:: Graph Theoretical Properties of Arcs in the Plane and Fashoda Meet Theorem
:: by Yatsuka Nakamura
::
:: Received August 21, 1998
:: Copyright (c) 1998-2011 Association of Mizar Users

begin

theorem :: JGRAPH_1:1

canceled;

theorem :: JGRAPH_1:2

canceled;

theorem :: JGRAPH_1:3

canceled;

theorem Th4: :: JGRAPH_1:4

for G being Graph
for IT being oriented Chain of G
for vs being FinSequence of the carrier of G st IT is Simple & vs is_oriented_vertex_seq_of IT holds
for n, m being Element of NAT st 1 <= n & n < m & m <= len vs & vs . n = vs . m holds
( n = 1 & m = len vs )

proof end;

definition

let X be set ;
func PGraph X -> MultiGraphStruct equals :: JGRAPH_1:def 1
MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #);
coherence
MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #) is MultiGraphStruct ;

end;

:: deftheorem defines PGraph JGRAPH_1:def 1 :
for X being set holds PGraph X = MultiGraphStruct(# X,[:X,X:],(pr1 (X,X)),(pr2 (X,X)) #);

theorem :: JGRAPH_1:5

canceled;

theorem :: JGRAPH_1:6

for X being set holds the carrier of (PGraph X) = X ;

definition

let f be FinSequence;
func PairF f -> FinSequence means :Def2: :: JGRAPH_1:def 2
( len it = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
it . i = [(f . i),(f . (i + 1))] ) );
existence
ex b₁ being FinSequence st
( len b₁ = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b₁ . i = [(f . i),(f . (i + 1))] ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence st len b₁ = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b₁ . i = [(f . i),(f . (i + 1))] ) & len b₂ = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b₂ . i = [(f . i),(f . (i + 1))] ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines PairF JGRAPH_1:def 2 :
for f, b₂ being FinSequence holds
( b₂ = PairF f iff ( len b₂ = (len f) -' 1 & ( for i being Element of NAT st 1 <= i & i < len f holds
b₂ . i = [(f . i),(f . (i + 1))] ) ) );

registration

let X be non empty set ;
cluster PGraph X -> non empty ;
coherence
not PGraph X is empty ;

end;

theorem :: JGRAPH_1:7

for X being non empty set
for f being FinSequence of X holds f is FinSequence of the carrier of (PGraph X) ;

theorem Th8: :: JGRAPH_1:8

for X being non empty set
for f being FinSequence of X holds PairF f is FinSequence of the carrier' of (PGraph X)

proof end;

definition

let X be non empty set ;
let f be FinSequence of X;
:: original: PairF
redefine func PairF f -> FinSequence of the carrier' of (PGraph X);
coherence
PairF f is FinSequence of the carrier' of (PGraph X) by Th8;

end;

theorem Th9: :: JGRAPH_1:9

for X being non empty set
for n being Element of NAT
for f being FinSequence of X st 1 <= n & n <= len (PairF f) holds
(PairF f) . n in the carrier' of (PGraph X)

proof end;

theorem Th10: :: JGRAPH_1:10

for X being non empty set
for f being FinSequence of X holds PairF f is oriented Chain of PGraph X

proof end;

definition

let X be non empty set ;
let f be FinSequence of X;
:: original: PairF
redefine func PairF f -> oriented Chain of PGraph X;
coherence
PairF f is oriented Chain of PGraph X by Th10;

end;

theorem Th11: :: JGRAPH_1:11

for X being non empty set
for f being FinSequence of X
for f1 being FinSequence of the carrier of (PGraph X) st len f >= 1 & f = f1 holds
f1 is_oriented_vertex_seq_of PairF f

proof end;

begin

definition

let X be non empty set ;
let f, g be FinSequence of X;
pred g is_Shortcut_of f means :Def3: :: JGRAPH_1:def 3
( f . 1 = g . 1 & f . (len f) = g . (len g) & ex fc being Subset of (PairF f) ex fvs being Subset of f ex sc being oriented simple Chain of PGraph X ex g1 being FinSequence of the carrier of (PGraph X) st
( Seq fc = sc & Seq fvs = g & g1 = g & g1 is_oriented_vertex_seq_of sc ) );

end;

:: deftheorem Def3 defines is_Shortcut_of JGRAPH_1:def 3 :
for X being non empty set
for f, g being FinSequence of X holds
( g is_Shortcut_of f iff ( f . 1 = g . 1 & f . (len f) = g . (len g) & ex fc being Subset of (PairF f) ex fvs being Subset of f ex sc being oriented simple Chain of PGraph X ex g1 being FinSequence of the carrier of (PGraph X) st
( Seq fc = sc & Seq fvs = g & g1 = g & g1 is_oriented_vertex_seq_of sc ) ) );

theorem Th12: :: JGRAPH_1:12

for X being non empty set
for f, g being FinSequence of X st g is_Shortcut_of f holds
( 1 <= len g & len g <= len f )

proof end;

theorem Th13: :: JGRAPH_1:13

for X being non empty set
for f being FinSequence of X st len f >= 1 holds
ex g being FinSequence of X st g is_Shortcut_of f

proof end;

theorem Th14: :: JGRAPH_1:14

for X being non empty set
for f, g being FinSequence of X st g is_Shortcut_of f holds
rng (PairF g) c= rng (PairF f)

proof end;

theorem Th15: :: JGRAPH_1:15

for X being non empty set
for f, g being FinSequence of X st f . 1 <> f . (len f) & g is_Shortcut_of f holds
g is one-to-one

proof end;

definition

let n be Element of NAT ;
let IT be FinSequence of (TOP-REAL n);
attr IT is nodic means :Def4: :: JGRAPH_1:def 4
for i, j being Element of NAT holds
( not LSeg (IT,i) meets LSeg (IT,j) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . i)} & ( IT . i = IT . j or IT . i = IT . (j + 1) ) ) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . (i + 1))} & ( IT . (i + 1) = IT . j or IT . (i + 1) = IT . (j + 1) ) ) or LSeg (IT,i) = LSeg (IT,j) );

end;

:: deftheorem Def4 defines nodic JGRAPH_1:def 4 :
for n being Element of NAT
for IT being FinSequence of (TOP-REAL n) holds
( IT is nodic iff for i, j being Element of NAT holds
( not LSeg (IT,i) meets LSeg (IT,j) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . i)} & ( IT . i = IT . j or IT . i = IT . (j + 1) ) ) or ( (LSeg (IT,i)) /\ (LSeg (IT,j)) = {(IT . (i + 1))} & ( IT . (i + 1) = IT . j or IT . (i + 1) = IT . (j + 1) ) ) or LSeg (IT,i) = LSeg (IT,j) ) );

theorem :: JGRAPH_1:16

for f being FinSequence of (TOP-REAL 2) st f is s.n.c. holds
f is s.c.c.

proof end;

theorem Th17: :: JGRAPH_1:17

for f being FinSequence of (TOP-REAL 2) st f is s.c.c. & LSeg (f,1) misses LSeg (f,((len f) -' 1)) holds
f is s.n.c.

proof end;

theorem Th18: :: JGRAPH_1:18

for f being FinSequence of (TOP-REAL 2) st f is nodic & PairF f is Simple holds
f is s.c.c.

proof end;

theorem Th19: :: JGRAPH_1:19

for f being FinSequence of (TOP-REAL 2) st f is nodic & PairF f is Simple & f . 1 <> f . (len f) holds
f is s.n.c.

proof end;

theorem Th20: :: JGRAPH_1:20

for n being Element of NAT
for p1, p2, p3 being Point of (TOP-REAL n) holds
( for x being set holds
( not x <> p2 or not x in (LSeg (p1,p2)) /\ (LSeg (p2,p3)) ) or p1 in LSeg (p2,p3) or p3 in LSeg (p1,p2) )

proof end;

theorem Th21: :: JGRAPH_1:21

for f being FinSequence of (TOP-REAL 2) st f is s.n.c. & (LSeg (f,1)) /\ (LSeg (f,(1 + 1))) c= {(f /. (1 + 1))} & (LSeg (f,((len f) -' 2))) /\ (LSeg (f,((len f) -' 1))) c= {(f /. ((len f) -' 1))} holds
f is unfolded

proof end;

theorem Th22: :: JGRAPH_1:22

for X being non empty set
for f being FinSequence of X st PairF f is Simple & f . 1 <> f . (len f) holds
( f is one-to-one & len f <> 1 )

proof end;

theorem :: JGRAPH_1:23

for X being non empty set
for f being FinSequence of X st f is one-to-one & len f > 1 holds
( PairF f is Simple & f . 1 <> f . (len f) )

proof end;

theorem :: JGRAPH_1:24

for f being FinSequence of (TOP-REAL 2) st f is nodic & PairF f is Simple & f . 1 <> f . (len f) holds
f is unfolded

proof end;

theorem Th25: :: JGRAPH_1:25

for f, g being FinSequence of (TOP-REAL 2)
for i being Element of NAT st g is_Shortcut_of f & 1 <= i & i + 1 <= len g holds
ex k1 being Element of NAT st
( 1 <= k1 & k1 + 1 <= len f & f /. k1 = g /. i & f /. (k1 + 1) = g /. (i + 1) & f . k1 = g . i & f . (k1 + 1) = g . (i + 1) )

proof end;

theorem Th26: :: JGRAPH_1:26

for f, g being FinSequence of (TOP-REAL 2) st g is_Shortcut_of f holds
rng g c= rng f

proof end;

theorem Th27: :: JGRAPH_1:27

for f, g being FinSequence of (TOP-REAL 2) st g is_Shortcut_of f holds
L~ g c= L~ f

proof end;

theorem Th28: :: JGRAPH_1:28

for f, g being FinSequence of (TOP-REAL 2) st f is special & g is_Shortcut_of f holds
g is special

proof end;

theorem Th29: :: JGRAPH_1:29

for f being FinSequence of (TOP-REAL 2) st f is special & 2 <= len f & f . 1 <> f . (len f) holds
ex g being FinSequence of (TOP-REAL 2) st
( 2 <= len g & g is special & g is one-to-one & L~ g c= L~ f & f . 1 = g . 1 & f . (len f) = g . (len g) & rng g c= rng f )

proof end;

theorem Th30: :: JGRAPH_1:30

for f1, f2 being FinSequence of (TOP-REAL 2) st f1 is special & f2 is special & 2 <= len f1 & 2 <= len f2 & f1 . 1 <> f1 . (len f1) & f2 . 1 <> f2 . (len f2) & X_axis f1 lies_between (X_axis f1) . 1,(X_axis f1) . (len f1) & X_axis f2 lies_between (X_axis f1) . 1,(X_axis f1) . (len f1) & Y_axis f1 lies_between (Y_axis f2) . 1,(Y_axis f2) . (len f2) & Y_axis f2 lies_between (Y_axis f2) . 1,(Y_axis f2) . (len f2) holds
L~ f1 meets L~ f2

proof end;

begin

theorem Th31: :: JGRAPH_1:31

for a, b, r1, r2 being Real st a <= r1 & r1 <= b & a <= r2 & r2 <= b holds
abs (r1 - r2) <= b - a

proof end;

theorem :: JGRAPH_1:32

canceled;

theorem :: JGRAPH_1:33

canceled;

theorem :: JGRAPH_1:34

canceled;

theorem :: JGRAPH_1:35

canceled;

theorem :: JGRAPH_1:36

canceled;

theorem :: JGRAPH_1:37

canceled;

theorem :: JGRAPH_1:38

canceled;

theorem :: JGRAPH_1:39

canceled;

theorem :: JGRAPH_1:40

canceled;

theorem :: JGRAPH_1:41

canceled;

theorem :: JGRAPH_1:42

canceled;

theorem :: JGRAPH_1:43

canceled;

theorem :: JGRAPH_1:44

canceled;

theorem Th45: :: JGRAPH_1:45

for n being Element of NAT
for p1, p2 being Point of (TOP-REAL n)
for x1, x2 being Point of (Euclid n) st x1 = p1 & x2 = p2 holds
|.(p1 - p2).| = dist (x1,x2)

proof end;

theorem Th46: :: JGRAPH_1:46

for p being Point of (TOP-REAL 2) holds |.p.| ^2 = ((p `1) ^2) + ((p `2) ^2)

proof end;

theorem Th47: :: JGRAPH_1:47

for p being Point of (TOP-REAL 2) holds |.p.| = sqrt (((p `1) ^2) + ((p `2) ^2))

proof end;

theorem Th48: :: JGRAPH_1:48

for p being Point of (TOP-REAL 2) holds |.p.| <= (abs (p `1)) + (abs (p `2))

proof end;

theorem Th49: :: JGRAPH_1:49

for p1, p2 being Point of (TOP-REAL 2) holds |.(p1 - p2).| <= (abs ((p1 `1) - (p2 `1))) + (abs ((p1 `2) - (p2 `2)))

proof end;

theorem Th50: :: JGRAPH_1:50

for p being Point of (TOP-REAL 2) holds
( abs (p `1) <= |.p.| & abs (p `2) <= |.p.| )

proof end;

theorem Th51: :: JGRAPH_1:51

for p1, p2 being Point of (TOP-REAL 2) holds
( abs ((p1 `1) - (p2 `1)) <= |.(p1 - p2).| & abs ((p1 `2) - (p2 `2)) <= |.(p1 - p2).| )

proof end;

theorem Th52: :: JGRAPH_1:52

for n being Element of NAT
for p, p1, p2 being Point of (TOP-REAL n) st p in LSeg (p1,p2) holds
ex r being Real st
( 0 <= r & r <= 1 & p = ((1 - r) * p1) + (r * p2) )

proof end;

theorem Th53: :: JGRAPH_1:53

for n being Element of NAT
for p, p1, p2 being Point of (TOP-REAL n) st p in LSeg (p1,p2) holds
( |.(p - p1).| <= |.(p1 - p2).| & |.(p - p2).| <= |.(p1 - p2).| )

proof end;

begin

theorem Th54: :: JGRAPH_1:54

for M being non empty MetrSpace
for P, Q being Subset of (TopSpaceMetr M) st P <> {} & P is compact & Q <> {} & Q is compact holds
min_dist_min (P,Q) >= 0

proof end;

theorem Th55: :: JGRAPH_1:55

for M being non empty MetrSpace
for P, Q being Subset of (TopSpaceMetr M) st P <> {} & P is compact & Q <> {} & Q is compact holds
( P misses Q iff min_dist_min (P,Q) > 0 )

proof end;

theorem Th56: :: JGRAPH_1:56

for f being FinSequence of (TOP-REAL 2)
for a, c, d being Real st 1 <= len f & X_axis f lies_between (X_axis f) . 1,(X_axis f) . (len f) & Y_axis f lies_between c,d & a > 0 & ( for i being Element of NAT st 1 <= i & i + 1 <= len f holds
|.((f /. i) - (f /. (i + 1))).| < a ) holds
ex g being FinSequence of (TOP-REAL 2) st
( g is special & g . 1 = f . 1 & g . (len g) = f . (len f) & len g >= len f & X_axis g lies_between (X_axis f) . 1,(X_axis f) . (len f) & Y_axis g lies_between c,d & ( for j being Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )

proof end;

theorem Th57: :: JGRAPH_1:57

for f being FinSequence of (TOP-REAL 2)
for a, c, d being Real st 1 <= len f & Y_axis f lies_between (Y_axis f) . 1,(Y_axis f) . (len f) & X_axis f lies_between c,d & a > 0 & ( for i being Element of NAT st 1 <= i & i + 1 <= len f holds
|.((f /. i) - (f /. (i + 1))).| < a ) holds
ex g being FinSequence of (TOP-REAL 2) st
( g is special & g . 1 = f . 1 & g . (len g) = f . (len f) & len g >= len f & Y_axis g lies_between (Y_axis f) . 1,(Y_axis f) . (len f) & X_axis g lies_between c,d & ( for j being Element of NAT st j in dom g holds
ex k being Element of NAT st
( k in dom f & |.((g /. j) - (f /. k)).| < a ) ) & ( for j being Element of NAT st 1 <= j & j + 1 <= len g holds
|.((g /. j) - (g /. (j + 1))).| < a ) )

proof end;

theorem :: JGRAPH_1:58

canceled;

theorem Th59: :: JGRAPH_1:59

for f being FinSequence of (TOP-REAL 2) st 1 <= len f holds
( len (X_axis f) = len f & (X_axis f) . 1 = (f /. 1) `1 & (X_axis f) . (len f) = (f /. (len f)) `1 )

proof end;

theorem Th60: :: JGRAPH_1:60

for f being FinSequence of (TOP-REAL 2) st 1 <= len f holds
( len (Y_axis f) = len f & (Y_axis f) . 1 = (f /. 1) `2 & (Y_axis f) . (len f) = (f /. (len f)) `2 )

proof end;

theorem Th61: :: JGRAPH_1:61

for i being Element of NAT
for f being FinSequence of (TOP-REAL 2) st i in dom f holds
( (X_axis f) . i = (f /. i) `1 & (Y_axis f) . i = (f /. i) `2 )

proof end;

theorem Th62: :: JGRAPH_1:62

for P, Q being non empty Subset of (TOP-REAL 2)
for p1, p2, q1, q2 being Point of (TOP-REAL 2) st P is_an_arc_of p1,p2 & Q is_an_arc_of q1,q2 & ( for p being Point of (TOP-REAL 2) st p in P holds
( p1 `1 <= p `1 & p `1 <= p2 `1 ) ) & ( for p being Point of (TOP-REAL 2) st p in Q holds
( p1 `1 <= p `1 & p `1 <= p2 `1 ) ) & ( for p being Point of (TOP-REAL 2) st p in P holds
( q1 `2 <= p `2 & p `2 <= q2 `2 ) ) & ( for p being Point of (TOP-REAL 2) st p in Q holds
( q1 `2 <= p `2 & p `2 <= q2 `2 ) ) holds
P meets Q

proof end;

theorem Th63: :: JGRAPH_1:63

for X, Y being non empty TopSpace
for f being Function of X,Y
for P being non empty Subset of Y
for f1 being Function of X,(Y | P) st f = f1 & f is continuous holds
f1 is continuous

proof end;

theorem Th64: :: JGRAPH_1:64

for X, Y being non empty TopSpace
for f being Function of X,Y
for P being non empty Subset of Y st X is compact & Y is T_2 & f is continuous & f is one-to-one & P = rng f holds
ex f1 being Function of X,(Y | P) st
( f = f1 & f1 is being_homeomorphism )

proof end;

theorem :: JGRAPH_1:65

for f, g being Function of I[01],(TOP-REAL 2)
for a, b, c, d being real number
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & (f . O) `1 = a & (f . I) `1 = b & (g . O) `2 = c & (g . I) `2 = d & ( for r being Point of I[01] holds
( a <= (f . r) `1 & (f . r) `1 <= b & a <= (g . r) `1 & (g . r) `1 <= b & c <= (f . r) `2 & (f . r) `2 <= d & c <= (g . r) `2 & (g . r) `2 <= d ) ) holds
rng f meets rng g

proof end;