:: NET_1 semantic presentation
:: deftheorem Def1 defines Petri NET_1:def 1 :
:: deftheorem Def2 defines Elements NET_1:def 2 :
theorem Th1: :: NET_1:1
canceled;
theorem Th2: :: NET_1:2
canceled;
theorem Th3: :: NET_1:3
canceled;
theorem Th4: :: NET_1:4
theorem Th5: :: NET_1:5
theorem Th6: :: NET_1:6
theorem Th7: :: NET_1:7
theorem Th8: :: NET_1:8
theorem Th9: :: NET_1:9
Lemma5:
Net(# {} ,{} ,{} #) is_Petri_net
theorem Th10: :: NET_1:10
canceled;
theorem Th11: :: NET_1:11
theorem Th12: :: NET_1:12
theorem Th13: :: NET_1:13
theorem Th14: :: NET_1:14
theorem Th15: :: NET_1:15
:: deftheorem Def3 NET_1:def 3 :
canceled;
:: deftheorem Def4 NET_1:def 4 :
canceled;
:: deftheorem Def5 defines pre NET_1:def 5 :
:: deftheorem Def6 defines post NET_1:def 6 :
definition
let c
1 be
Net ;
let c
2 be
Element of
Elements c
1;
func Pre c
1,c
2 -> set means :
Def7:
:: NET_1:def 7
for b
1 being
set holds
( b
1 in a
3 iff ( b
1 in Elements a
1 &
[b1,a2] in the
Flow of a
1 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ( b2 in Elements c1 & [b2,c2] in the Flow of c1 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 in Elements c1 & [b3,c2] in the Flow of c1 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 in Elements c1 & [b3,c2] in the Flow of c1 ) ) ) implies b1 = b2 )
func Post c
1,c
2 -> set means :
Def8:
:: NET_1:def 8
for b
1 being
set holds
( b
1 in a
3 iff ( b
1 in Elements a
1 &
[a2,b1] in the
Flow of a
1 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ( b2 in Elements c1 & [c2,b2] in the Flow of c1 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 in Elements c1 & [c2,b3] in the Flow of c1 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 in Elements c1 & [c2,b3] in the Flow of c1 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def7 defines Pre NET_1:def 7 :
:: deftheorem Def8 defines Post NET_1:def 8 :
theorem Th16: :: NET_1:16
theorem Th17: :: NET_1:17
theorem Th18: :: NET_1:18
theorem Th19: :: NET_1:19
theorem Th20: :: NET_1:20
theorem Th21: :: NET_1:21
:: deftheorem Def9 defines enter NET_1:def 9 :
theorem Th22: :: NET_1:22
theorem Th23: :: NET_1:23
theorem Th24: :: NET_1:24
:: deftheorem Def10 defines exit NET_1:def 10 :
theorem Th25: :: NET_1:25
theorem Th26: :: NET_1:26
theorem Th27: :: NET_1:27
:: deftheorem Def11 defines field NET_1:def 11 :
definition
let c
1 be
Net ;
let c
2 be
Element of the
Transitions of c
1;
func Prec c
1,c
2 -> set means :: NET_1:def 12
for b
1 being
set holds
( b
1 in a
3 iff ( b
1 in the
Places of a
1 &
[b1,a2] in the
Flow of a
1 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ( b2 in the Places of c1 & [b2,c2] in the Flow of c1 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 in the Places of c1 & [b3,c2] in the Flow of c1 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 in the Places of c1 & [b3,c2] in the Flow of c1 ) ) ) implies b1 = b2 )
func Postc c
1,c
2 -> set means :: NET_1:def 13
for b
1 being
set holds
( b
1 in a
3 iff ( b
1 in the
Places of a
1 &
[a2,b1] in the
Flow of a
1 ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ( b2 in the Places of c1 & [c2,b2] in the Flow of c1 ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 in the Places of c1 & [c2,b3] in the Flow of c1 ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 in the Places of c1 & [c2,b3] in the Flow of c1 ) ) ) implies b1 = b2 )
end;
:: deftheorem Def12 defines Prec NET_1:def 12 :
:: deftheorem Def13 defines Postc NET_1:def 13 :
definition
let c
1 be
Pnet;
let c
2 be
set ;
func Entr c
1,c
2 -> set means :
Def14:
:: NET_1:def 14
for b
1 being
set holds
( b
1 in a
3 iff ( b
1 c= Elements a
1 & ex b
2 being
Element of
Elements a
1 st
( b
2 in a
2 & b
1 = enter a
1,b
2 ) ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ( b2 c= Elements c1 & ex b3 being Element of Elements c1 st
( b3 in c2 & b2 = enter c1,b3 ) ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 c= Elements c1 & ex b4 being Element of Elements c1 st
( b4 in c2 & b3 = enter c1,b4 ) ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 c= Elements c1 & ex b4 being Element of Elements c1 st
( b4 in c2 & b3 = enter c1,b4 ) ) ) ) implies b1 = b2 )
func Ext c
1,c
2 -> set means :
Def15:
:: NET_1:def 15
for b
1 being
set holds
( b
1 in a
3 iff ( b
1 c= Elements a
1 & ex b
2 being
Element of
Elements a
1 st
( b
2 in a
2 & b
1 = exit a
1,b
2 ) ) );
existence
ex b1 being set st
for b2 being set holds
( b2 in b1 iff ( b2 c= Elements c1 & ex b3 being Element of Elements c1 st
( b3 in c2 & b2 = exit c1,b3 ) ) )
uniqueness
for b1, b2 being set holds
( ( for b3 being set holds
( b3 in b1 iff ( b3 c= Elements c1 & ex b4 being Element of Elements c1 st
( b4 in c2 & b3 = exit c1,b4 ) ) ) ) & ( for b3 being set holds
( b3 in b2 iff ( b3 c= Elements c1 & ex b4 being Element of Elements c1 st
( b4 in c2 & b3 = exit c1,b4 ) ) ) ) implies b1 = b2 )
end;
:: deftheorem Def14 defines Entr NET_1:def 14 :
:: deftheorem Def15 defines Ext NET_1:def 15 :
theorem Th28: :: NET_1:28
theorem Th29: :: NET_1:29
:: deftheorem Def16 defines Input NET_1:def 16 :
:: deftheorem Def17 defines Output NET_1:def 17 :
theorem Th30: :: NET_1:30
theorem Th31: :: NET_1:31
theorem Th32: :: NET_1:32
theorem Th33: :: NET_1:33