:: SGRAPH1 semantic presentation
Lemma15:
for e being set
for n being Element of NAT st e in Seg n holds
ex i being Element of NAT st
( e = i & 1 <= i & i <= n )
:: deftheorem Def1 defines nat_interval SGRAPH1:def 1 :
theorem Th1: :: SGRAPH1:1
canceled;
theorem Th2: :: SGRAPH1:2
theorem Th3: :: SGRAPH1:3
theorem Th4: :: SGRAPH1:4
theorem Th5: :: SGRAPH1:5
theorem Th6: :: SGRAPH1:6
theorem Th7: :: SGRAPH1:7
Lemma30:
for A being set
for s being Subset of A
for n being set st n in A holds
s \/ {n} is Subset of A
:: deftheorem Def2 SGRAPH1:def 2 :
canceled;
:: deftheorem Def3 SGRAPH1:def 3 :
canceled;
:: deftheorem Def4 defines TWOELEMENTSETS SGRAPH1:def 4 :
theorem Th8: :: SGRAPH1:8
canceled;
theorem Th9: :: SGRAPH1:9
theorem Th10: :: SGRAPH1:10
theorem Th11: :: SGRAPH1:11
theorem Th12: :: SGRAPH1:12
theorem Th13: :: SGRAPH1:13
theorem Th14: :: SGRAPH1:14
theorem Th15: :: SGRAPH1:15
theorem Th16: :: SGRAPH1:16
theorem Th17: :: SGRAPH1:17
:: deftheorem Def5 SGRAPH1:def 5 :
canceled;
:: deftheorem Def6 defines SIMPLEGRAPHS SGRAPH1:def 6 :
theorem Th18: :: SGRAPH1:18
canceled;
theorem Th19: :: SGRAPH1:19
:: deftheorem Def7 defines SimpleGraph SGRAPH1:def 7 :
theorem Th20: :: SGRAPH1:20
canceled;
theorem Th21: :: SGRAPH1:21
theorem Th22: :: SGRAPH1:22
canceled;
theorem Th23: :: SGRAPH1:23
theorem Th24: :: SGRAPH1:24
canceled;
theorem Th25: :: SGRAPH1:25
theorem Th26: :: SGRAPH1:26
theorem Th27: :: SGRAPH1:27
:: deftheorem Def8 defines is_isomorphic_to SGRAPH1:def 8 :
theorem Th28: :: SGRAPH1:28
theorem Th29: :: SGRAPH1:29
canceled;
theorem Th30: :: SGRAPH1:30
theorem Th31: :: SGRAPH1:31
:: deftheorem Def9 defines is_SetOfSimpleGraphs_of SGRAPH1:def 9 :
theorem Th32: :: SGRAPH1:32
canceled;
theorem Th33: :: SGRAPH1:33
canceled;
theorem Th34: :: SGRAPH1:34
canceled;
theorem Th35: :: SGRAPH1:35
theorem Th36: :: SGRAPH1:36
theorem Th37: :: SGRAPH1:37
:: deftheorem Def10 defines SubGraph SGRAPH1:def 10 :
:: deftheorem Def11 defines degree SGRAPH1:def 11 :
theorem Th38: :: SGRAPH1:38
canceled;
theorem Th39: :: SGRAPH1:39
theorem Th40: :: SGRAPH1:40
theorem Th41: :: SGRAPH1:41
theorem Th42: :: SGRAPH1:42
:: deftheorem Def12 defines is_path_of SGRAPH1:def 12 :
:: deftheorem Def13 defines PATHS SGRAPH1:def 13 :
theorem Th43: :: SGRAPH1:43
canceled;
theorem Th44: :: SGRAPH1:44
theorem Th45: :: SGRAPH1:45
:: deftheorem Def14 defines is_cycle_of SGRAPH1:def 14 :
definition
let n be
Element of
NAT ,
m be
Element of
NAT ;
canceled;func K_ c2,
c1 -> SimpleGraph of
NAT means :: SGRAPH1:def 16
ex
ee being
Subset of
(TWOELEMENTSETS (Seg (m + n))) st
(
ee = { {i,j} where i is Element of NAT , j is Element of NAT : ( i in Seg m & j in nat_interval (m + 1),(m + n) ) } &
it = SimpleGraphStruct(#
(Seg (m + n)),
ee #) );
existence
ex b1 being SimpleGraph of NAT ex ee being Subset of (TWOELEMENTSETS (Seg (m + n))) st
( ee = { {i,j} where i is Element of NAT , j is Element of NAT : ( i in Seg m & j in nat_interval (m + 1),(m + n) ) } & b1 = SimpleGraphStruct(# (Seg (m + n)),ee #) )
uniqueness
for b1, b2 being SimpleGraph of NAT st ex ee being Subset of (TWOELEMENTSETS (Seg (m + n))) st
( ee = { {i,j} where i is Element of NAT , j is Element of NAT : ( i in Seg m & j in nat_interval (m + 1),(m + n) ) } & b1 = SimpleGraphStruct(# (Seg (m + n)),ee #) ) & ex ee being Subset of (TWOELEMENTSETS (Seg (m + n))) st
( ee = { {i,j} where i is Element of NAT , j is Element of NAT : ( i in Seg m & j in nat_interval (m + 1),(m + n) ) } & b2 = SimpleGraphStruct(# (Seg (m + n)),ee #) ) holds
b1 = b2
;
end;
:: deftheorem Def15 SGRAPH1:def 15 :
canceled;
:: deftheorem Def16 defines K_ SGRAPH1:def 16 :
definition
let n be
Element of
NAT ;
func K_ c1 -> SimpleGraph of
NAT means :
Def17:
:: SGRAPH1:def 17
ex
ee being
finite Subset of
(TWOELEMENTSETS (Seg n)) st
(
ee = { {i,j} where i is Element of NAT , j is Element of NAT : ( i in Seg n & j in Seg n & i <> j ) } &
it = SimpleGraphStruct(#
(Seg n),
ee #) );
existence
ex b1 being SimpleGraph of NAT ex ee being finite Subset of (TWOELEMENTSETS (Seg n)) st
( ee = { {i,j} where i is Element of NAT , j is Element of NAT : ( i in Seg n & j in Seg n & i <> j ) } & b1 = SimpleGraphStruct(# (Seg n),ee #) )
uniqueness
for b1, b2 being SimpleGraph of NAT st ex ee being finite Subset of (TWOELEMENTSETS (Seg n)) st
( ee = { {i,j} where i is Element of NAT , j is Element of NAT : ( i in Seg n & j in Seg n & i <> j ) } & b1 = SimpleGraphStruct(# (Seg n),ee #) ) & ex ee being finite Subset of (TWOELEMENTSETS (Seg n)) st
( ee = { {i,j} where i is Element of NAT , j is Element of NAT : ( i in Seg n & j in Seg n & i <> j ) } & b2 = SimpleGraphStruct(# (Seg n),ee #) ) holds
b1 = b2
;
end;
:: deftheorem Def17 defines K_ SGRAPH1:def 17 :
:: deftheorem Def18 defines TriangleGraph SGRAPH1:def 18 :
theorem Th46: :: SGRAPH1:46
theorem Th47: :: SGRAPH1:47
theorem Th48: :: SGRAPH1:48
theorem Th49: :: SGRAPH1:49