:: LEXBFS semantic presentation
theorem :: LEXBFS:1
Lm1:
for a, b, c being real number st a < b holds
(c - b) + 1 < (c - a) + 1
theorem Th2: :: LEXBFS:2
theorem Th3: :: LEXBFS:3
theorem Th4: :: LEXBFS:4
:: deftheorem Def1 defines with_finite-elements LEXBFS:def 1 :
:: deftheorem Def2 defines .\/ LEXBFS:def 2 :
theorem Th5: :: LEXBFS:5
theorem Th6: :: LEXBFS:6
theorem Th7: :: LEXBFS:7
:: deftheorem Def3 defines natsubset-yielding LEXBFS:def 3 :
Lm2:
for F being Function st ( for x being set st x in rng F holds
x is finite ) holds
F is finite-yielding
Lm3:
for F being finite-yielding Function
for x being set st x in rng F holds
x is finite
theorem Th8: :: LEXBFS:8
:: deftheorem Def4 defines .incSubset LEXBFS:def 4 :
:: deftheorem Def5 defines max LEXBFS:def 5 :
:: deftheorem Def6 defines Walk LEXBFS:def 6 :
Lm4:
for G being _Graph
for W being Walk of G
for e, v being set st e Joins W .last() ,v,G holds
(W .addEdge e) .length() = (W .length() ) + 1
Lm5:
for G being _Graph
for W being Walk of G holds W .length() = (W .reverse() ) .length()
Lm6:
for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
for n being natural number st n in dom W holds
( (W .addEdge e) . n = W . n & n in dom (W .addEdge e) )
theorem :: LEXBFS:9
theorem :: LEXBFS:10
:: deftheorem Def7 defines natural-vlabeled LEXBFS:def 7 :
:: deftheorem defines V2LabelSelector LEXBFS:def 8 :
:: deftheorem Def9 defines [V2Labeled] LEXBFS:def 9 :
:: deftheorem defines the_V2Label_of LEXBFS:def 10 :
theorem Th11: :: LEXBFS:11
theorem Th12: :: LEXBFS:12
:: deftheorem Def11 defines natural-v2labeled LEXBFS:def 11 :
:: deftheorem Def12 defines finite-v2labeled LEXBFS:def 12 :
:: deftheorem Def13 defines natsubset-v2labeled LEXBFS:def 13 :
theorem Th13: :: LEXBFS:13
:: deftheorem Def14 defines v2label-inheriting LEXBFS:def 14 :
:: deftheorem Def15 defines iterative LEXBFS:def 15 :
:: deftheorem Def16 defines eventually-constant LEXBFS:def 16 :
theorem Th14: :: LEXBFS:14
theorem Th15: :: LEXBFS:15
theorem Th16: :: LEXBFS:16
theorem Th17: :: LEXBFS:17
:: deftheorem Def17 defines natural-vlabeled LEXBFS:def 17 :
:: deftheorem Def18 defines chordal LEXBFS:def 18 :
:: deftheorem Def19 defines fixed-vertices LEXBFS:def 19 :
:: deftheorem Def20 defines [V2Labeled] LEXBFS:def 20 :
:: deftheorem Def21 defines natural-v2labeled LEXBFS:def 21 :
:: deftheorem Def22 defines finite-v2labeled LEXBFS:def 22 :
:: deftheorem Def23 defines natsubset-v2labeled LEXBFS:def 23 :
:: deftheorem Def24 defines vlabel-initially-empty LEXBFS:def 24 :
:: deftheorem Def25 defines adds-one-at-a-step LEXBFS:def 25 :
:: deftheorem Def26 defines vlabel-numbering LEXBFS:def 26 :
Lm7:
ex GS being VGraphSeq st
( GS is iterative & GS is eventually-constant & GS is finite & GS is fixed-vertices & GS is natural-vlabeled & GS is vlabel-initially-empty & GS is adds-one-at-a-step )
:: deftheorem Def27 defines .PickedAt LEXBFS:def 27 :
theorem Th18: :: LEXBFS:18
theorem Th19: :: LEXBFS:19
theorem :: LEXBFS:20
theorem Th21: :: LEXBFS:21
theorem Th22: :: LEXBFS:22
theorem Th23: :: LEXBFS:23
theorem Th24: :: LEXBFS:24
theorem Th25: :: LEXBFS:25
theorem Th26: :: LEXBFS:26
theorem Th27: :: LEXBFS:27
theorem Th28: :: LEXBFS:28
theorem Th29: :: LEXBFS:29
theorem Th30: :: LEXBFS:30
theorem Th31: :: LEXBFS:31
:: deftheorem defines LexBFS:Init LEXBFS:def 28 :
definition
let G be
finite finite-v2labeled natsubset-v2labeled VVGraph;
assume A1:
dom (the_V2Label_of G) = the_Vertices_of G
;
func LexBFS:PickUnnumbered G -> Vertex of
G means :
Def29:
:: LEXBFS:def 29
it = choose (the_Vertices_of G) if dom (the_VLabel_of G) = the_Vertices_of G otherwise ex
S being non
empty finite Subset of
(bool NAT ) ex
B being non
empty finite Subset of
(Bags NAT ) ex
F being
Function st
(
S = rng F &
F = (the_V2Label_of G) | ((the_Vertices_of G) \ (dom (the_VLabel_of G))) & ( for
x being
finite Subset of
NAT st
x in S holds
x,1
-bag in B ) & ( for
x being
set st
x in B holds
ex
y being
finite Subset of
NAT st
(
y in S &
x = y,1
-bag ) ) &
it = choose (F " {(support (max B,(InvLexOrder NAT )))}) );
existence
( ( dom (the_VLabel_of G) = the_Vertices_of G implies ex b1 being Vertex of G st b1 = choose (the_Vertices_of G) ) & ( not dom (the_VLabel_of G) = the_Vertices_of G implies ex b1 being Vertex of G ex S being non empty finite Subset of (bool NAT ) ex B being non empty finite Subset of (Bags NAT ) ex F being Function st
( S = rng F & F = (the_V2Label_of G) | ((the_Vertices_of G) \ (dom (the_VLabel_of G))) & ( for x being finite Subset of NAT st x in S holds
x,1 -bag in B ) & ( for x being set st x in B holds
ex y being finite Subset of NAT st
( y in S & x = y,1 -bag ) ) & b1 = choose (F " {(support (max B,(InvLexOrder NAT )))}) ) ) )
uniqueness
for b1, b2 being Vertex of G holds
( ( dom (the_VLabel_of G) = the_Vertices_of G & b1 = choose (the_Vertices_of G) & b2 = choose (the_Vertices_of G) implies b1 = b2 ) & ( not dom (the_VLabel_of G) = the_Vertices_of G & ex S being non empty finite Subset of (bool NAT ) ex B being non empty finite Subset of (Bags NAT ) ex F being Function st
( S = rng F & F = (the_V2Label_of G) | ((the_Vertices_of G) \ (dom (the_VLabel_of G))) & ( for x being finite Subset of NAT st x in S holds
x,1 -bag in B ) & ( for x being set st x in B holds
ex y being finite Subset of NAT st
( y in S & x = y,1 -bag ) ) & b1 = choose (F " {(support (max B,(InvLexOrder NAT )))}) ) & ex S being non empty finite Subset of (bool NAT ) ex B being non empty finite Subset of (Bags NAT ) ex F being Function st
( S = rng F & F = (the_V2Label_of G) | ((the_Vertices_of G) \ (dom (the_VLabel_of G))) & ( for x being finite Subset of NAT st x in S holds
x,1 -bag in B ) & ( for x being set st x in B holds
ex y being finite Subset of NAT st
( y in S & x = y,1 -bag ) ) & b2 = choose (F " {(support (max B,(InvLexOrder NAT )))}) ) implies b1 = b2 ) )
consistency
for b1 being Vertex of G holds verum
;
end;
:: deftheorem Def29 defines LexBFS:PickUnnumbered LEXBFS:def 29 :
:: deftheorem defines LexBFS:LabelAdjacent LEXBFS:def 30 :
theorem Th32: :: LEXBFS:32
theorem Th33: :: LEXBFS:33
theorem Th34: :: LEXBFS:34
theorem Th35: :: LEXBFS:35
:: deftheorem defines LexBFS:Update LEXBFS:def 31 :
:: deftheorem Def32 defines LexBFS:Step LEXBFS:def 32 :
:: deftheorem Def33 defines LexBFS:CSeq LEXBFS:def 33 :
theorem Th36: :: LEXBFS:36
theorem Th37: :: LEXBFS:37
theorem Th38: :: LEXBFS:38
theorem Th39: :: LEXBFS:39
theorem Th40: :: LEXBFS:40
theorem Th41: :: LEXBFS:41
theorem Th42: :: LEXBFS:42
theorem Th43: :: LEXBFS:43
theorem Th44: :: LEXBFS:44
theorem Th45: :: LEXBFS:45
theorem Th46: :: LEXBFS:46
theorem Th47: :: LEXBFS:47
theorem Th48: :: LEXBFS:48
theorem Th49: :: LEXBFS:49
theorem Th50: :: LEXBFS:50
theorem Th51: :: LEXBFS:51
theorem Th52: :: LEXBFS:52
theorem Th53: :: LEXBFS:53
theorem Th54: :: LEXBFS:54
theorem Th55: :: LEXBFS:55
theorem Th56: :: LEXBFS:56
theorem Th57: :: LEXBFS:57
theorem Th58: :: LEXBFS:58
theorem Th59: :: LEXBFS:59
theorem Th60: :: LEXBFS:60
theorem Th61: :: LEXBFS:61
theorem Th62: :: LEXBFS:62
theorem Th63: :: LEXBFS:63
theorem Th64: :: LEXBFS:64
:: deftheorem Def34 defines with_property_L3 LEXBFS:def 34 :
theorem Th65: :: LEXBFS:65
theorem Th66: :: LEXBFS:66
theorem :: LEXBFS:67
:: deftheorem defines MCS:Init LEXBFS:def 35 :
:: deftheorem Def36 defines MCS:PickUnnumbered LEXBFS:def 36 :
:: deftheorem defines MCS:LabelAdjacent LEXBFS:def 37 :
:: deftheorem defines MCS:Update LEXBFS:def 38 :
:: deftheorem Def39 defines MCS:Step LEXBFS:def 39 :
:: deftheorem Def40 defines MCS:CSeq LEXBFS:def 40 :
theorem Th68: :: LEXBFS:68
theorem Th69: :: LEXBFS:69
theorem Th70: :: LEXBFS:70
theorem Th71: :: LEXBFS:71
theorem Th72: :: LEXBFS:72
theorem Th73: :: LEXBFS:73
theorem Th74: :: LEXBFS:74
theorem Th75: :: LEXBFS:75
theorem Th76: :: LEXBFS:76
theorem Th77: :: LEXBFS:77
theorem Th78: :: LEXBFS:78
theorem Th79: :: LEXBFS:79
theorem Th80: :: LEXBFS:80
theorem Th81: :: LEXBFS:81
theorem Th82: :: LEXBFS:82
theorem Th83: :: LEXBFS:83
theorem Th84: :: LEXBFS:84
theorem Th85: :: LEXBFS:85
theorem Th86: :: LEXBFS:86
theorem Th87: :: LEXBFS:87
theorem Th88: :: LEXBFS:88
theorem Th89: :: LEXBFS:89
theorem Th90: :: LEXBFS:90
theorem Th91: :: LEXBFS:91
:: deftheorem Def41 defines with_property_T LEXBFS:def 41 :
theorem :: LEXBFS:92
theorem :: LEXBFS:93
theorem :: LEXBFS:94