:: GRAPH_2 semantic presentation
theorem Th1: :: GRAPH_2:1
for b
1, b
2, b
3 being
Nat holds
( ( b
1 + 1
<= b
2 & b
2 <= b
3 ) iff ex b
4 being
Nat st
( b
1 <= b
4 & b
4 < b
3 & b
2 = b
4 + 1 ) )
theorem Th2: :: GRAPH_2:2
theorem Th3: :: GRAPH_2:3
Lemma4:
for b1, b2 being Nat
for b3 being finite set holds
( b3 = { b4 where B is Nat : ( b1 <= b4 & b4 <= b1 + b2 ) } implies card b3 = b2 + 1 )
theorem Th4: :: GRAPH_2:4
for b
1, b
2 being
Nat holds
Card { b3 where B is Nat : ( b1 <= b3 & b3 <= b1 + b2 ) } = b
2 + 1
theorem Th5: :: GRAPH_2:5
for b
1, b
2, b
3 being
Nat holds
( 1
<= b
3 & b
3 <= b
1 implies
(Sgm { b4 where B is Nat : ( b2 + 1 <= b4 & b4 <= b2 + b1 ) } ) . b
3 = b
2 + b
3 )
:: deftheorem Def1 defines -cut GRAPH_2:def 1 :
Lemma8:
for b1 being FinSequence
for b2, b3 being Nat holds
( 1 <= b2 & b2 <= b3 + 1 & b3 <= len b1 implies ( (len (b2,b3 -cut b1)) + b2 = b3 + 1 & ( for b4 being Nat holds
( b4 < len (b2,b3 -cut b1) implies (b2,b3 -cut b1) . (b4 + 1) = b1 . (b2 + b4) ) ) ) )
theorem Th6: :: GRAPH_2:6
theorem Th7: :: GRAPH_2:7
theorem Th8: :: GRAPH_2:8
theorem Th9: :: GRAPH_2:9
theorem Th10: :: GRAPH_2:10
theorem Th11: :: GRAPH_2:11
theorem Th12: :: GRAPH_2:12
:: deftheorem Def2 defines ^' GRAPH_2:def 2 :
theorem Th13: :: GRAPH_2:13
theorem Th14: :: GRAPH_2:14
theorem Th15: :: GRAPH_2:15
theorem Th16: :: GRAPH_2:16
theorem Th17: :: GRAPH_2:17
theorem Th18: :: GRAPH_2:18
:: deftheorem Def3 defines TwoValued GRAPH_2:def 3 :
theorem Th19: :: GRAPH_2:19
then Lemma22:
<*1,2*> is TwoValued
by Lemma20;
:: deftheorem Def4 defines Alternating GRAPH_2:def 4 :
Lemma25:
<*1,2*> is Alternating
by Def4, Lemma23;
theorem Th20: :: GRAPH_2:20
theorem Th21: :: GRAPH_2:21
theorem Th22: :: GRAPH_2:22
theorem Th23: :: GRAPH_2:23
:: deftheorem Def5 defines FinSubsequence GRAPH_2:def 5 :
theorem Th24: :: GRAPH_2:24
theorem Th25: :: GRAPH_2:25
canceled;
theorem Th26: :: GRAPH_2:26
theorem Th27: :: GRAPH_2:27
theorem Th28: :: GRAPH_2:28
theorem Th29: :: GRAPH_2:29
theorem Th30: :: GRAPH_2:30
theorem Th31: :: GRAPH_2:31
theorem Th32: :: GRAPH_2:32
for b
1 being
Graphfor b
2, b
3, b
4, b
5 being
Element of the
Vertices of b
1for b
6 being
set holds
not ( b
6 joins b
2,b
3 & b
6 joins b
4,b
5 & not ( b
2 = b
4 & b
3 = b
5 ) & not ( b
2 = b
5 & b
3 = b
4 ) )
:: deftheorem Def6 defines -VSet GRAPH_2:def 6 :
:: deftheorem Def7 defines is_vertex_seq_of GRAPH_2:def 7 :
theorem Th33: :: GRAPH_2:33
canceled;
theorem Th34: :: GRAPH_2:34
theorem Th35: :: GRAPH_2:35
theorem Th36: :: GRAPH_2:36
theorem Th37: :: GRAPH_2:37
:: deftheorem Def8 defines alternates_vertices_in GRAPH_2:def 8 :
theorem Th38: :: GRAPH_2:38
theorem Th39: :: GRAPH_2:39
theorem Th40: :: GRAPH_2:40
theorem Th41: :: GRAPH_2:41
Lemma49:
for b1 being non empty set holds
( ( for b2, b3 being set holds
( b2 in b1 & b3 in b1 implies b2 = b3 ) ) implies Card b1 = 1 )
theorem Th42: :: GRAPH_2:42
:: deftheorem Def9 defines vertex-seq GRAPH_2:def 9 :
theorem Th43: :: GRAPH_2:43
theorem Th44: :: GRAPH_2:44
theorem Th45: :: GRAPH_2:45
theorem Th46: :: GRAPH_2:46
theorem Th47: :: GRAPH_2:47
Lemma56:
for b1 being Graph
for b2 being Element of the Vertices of b1 holds <*b2*> is_vertex_seq_of {}
:: deftheorem Def10 defines simple GRAPH_2:def 10 :
theorem Th48: :: GRAPH_2:48
canceled;
theorem Th49: :: GRAPH_2:49
theorem Th50: :: GRAPH_2:50
theorem Th51: :: GRAPH_2:51
theorem Th52: :: GRAPH_2:52
theorem Th53: :: GRAPH_2:53
theorem Th54: :: GRAPH_2:54
theorem Th55: :: GRAPH_2:55
theorem Th56: :: GRAPH_2:56
:: deftheorem Def11 defines vertex-seq GRAPH_2:def 11 :
theorem Th57: :: GRAPH_2:57
theorem Th58: :: GRAPH_2:58
theorem Th59: :: GRAPH_2:59
theorem Th60: :: GRAPH_2:60
theorem Th61: :: GRAPH_2:61
theorem Th62: :: GRAPH_2:62