:: Processes in {P}etri nets
:: by Grzegorz Bancerek , Mitsuru Aoki , Akio Matsumoto and Yasunari Shidama
::
:: Received December 20, 2002
:: Copyright (c) 2002-2012 Association of Mizar Users


begin

theorem Th1: :: PNPROC_1:1
for i being Element of NAT
for x being set st i > 0 holds
{[i,x]} is FinSubsequence
proofend;

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

theorem :: PNPROC_1:2
for q being FinSubsequence holds
( q = {} iff Seq q = {} )
proofend;

theorem Th3: :: PNPROC_1:3
for i being Element of NAT
for x being set
for q being FinSubsequence st q = {[i,x]} holds
Seq q = <*x*>
proofend;

registration
cluster Relation-like Function-like FinSubsequence-like  ->  finite for set ;
coherence
 for b1 being FinSubsequence holds b1 is finite
proofend;
end;

theorem Th4: :: PNPROC_1:4
for x being set
for q being FinSubsequence st Seq q = <*x*> holds
ex i being Element of NAT st q = {[i,x]}
proofend;

theorem Th5: :: PNPROC_1:5
for x1, y1, x2, y2 being set holds
( not {[x1,y1],[x2,y2]} is FinSequence or ( x1 = 1 &amp; x2 = 1 &amp; y1 = y2 ) or ( x1 = 1 &amp; x2 = 2 ) or ( x1 = 2 &amp; x2 = 1 ) )
proofend;

theorem Th6: :: PNPROC_1:6
for x1, x2 being set holds <*x1,x2*> = {[1,x1],[2,x2]}
proofend;

Lm2:  for j, k, l being Element of NAT st ( ( 1 <= j &amp; j <= l ) or ( l + 1 <= j &amp; j <= l + k ) ) holds
( 1 <= j &amp; j <= l + k )
proofend;

theorem Th7: :: PNPROC_1:7
for p being FinSubsequence holds card p = len (Seq p)
proofend;

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

Lm4:  for P, R being Relation st dom P misses dom R holds
P misses R
by RELAT_1:179;

theorem Th8: :: PNPROC_1:8
for X, Y being set
for P, R being Relation st X misses Y holds
P | X misses R | Y
proofend;

Lm5:  for q being FinSubsequence holds dom (Seq q) = dom (Sgm (dom q))
proofend;

Lm6:  for q being FinSubsequence holds rng q = rng (Seq q)
proofend;

theorem Th9: :: PNPROC_1:9
for f, g, h being Function st f c= h &amp; g c= h &amp; f misses g holds
dom f misses dom g
proofend;

theorem :: PNPROC_1:10
for Y being set
for R being Relation holds Y |` R c= R | (R " Y)
proofend;

theorem Th11: :: PNPROC_1:11
for Y being set
for f being Function holds Y |` f = f | (f " Y)
proofend;

begin

definition
let P be set ;
mode marking of P is Function of P,NAT;
end;

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

scheme :: PNPROC_1:sch 1
MarkingLambda{ F1() ->  set , F2( set ) ->  Element of NAT } :
 ex m being Function of F1(),NAT st
for p being set st p in F1() holds
m . p = F2(p)
proofend;

definition
let P be set ;
let m1, m2 be marking of P;
:: original:=
redefine predm1 = m2 means :: PNPROC_1:def 1
 for p being set st p in P holds
p multitude_of = p multitude_of ;
compatibility
 ( m1 = m2 iff for p being set st p in P holds
p multitude_of = p multitude_of )
proofend;
end;

:: deftheorem  defines = PNPROC_1:def_1_:_
 for P being set
for m1, m2 being marking of P holds
( m1 = m2 iff for p being set st p in P holds
p multitude_of = p multitude_of );

definition
let P be set ;
func {$} P ->  marking of P equals :: PNPROC_1:def 2
P --> 0;
coherence
 P --> 0 is marking of P ;
end;

:: deftheorem  defines {$} PNPROC_1:def_2_:_
 for P being set holds {$} P = P --> 0;

definition
let P be set ;
let m1, m2 be marking of P;
predm1 c= m2 means :Def3: :: PNPROC_1:def 3
 for p being set st p in P holds
p multitude_of <= p multitude_of ;
reflexivity
 for m1 being marking of P
for p being set st p in P holds
p multitude_of <= p multitude_of ;
end;

:: deftheorem Def3 defines c= PNPROC_1:def_3_:_
 for P being set
for m1, m2 being marking of P holds
( m1 c= m2 iff for p being set st p in P holds
p multitude_of <= p multitude_of );

Lm7:  for p, P being set st p in P holds
({$} P) . p = 0
by FUNCOP_1:7;

theorem Th12: :: PNPROC_1:12
for P being set
for m being marking of P holds {$} P c= m
proofend;

theorem Th13: :: PNPROC_1:13
for P being set
for m1, m2, m3 being marking of P st m1 c= m2 &amp; m2 c= m3 holds
m1 c= m3
proofend;

definition
let P be set ;
let m1, m2 be marking of P;
:: original:+
redefine funcm1 + m2 ->  marking of P means :Def4: :: PNPROC_1:def 4
 for p being set st p in P holds
p multitude_of = (p multitude_of ) + (p multitude_of );
coherence
 m1 + m2 is marking of P
proofend;
compatibility
 for b1 being marking of P holds
( b1 = m1 + m2 iff for p being set st p in P holds
p multitude_of = (p multitude_of ) + (p multitude_of ) )
proofend;
end;

:: deftheorem Def4 defines + PNPROC_1:def_4_:_
 for P being set
for m1, m2, b4 being marking of P holds
( b4 = m1 + m2 iff for p being set st p in P holds
p multitude_of = (p multitude_of ) + (p multitude_of ) );

theorem :: PNPROC_1:14
for P being set
for m being marking of P holds m + ({$} P) = m
proofend;

definition
let P be set ;
let m1, m2 be marking of P;
assume B1: m2 c= m1 ;
funcm1 - m2 ->  marking of P means :Def5: :: PNPROC_1:def 5
 for p being set st p in P holds
p multitude_of = (p multitude_of ) - (p multitude_of );
existence
 ex b1 being marking of P st
for p being set st p in P holds
p multitude_of = (p multitude_of ) - (p multitude_of )
proofend;
uniqueness
 for b1, b2 being marking of P st ( for p being set st p in P holds
p multitude_of = (p multitude_of ) - (p multitude_of ) ) &amp; ( for p being set st p in P holds
p multitude_of = (p multitude_of ) - (p multitude_of ) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def5 defines - PNPROC_1:def_5_:_
 for P being set
for m1, m2 being marking of P st m2 c= m1 holds
for b4 being marking of P holds
( b4 = m1 - m2 iff for p being set st p in P holds
p multitude_of = (p multitude_of ) - (p multitude_of ) );

theorem Th15: :: PNPROC_1:15
for P being set
for m1, m2 being marking of P holds m1 c= m1 + m2
proofend;

theorem :: PNPROC_1:16
for P being set
for m being marking of P holds m - ({$} P) = m
proofend;

theorem :: PNPROC_1:17
for P being set
for m1, m2, m3 being marking of P st m1 c= m2 &amp; m2 c= m3 holds
m3 - m2 c= m3 - m1
proofend;

theorem Th18: :: PNPROC_1:18
for P being set
for m1, m2 being marking of P holds (m1 + m2) - m2 = m1
proofend;

theorem Th19: :: PNPROC_1:19
for P being set
for m, m1, m2 being marking of P st m c= m1 &amp; m1 c= m2 holds
m1 - m c= m2 - m
proofend;

theorem Th20: :: PNPROC_1:20
for P being set
for m1, m2, m3 being marking of P st m1 c= m2 holds
(m2 + m3) - m1 = (m2 - m1) + m3
proofend;

theorem :: PNPROC_1:21
for P being set
for m1, m2 being marking of P st m1 c= m2 &amp; m2 c= m1 holds
m1 = m2
proofend;

theorem Th22: :: PNPROC_1:22
for P being set
for m1, m2, m3 being marking of P holds (m1 + m2) + m3 = m1 + (m2 + m3)
proofend;

theorem :: PNPROC_1:23
for P being set
for m1, m2, m3, m4 being marking of P st m1 c= m2 &amp; m3 c= m4 holds
m1 + m3 c= m2 + m4
proofend;

theorem :: PNPROC_1:24
for P being set
for m1, m2 being marking of P st m1 c= m2 holds
m2 - m1 c= m2
proofend;

theorem Th25: :: PNPROC_1:25
for P being set
for m1, m2, m3, m4 being marking of P st m1 c= m2 &amp; m3 c= m4 &amp; m4 c= m1 holds
m1 - m4 c= m2 - m3
proofend;

theorem Th26: :: PNPROC_1:26
for P being set
for m1, m2 being marking of P st m1 c= m2 holds
m2 = (m2 - m1) + m1
proofend;

theorem Th27: :: PNPROC_1:27
for P being set
for m1, m2 being marking of P holds (m1 + m2) - m1 = m2
proofend;

theorem Th28: :: PNPROC_1:28
for P being set
for m2, m3, m1 being marking of P st m2 + m3 c= m1 holds
(m1 - m2) - m3 = m1 - (m2 + m3)
proofend;

theorem :: PNPROC_1:29
for P being set
for m3, m2, m1 being marking of P st m3 c= m2 &amp; m2 c= m1 holds
m1 - (m2 - m3) = (m1 - m2) + m3
proofend;

begin

definition
let P be set ;
mode transition of P ->  set  means :Def6: :: PNPROC_1:def 6
 ex m1, m2 being marking of P st it = [m1,m2];
existence
 ex b1 being set ex m1, m2 being marking of P st b1 = [m1,m2]
proofend;
end;

:: deftheorem Def6 defines transition PNPROC_1:def_6_:_
 for P being set
for b2 being set holds
( b2 is transition of P iff ex m1, m2 being marking of P st b2 = [m1,m2] );

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
 Pre is marking of P
proofend;
:: original:Post
redefine func Post t ->  marking of P;
coherence
 Post is marking of P
proofend;
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 :Def7: :: PNPROC_1:def 7
(m - (Pre t)) + (Post t) if  Pre t c= m
otherwise m;
coherence
 ( ( Pre t c= m implies (m - (Pre t)) + (Post t) is marking of P ) &amp; ( not Pre t c= m implies m is marking of P ) ) ;
consistency
 for b1 being marking of P holds verum ;
end;

:: deftheorem Def7 defines fire PNPROC_1:def_7_:_
 for P being set
for m being marking of P
for t being transition of P holds
( ( Pre t c= m implies fire (t,m) = (m - (Pre t)) + (Post t) ) &amp; ( not Pre t c= m implies fire (t,m) = m ) );

theorem :: PNPROC_1:30
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)
proofend;

definition
let P be set ;
let t be transition of P;
func fire t ->  Function means :Def8: :: PNPROC_1:def 8
( dom it = Funcs (P,NAT) &amp; ( for m being marking of P holds it . m = fire (t,m) ) );
existence
 ex b1 being Function st
( dom b1 = Funcs (P,NAT) &amp; ( for m being marking of P holds b1 . m = fire (t,m) ) )
proofend;
uniqueness
 for b1, b2 being Function st dom b1 = Funcs (P,NAT) &amp; ( for m being marking of P holds b1 . m = fire (t,m) ) &amp; dom b2 = Funcs (P,NAT) &amp; ( for m being marking of P holds b2 . m = fire (t,m) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def8 defines fire PNPROC_1:def_8_:_
 for P being set
for t being transition of P
for b3 being Function holds
( b3 = fire t iff ( dom b3 = Funcs (P,NAT) &amp; ( for m being marking of P holds b3 . m = fire (t,m) ) ) );

theorem Th31: :: PNPROC_1:31
for P being set
for t being transition of P holds rng (fire t) c= Funcs (P,NAT)
proofend;

theorem :: PNPROC_1:32
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
proofend;

definition
let P be set ;
mode Petri_net of P ->  non empty set  means :Def9: :: PNPROC_1:def 9
 for x being set st x in it holds
x is transition of P;
existence
 ex b1 being non empty set st
for x being set st x in b1 holds
x is transition of P
proofend;
end;

:: deftheorem Def9 defines Petri_net PNPROC_1:def_9_:_
 for P being set
for b2 being non empty set holds
( b2 is Petri_net of P iff for x being set st x in b2 holds
x is transition of P );

definition
let P be set ;
let N be Petri_net of P;
:: original:Element
redefine mode Element of N ->  transition of P;
coherence
 for b1 being Element of N holds b1 is transition of P by Def9;
end;

begin

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 :Def10: :: PNPROC_1:def 10
 ex F being Function-yielding FinSequence st
( it = compose (F,(Funcs (P,NAT))) &amp; len F = len C &amp; ( for i being Element of NAT st i in dom C holds
F . i = fire (C /. i) ) );
existence
 ex b1 being Function ex F being Function-yielding FinSequence st
( b1 = compose (F,(Funcs (P,NAT))) &amp; len F = len C &amp; ( for i being Element of NAT st i in dom C holds
F . i = fire (C /. i) ) )
proofend;
uniqueness
 for b1, b2 being Function st ex F being Function-yielding FinSequence st
( b1 = compose (F,(Funcs (P,NAT))) &amp; len F = len C &amp; ( for i being Element of NAT st i in dom C holds
F . i = fire (C /. i) ) ) &amp; ex F being Function-yielding FinSequence st
( b2 = compose (F,(Funcs (P,NAT))) &amp; len F = len C &amp; ( for i being Element of NAT st i in dom C holds
F . i = fire (C /. i) ) ) holds
b1 = b2
proofend;
end;

:: deftheorem Def10 defines fire PNPROC_1:def_10_:_
 for P being set
for N being Petri_net of P
for C being firing-sequence of N
for b4 being Function holds
( b4 = fire C iff ex F being Function-yielding FinSequence st
( b4 = compose (F,(Funcs (P,NAT))) &amp; len F = len C &amp; ( for i being Element of NAT st i in dom C holds
F . i = fire (C /. i) ) ) );

:: Firing of empty firing-sequence <*>N
theorem :: PNPROC_1:33
for P being set
for N being Petri_net of P holds fire (<*> N) = id (Funcs (P,NAT))
proofend;

:: Firing of firing-sequence with one translation <*e*>
theorem Th34: :: PNPROC_1:34
for P being set
for N being Petri_net of P
for e being Element of N holds fire <*e*> = fire e
proofend;

theorem Th35: :: PNPROC_1:35
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
proofend;

:: Firing of firing-sequence with two translations <*e1,e2*>
theorem :: PNPROC_1:36
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)
proofend;

theorem Th37: :: PNPROC_1:37
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) &amp; rng (fire C) c= Funcs (P,NAT) )
proofend;

:: Firing of compound firing-sequence
theorem Th38: :: PNPROC_1:38
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)
proofend;

theorem :: PNPROC_1:39
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)
proofend;

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 11
(fire C) . m;
coherence
 (fire C) . m is marking of P
proofend;
end;

:: deftheorem  defines fire PNPROC_1:def_11_:_
 for P being set
for N being Petri_net of P
for C being firing-sequence of N
for m being marking of P holds fire (C,m) = (fire C) . m;

begin

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

definition
let P be set ;
let N be Petri_net of P;
let R1, R2 be process of N;
funcR1 before R2 ->  process of N equals :: PNPROC_1:def 12
 {  (C1 ^ C2) where C1, C2 is firing-sequence of N : ( C1 in R1 &amp; C2 in R2 )  }  ;
coherence
  {  (C1 ^ C2) where C1, C2 is firing-sequence of N : ( C1 in R1 &amp; C2 in R2 )  }  is process of N
proofend;
end;

:: deftheorem  defines before PNPROC_1:def_12_:_
 for P being set
for N being Petri_net of P
for R1, R2 being process of N holds R1 before R2 =  {  (C1 ^ C2) where C1, C2 is firing-sequence of N : ( C1 in R1 &amp; C2 in R2 )  }  ;

registration
let P be set ;
let N be Petri_net of P;
let R1, R2 be non empty process of N;
clusterR1 before R2 ->  non empty ;
coherence
 not R1 before R2 is empty
proofend;
end;

theorem Th40: :: PNPROC_1:40
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)
proofend;

theorem Th41: :: PNPROC_1:41
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)
proofend;

theorem Th42: :: PNPROC_1:42
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)}
proofend;

theorem :: PNPROC_1:43
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)}
proofend;

theorem :: PNPROC_1:44
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)}
proofend;

begin

definition
let P be set ;
let N be Petri_net of P;
let R1, R2 be process of N;
funcR1 concur R2 ->  process of N equals :: PNPROC_1:def 13
 {  C where C is firing-sequence of N : ex q1, q2 being FinSubsequence st
( C = q1 \/ q2 &amp; q1 misses q2 &amp; Seq q1 in R1 &amp; Seq q2 in R2 )  }  ;
coherence
  {  C where C is firing-sequence of N : ex q1, q2 being FinSubsequence st
( C = q1 \/ q2 &amp; q1 misses q2 &amp; Seq q1 in R1 &amp; Seq q2 in R2 )  }  is process of N
proofend;
commutativity
 for b1, R1, R2 being process of N st b1 =  {  C where C is firing-sequence of N : ex q1, q2 being FinSubsequence st
( C = q1 \/ q2 &amp; q1 misses q2 &amp; Seq q1 in R1 &amp; Seq q2 in R2 )  }  holds
b1 =  {  C where C is firing-sequence of N : ex q1, q2 being FinSubsequence st
( C = q1 \/ q2 &amp; q1 misses q2 &amp; Seq q1 in R2 &amp; Seq q2 in R1 )  }
proofend;
end;

:: deftheorem  defines concur PNPROC_1:def_13_:_
 for P being set
for N being Petri_net of P
for R1, R2 being process of N holds R1 concur R2 =  {  C where C is firing-sequence of N : ex q1, q2 being FinSubsequence st
( C = q1 \/ q2 &amp; q1 misses q2 &amp; Seq q1 in R1 &amp; Seq q2 in R2 )  }  ;

theorem Th45: :: PNPROC_1:45
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)
proofend;

theorem Th46: :: PNPROC_1:46
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*>}
proofend;

theorem :: PNPROC_1:47
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*>}
proofend;

theorem :: PNPROC_1:48
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)
proofend;

definition
let p be FinSubsequence;
let i be Element of NAT ;
set X =  {  (i + k) where k is Element of NAT : k in dom p  }  ;
funci Shift p ->  FinSubsequence means :Def14: :: PNPROC_1:def 14
( dom it =  {  (i + k) where k is Element of NAT : k in dom p  }  &amp; ( for j being Nat st j in dom p holds
it . (i + j) = p . j ) );
existence
 ex b1 being FinSubsequence st
( dom b1 =  {  (i + k) where k is Element of NAT : k in dom p  }  &amp; ( for j being Nat st j in dom p holds
b1 . (i + j) = p . j ) )
proofend;
uniqueness
 for b1, b2 being FinSubsequence st dom b1 =  {  (i + k) where k is Element of NAT : k in dom p  }  &amp; ( for j being Nat st j in dom p holds
b1 . (i + j) = p . j ) &amp; dom b2 =  {  (i + k) where k is Element of NAT : k in dom p  }  &amp; ( for j being Nat st j in dom p holds
b2 . (i + j) = p . j ) holds
b1 = b2
proofend;
end;

:: deftheorem Def14 defines Shift PNPROC_1:def_14_:_
 for p being FinSubsequence
for i being Element of NAT
for b3 being FinSubsequence holds
( b3 = i Shift p iff ( dom b3 =  {  (i + k) where k is Element of NAT : k in dom p  }  &amp; ( for j being Nat st j in dom p holds
b3 . (i + j) = p . j ) ) );

theorem :: PNPROC_1:49
for q being FinSubsequence holds 0 Shift q = q
proofend;

theorem :: PNPROC_1:50
for i, j being Element of NAT
for q being FinSubsequence holds (i + j) Shift q = i Shift (j Shift q)
proofend;

theorem Th51: :: PNPROC_1:51
for i being Element of NAT
for p being FinSequence holds dom (i Shift p) =  {  j1 where j1 is Element of NAT : ( i + 1 <= j1 &amp; j1 <= i + (len p) )  }
proofend;

theorem :: PNPROC_1:52
for i being Element of NAT
for q being FinSubsequence holds
( q = {} iff i Shift q = {} )
proofend;

theorem Th53: :: PNPROC_1:53
for i being Element of NAT
for q being FinSubsequence ex ss being FinSubsequence st
( dom ss = dom q &amp; rng ss = dom (i Shift q) &amp; ( for k being Element of NAT st k in dom q holds
ss . k = i + k ) &amp; ss is one-to-one )
proofend;

Lm8:  for i being Element of NAT
for p being FinSequence ex fs being FinSequence st
( dom fs = dom p &amp; rng fs = dom (i Shift p) &amp; ( for k being Element of NAT st k in dom p holds
fs . k = i + k ) &amp; fs is one-to-one )
proofend;

theorem Th54: :: PNPROC_1:54
for i being Element of NAT
for q being FinSubsequence holds card q = card (i Shift q)
proofend;

theorem Th55: :: PNPROC_1:55
for i being Element of NAT
for p being FinSequence holds dom p = dom (Seq (i Shift p))
proofend;

theorem Th56: :: PNPROC_1:56
for k, i being Element of NAT
for p being FinSequence st k in dom p holds
(Sgm (dom (i Shift p))) . k = i + k
proofend;

theorem Th57: :: PNPROC_1:57
for k, i being Element of NAT
for p being FinSequence st k in dom p holds
(Seq (i Shift p)) . k = p . k
proofend;

theorem Th58: :: PNPROC_1:58
for i being Element of NAT
for p being FinSequence holds Seq (i Shift p) = p
proofend;

theorem Th59: :: PNPROC_1:59
for p1, p2 being FinSequence holds dom (p1 \/ ((len p1) Shift p2)) = Seg ((len p1) + (len p2))
proofend;

theorem Th60: :: PNPROC_1:60
for i being Element of NAT
for p1 being FinSequence
for p2 being FinSubsequence st len p1 <= i holds
dom p1 misses dom (i Shift p2)
proofend;

theorem Th61: :: PNPROC_1:61
for p1, p2 being FinSequence holds p1 ^ p2 = p1 \/ ((len p1) Shift p2)
proofend;

theorem Th62: :: PNPROC_1:62
for i being Element of NAT
for p1 being FinSequence
for p2 being FinSubsequence st i >= len p1 holds
p1 misses i Shift p2
proofend;

theorem :: PNPROC_1:63
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)
proofend;

theorem :: PNPROC_1:64
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
proofend;

theorem :: PNPROC_1:65
for P being set
for N being Petri_net of P
for R1, P1, R2, P2 being process of N st R1 c= P1 &amp; R2 c= P2 holds
R1 before R2 c= P1 before P2
proofend;

theorem :: PNPROC_1:66
for P being set
for N being Petri_net of P
for R1, P1, R2, P2 being process of N st R1 c= P1 &amp; R2 c= P2 holds
R1 concur R2 c= P1 concur P2
proofend;

Lm9:  for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 &amp; q2 c= p2 holds
dom (q1 \/ ((len p1) Shift q2)) c= dom (p1 ^ p2)
proofend;

Lm10:  for p1 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 holds
q1 misses (len p1) Shift q2
by Th62, XBOOLE_1:63;

Lm11:  for i being Element of NAT
for p, q being FinSubsequence st q c= p holds
dom (i Shift q) c= dom (i Shift p)
proofend;

theorem Th67: :: PNPROC_1:67
for i being Element of NAT
for p, q being FinSubsequence st q c= p holds
i Shift q c= i Shift p
proofend;

theorem Th68: :: PNPROC_1:68
for p1, p2 being FinSequence holds (len p1) Shift p2 c= p1 ^ p2
proofend;

theorem Th69: :: PNPROC_1:69
for i being Element of NAT
for q1, q2 being FinSubsequence st dom q1 misses dom q2 holds
dom (i Shift q1) misses dom (i Shift q2)
proofend;

theorem Th70: :: PNPROC_1:70
for i being Element of NAT
for q, q1, q2 being FinSubsequence st q = q1 \/ q2 &amp; q1 misses q2 holds
(i Shift q1) \/ (i Shift q2) = i Shift q
proofend;

theorem Th71: :: PNPROC_1:71
for i being Element of NAT
for q being FinSubsequence holds dom (Seq q) = dom (Seq (i Shift q))
proofend;

theorem Th72: :: PNPROC_1:72
for k, i being Element of NAT
for q being FinSubsequence st k in dom (Seq q) holds
ex j being Element of NAT st
( j = (Sgm (dom q)) . k &amp; (Sgm (dom (i Shift q))) . k = i + j )
proofend;

theorem Th73: :: PNPROC_1:73
for k, i being Element of NAT
for q being FinSubsequence st k in dom (Seq q) holds
(Seq (i Shift q)) . k = (Seq q) . k
proofend;

theorem :: PNPROC_1:74
for i being Element of NAT
for q being FinSubsequence holds Seq q = Seq (i Shift q)
proofend;

theorem Th75: :: PNPROC_1:75
for k, i being Element of NAT
for q being FinSubsequence st dom q c= Seg k holds
dom (i Shift q) c= Seg (i + k)
proofend;

theorem Th76: :: PNPROC_1:76
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)
proofend;

Lm12:  for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 &amp; q2 c= p2 holds
Sgm ((dom q1) \/ (dom ((len p1) Shift q2))) = (Sgm (dom q1)) ^ (Sgm (dom ((len p1) Shift q2)))
proofend;

theorem Th77: :: PNPROC_1:77
for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 &amp; q2 c= p2 holds
ex ss being FinSubsequence st
( ss = q1 \/ ((len p1) Shift q2) &amp; dom (Seq ss) = Seg ((len (Seq q1)) + (len (Seq q2))) )
proofend;

theorem Th78: :: PNPROC_1:78
for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 &amp; q2 c= p2 holds
ex ss being FinSubsequence st
( ss = q1 \/ ((len p1) Shift q2) &amp; dom (Seq ss) = Seg ((len (Seq q1)) + (len (Seq q2))) &amp; Seq ss = (Seq q1) \/ ((len (Seq q1)) Shift (Seq q2)) )
proofend;

theorem Th79: :: PNPROC_1:79
for p1, p2 being FinSequence
for q1, q2 being FinSubsequence st q1 c= p1 &amp; q2 c= p2 holds
ex ss being FinSubsequence st
( ss = q1 \/ ((len p1) Shift q2) &amp; (Seq q1) ^ (Seq q2) = Seq ss )
proofend;

theorem :: PNPROC_1:80
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)
proofend;

registration
let P be set ;
let N be Petri_net of P;
let R1, R2 be non empty process of N;
clusterR1 concur R2 ->  non empty ;
coherence
 not R1 concur R2 is empty
proofend;
end;

begin

definition
let P be set ;
let N be Petri_net of P;
func NeutralProcess N ->  non empty process of N equals :: PNPROC_1:def 15
{(<*> N)};
coherence
 {(<*> N)} is non empty process of N ;
end;

:: deftheorem  defines NeutralProcess PNPROC_1:def_15_:_
 for P being set
for N being Petri_net of P holds NeutralProcess N = {(<*> N)};

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 16
{<*t*>};
coherence
 {<*t*>} is non empty process of N ;
end;

:: deftheorem  defines ElementaryProcess PNPROC_1:def_16_:_
 for P being set
for N being Petri_net of P
for t being Element of N holds ElementaryProcess t = {<*t*>};

theorem :: PNPROC_1:81
for P being set
for N being Petri_net of P
for R being process of N holds (NeutralProcess N) before R = R
proofend;

theorem :: PNPROC_1:82
for P being set
for N being Petri_net of P
for R being process of N holds R before (NeutralProcess N) = R
proofend;

theorem :: PNPROC_1:83
for P being set
for N being Petri_net of P
for R being process of N holds (NeutralProcess N) concur R = R
proofend;