P;
let t be Element of N;
func ElementaryProcess t -> non empty process of N equals :: PNPROC_1:def 17
{<*t*>};
coherence ;

end;

:: deftheorem defines ElementaryProcess PNPROC_1:def 17 :

theorem :: PNPROC_1:84

for P being set
for N being Petri_net of P
for R being process of N holds (NeutralProcess N) before R = R

proof end;

theorem :: PNPROC_1:85

for P being set
for N being Petri_net of P
for R being process of N holds R before (NeutralProcess N) = R

proof end;

theorem :: PNPROC_1:86

for P being set
for N being Petri_net of P
for R being process of N holds (NeutralProcess N) concur R = R

proof end;