:: GRAPH_1 semantic presentation
set c1 = {1,2};
reconsider c2 = {} as Function of {} ,{1,2} by FUNCT_2:55, RELAT_1:60;
:: deftheorem Def1 defines Graph-like GRAPH_1:def 1 :
Lemma2:
for b1 being Graph holds
( dom the Source of b1 = the Edges of b1 & dom the Target of b1 = the Edges of b1 )
Lemma3:
for b1 being Graph
for b2 being Element of the Vertices of b1 holds b2 in the Vertices of b1
:: deftheorem Def2 defines \/ GRAPH_1:def 2 :
:: deftheorem Def3 defines is_sum_of GRAPH_1:def 3 :
:: deftheorem Def4 defines oriented GRAPH_1:def 4 :
:: deftheorem Def5 defines non-multi GRAPH_1:def 5 :
:: deftheorem Def6 defines simple GRAPH_1:def 6 :
:: deftheorem Def7 defines connected GRAPH_1:def 7 :
:: deftheorem Def8 defines finite GRAPH_1:def 8 :
:: deftheorem Def9 defines joins GRAPH_1:def 9 :
:: deftheorem Def10 defines are_incydent GRAPH_1:def 10 :
:: deftheorem Def11 defines Chain GRAPH_1:def 11 :
Lemma12:
for b1 being Graph holds
{} is Chain of b1
:: deftheorem Def12 defines oriented GRAPH_1:def 12 :
Lemma14:
for b1 being Graph holds
{} is oriented Chain of b1
:: deftheorem Def13 defines one-to-one GRAPH_1:def 13 :
:: deftheorem Def14 GRAPH_1:def 14 :
canceled;
:: deftheorem Def15 defines cyclic GRAPH_1:def 15 :
Lemma16:
for b1 being Graph holds
{} is Cycle of b1
Lemma17:
for b1 being Graph
for b2 being set holds
( b2 in the Edges of b1 implies ( the Source of b1 . b2 in the Vertices of b1 & the Target of b1 . b2 in the Vertices of b1 ) )
:: deftheorem Def16 GRAPH_1:def 16 :
canceled;
:: deftheorem Def17 defines Subgraph GRAPH_1:def 17 :
:: deftheorem Def18 defines VerticesCount GRAPH_1:def 18 :
:: deftheorem Def19 defines EdgesCount GRAPH_1:def 19 :
:: deftheorem Def20 defines EdgesIn GRAPH_1:def 20 :
:: deftheorem Def21 defines EdgesOut GRAPH_1:def 21 :
:: deftheorem Def22 defines Degree GRAPH_1:def 22 :
Lemma19:
for b1 being Nat
for b2 being Graph
for b3 being Chain of b2 holds
b3 | (Seg b1) is Chain of b2
Lemma20:
for b1 being Graph
for b2, b3 being strict Subgraph of b1 holds
( the Vertices of b2 = the Vertices of b3 & the Edges of b2 = the Edges of b3 implies b2 = b3 )
:: deftheorem Def23 defines c= GRAPH_1:def 23 :
Lemma22:
for b1 being Graph
for b2 being Subgraph of b1 holds
( the Source of b2 in PFuncs the Edges of b1,the Vertices of b1 & the Target of b2 in PFuncs the Edges of b1,the Vertices of b1 )
:: deftheorem Def24 defines bool GRAPH_1:def 24 :
theorem Th1: :: GRAPH_1:1
theorem Th2: :: GRAPH_1:2
theorem Th3: :: GRAPH_1:3
theorem Th4: :: GRAPH_1:4
theorem Th5: :: GRAPH_1:5
theorem Th6: :: GRAPH_1:6
theorem Th7: :: GRAPH_1:7
theorem Th8: :: GRAPH_1:8
theorem Th9: :: GRAPH_1:9
theorem Th10: :: GRAPH_1:10
theorem Th11: :: GRAPH_1:11
theorem Th12: :: GRAPH_1:12
for b
1, b
2 being
Graph holds
( ex b
3 being
Graph st
( b
1 c= b
3 & b
2 c= b
3 ) implies b
1 \/ b
2 = b
2 \/ b
1 )
theorem Th13: :: GRAPH_1:13
for b
1, b
2, b
3 being
Graph holds
( ex b
4 being
Graph st
( b
1 c= b
4 & b
2 c= b
4 & b
3 c= b
4 ) implies
(b1 \/ b2) \/ b
3 = b
1 \/ (b2 \/ b3) )
theorem Th14: :: GRAPH_1:14
theorem Th15: :: GRAPH_1:15
theorem Th16: :: GRAPH_1:16
theorem Th17: :: GRAPH_1:17
for b
1, b
2, b
3 being
Graph holds
( b
1 c= b
2 & b
2 c= b
3 implies b
1 c= b
3 )
theorem Th18: :: GRAPH_1:18
theorem Th19: :: GRAPH_1:19
theorem Th20: :: GRAPH_1:20
for b
1, b
2 being
Graph holds
( ex b
3 being
Graph st
( b
1 c= b
3 & b
2 c= b
3 ) implies ( b
1 c= b
1 \/ b
2 & b
2 c= b
1 \/ b
2 ) )
theorem Th21: :: GRAPH_1:21
theorem Th22: :: GRAPH_1:22
for b
1, b
2, b
3 being
Graph holds
( b
1 c= b
2 & b
3 c= b
2 implies b
1 \/ b
3 c= b
2 )
theorem Th23: :: GRAPH_1:23
theorem Th24: :: GRAPH_1:24
theorem Th25: :: GRAPH_1:25
canceled;
theorem Th26: :: GRAPH_1:26
canceled;
theorem Th27: :: GRAPH_1:27
theorem Th28: :: GRAPH_1:28
theorem Th29: :: GRAPH_1:29
theorem Th30: :: GRAPH_1:30
theorem Th31: :: GRAPH_1:31
theorem Th32: :: GRAPH_1:32
theorem Th33: :: GRAPH_1:33
canceled;
theorem Th34: :: GRAPH_1:34
theorem Th35: :: GRAPH_1:35
theorem Th36: :: GRAPH_1:36
theorem Th37: :: GRAPH_1:37