GLIB_001: Walks in a Graph

:: Walks in a Graph
:: by Gilbert Lee
::
:: Received February 22, 2005
:: Copyright (c) 2005-2011 Association of Mizar Users

begin

theorem Th1: :: GLIB_001:1

for x, y being odd Element of NAT holds
( x < y iff x + 2 <= y )

proof end;

theorem Th2: :: GLIB_001:2

for X being set
for k being Element of NAT st X c= Seg k holds
for m, n being Element of NAT st m in dom (Sgm X) & n = (Sgm X) . m holds
m <= n

proof end;

theorem Th3: :: GLIB_001:3

for X being set
for fs being FinSequence of X
for fss being Subset of fs holds len (Seq fss) <= len fs

proof end;

theorem Th4: :: GLIB_001:4

for X being set
for fs being FinSequence of X
for fss being Subset of fs
for m being Element of NAT st m in dom (Seq fss) holds
ex n being Element of NAT st
( n in dom fs & m <= n & (Seq fss) . m = fs . n )

proof end;

theorem Th5: :: GLIB_001:5

for X being set
for fs being FinSequence of X
for fss being Subset of fs holds len (Seq fss) = card fss

proof end;

theorem Th6: :: GLIB_001:6

for X being set
for fs being FinSequence of X
for fss being Subset of fs holds dom (Seq fss) = dom (Sgm (dom fss))

proof end;

begin

definition

let G be _Graph;
mode VertexSeq of G -> FinSequence of the_Vertices_of G means :Def1: :: GLIB_001:def 1
for n being Element of NAT st 1 <= n & n < len it holds
ex e being set st e Joins it . n,it . (n + 1),G;
existence

proof end;

end;

:: deftheorem Def1 defines VertexSeq GLIB_001:def 1 :

definition

let G be _Graph;
mode EdgeSeq of G -> FinSequence of the_Edges_of G means :Def2: :: GLIB_001:def 2
ex vs being FinSequence of the_Vertices_of G st
( len vs = (len it) + 1 & ( for n being Element of NAT st 1 <= n & n <= len it holds
it . n Joins vs . n,vs . (n + 1),G ) );
existence

proof end;

end;

:: deftheorem Def2 defines EdgeSeq GLIB_001:def 2 :

definition

let G be _Graph;
mode Walk of G -> FinSequence of (the_Vertices_of G) \/ (the_Edges_of G) means :Def3: :: GLIB_001:def 3
( not len it is even & it . 1 in the_Vertices_of G & ( for n being odd Element of NAT st n < len it holds
it . (n + 1) Joins it . n,it . (n + 2),G ) );
existence

proof end;

end;

:: deftheorem Def3 defines Walk GLIB_001:def 3 :

registration

let G be _Graph;
let W be Walk of G;
cluster card W -> non empty odd ;
correctness

proof end;

end;

definition

let G be _Graph;
let v be Vertex of G;
func G .walkOf v -> Walk of G equals :: GLIB_001:def 4
<*v*>;
coherence

proof end;

end;

:: deftheorem defines .walkOf GLIB_001:def 4 :

definition

let G be _Graph;
let x, y, e be set ;
func G .walkOf (x,e,y) -> Walk of G equals :Def5: :: GLIB_001:def 5
<*x,e,y*> if e Joins x,y,G
otherwise G .walkOf (choose (the_Vertices_of G));
coherence

proof end;

consistency ;

end;

:: deftheorem Def5 defines .walkOf GLIB_001:def 5 :

definition

let G be _Graph;
let W be Walk of G;
func W .first() -> Vertex of G equals :: GLIB_001:def 6
W . 1;
coherence by Def3;
func W .last() -> Vertex of G equals :: GLIB_001:def 7
W . (len W);
coherence

proof end;

end;

:: deftheorem defines .first() GLIB_001:def 6 :

:: deftheorem defines .last() GLIB_001:def 7 :

definition

let G be _Graph;
let W be Walk of G;
let n be Nat;
func W .vertexAt n -> Vertex of G equals :Def8: :: GLIB_001:def 8
W . n if ( not n is even & n <= len W )
otherwise W .first() ;
correctness

proof end;

end;

:: deftheorem Def8 defines .vertexAt GLIB_001:def 8 :

definition

let G be _Graph;
let W be Walk of G;
func W .reverse() -> Walk of G equals :: GLIB_001:def 9
Rev W;
coherence

proof end;

end;

:: deftheorem defines .reverse() GLIB_001:def 9 :

definition

let G be _Graph;
let W1, W2 be Walk of G;
func W1 .append W2 -> Walk of G equals :Def10: :: GLIB_001:def 10
W1 ^' W2 if W1 .last() = W2 .first()
otherwise W1;
correctness

proof end;

end;

:: deftheorem Def10 defines .append GLIB_001:def 10 :

definition

let G be _Graph;
let W be Walk of G;
let m, n be Nat;
func W .cut (m,n) -> Walk of G equals :Def11: :: GLIB_001:def 11
(m,n) -cut W if ( not m is even & not n is even & m <= n & n <= len W )
otherwise W;
correctness

proof end;

end;

:: deftheorem Def11 defines .cut GLIB_001:def 11 :

definition

let G be _Graph;
let W be Walk of G;
let m, n be Element of NAT ;
func W .remove (m,n) -> Walk of G equals :Def12: :: GLIB_001:def 12
(W .cut (1,m)) .append (W .cut (n,(len W))) if ( not m is even & not n is even & m <= n & n <= len W & W . m = W . n )
otherwise W;
correctness ;

end;

:: deftheorem Def12 defines .remove GLIB_001:def 12 :

definition

let G be _Graph;
let W be Walk of G;
let e be set ;
func W .addEdge e -> Walk of G equals :: GLIB_001:def 13
W .append (G .walkOf ((W .last()),e,((W .last()) .adj e)));
coherence ;

end;

:: deftheorem defines .addEdge GLIB_001:def 13 :

definition

let G be _Graph;
let W be Walk of G;
func W .vertexSeq() -> VertexSeq of G means :Def14: :: GLIB_001:def 14
( (len W) + 1 = 2 * (len it) & ( for n being Nat st 1 <= n & n <= len it holds
it . n = W . ((2 * n) - 1) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def14 defines .vertexSeq() GLIB_001:def 14 :

definition

let G be _Graph;
let W be Walk of G;
func W .edgeSeq() -> EdgeSeq of G means :Def15: :: GLIB_001:def 15
( len W = (2 * (len it)) + 1 & ( for n being Nat st 1 <= n & n <= len it holds
it . n = W . (2 * n) ) );
existence

proof end;

uniqueness

proof end;

end;

:: deftheorem Def15 defines .edgeSeq() GLIB_001:def 15 :

definition

let G be _Graph;
let W be Walk of G;
func W .vertices() -> finite Subset of (the_Vertices_of G) equals :: GLIB_001:def 16
rng (W .vertexSeq());
coherence ;

end;

:: deftheorem defines .vertices() GLIB_001:def 16 :

definition

let G be _Graph;
let W be Walk of G;
func W .edges() -> finite Subset of (the_Edges_of G) equals :: GLIB_001:def 17
rng (W .edgeSeq());
coherence ;

end;

:: deftheorem defines .edges() GLIB_001:def 17 :

definition

let G be _Graph;
let W be Walk of G;
func W .length() -> Element of NAT equals :: GLIB_001:def 18
len (W .edgeSeq());
coherence ;

end;

:: deftheorem defines .length() GLIB_001:def 18 :

definition

let G be _Graph;
let W be Walk of G;
let v be set ;
func W .find v -> odd Element of NAT means :Def19: :: GLIB_001:def 19
( it <= len W & W . it = v & ( for n being odd Nat st n <= len W & W . n = v holds
it <= n ) ) if v in W .vertices()
otherwise it = len W;
existence

proof end;

uniqueness

proof end;

consistency ;

end;

:: deftheorem Def19 defines .find GLIB_001:def 19 :

definition

let G be _Graph;
let W be Walk of G;
let n be Element of NAT ;
func W .find n -> odd Element of NAT means :Def20: :: GLIB_001:def 20
( it <= len W & W . it = W . n & ( for k being odd Nat st k <= len W & W . k = W . n holds
it <= k ) ) if ( not n is even & n <= len W )
otherwise it = len W;
existence

proof end;

uniqueness

proof end;

consistency ;

end;

:: deftheorem Def20 defines .find GLIB_001:def 20 :

definition

let G be _Graph;
let W be Walk of G;
let v be set ;
func W .rfind v -> odd Element of NAT means :Def21: :: GLIB_001:def 21
( it <= len W & W . it = v & ( for n being odd Element of NAT st n <= len W & W . n = v holds
n <= it ) ) if v in W .vertices()
otherwise it = len W;
existence

proof end;

uniqueness

proof end;

consistency ;

end;

:: deftheorem Def21 defines .rfind GLIB_001:def 21 :

definition

let G be _Graph;
let W be Walk of G;
let n be Element of NAT ;
func W .rfind n -> odd Element of NAT means :Def22: :: GLIB_001:def 22
( it <= len W & W . it = W . n & ( for k being odd Element of NAT st k <= len W & W . k = W . n holds
k <= it ) ) if ( not n is even & n <= len W )
otherwise it = len W;
existence

proof end;

uniqueness

proof end;

consistency ;

end;

:: deftheorem Def22 defines .rfind GLIB_001:def 22 :

definition

let G be _Graph;
let u, v be set ;
let W be Walk of G;
pred W is_Walk_from u,v means :Def23: :: GLIB_001:def 23
( W .first() = u & W .last() = v );

end;

:: deftheorem Def23 defines is_Walk_from GLIB_001:def 23 :

definition

let G be _Graph;
let W be Walk of G;
attr W is closed means :Def24: :: GLIB_001:def 24
W .first() = W .last() ;
attr W is directed means :Def25: :: GLIB_001:def 25
for n being odd Element of NAT st n < len W holds
(the_Source_of G) . (W . (n + 1)) = W . n;
attr W is trivial means :Def26: :: GLIB_001:def 26
W .length() = 0 ;
attr W is Trail-like means :Def27: :: GLIB_001:def 27
W .edgeSeq() is one-to-one ;

end;

:: deftheorem Def24 defines closed GLIB_001:def 24 :

:: deftheorem Def25 defines directed GLIB_001:def 25 :

:: deftheorem Def26 defines trivial GLIB_001:def 26 :

:: deftheorem Def27 defines Trail-like GLIB_001:def 27 :

notation

let G be _Graph;
let W be Walk of G;
antonym open W for closed ;

end;

definition

let G be _Graph;
let W be Walk of G;
attr W is Path-like means :Def28: :: GLIB_001:def 28
( W is Trail-like & ( for m, n being odd Element of NAT st m < n & n <= len W & W . m = W . n holds
( m = 1 & n = len W ) ) );

end;

:: deftheorem Def28 defines Path-like GLIB_001:def 28 :

definition

let G be _Graph;
let W be Walk of G;
attr W is vertex-distinct means :Def29: :: GLIB_001:def 29
for m, n being odd Element of NAT st m <= len W & n <= len W & W . m = W . n holds
m = n;

end;

:: deftheorem Def29 defines vertex-distinct GLIB_001:def 29 :

definition

let G be _Graph;
let W be Walk of G;
attr W is Circuit-like means :Def30: :: GLIB_001:def 30
( W is closed & W is Trail-like & not W is trivial );
attr W is Cycle-like means :Def31: :: GLIB_001:def 31
( W is closed & W is Path-like & not W is trivial );

end;

:: deftheorem Def30 defines Circuit-like GLIB_001:def 30 :

:: deftheorem Def31 defines Cycle-like GLIB_001:def 31 :

Lm1: for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W . n in the_Vertices_of G

proof end;

Lm2: for G being _Graph
for W being Walk of G
for n being even Element of NAT st n in dom W holds
ex naa1 being odd Element of NAT st
( naa1 = n - 1 & n - 1 in dom W & n + 1 in dom W & W . n Joins W . naa1,W . (n + 1),G )

proof end;

Lm3: for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n < len W holds
( n in dom W & n + 1 in dom W & n + 2 in dom W )

proof end;

Lm4: for G being _Graph
for v being Vertex of G holds
( G .walkOf v is closed & G .walkOf v is directed & G .walkOf v is trivial & G .walkOf v is Trail-like & G .walkOf v is Path-like )

proof end;

Lm5: for G being _Graph
for x, e, y being set st e Joins x,y,G holds
len (G .walkOf (x,e,y)) = 3

proof end;

Lm6: for G being _Graph
for x, e, y being set st e Joins x,y,G holds
( (G .walkOf (x,e,y)) .first() = x & (G .walkOf (x,e,y)) .last() = y & G .walkOf (x,e,y) is_Walk_from x,y )

proof end;

Lm7: for G being _Graph
for W being Walk of G holds
( W .first() = (W .reverse()) .last() & W .last() = (W .reverse()) .first() )

proof end;

Lm8: for G being _Graph
for W being Walk of G
for n being Element of NAT st n in dom (W .reverse()) holds
( (W .reverse()) . n = W . (((len W) - n) + 1) & ((len W) - n) + 1 in dom W )

proof end;

Lm9: for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
(len (W1 .append W2)) + 1 = (len W1) + (len W2)

proof end;

Lm10: for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
( len W1 <= len (W1 .append W2) & len W2 <= len (W1 .append W2) )

proof end;

Lm11: for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
( (W1 .append W2) .first() = W1 .first() & (W1 .append W2) .last() = W2 .last() & W1 .append W2 is_Walk_from W1 .first() ,W2 .last() )

proof end;

Lm12: for G being _Graph
for W1, W2 being Walk of G
for n being Element of NAT st n in dom W1 holds
( (W1 .append W2) . n = W1 . n & n in dom (W1 .append W2) )

proof end;

Lm13: for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
for n being Element of NAT st n < len W2 holds
( (W1 .append W2) . ((len W1) + n) = W2 . (n + 1) & (len W1) + n in dom (W1 .append W2) )

proof end;

Lm14: for G being _Graph
for W1, W2 being Walk of G
for n being Element of NAT holds
( not n in dom (W1 .append W2) or n in dom W1 or ex k being Element of NAT st
( k < len W2 & n = (len W1) + k ) )

proof end;

Lm15: for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W holds
( (len (W .cut (m,n))) + m = n + 1 & ( for i being Element of NAT st i < len (W .cut (m,n)) holds
( (W .cut (m,n)) . (i + 1) = W . (m + i) & m + i in dom W ) ) )

proof end;

Lm16: for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W holds
( (W .cut (m,n)) .first() = W . m & (W .cut (m,n)) .last() = W . n & W .cut (m,n) is_Walk_from W . m,W . n )

proof end;

Lm17: for G being _Graph
for W being Walk of G
for m, n, o being odd Element of NAT st m <= n & n <= o & o <= len W holds
(W .cut (m,n)) .append (W .cut (n,o)) = W .cut (m,o)

proof end;

Lm18: for G being _Graph
for W being Walk of G holds W .cut (1,(len W)) = W

proof end;

Lm19: for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .cut (n,n) = <*(W .vertexAt n)*>

proof end;

Lm20: for G being _Graph
for W being Walk of G
for m, n being Element of NAT st not m is even & m <= n holds
(W .cut (1,n)) .cut (1,m) = W .cut (1,m)

proof end;

Lm21: for G being _Graph
for W1, W2 being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W1 & W1 .last() = W2 .first() holds
(W1 .append W2) .cut (m,n) = W1 .cut (m,n)

proof end;

Lm22: for G being _Graph
for W being Walk of G
for m being odd Element of NAT st m <= len W holds
len (W .cut (1,m)) = m

proof end;

Lm23: for G being _Graph
for W being Walk of G
for m being odd Element of NAT
for x being Element of NAT st x in dom (W .cut (1,m)) & m <= len W holds
(W .cut (1,m)) . x = W . x

proof end;

Lm24: for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
(len (W .remove (m,n))) + n = (len W) + m

proof end;

Lm25: for G being _Graph
for W being Walk of G
for m, n being Element of NAT
for x, y being set st W is_Walk_from x,y holds
W .remove (m,n) is_Walk_from x,y

proof end;

Lm26: for G being _Graph
for W being Walk of G
for m, n being Element of NAT holds len (W .remove (m,n)) <= len W

proof end;

Lm27: for G being _Graph
for W being Walk of G
for m being Element of NAT holds W .remove (m,m) = W

proof end;

Lm28: for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
(W .cut (1,m)) .last() = (W .cut (n,(len W))) .first()

proof end;

Lm29: for G being _Graph
for W being Walk of G
for x, y being set
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
for x being Element of NAT st x in Seg m holds
(W .remove (m,n)) . x = W . x

proof end;

Lm30: for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
for x being Element of NAT st m <= x & x <= len (W .remove (m,n)) holds
( (W .remove (m,n)) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W )

proof end;

Lm31: for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
len (W .remove (m,n)) = ((len W) + m) - n

proof end;

Lm32: for G being _Graph
for W being Walk of G
for m being Element of NAT st W .first() = W . m holds
W .remove (1,m) = W .cut (m,(len W))

proof end;

Lm33: for G being _Graph
for W being Walk of G
for m, n being Element of NAT holds
( (W .remove (m,n)) .first() = W .first() & (W .remove (m,n)) .last() = W .last() )

proof end;

Lm34: for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT
for x being Element of NAT holds
( not x in dom (W .remove (m,n)) or x in Seg m or ( m <= x & x <= len (W .remove (m,n)) ) )

proof end;

Lm35: for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
W .addEdge e = W ^ <*e,x*>

proof end;

Lm36: for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
( (W .addEdge e) .first() = W .first() & (W .addEdge e) .last() = x & W .addEdge e is_Walk_from W .first() ,x )

proof end;

Lm37: for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
len (W .addEdge e) = (len W) + 2

proof end;

Lm38: for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
( (W .addEdge e) . ((len W) + 1) = e & (W .addEdge e) . ((len W) + 2) = x & ( for n being Element of NAT st n in dom W holds
(W .addEdge e) . n = W . n ) )

proof end;

Lm39: for G being _Graph
for W being Walk of G
for e, x, y, z being set st W is_Walk_from x,y & e Joins y,z,G holds
W .addEdge e is_Walk_from x,z

proof end;

Lm40: for G being _Graph
for W being Walk of G
for n being even Element of NAT st 1 <= n & n <= len W holds
( n div 2 in dom (W .edgeSeq()) & W . n = (W .edgeSeq()) . (n div 2) )

proof end;

Lm41: for G being _Graph
for W being Walk of G
for n being Element of NAT holds
( n in dom (W .edgeSeq()) iff 2 * n in dom W )

proof end;

Lm42: for G being _Graph
for W being Walk of G ex lenWaa1 being even Element of NAT st
( lenWaa1 = (len W) - 1 & len (W .edgeSeq()) = lenWaa1 div 2 )

proof end;

Lm43: for G being _Graph
for W being Walk of G
for n being Element of NAT holds (W .cut (1,n)) .edgeSeq() c= W .edgeSeq()

proof end;

Lm44: for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
(W .addEdge e) .edgeSeq() = (W .edgeSeq()) ^ <*e*>

proof end;

Lm45: for G being _Graph
for W being Walk of G
for x being set holds
( x in W .vertices() iff ex n being odd Element of NAT st
( n <= len W & W . n = x ) )

proof end;

Lm46: for G being _Graph
for W being Walk of G
for e being set holds
( e in W .edges() iff ex n being even Element of NAT st
( 1 <= n & n <= len W & W . n = e ) )

proof end;

Lm47: for G being _Graph
for W being Walk of G
for e being set st e in W .edges() holds
ex v1, v2 being Vertex of G ex n being odd Element of NAT st
( n + 2 <= len W & v1 = W . n & e = W . (n + 1) & v2 = W . (n + 2) & e Joins v1,v2,G )

proof end;

Lm48: for G being _Graph
for W being Walk of G
for e, x, y being set st e in W .edges() & e Joins x,y,G holds
( x in W .vertices() & y in W .vertices() )

proof end;

Lm49: for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .find n <= n

proof end;

Lm50: for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .rfind n >= n

proof end;

Lm51: for G being _Graph
for W being Walk of G holds
( W is directed iff for n being odd Element of NAT st n < len W holds
W . (n + 1) DJoins W . n,W . (n + 2),G )

proof end;

Lm52: for G being _Graph
for W being Walk of G
for x, e, y, z being set st W is directed & W is_Walk_from x,y & e DJoins y,z,G holds
( W .addEdge e is directed & W .addEdge e is_Walk_from x,z )

proof end;

Lm53: for G being _Graph
for W being Walk of G
for m, n being Element of NAT st W is directed holds
W .cut (m,n) is directed

proof end;

Lm54: for G being _Graph
for W being Walk of G holds
( not W is trivial iff 3 <= len W )

proof end;

Lm55: for G being _Graph
for W being Walk of G holds
( not W is trivial iff len W <> 1 )

proof end;

Lm56: for G being _Graph
for W being Walk of G holds
( W is trivial iff ex v being Vertex of G st W = G .walkOf v )

proof end;

Lm57: for G being _Graph
for W being Walk of G holds
( W is Trail-like iff for m, n being even Element of NAT st 1 <= m & m < n & n <= len W holds
W . m <> W . n )

proof end;

Lm58: for G being _Graph
for W being Walk of G holds
( W is Trail-like iff W .reverse() is Trail-like )

proof end;

Lm59: for G being _Graph
for W being Walk of G
for m, n being Element of NAT st W is Trail-like holds
W .cut (m,n) is Trail-like

proof end;

Lm60: for G being _Graph
for W being Walk of G
for e being set st W is Trail-like & e in (W .last()) .edgesInOut() & not e in W .edges() holds
W .addEdge e is Trail-like

proof end;

Lm61: for G being _Graph
for W being Walk of G st len W <= 3 holds
W is Trail-like

proof end;

Lm62: for G being _Graph
for x, e, y being set st e Joins x,y,G holds
G .walkOf (x,e,y) is Path-like

proof end;

Lm63: for G being _Graph
for W being Walk of G holds
( W is Path-like iff W .reverse() is Path-like )

proof end;

Lm64: for G being _Graph
for W being Walk of G
for m, n being Element of NAT st W is Path-like holds
W .cut (m,n) is Path-like

proof end;

Lm65: for G being _Graph
for W being Walk of G
for e, v being set st W is Path-like & e Joins W .last() ,v,G & not e in W .edges() & ( W is trivial or not W is closed ) & ( for n being odd Element of NAT st 1 < n & n <= len W holds
W . n <> v ) holds
W .addEdge e is Path-like

proof end;

Lm66: for G being _Graph
for W being Walk of G st ( for m, n being odd Element of NAT st m <= len W & n <= len W & W . m = W . n holds
m = n ) holds
W is Path-like

proof end;

Lm67: for G being _Graph
for W being Walk of G st ( for n being odd Element of NAT st n <= len W holds
W .rfind n = n ) holds
W is Path-like

proof end;

Lm68: for G being _Graph
for W being Walk of G
for e, v being set st e Joins W .last() ,v,G & W is Path-like & not v in W .vertices() & ( W is trivial or not W is closed ) holds
W .addEdge e is Path-like

proof end;

Lm69: for G being _Graph
for W being Walk of G st len W <= 3 holds
W is Path-like

proof end;

registration

let G be _Graph;
cluster Path-like -> Trail-like Walk of G;
correctness by Def28;
cluster trivial -> Path-like Walk of G;
correctness

proof end;

cluster trivial -> vertex-distinct Walk of G;
coherence

proof end;

cluster vertex-distinct -> Path-like Walk of G;
coherence

proof end;

cluster Circuit-like -> closed non trivial Trail-like Walk of G;
correctness by Def30;
cluster Cycle-like -> closed non trivial Path-like Walk of G;
correctness by Def31;

end;

registration

let G be _Graph;
cluster Relation-like (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like closed directed trivial Walk of G;
existence

proof end;

end;

registration

let G be _Graph;
cluster Relation-like (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like vertex-distinct Walk of G;
existence

proof end;

end;

definition

let G be _Graph;
mode Trail of G is Trail-like Walk of G;
mode Path of G is Path-like Walk of G;

end;

definition

let G be _Graph;
mode DWalk of G is directed Walk of G;
mode DTrail of G is directed Trail of G;
mode DPath of G is directed Path of G;

end;

registration

let G be _Graph;
let v be Vertex of G;
cluster G .walkOf v -> closed directed trivial ;
coherence by Lm4;

end;

registration

let G be _Graph;
let x, e, y be set ;
cluster G .walkOf (x,e,y) -> Path-like ;
coherence

proof end;

end;

registration

let G be _Graph;
let x, e be set ;
cluster G .walkOf (x,e,x) -> closed ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be closed Walk of G;
cluster W .reverse() -> closed ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be trivial Walk of G;
cluster W .reverse() -> trivial ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be Trail of G;
cluster W .reverse() -> Trail-like ;
coherence by Lm58;

end;

registration

let G be _Graph;
let W be Path of G;
cluster W .reverse() -> Path-like ;
coherence by Lm63;

end;

registration

let G be _Graph;
let W1, W2 be closed Walk of G;
cluster W1 .append W2 -> closed ;
coherence

proof end;

end;

registration

let G be _Graph;
let W1, W2 be DWalk of G;
cluster W1 .append W2 -> directed ;
coherence

proof end;

end;

registration

let G be _Graph;
let W1, W2 be trivial Walk of G;
cluster W1 .append W2 -> trivial ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be DWalk of G;
let m, n be Element of NAT ;
cluster W .cut (m,n) -> directed ;
coherence by Lm53;

end;

registration

let G be _Graph;
let W be trivial Walk of G;
let m, n be Element of NAT ;
cluster W .cut (m,n) -> trivial ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be Trail of G;
let m, n be Element of NAT ;
cluster W .cut (m,n) -> Trail-like ;
coherence by Lm59;

end;

registration

let G be _Graph;
let W be Path of G;
let m, n be Element of NAT ;
cluster W .cut (m,n) -> Path-like ;
coherence by Lm64;

end;

registration

let G be _Graph;
let W be vertex-distinct Walk of G;
let m, n be Element of NAT ;
cluster W .cut (m,n) -> vertex-distinct ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be closed Walk of G;
let m, n be Element of NAT ;
cluster W .remove (m,n) -> closed ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be DWalk of G;
let m, n be Element of NAT ;
cluster W .remove (m,n) -> directed ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be trivial Walk of G;
let m, n be Element of NAT ;
cluster W .remove (m,n) -> trivial ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be Trail of G;
let m, n be Element of NAT ;
cluster W .remove (m,n) -> Trail-like ;
coherence

proof end;

end;

registration

let G be _Graph;
let W be Path of G;
let m, n be Element of NAT ;
cluster W .remove (m,n) -> Path-like ;
coherence

proof end;

end;

definition

let G be _Graph;
let W be Walk of G;
mode Subwalk of W -> Walk of G means :Def32: :: GLIB_001:def 32
( it is_Walk_from W .first() ,W .last() & ex es being Subset of (W .edgeSeq()) st it .edgeSeq() = Seq es );
existence

proof end;

end;

:: deftheorem Def32 defines Subwalk GLIB_001:def 32 :

Lm70: for G being _Graph
for W being Walk of G holds W is Subwalk of W

proof end;

Lm71: for G being _Graph
for W1 being Walk of G
for W2 being Subwalk of W1
for W3 being Subwalk of W2 holds W3 is Subwalk of W1

proof end;

Lm72: for G being _Graph
for W1, W2 being Walk of G st W1 is Subwalk of W2 holds
len W1 <= len W2

proof end;

definition

let G be _Graph;
let W be Walk of G;
let m, n be Element of NAT ;
:: original: .remove
redefine func W .remove (m,n) -> Subwalk of W;
coherence

proof end;

end;

registration

let G be _Graph;
let W be Walk of G;
cluster Relation-like (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like Trail-like Path-like Subwalk of W;
existence

proof end;

end;

definition

let G be _Graph;
let W be Walk of G;
mode Trail of W is Trail-like Subwalk of W;
mode Path of W is Path-like Subwalk of W;

end;

registration

let G be _Graph;
let W be DWalk of G;
cluster Relation-like (the_Vertices_of G) \/ (the_Edges_of G) -valued Function-like finite FinSequence-like FinSubsequence-like directed Trail-like Path-like Subwalk of W;
existence

proof end;

end;

definition

let G be _Graph;
let W be DWalk of G;
mode DWalk of W is directed Subwalk of W;
mode DTrail of W is directed Trail of W;
mode DPath of W is directed Path of W;

end;

definition

let G be _Graph;
func G .allWalks() -> non empty Subset of (((the_Vertices_of G) \/ (the_Edges_of G)) *) equals :: GLIB_001:def 33
{ W where W is Walk of G : verum } ;
coherence

proof end;

end;

:: deftheorem defines .allWalks() GLIB_001:def 33 :

definition

let G be _Graph;
func G .allTrails() -> non empty Subset of (G .allWalks()) equals :: GLIB_001:def 34
{ W where W is Trail of G : verum } ;
coherence

proof end;

end;

:: deftheorem defines .allTrails() GLIB_001:def 34 :

definition

let G be _Graph;
func G .allPaths() -> non empty Subset of (G .allTrails()) equals :: GLIB_001:def 35
{ W where W is Path of G : verum } ;
coherence

proof end;

end;

:: deftheorem defines .allPaths() GLIB_001:def 35 :

definition

let G be _Graph;
func G .allDWalks() -> non empty Subset of (G .allWalks()) equals :: GLIB_001:def 36
{ W where W is DWalk of G : verum } ;
coherence

proof end;

end;

:: deftheorem defines .allDWalks() GLIB_001:def 36 :

definition

let G be _Graph;
func G .allDTrails() -> non empty Subset of (G .allTrails()) equals :: GLIB_001:def 37
{ W where W is DTrail of G : verum } ;
coherence

proof end;

end;

:: deftheorem defines .allDTrails() GLIB_001:def 37 :

definition

let G be _Graph;
func G .allDPaths() -> non empty Subset of (G .allDTrails()) equals :: GLIB_001:def 38
{ W where W is directed Path of G : verum } ;
coherence

proof end;

end;

:: deftheorem defines .allDPaths() GLIB_001:def 38 :

registration

let G be finite _Graph;
cluster G .allTrails() -> non empty finite ;
correctness

proof end;

end;

definition

let G be _Graph;
let X be non empty Subset of (G .allWalks());
:: original: Element
redefine mode Element of X -> Walk of G;
coherence

proof end;

end;

definition

let G be _Graph;
let X be non empty Subset of (G .allTrails());
:: original: Element
redefine mode Element of X -> Trail of G;
coherence

proof end;

end;

definition

let G be _Graph;
let X be non empty Subset of (G .allPaths());
:: original: Element
redefine mode Element of X -> Path of G;
coherence

proof end;

end;

definition

let G be _Graph;
let X be non empty Subset of (G .allDWalks());
:: original: Element
redefine mode Element of X -> DWalk of G;
coherence

proof end;

end;

definition

let G be _Graph;
let X be non empty Subset of (G .allDTrails());
:: original: Element
redefine mode Element of X -> DTrail of G;
coherence

proof end;

end;

definition

let G be _Graph;
let X be non empty Subset of (G .allDPaths());
:: original: Element
redefine mode Element of X -> DPath of G;
coherence

proof end;

end;

begin

theorem :: GLIB_001:7

canceled;

theorem :: GLIB_001:8

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W . n in the_Vertices_of G by Lm1;

theorem Th9: :: GLIB_001:9

for G being _Graph
for W being Walk of G
for n being even Element of NAT st n in dom W holds
W . n in the_Edges_of G

proof end;

theorem :: GLIB_001:10

for G being _Graph
for W being Walk of G
for n being even Element of NAT st n in dom W holds
ex naa1 being odd Element of NAT st
( naa1 = n - 1 & n - 1 in dom W & n + 1 in dom W & W . n Joins W . naa1,W . (n + 1),G ) by Lm2;

theorem Th11: :: GLIB_001:11

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n < len W holds
W . (n + 1) in (W .vertexAt n) .edgesInOut()

proof end;

theorem Th12: :: GLIB_001:12

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st 1 < n & n <= len W holds
W . (n - 1) in (W .vertexAt n) .edgesInOut()

proof end;

theorem :: GLIB_001:13

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n < len W holds
( n in dom W & n + 1 in dom W & n + 2 in dom W )

proof end;

theorem Th14: :: GLIB_001:14

for G being _Graph
for v being Vertex of G holds
( len (G .walkOf v) = 1 & (G .walkOf v) . 1 = v & (G .walkOf v) .first() = v & (G .walkOf v) .last() = v & G .walkOf v is_Walk_from v,v )

proof end;

theorem Th15: :: GLIB_001:15

for G being _Graph
for e, x, y being set st e Joins x,y,G holds
len (G .walkOf (x,e,y)) = 3

proof end;

theorem Th16: :: GLIB_001:16

for G being _Graph
for e, x, y being set st e Joins x,y,G holds
( (G .walkOf (x,e,y)) .first() = x & (G .walkOf (x,e,y)) .last() = y & G .walkOf (x,e,y) is_Walk_from x,y )

proof end;

theorem :: GLIB_001:17

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
( W1 .first() = W2 .first() & W1 .last() = W2 .last() ) ;

theorem Th18: :: GLIB_001:18

for G being _Graph
for W being Walk of G
for x, y being set holds
( W is_Walk_from x,y iff ( W . 1 = x & W . (len W) = y ) )

proof end;

theorem :: GLIB_001:19

for G being _Graph
for W being Walk of G
for x, y being set st W is_Walk_from x,y holds
( x is Vertex of G & y is Vertex of G )

proof end;

theorem :: GLIB_001:20

for G1, G2 being _Graph
for x, y being set
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
( W1 is_Walk_from x,y iff W2 is_Walk_from x,y )

proof end;

theorem :: GLIB_001:21

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
for n being Element of NAT holds W1 .vertexAt n = W2 .vertexAt n

proof end;

theorem :: GLIB_001:22

for G being _Graph
for W being Walk of G holds
( len W = len (W .reverse()) & dom W = dom (W .reverse()) & rng W = rng (W .reverse()) ) by FINSEQ_5:60, FINSEQ_5:def 3;

theorem Th23: :: GLIB_001:23

for G being _Graph
for W being Walk of G holds
( W .first() = (W .reverse()) .last() & W .last() = (W .reverse()) .first() )

proof end;

theorem Th24: :: GLIB_001:24

for G being _Graph
for W being Walk of G
for x, y being set holds
( W is_Walk_from x,y iff W .reverse() is_Walk_from y,x )

proof end;

theorem Th25: :: GLIB_001:25

for G being _Graph
for W being Walk of G
for n being Element of NAT st n in dom W holds
( W . n = (W .reverse()) . (((len W) - n) + 1) & ((len W) - n) + 1 in dom (W .reverse()) )

proof end;

theorem :: GLIB_001:26

for G being _Graph
for W being Walk of G
for n being Element of NAT st n in dom (W .reverse()) holds
( (W .reverse()) . n = W . (((len W) - n) + 1) & ((len W) - n) + 1 in dom W ) by Lm8;

theorem :: GLIB_001:27

for G being _Graph
for W being Walk of G holds (W .reverse()) .reverse() = W by FINSEQ_6:29;

theorem :: GLIB_001:28

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
W1 .reverse() = W2 .reverse() ;

theorem :: GLIB_001:29

for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
(len (W1 .append W2)) + 1 = (len W1) + (len W2) by Lm9;

theorem :: GLIB_001:30

for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
( len W1 <= len (W1 .append W2) & len W2 <= len (W1 .append W2) ) by Lm10;

theorem :: GLIB_001:31

for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
( (W1 .append W2) .first() = W1 .first() & (W1 .append W2) .last() = W2 .last() & W1 .append W2 is_Walk_from W1 .first() ,W2 .last() ) by Lm11;

theorem Th32: :: GLIB_001:32

for G being _Graph
for W1, W2 being Walk of G
for x, y, z being set st W1 is_Walk_from x,y & W2 is_Walk_from y,z holds
W1 .append W2 is_Walk_from x,z

proof end;

theorem :: GLIB_001:33

for G being _Graph
for W1, W2 being Walk of G
for n being Element of NAT st n in dom W1 holds
( (W1 .append W2) . n = W1 . n & n in dom (W1 .append W2) ) by Lm12;

theorem :: GLIB_001:34

for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
for n being Element of NAT st n < len W2 holds
( (W1 .append W2) . ((len W1) + n) = W2 . (n + 1) & (len W1) + n in dom (W1 .append W2) ) by Lm13;

theorem :: GLIB_001:35

for G being _Graph
for W1, W2 being Walk of G
for n being Element of NAT holds
( not n in dom (W1 .append W2) or n in dom W1 or ex k being Element of NAT st
( k < len W2 & n = (len W1) + k ) ) by Lm14;

theorem Th36: :: GLIB_001:36

for G1, G2 being _Graph
for W1A, W1B being Walk of G1
for W2A, W2B being Walk of G2 st W1A = W2A & W1B = W2B holds
W1A .append W1B = W2A .append W2B

proof end;

theorem :: GLIB_001:37

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W holds
( (len (W .cut (m,n))) + m = n + 1 & ( for i being Element of NAT st i < len (W .cut (m,n)) holds
( (W .cut (m,n)) . (i + 1) = W . (m + i) & m + i in dom W ) ) ) by Lm15;

theorem :: GLIB_001:38

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W holds
( (W .cut (m,n)) .first() = W . m & (W .cut (m,n)) .last() = W . n & W .cut (m,n) is_Walk_from W . m,W . n ) by Lm16;

theorem :: GLIB_001:39

for G being _Graph
for W being Walk of G
for m, n, o being odd Element of NAT st m <= n & n <= o & o <= len W holds
(W .cut (m,n)) .append (W .cut (n,o)) = W .cut (m,o) by Lm17;

theorem :: GLIB_001:40

for G being _Graph
for W being Walk of G holds W .cut (1,(len W)) = W by Lm18;

theorem Th41: :: GLIB_001:41

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n < len W holds
G .walkOf ((W . n),(W . (n + 1)),(W . (n + 2))) = W .cut (n,(n + 2))

proof end;

theorem Th42: :: GLIB_001:42

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n < len W holds
(W .cut (m,n)) .addEdge (W . (n + 1)) = W .cut (m,(n + 2))

proof end;

theorem :: GLIB_001:43

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .cut (n,n) = <*(W .vertexAt n)*> by Lm19;

theorem :: GLIB_001:44

for G being _Graph
for W being Walk of G
for m, n being Element of NAT st not m is even & m <= n holds
(W .cut (1,n)) .cut (1,m) = W .cut (1,m) by Lm20;

theorem :: GLIB_001:45

for G being _Graph
for W1, W2 being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W1 & W1 .last() = W2 .first() holds
(W1 .append W2) .cut (m,n) = W1 .cut (m,n) by Lm21;

theorem :: GLIB_001:46

for G being _Graph
for W being Walk of G
for m being odd Element of NAT st m <= len W holds
len (W .cut (1,m)) = m by Lm22;

theorem :: GLIB_001:47

for G being _Graph
for W being Walk of G
for m being odd Element of NAT
for x being Element of NAT st x in dom (W .cut (1,m)) & m <= len W holds
(W .cut (1,m)) . x = W . x by Lm23;

theorem :: GLIB_001:48

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT
for i being Element of NAT st m <= n & n <= len W & i in dom (W .cut (m,n)) holds
( (W .cut (m,n)) . i = W . ((m + i) - 1) & (m + i) - 1 in dom W )

proof end;

theorem Th49: :: GLIB_001:49

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2
for m, n being Element of NAT st W1 = W2 holds
W1 .cut (m,n) = W2 .cut (m,n)

proof end;

theorem :: GLIB_001:50

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
(len (W .remove (m,n))) + n = (len W) + m by Lm24;

theorem :: GLIB_001:51

for G being _Graph
for W being Walk of G
for x, y being set
for m, n being Element of NAT st W is_Walk_from x,y holds
W .remove (m,n) is_Walk_from x,y by Lm25;

theorem :: GLIB_001:52

for G being _Graph
for W being Walk of G
for m, n being Element of NAT holds len (W .remove (m,n)) <= len W by Lm26;

theorem :: GLIB_001:53

for G being _Graph
for W being Walk of G
for m being Element of NAT holds W .remove (m,m) = W by Lm27;

theorem :: GLIB_001:54

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
(W .cut (1,m)) .last() = (W .cut (n,(len W))) .first() by Lm28;

theorem :: GLIB_001:55

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
for x being Element of NAT st x in Seg m holds
(W .remove (m,n)) . x = W . x by Lm29;

theorem :: GLIB_001:56

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
for x being Element of NAT st m <= x & x <= len (W .remove (m,n)) holds
( (W .remove (m,n)) . x = W . ((x - m) + n) & (x - m) + n is Element of NAT & (x - m) + n <= len W ) by Lm30;

theorem :: GLIB_001:57

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W & W . m = W . n holds
len (W .remove (m,n)) = ((len W) + m) - n by Lm31;

theorem Th58: :: GLIB_001:58

for G being _Graph
for W being Walk of G
for m being Element of NAT st W . m = W .last() holds
W .remove (m,(len W)) = W .cut (1,m)

proof end;

theorem :: GLIB_001:59

for G being _Graph
for W being Walk of G
for m being Element of NAT st W .first() = W . m holds
W .remove (1,m) = W .cut (m,(len W)) by Lm32;

theorem :: GLIB_001:60

for G being _Graph
for W being Walk of G
for m, n being Element of NAT holds
( (W .remove (m,n)) .first() = W .first() & (W .remove (m,n)) .last() = W .last() ) by Lm33;

theorem :: GLIB_001:61

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT
for x being Element of NAT st m <= n & n <= len W & W . m = W . n & x in dom (W .remove (m,n)) & not x in Seg m holds
( m <= x & x <= len (W .remove (m,n)) ) by Lm34;

theorem :: GLIB_001:62

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2
for m, n being Element of NAT st W1 = W2 holds
W1 .remove (m,n) = W2 .remove (m,n)

proof end;

theorem :: GLIB_001:63

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
W .addEdge e = W ^ <*e,x*> by Lm35;

theorem :: GLIB_001:64

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
( (W .addEdge e) .first() = W .first() & (W .addEdge e) .last() = x & W .addEdge e is_Walk_from W .first() ,x ) by Lm36;

theorem :: GLIB_001:65

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
len (W .addEdge e) = (len W) + 2 by Lm37;

theorem :: GLIB_001:66

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
( (W .addEdge e) . ((len W) + 1) = e & (W .addEdge e) . ((len W) + 2) = x & ( for n being Element of NAT st n in dom W holds
(W .addEdge e) . n = W . n ) ) by Lm38;

theorem :: GLIB_001:67

for G being _Graph
for W being Walk of G
for x, y, e, z being set st W is_Walk_from x,y & e Joins y,z,G holds
W .addEdge e is_Walk_from x,z by Lm39;

theorem Th68: :: GLIB_001:68

for G being _Graph
for W being Walk of G holds 1 <= len (W .vertexSeq())

proof end;

theorem Th69: :: GLIB_001:69

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
( (2 * ((n + 1) div 2)) - 1 = n & 1 <= (n + 1) div 2 & (n + 1) div 2 <= len (W .vertexSeq()) )

proof end;

theorem :: GLIB_001:70

for G being _Graph
for v being Vertex of G holds (G .walkOf v) .vertexSeq() = <*v*>

proof end;

theorem Th71: :: GLIB_001:71

for G being _Graph
for e, x, y being set st e Joins x,y,G holds
(G .walkOf (x,e,y)) .vertexSeq() = <*x,y*>

proof end;

theorem :: GLIB_001:72

for G being _Graph
for W being Walk of G holds
( W .first() = (W .vertexSeq()) . 1 & W .last() = (W .vertexSeq()) . (len (W .vertexSeq())) )

proof end;

theorem :: GLIB_001:73

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .vertexAt n = (W .vertexSeq()) . ((n + 1) div 2)

proof end;

theorem Th74: :: GLIB_001:74

for G being _Graph
for W being Walk of G
for n being Element of NAT holds
( n in dom (W .vertexSeq()) iff (2 * n) - 1 in dom W )

proof end;

theorem :: GLIB_001:75

for G being _Graph
for W being Walk of G
for n being Element of NAT holds (W .cut (1,n)) .vertexSeq() c= W .vertexSeq()

proof end;

theorem Th76: :: GLIB_001:76

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
(W .addEdge e) .vertexSeq() = (W .vertexSeq()) ^ <*x*>

proof end;

theorem Th77: :: GLIB_001:77

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
W1 .vertexSeq() = W2 .vertexSeq()

proof end;

theorem :: GLIB_001:78

for G being _Graph
for W being Walk of G
for n being even Element of NAT st 1 <= n & n <= len W holds
( n div 2 in dom (W .edgeSeq()) & W . n = (W .edgeSeq()) . (n div 2) ) by Lm40;

theorem :: GLIB_001:79

for G being _Graph
for W being Walk of G
for n being Element of NAT holds
( n in dom (W .edgeSeq()) iff 2 * n in dom W ) by Lm41;

theorem :: GLIB_001:80

for G being _Graph
for W being Walk of G
for n being Element of NAT st n in dom (W .edgeSeq()) holds
(W .edgeSeq()) . n in the_Edges_of G

proof end;

theorem :: GLIB_001:81

for G being _Graph
for W being Walk of G ex lenWaa1 being even Element of NAT st
( lenWaa1 = (len W) - 1 & len (W .edgeSeq()) = lenWaa1 div 2 ) by Lm42;

theorem :: GLIB_001:82

for G being _Graph
for W being Walk of G
for n being Element of NAT holds (W .cut (1,n)) .edgeSeq() c= W .edgeSeq() by Lm43;

theorem :: GLIB_001:83

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
(W .addEdge e) .edgeSeq() = (W .edgeSeq()) ^ <*e*> by Lm44;

theorem Th84: :: GLIB_001:84

for G being _Graph
for e, x, y being set holds
( e Joins x,y,G iff (G .walkOf (x,e,y)) .edgeSeq() = <*e*> )

proof end;

theorem :: GLIB_001:85

for G being _Graph
for W being Walk of G holds (W .reverse()) .edgeSeq() = Rev (W .edgeSeq())

proof end;

theorem :: GLIB_001:86

for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
(W1 .append W2) .edgeSeq() = (W1 .edgeSeq()) ^ (W2 .edgeSeq())

proof end;

theorem Th87: :: GLIB_001:87

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
W1 .edgeSeq() = W2 .edgeSeq()

proof end;

theorem :: GLIB_001:88

for G being _Graph
for W being Walk of G
for x being set holds
( x in W .vertices() iff ex n being odd Element of NAT st
( n <= len W & W . n = x ) ) by Lm45;

theorem Th89: :: GLIB_001:89

for G being _Graph
for W being Walk of G holds
( W .first() in W .vertices() & W .last() in W .vertices() )

proof end;

theorem Th90: :: GLIB_001:90

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
W .vertexAt n in W .vertices()

proof end;

theorem :: GLIB_001:91

for G being _Graph
for v being Vertex of G holds (G .walkOf v) .vertices() = {v}

proof end;

theorem Th92: :: GLIB_001:92

for G being _Graph
for e, x, y being set st e Joins x,y,G holds
(G .walkOf (x,e,y)) .vertices() = {x,y}

proof end;

theorem :: GLIB_001:93

for G being _Graph
for W being Walk of G holds W .vertices() = (W .reverse()) .vertices()

proof end;

theorem Th94: :: GLIB_001:94

for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
(W1 .append W2) .vertices() = (W1 .vertices()) \/ (W2 .vertices())

proof end;

theorem :: GLIB_001:95

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W holds
(W .cut (m,n)) .vertices() c= W .vertices()

proof end;

theorem Th96: :: GLIB_001:96

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
(W .addEdge e) .vertices() = (W .vertices()) \/ {x}

proof end;

theorem :: GLIB_001:97

for G being finite _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G & not x in W .vertices() holds
card ((W .addEdge e) .vertices()) = (card (W .vertices())) + 1

proof end;

theorem :: GLIB_001:98

for G being _Graph
for W being Walk of G
for x, y being set st x in W .vertices() & y in W .vertices() holds
ex W9 being Walk of G st W9 is_Walk_from x,y

proof end;

theorem :: GLIB_001:99

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
W1 .vertices() = W2 .vertices() by Th77;

theorem :: GLIB_001:100

for G being _Graph
for W being Walk of G
for e being set holds
( e in W .edges() iff ex n being even Element of NAT st
( 1 <= n & n <= len W & W . n = e ) ) by Lm46;

theorem Th101: :: GLIB_001:101

for G being _Graph
for W being Walk of G
for e being set holds
( e in W .edges() iff ex n being odd Element of NAT st
( n < len W & W . (n + 1) = e ) )

proof end;

theorem Th102: :: GLIB_001:102

for G being _Graph
for W being Walk of G holds rng W = (W .vertices()) \/ (W .edges())

proof end;

theorem Th103: :: GLIB_001:103

for G being _Graph
for W1, W2 being Walk of G st W1 .last() = W2 .first() holds
(W1 .append W2) .edges() = (W1 .edges()) \/ (W2 .edges())

proof end;

theorem :: GLIB_001:104

for G being _Graph
for W being Walk of G
for e being set st e in W .edges() holds
ex v1, v2 being Vertex of G ex n being odd Element of NAT st
( n + 2 <= len W & v1 = W . n & e = W . (n + 1) & v2 = W . (n + 2) & e Joins v1,v2,G ) by Lm47;

theorem Th105: :: GLIB_001:105

for G being _Graph
for W being Walk of G
for e being set holds
( e in W .edges() iff ex n being Element of NAT st
( n in dom (W .edgeSeq()) & (W .edgeSeq()) . n = e ) )

proof end;

theorem :: GLIB_001:106

for G being _Graph
for W being Walk of G
for e, x, y being set st e in W .edges() & e Joins x,y,G holds
( x in W .vertices() & y in W .vertices() ) by Lm48;

theorem :: GLIB_001:107

for G being _Graph
for W being Walk of G
for m, n being Element of NAT holds (W .cut (m,n)) .edges() c= W .edges()

proof end;

theorem Th108: :: GLIB_001:108

for G being _Graph
for W being Walk of G holds W .edges() = (W .reverse()) .edges()

proof end;

theorem Th109: :: GLIB_001:109

for G being _Graph
for e, x, y being set holds
( e Joins x,y,G iff (G .walkOf (x,e,y)) .edges() = {e} )

proof end;

theorem :: GLIB_001:110

for G being _Graph
for W being Walk of G holds W .edges() c= G .edgesBetween (W .vertices())

proof end;

theorem :: GLIB_001:111

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
W1 .edges() = W2 .edges() by Th87;

theorem :: GLIB_001:112

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
(W .addEdge e) .edges() = (W .edges()) \/ {e}

proof end;

theorem :: GLIB_001:113

for G being _Graph
for W being Walk of G holds len W = (2 * (W .length())) + 1 by Def15;

theorem :: GLIB_001:114

for G being _Graph
for W1, W2 being Walk of G holds
( len W1 = len W2 iff W1 .length() = W2 .length() )

proof end;

theorem :: GLIB_001:115

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
W1 .length() = W2 .length() by Th87;

theorem Th116: :: GLIB_001:116

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
( W .find (W . n) <= n & W .rfind (W . n) >= n )

proof end;

theorem :: GLIB_001:117

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2
for v being set st W1 = W2 holds
( W1 .find v = W2 .find v & W1 .rfind v = W2 .rfind v )

proof end;

theorem :: GLIB_001:118

for G being _Graph
for W being Walk of G
for n being odd Element of NAT st n <= len W holds
( W .find n <= n & W .rfind n >= n ) by Lm49, Lm50;

theorem Th119: :: GLIB_001:119

for G being _Graph
for W being Walk of G holds
( W is closed iff W . 1 = W . (len W) )

proof end;

theorem :: GLIB_001:120

for G being _Graph
for W being Walk of G holds
( W is closed iff ex x being set st W is_Walk_from x,x )

proof end;

theorem :: GLIB_001:121

for G being _Graph
for W being Walk of G holds
( W is closed iff W .reverse() is closed )

proof end;

theorem :: GLIB_001:122

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 & W1 is closed holds
W2 is closed

proof end;

theorem :: GLIB_001:123

for G being _Graph
for W being Walk of G holds
( W is directed iff for n being odd Element of NAT st n < len W holds
W . (n + 1) DJoins W . n,W . (n + 2),G ) by Lm51;

theorem :: GLIB_001:124

for G being _Graph
for W being Walk of G
for x, y, e, z being set st W is directed & W is_Walk_from x,y & e DJoins y,z,G holds
( W .addEdge e is directed & W .addEdge e is_Walk_from x,z ) by Lm52;

theorem :: GLIB_001:125

for G being _Graph
for W being DWalk of G
for m, n being Element of NAT holds W .cut (m,n) is directed ;

theorem :: GLIB_001:126

for G being _Graph
for W being Walk of G holds
( not W is trivial iff 3 <= len W ) by Lm54;

theorem :: GLIB_001:127

for G being _Graph
for W being Walk of G holds
( not W is trivial iff len W <> 1 ) by Lm55;

theorem :: GLIB_001:128

for G being _Graph
for W being Walk of G st W .first() <> W .last() holds
not W is trivial by Lm55;

theorem :: GLIB_001:129

for G being _Graph
for W being Walk of G holds
( W is trivial iff ex v being Vertex of G st W = G .walkOf v ) by Lm56;

theorem :: GLIB_001:130

for G being _Graph
for W being Walk of G holds
( W is trivial iff W .reverse() is trivial )

proof end;

theorem :: GLIB_001:131

for G being _Graph
for W2, W1 being Walk of G st W2 is trivial holds
W1 .append W2 = W1

proof end;

theorem :: GLIB_001:132

for G being _Graph
for W being Walk of G
for m, n being odd Element of NAT st m <= n & n <= len W holds
( W .cut (m,n) is trivial iff m = n )

proof end;

theorem Th133: :: GLIB_001:133

for G being _Graph
for W being Walk of G
for e, x being set st e Joins W .last() ,x,G holds
not W .addEdge e is trivial

proof end;

theorem Th134: :: GLIB_001:134

for G being _Graph
for W being Walk of G st not W is trivial holds
ex lenW2 being odd Element of NAT st
( lenW2 = (len W) - 2 & (W .cut (1,lenW2)) .addEdge (W . (lenW2 + 1)) = W )

proof end;

theorem Th135: :: GLIB_001:135

for G being _Graph
for W2, W1 being Walk of G st not W2 is trivial & W2 .edges() c= W1 .edges() holds
W2 .vertices() c= W1 .vertices()

proof end;

theorem :: GLIB_001:136

for G being _Graph
for W being Walk of G st not W is trivial holds
for v being Vertex of G st v in W .vertices() holds
not v is isolated

proof end;

theorem :: GLIB_001:137

for G being _Graph
for W being Walk of G holds
( W is trivial iff W .edges() = {} )

proof end;

theorem :: GLIB_001:138

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 & W1 is trivial holds
W2 is trivial

proof end;

theorem :: GLIB_001:139

for G being _Graph
for W being Walk of G holds
( W is Trail-like iff for m, n being even Element of NAT st 1 <= m & m < n & n <= len W holds
W . m <> W . n ) by Lm57;

theorem :: GLIB_001:140

for G being _Graph
for W being Walk of G st len W <= 3 holds
W is Trail-like by Lm61;

theorem :: GLIB_001:141

for G being _Graph
for W being Walk of G holds
( W is Trail-like iff W .reverse() is Trail-like ) by Lm58;

theorem :: GLIB_001:142

for G being _Graph
for W being Trail of G
for m, n being Element of NAT holds W .cut (m,n) is Trail-like ;

theorem :: GLIB_001:143

for G being _Graph
for W being Trail of G
for e being set st e in (W .last()) .edgesInOut() & not e in W .edges() holds
W .addEdge e is Trail-like by Lm60;

theorem :: GLIB_001:144

for G being _Graph
for W being Trail of G
for v being Vertex of G st v in W .vertices() & v is endvertex & not v = W .first() holds
v = W .last()

proof end;

theorem :: GLIB_001:145

for G being finite _Graph
for W being Trail of G holds len (W .edgeSeq()) <= G .size()

proof end;

theorem :: GLIB_001:146

for G being _Graph
for W being Walk of G st len W <= 3 holds
W is Path-like by Lm69;

theorem :: GLIB_001:147

for G being _Graph
for W being Walk of G st ( for m, n being odd Element of NAT st m <= len W & n <= len W & W . m = W . n holds
m = n ) holds
W is Path-like by Lm66;

theorem :: GLIB_001:148

for G being _Graph
for W being Path of G st not W is closed holds
for m, n being odd Element of NAT st m < n & n <= len W holds
W . m <> W . n

proof end;

theorem :: GLIB_001:149

for G being _Graph
for W being Walk of G holds
( W is Path-like iff W .reverse() is Path-like ) by Lm63;

theorem :: GLIB_001:150

for G being _Graph
for W being Path of G
for m, n being Element of NAT holds W .cut (m,n) is Path-like ;

theorem Th151: :: GLIB_001:151

for G being _Graph
for W being Path of G
for e, v being set st e Joins W .last() ,v,G & not e in W .edges() & ( W is trivial or not W is closed ) & ( for n being odd Element of NAT st 1 < n & n <= len W holds
W . n <> v ) holds
W .addEdge e is Path-like

proof end;

theorem :: GLIB_001:152

for G being _Graph
for W being Path of G
for e, v being set st e Joins W .last() ,v,G & not v in W .vertices() & ( W is trivial or not W is closed ) holds
W .addEdge e is Path-like by Lm68;

theorem :: GLIB_001:153

for G being _Graph
for W being Walk of G st ( for n being odd Element of NAT st n <= len W holds
W .find (W . n) = W .rfind (W . n) ) holds
W is Path-like

proof end;

theorem :: GLIB_001:154

for G being _Graph
for W being Walk of G st ( for n being odd Element of NAT st n <= len W holds
W .rfind n = n ) holds
W is Path-like by Lm67;

theorem :: GLIB_001:155

for G being finite _Graph
for W being Path of G holds len (W .vertexSeq()) <= (G .order()) + 1

proof end;

theorem :: GLIB_001:156

for G being _Graph
for W being vertex-distinct Walk of G
for e, v being set st e Joins W .last() ,v,G & not v in W .vertices() holds
W .addEdge e is vertex-distinct

proof end;

theorem :: GLIB_001:157

for G being _Graph
for e, x being set st e Joins x,x,G holds
G .walkOf (x,e,x) is Cycle-like

proof end;

theorem :: GLIB_001:158

for G being _Graph
for W1 being Walk of G
for e, x, y being set st e Joins x,y,G & e in W1 .edges() & W1 is Cycle-like holds
ex W2 being Walk of G st
( W2 is_Walk_from x,y & not e in W2 .edges() )

proof end;

theorem :: GLIB_001:159

for G being _Graph
for W being Walk of G holds W is Subwalk of W by Lm70;

theorem :: GLIB_001:160

for G being _Graph
for W1 being Walk of G
for W2 being Subwalk of W1
for W3 being Subwalk of W2 holds W3 is Subwalk of W1 by Lm71;

theorem :: GLIB_001:161

for G being _Graph
for W1, W2 being Walk of G
for x, y being set st W1 is Subwalk of W2 holds
( W1 is_Walk_from x,y iff W2 is_Walk_from x,y )

proof end;

theorem Th162: :: GLIB_001:162

for G being _Graph
for W1, W2 being Walk of G st W1 is Subwalk of W2 holds
( W1 .first() = W2 .first() & W1 .last() = W2 .last() )

proof end;

theorem :: GLIB_001:163

for G being _Graph
for W1, W2 being Walk of G st W1 is Subwalk of W2 holds
len W1 <= len W2 by Lm72;

theorem Th164: :: GLIB_001:164

for G being _Graph
for W1, W2 being Walk of G st W1 is Subwalk of W2 holds
( W1 .edges() c= W2 .edges() & W1 .vertices() c= W2 .vertices() )

proof end;

theorem Th165: :: GLIB_001:165

for G being _Graph
for W1, W2 being Walk of G st W1 is Subwalk of W2 holds
for m being odd Element of NAT st m <= len W1 holds
ex n being odd Element of NAT st
( m <= n & n <= len W2 & W1 . m = W2 . n )

proof end;

theorem :: GLIB_001:166

for G being _Graph
for W1, W2 being Walk of G st W1 is Subwalk of W2 holds
for m being even Element of NAT st 1 <= m & m <= len W1 holds
ex n being even Element of NAT st
( m <= n & n <= len W2 & W1 . m = W2 . n )

proof end;

theorem :: GLIB_001:167

for G being _Graph
for W1 being Trail of G st not W1 is trivial holds
ex W2 being Path of W1 st not W2 is trivial

proof end;

theorem Th168: :: GLIB_001:168

for G1 being _Graph
for G2 being Subgraph of G1
for W being Walk of G2 holds W is Walk of G1

proof end;

theorem Th169: :: GLIB_001:169

for G1 being _Graph
for G2 being Subgraph of G1
for W being Walk of G1 st W is trivial & W .first() in the_Vertices_of G2 holds
W is Walk of G2

proof end;

theorem Th170: :: GLIB_001:170

for G1 being _Graph
for G2 being Subgraph of G1
for W being Walk of G1 st not W is trivial & W .edges() c= the_Edges_of G2 holds
W is Walk of G2

proof end;

theorem Th171: :: GLIB_001:171

for G1 being _Graph
for G2 being Subgraph of G1
for W being Walk of G1 st W .vertices() c= the_Vertices_of G2 & W .edges() c= the_Edges_of G2 holds
W is Walk of G2

proof end;

theorem :: GLIB_001:172

for G1 being non trivial _Graph
for W being Walk of G1
for v being Vertex of G1
for G2 being removeVertex of G1,v st not v in W .vertices() holds
W is Walk of G2

proof end;

theorem :: GLIB_001:173

for G1 being _Graph
for W being Walk of G1
for e being set
for G2 being removeEdge of G1,e st not e in W .edges() holds
W is Walk of G2

proof end;

theorem Th174: :: GLIB_001:174

for G1 being _Graph
for G2 being Subgraph of G1
for x, y, e being set st e Joins x,y,G2 holds
G1 .walkOf (x,e,y) = G2 .walkOf (x,e,y)

proof end;

theorem :: GLIB_001:175

for G1 being _Graph
for G2 being Subgraph of G1
for W1 being Walk of G1
for W2 being Walk of G2
for e being set st W1 = W2 & e in (W2 .last()) .edgesInOut() holds
W1 .addEdge e = W2 .addEdge e

proof end;

theorem Th176: :: GLIB_001:176

for G1 being _Graph
for G2 being Subgraph of G1
for W being Walk of G2 holds
( ( W is closed implies W is closed Walk of G1 ) & ( W is directed implies W is directed Walk of G1 ) & ( W is trivial implies W is trivial Walk of G1 ) & ( W is Trail-like implies W is Trail-like Walk of G1 ) & ( W is Path-like implies W is Path-like Walk of G1 ) & ( W is vertex-distinct implies W is vertex-distinct Walk of G1 ) )

proof end;

theorem Th177: :: GLIB_001:177

for G1 being _Graph
for G2 being Subgraph of G1
for W1 being Walk of G1
for W2 being Walk of G2 st W1 = W2 holds
( ( W1 is closed implies W2 is closed ) & ( W2 is closed implies W1 is closed ) & ( W1 is directed implies W2 is directed ) & ( W2 is directed implies W1 is directed ) & ( W1 is trivial implies W2 is trivial ) & ( W2 is trivial implies W1 is trivial ) & ( W1 is Trail-like implies W2 is Trail-like ) & ( W2 is Trail-like implies W1 is Trail-like ) & ( W1 is Path-like implies W2 is Path-like ) & ( W2 is Path-like implies W1 is Path-like ) & ( W1 is vertex-distinct implies W2 is vertex-distinct ) & ( W2 is vertex-distinct implies W1 is vertex-distinct ) )

proof end;

theorem :: GLIB_001:178

for G1, G2 being _Graph
for x being set st G1 == G2 & x is VertexSeq of G1 holds
x is VertexSeq of G2

proof end;

theorem :: GLIB_001:179

for G1, G2 being _Graph
for x being set st G1 == G2 & x is EdgeSeq of G1 holds
x is EdgeSeq of G2

proof end;

theorem :: GLIB_001:180

for G1, G2 being _Graph
for x being set st G1 == G2 & x is Walk of G1 holds
x is Walk of G2

proof end;

theorem :: GLIB_001:181

for G1, G2 being _Graph
for x, e, y being set st G1 == G2 holds
G1 .walkOf (x,e,y) = G2 .walkOf (x,e,y)

proof end;

theorem :: GLIB_001:182

for G1, G2 being _Graph
for W1 being Walk of G1
for W2 being Walk of G2 st G1 == G2 & W1 = W2 holds
( ( W1 is closed implies W2 is closed ) & ( W2 is closed implies W1 is closed ) & ( W1 is directed implies W2 is directed ) & ( W2 is directed implies W1 is directed ) & ( W1 is trivial implies W2 is trivial ) & ( W2 is trivial implies W1 is trivial ) & ( W1 is Trail-like implies W2 is Trail-like ) & ( W2 is Trail-like implies W1 is Trail-like ) & ( W1 is Path-like implies W2 is Path-like ) & ( W2 is Path-like implies W1 is Path-like ) & ( W1 is vertex-distinct implies W2 is vertex-distinct ) & ( W2 is vertex-distinct implies W1 is vertex-distinct ) )

proof end;