:: PNPROC_1 semantic presentation
theorem Th1: :: PNPROC_1:1
Lemma2:
for b1 being FinSubsequence holds
( Seq b1 = {} implies b1 = {} )
theorem Th2: :: PNPROC_1:2
theorem Th3: :: PNPROC_1:3
theorem Th4: :: PNPROC_1:4
theorem Th5: :: PNPROC_1:5
for b
1, b
2, b
3, b
4 being
set holds
not (
{[b1,b2],[b3,b4]} is
FinSequence & not ( b
1 = 1 & b
3 = 1 & b
2 = b
4 ) & not ( b
1 = 1 & b
3 = 2 ) & not ( b
1 = 2 & b
3 = 1 ) )
theorem Th6: :: PNPROC_1:6
Lemma8:
for b1, b2, b3 being Nat holds
( ( ( 1 <= b1 & b1 <= b3 ) or ( b3 + 1 <= b1 & b1 <= b3 + b2 ) ) implies ( 1 <= b1 & b1 <= b3 + b2 ) )
theorem Th7: :: PNPROC_1:7
Lemma10:
for b1, b2 being set holds
( ( for b3 being set holds
not ( b3 in b1 & b3 in b2 ) ) iff b1 misses b2 )
theorem Th8: :: PNPROC_1:8
theorem Th9: :: PNPROC_1:9
Lemma13:
for b1 being FinSubsequence holds dom (Seq b1) = dom (Sgm (dom b1))
Lemma14:
for b1 being FinSubsequence holds rng b1 = rng (Seq b1)
theorem Th10: :: PNPROC_1:10
theorem Th11: :: PNPROC_1:11
theorem Th12: :: PNPROC_1:12
:: deftheorem Def1 defines marking PNPROC_1:def 1 :
:: deftheorem Def2 defines = PNPROC_1:def 2 :
:: deftheorem Def3 defines {$} PNPROC_1:def 3 :
:: deftheorem Def4 defines c= PNPROC_1:def 4 :
Lemma19:
for b1, b2 being set holds
( b1 in b2 implies ({$} b2) multitude_of b1 = 0 )
theorem Th13: :: PNPROC_1:13
theorem Th14: :: PNPROC_1:14
for b
1 being
set for b
2, b
3, b
4 being
marking of b
1 holds
( b
2 c= b
3 & b
3 c= b
4 implies b
2 c= b
4 )
:: deftheorem Def5 defines + PNPROC_1:def 5 :
theorem Th15: :: PNPROC_1:15
:: deftheorem Def6 defines - PNPROC_1:def 6 :
theorem Th16: :: PNPROC_1:16
for b
1 being
set for b
2, b
3 being
marking of b
1 holds b
2 c= b
2 + b
3
theorem Th17: :: PNPROC_1:17
theorem Th18: :: PNPROC_1:18
for b
1 being
set for b
2, b
3, b
4 being
marking of b
1 holds
( b
2 c= b
3 & b
3 c= b
4 implies b
4 - b
3 c= b
4 - b
2 )
theorem Th19: :: PNPROC_1:19
for b
1 being
set for b
2, b
3 being
marking of b
1 holds
(b2 + b3) - b
3 = b
2
theorem Th20: :: PNPROC_1:20
for b
1 being
set for b
2, b
3, b
4 being
marking of b
1 holds
( b
2 c= b
3 & b
3 c= b
4 implies b
3 - b
2 c= b
4 - b
2 )
theorem Th21: :: PNPROC_1:21
for b
1 being
set for b
2, b
3, b
4 being
marking of b
1 holds
( b
2 c= b
3 implies
(b3 + b4) - b
2 = (b3 - b2) + b
4 )
theorem Th22: :: PNPROC_1:22
for b
1 being
set for b
2, b
3 being
marking of b
1 holds
( b
2 c= b
3 & b
3 c= b
2 implies b
2 = b
3 )
theorem Th23: :: PNPROC_1:23
for b
1 being
set for b
2, b
3, b
4 being
marking of b
1 holds
(b2 + b3) + b
4 = b
2 + (b3 + b4)
theorem Th24: :: PNPROC_1:24
for b
1 being
set for b
2, b
3, b
4, b
5 being
marking of b
1 holds
( b
2 c= b
3 & b
4 c= b
5 implies b
2 + b
4 c= b
3 + b
5 )
theorem Th25: :: PNPROC_1:25
for b
1 being
set for b
2, b
3 being
marking of b
1 holds
( b
2 c= b
3 implies b
3 - b
2 c= b
3 )
theorem Th26: :: PNPROC_1:26
for b
1 being
set for b
2, b
3, b
4, b
5 being
marking of b
1 holds
( b
2 c= b
3 & b
4 c= b
5 & b
5 c= b
2 implies b
2 - b
5 c= b
3 - b
4 )
theorem Th27: :: PNPROC_1:27
for b
1 being
set for b
2, b
3 being
marking of b
1 holds
( b
2 c= b
3 implies b
3 = (b3 - b2) + b
2 )
theorem Th28: :: PNPROC_1:28
for b
1 being
set for b
2, b
3 being
marking of b
1 holds
(b2 + b3) - b
2 = b
3
theorem Th29: :: PNPROC_1:29
for b
1 being
set for b
2, b
3, b
4 being
marking of b
1 holds
( b
2 + b
3 c= b
4 implies
(b4 - b2) - b
3 = b
4 - (b2 + b3) )
theorem Th30: :: PNPROC_1:30
for b
1 being
set for b
2, b
3, b
4 being
marking of b
1 holds
( b
2 c= b
3 & b
3 c= b
4 implies b
4 - (b3 - b2) = (b4 - b3) + b
2 )
theorem Th31: :: PNPROC_1:31
theorem Th32: :: PNPROC_1:32
:: deftheorem Def7 defines transition PNPROC_1:def 7 :
:: deftheorem Def8 defines fire PNPROC_1:def 8 :
theorem Th33: :: PNPROC_1:33
:: deftheorem Def9 defines fire PNPROC_1:def 9 :
theorem Th34: :: PNPROC_1:34
theorem Th35: :: PNPROC_1:35
:: deftheorem Def10 defines Petri_net PNPROC_1:def 10 :
:: deftheorem Def11 defines fire PNPROC_1:def 11 :
theorem Th36: :: PNPROC_1:36
theorem Th37: :: PNPROC_1:37
theorem Th38: :: PNPROC_1:38
theorem Th39: :: PNPROC_1:39
theorem Th40: :: PNPROC_1:40
theorem Th41: :: PNPROC_1:41
theorem Th42: :: PNPROC_1:42
:: deftheorem Def12 defines fire PNPROC_1:def 12 :
:: deftheorem Def13 defines before PNPROC_1:def 13 :
theorem Th43: :: PNPROC_1:43
theorem Th44: :: PNPROC_1:44
theorem Th45: :: PNPROC_1:45
theorem Th46: :: PNPROC_1:46
theorem Th47: :: PNPROC_1:47
:: deftheorem Def14 defines concur PNPROC_1:def 14 :
theorem Th48: :: PNPROC_1:48
theorem Th49: :: PNPROC_1:49
theorem Th50: :: PNPROC_1:50
for b
1 being
set for b
2 being
Petri_net of b
1for b
3, b
4, b
5 being
Element of b
2 holds
{<*b3*>,<*b4*>} concur {<*b5*>} = {<*b3,b5*>,<*b4,b5*>,<*b5,b3*>,<*b5,b4*>}
theorem Th51: :: PNPROC_1:51
:: deftheorem Def15 defines Shift PNPROC_1:def 15 :
theorem Th52: :: PNPROC_1:52
theorem Th53: :: PNPROC_1:53
theorem Th54: :: PNPROC_1:54
theorem Th55: :: PNPROC_1:55
theorem Th56: :: PNPROC_1:56
Lemma55:
for b1 being Nat
for b2 being FinSequence holds
ex b3 being FinSequence st
( dom b3 = dom b2 & rng b3 = dom (b1 Shift b2) & ( for b4 being Nat holds
( b4 in dom b2 implies b3 . b4 = b1 + b4 ) ) & b3 is one-to-one )
theorem Th57: :: PNPROC_1:57
theorem Th58: :: PNPROC_1:58
theorem Th59: :: PNPROC_1:59
theorem Th60: :: PNPROC_1:60
theorem Th61: :: PNPROC_1:61
theorem Th62: :: PNPROC_1:62
theorem Th63: :: PNPROC_1:63
theorem Th64: :: PNPROC_1:64
theorem Th65: :: PNPROC_1:65
theorem Th66: :: PNPROC_1:66
theorem Th67: :: PNPROC_1:67
theorem Th68: :: PNPROC_1:68
theorem Th69: :: PNPROC_1:69
Lemma65:
for b1, b2 being FinSequence
for b3, b4 being FinSubsequence holds
( b3 c= b1 & b4 c= b2 implies dom (b3 \/ ((len b1) Shift b4)) c= dom (b1 ^ b2) )
Lemma66:
for b1 being FinSequence
for b2, b3 being FinSubsequence holds
( b2 c= b1 implies b2 misses (len b1) Shift b3 )
Lemma67:
for b1 being Nat
for b2, b3 being FinSubsequence holds
( b3 c= b2 implies dom (b1 Shift b3) c= dom (b1 Shift b2) )
theorem Th70: :: PNPROC_1:70
theorem Th71: :: PNPROC_1:71
theorem Th72: :: PNPROC_1:72
theorem Th73: :: PNPROC_1:73
theorem Th74: :: PNPROC_1:74
theorem Th75: :: PNPROC_1:75
theorem Th76: :: PNPROC_1:76
theorem Th77: :: PNPROC_1:77
theorem Th78: :: PNPROC_1:78
theorem Th79: :: PNPROC_1:79
Lemma78:
for b1, b2 being FinSequence
for b3, b4 being FinSubsequence holds
( b4 <> {} & b3 c= b1 & b4 c= b2 implies Sgm ((dom b3) \/ (dom ((len b1) Shift b4))) = (Sgm (dom b3)) ^ (Sgm (dom ((len b1) Shift b4))) )
theorem Th80: :: PNPROC_1:80
theorem Th81: :: PNPROC_1:81
theorem Th82: :: PNPROC_1:82
theorem Th83: :: PNPROC_1:83
:: deftheorem Def16 defines NeutralProcess PNPROC_1:def 16 :
:: deftheorem Def17 defines ElementaryProcess PNPROC_1:def 17 :
theorem Th84: :: PNPROC_1:84
theorem Th85: :: PNPROC_1:85
theorem Th86: :: PNPROC_1:86