:: GRAPH_5 semantic presentation
theorem Th1: :: GRAPH_5:1
theorem Th2: :: GRAPH_5:2
theorem Th3: :: GRAPH_5:3
theorem Th4: :: GRAPH_5:4
theorem Th5: :: GRAPH_5:5
theorem Th6: :: GRAPH_5:6
theorem Th7: :: GRAPH_5:7
theorem Th8: :: GRAPH_5:8
Lemma9:
for b1 being Nat
for b2, b3 being FinSequence holds
( 1 <= b1 & b1 <= len b2 implies b2 . b1 = (b2 ^ b3) . b1 )
Lemma10:
for b1 being Nat
for b2, b3 being FinSequence holds
( 1 <= b1 & b1 <= len b2 implies b2 . b1 = (b3 ^ b2) . ((len b3) + b1) )
theorem Th9: :: GRAPH_5:9
theorem Th10: :: GRAPH_5:10
theorem Th11: :: GRAPH_5:11
theorem Th12: :: GRAPH_5:12
theorem Th13: :: GRAPH_5:13
theorem Th14: :: GRAPH_5:14
theorem Th15: :: GRAPH_5:15
theorem Th16: :: GRAPH_5:16
theorem Th17: :: GRAPH_5:17
theorem Th18: :: GRAPH_5:18
theorem Th19: :: GRAPH_5:19
theorem Th20: :: GRAPH_5:20
:: deftheorem Def1 defines vertices GRAPH_5:def 1 :
:: deftheorem Def2 defines vertices GRAPH_5:def 2 :
theorem Th21: :: GRAPH_5:21
theorem Th22: :: GRAPH_5:22
theorem Th23: :: GRAPH_5:23
theorem Th24: :: GRAPH_5:24
theorem Th25: :: GRAPH_5:25
theorem Th26: :: GRAPH_5:26
theorem Th27: :: GRAPH_5:27
theorem Th28: :: GRAPH_5:28
theorem Th29: :: GRAPH_5:29
theorem Th30: :: GRAPH_5:30
theorem Th31: :: GRAPH_5:31
theorem Th32: :: GRAPH_5:32
:: deftheorem Def3 defines is_orientedpath_of GRAPH_5:def 3 :
:: deftheorem Def4 defines is_orientedpath_of GRAPH_5:def 4 :
:: deftheorem Def5 defines OrientedPaths GRAPH_5:def 5 :
theorem Th33: :: GRAPH_5:33
theorem Th34: :: GRAPH_5:34
theorem Th35: :: GRAPH_5:35
theorem Th36: :: GRAPH_5:36
theorem Th37: :: GRAPH_5:37
theorem Th38: :: GRAPH_5:38
:: deftheorem Def6 defines is_acyclicpath_of GRAPH_5:def 6 :
:: deftheorem Def7 defines is_acyclicpath_of GRAPH_5:def 7 :
:: deftheorem Def8 defines AcyclicPaths GRAPH_5:def 8 :
:: deftheorem Def9 defines AcyclicPaths GRAPH_5:def 9 :
:: deftheorem Def10 defines AcyclicPaths GRAPH_5:def 10 :
:: deftheorem Def11 defines AcyclicPaths GRAPH_5:def 11 :
theorem Th39: :: GRAPH_5:39
theorem Th40: :: GRAPH_5:40
theorem Th41: :: GRAPH_5:41
theorem Th42: :: GRAPH_5:42
theorem Th43: :: GRAPH_5:43
theorem Th44: :: GRAPH_5:44
theorem Th45: :: GRAPH_5:45
theorem Th46: :: GRAPH_5:46
theorem Th47: :: GRAPH_5:47
theorem Th48: :: GRAPH_5:48
theorem Th49: :: GRAPH_5:49
:: deftheorem Def12 defines Real>=0 GRAPH_5:def 12 :
:: deftheorem Def13 defines is_weight>=0of GRAPH_5:def 13 :
:: deftheorem Def14 defines is_weight_of GRAPH_5:def 14 :
:: deftheorem Def15 defines RealSequence GRAPH_5:def 15 :
:: deftheorem Def16 defines cost GRAPH_5:def 16 :
theorem Th50: :: GRAPH_5:50
theorem Th51: :: GRAPH_5:51
theorem Th52: :: GRAPH_5:52
Lemma57:
for b1 being FinSequence of REAL holds
( ( for b2 being Real holds
( b2 in rng b1 implies b2 >= 0 ) ) iff for b2 being Nat holds
( b2 in dom b1 implies b1 . b2 >= 0 ) )
Lemma58:
for b1, b2, b3 being FinSequence of REAL holds
( b3 = b1 ^ b2 & ( for b4 being Real holds
( b4 in rng b3 implies b4 >= 0 ) ) implies ( ( for b4 being Nat holds
( b4 in dom b1 implies b1 . b4 >= 0 ) ) & ( for b4 being Nat holds
( b4 in dom b2 implies b2 . b4 >= 0 ) ) ) )
theorem Th53: :: GRAPH_5:53
theorem Th54: :: GRAPH_5:54
theorem Th55: :: GRAPH_5:55
theorem Th56: :: GRAPH_5:56
theorem Th57: :: GRAPH_5:57
theorem Th58: :: GRAPH_5:58
theorem Th59: :: GRAPH_5:59
theorem Th60: :: GRAPH_5:60
:: deftheorem Def17 defines is_shortestpath_of GRAPH_5:def 17 :
definition
let c
1 be
Graph;
let c
2, c
3 be
Vertex of c
1;
let c
4 be
oriented Chain of c
1;
let c
5 be
set ;
let c
6 be
Function;
pred c
4 is_shortestpath_of c
2,c
3,c
5,c
6 means :
Def18:
:: GRAPH_5:def 18
( a
4 is_orientedpath_of a
2,a
3,a
5 & ( for b
1 being
oriented Chain of a
1 holds
( b
1 is_orientedpath_of a
2,a
3,a
5 implies
cost a
4,a
6 <= cost b
1,a
6 ) ) );
end;
:: deftheorem Def18 defines is_shortestpath_of GRAPH_5:def 18 :
for b
1 being
Graphfor b
2, b
3 being
Vertex of b
1for b
4 being
oriented Chain of b
1for b
5 being
set for b
6 being
Function holds
( b
4 is_shortestpath_of b
2,b
3,b
5,b
6 iff ( b
4 is_orientedpath_of b
2,b
3,b
5 & ( for b
7 being
oriented Chain of b
1 holds
( b
7 is_orientedpath_of b
2,b
3,b
5 implies
cost b
4,b
6 <= cost b
7,b
6 ) ) ) );
theorem Th61: :: GRAPH_5:61
theorem Th62: :: GRAPH_5:62
Lemma69:
for b1 being finite Graph holds AcyclicPaths b1 is finite
Lemma70:
for b1 being finite Graph
for b2 being oriented Chain of b1 holds AcyclicPaths b2 is finite
Lemma71:
for b1 being finite Graph
for b2, b3 being Element of the Vertices of b1 holds AcyclicPaths b2,b3 is finite
Lemma72:
for b1 being set
for b2 being finite Graph
for b3, b4 being Element of the Vertices of b2 holds AcyclicPaths b3,b4,b1 is finite
theorem Th63: :: GRAPH_5:63
theorem Th64: :: GRAPH_5:64
theorem Th65: :: GRAPH_5:65
theorem Th66: :: GRAPH_5:66
theorem Th67: :: GRAPH_5:67
for b
1 being
set for b
2 being
Functionfor b
3 being
finite Graphfor b
4 being
oriented Chain of b
3for b
5, b
6 being
Element of the
Vertices of b
3 holds
( b
2 is_weight>=0of b
3 & b
4 is_shortestpath_of b
5,b
6,b
1,b
2 & b
5 <> b
6 & ( for b
7 being
oriented Chain of b
3for b
8 being
Element of the
Vertices of b
3 holds
( not b
8 in b
1 & b
7 is_shortestpath_of b
5,b
8,b
1,b
2 implies
cost b
4,b
2 <= cost b
7,b
2 ) ) implies b
4 is_shortestpath_of b
5,b
6,b
2 )
theorem Th68: :: GRAPH_5:68
for b
1, b
2 being
set for b
3 being
Functionfor b
4 being
finite Graphfor b
5 being
oriented Chain of b
4for b
6, b
7 being
Element of the
Vertices of b
4 holds
( b
3 is_weight>=0of b
4 & b
5 is_shortestpath_of b
6,b
7,b
1,b
3 & b
6 <> b
7 & b
1 c= b
2 & ( for b
8 being
oriented Chain of b
4for b
9 being
Element of the
Vertices of b
4 holds
( not b
9 in b
1 & b
8 is_shortestpath_of b
6,b
9,b
1,b
3 implies
cost b
5,b
3 <= cost b
8,b
3 ) ) implies b
5 is_shortestpath_of b
6,b
7,b
2,b
3 )
:: deftheorem Def19 defines islongestInShortestpath GRAPH_5:def 19 :
theorem Th69: :: GRAPH_5:69
for b
1, b
2 being
set for b
3 being
Functionfor b
4 being
oriented finite Graphfor b
5, b
6, b
7 being
oriented Chain of b
4for b
8, b
9, b
10 being
Element of the
Vertices of b
4 holds
( b
1 in the
Edges of b
4 & b
3 is_weight>=0of b
4 &
len b
5 >= 1 & b
5 is_shortestpath_of b
8,b
9,b
2,b
3 & b
8 <> b
9 & b
8 <> b
10 & b
7 = b
5 ^ <*b1*> & b
6 is_shortestpath_of b
8,b
10,b
2,b
3 & b
1 orientedly_joins b
9,b
10 & b
5 islongestInShortestpath b
2,b
8,b
3 implies ( (
cost b
6,b
3 <= cost b
7,b
3 implies b
6 is_shortestpath_of b
8,b
10,b
2 \/ {b9},b
3 ) & (
cost b
6,b
3 >= cost b
7,b
3 implies b
7 is_shortestpath_of b
8,b
10,b
2 \/ {b9},b
3 ) ) )