:: On the Segmentation of a Simple Closed Curve
:: by Andrzej Trybulec
::
:: Received August 18, 2003
:: Copyright (c) 2003-2021 Association of Mizar Users

theorem Th1: :: JORDAN_A:1

for T being non empty TopSpace
for f being continuous RealMap of T
for A being compact Subset of T holds f .: A is compact

proof end;

theorem Th2: :: JORDAN_A:2

for A being compact Subset of REAL
for B being non empty Subset of REAL st B c= A holds
lower_bound B in A

proof end;

theorem :: JORDAN_A:3

for n being Nat
for A, B being non empty compact Subset of (TOP-REAL n)
for f being continuous RealMap of [:(TOP-REAL n),(TOP-REAL n):]
for g being RealMap of (TOP-REAL n) st ( for p being Point of (TOP-REAL n) ex G being Subset of REAL st
( G = { (f . (p,q)) where q is Point of (TOP-REAL n) : q in B } & g . p = lower_bound G ) ) holds
lower_bound (f .: [:A,B:]) = lower_bound (g .: A)

proof end;

theorem :: JORDAN_A:4

for n being Nat
for A, B being non empty compact Subset of (TOP-REAL n)
for f being continuous RealMap of [:(TOP-REAL n),(TOP-REAL n):]
for g being RealMap of (TOP-REAL n) st ( for q being Point of (TOP-REAL n) ex G being Subset of REAL st
( G = { (f . (p,q)) where p is Point of (TOP-REAL n) : p in A } & g . q = lower_bound G ) ) holds
lower_bound (f .: [:A,B:]) = lower_bound (g .: B)

proof end;

theorem Th5: :: JORDAN_A:5

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st q in Lower_Arc C & q <> W-min C holds
LE E-max C,q,C

proof end;

theorem Th6: :: JORDAN_A:6

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st q in Upper_Arc C holds
LE q, E-max C,C

proof end;

definition

let n be Nat;
A1: [: the carrier of (TOP-REAL n), the carrier of (TOP-REAL n):] = the carrier of [:(TOP-REAL n),(TOP-REAL n):] by BORSUK_1:def 2;

func Eucl_dist n -> RealMap of [:(TOP-REAL n),(TOP-REAL n):] means :Def1: :: JORDAN_A:def 1
for p, q being Point of (TOP-REAL n) holds it . (p,q) = |.(p - q).|;

existence
ex b₁ being RealMap of [:(TOP-REAL n),(TOP-REAL n):] st
for p, q being Point of (TOP-REAL n) holds b₁ . (p,q) = |.(p - q).|

proof end;

uniqueness
for b₁, b₂ being RealMap of [:(TOP-REAL n),(TOP-REAL n):] st ( for p, q being Point of (TOP-REAL n) holds b₁ . (p,q) = |.(p - q).| ) & ( for p, q being Point of (TOP-REAL n) holds b₂ . (p,q) = |.(p - q).| ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines Eucl_dist JORDAN_A:def 1 :
for n being Nat
for b₂ being RealMap of [:(TOP-REAL n),(TOP-REAL n):] holds
( b₂ = Eucl_dist n iff for p, q being Point of (TOP-REAL n) holds b₂ . (p,q) = |.(p - q).| );

definition

let T be non empty TopSpace;
let f be RealMap of T;

redefine attr f is continuous means :: JORDAN_A:def 2
for p being Point of T
for N being Neighbourhood of f . p ex V being a_neighborhood of p st f .: V c= N;

compatibility
( f is continuous iff for p being Point of T
for N being Neighbourhood of f . p ex V being a_neighborhood of p st f .: V c= N )

proof end;

end;

:: deftheorem defines continuous JORDAN_A:def 2 :
for T being non empty TopSpace
for f being RealMap of T holds
( f is continuous iff for p being Point of T
for N being Neighbourhood of f . p ex V being a_neighborhood of p st f .: V c= N );

Lm1: for X, Y being TopSpace holds TopStruct(# the carrier of [:X,Y:], the topology of [:X,Y:] #) = [:TopStruct(# the carrier of X, the topology of X #),TopStruct(# the carrier of Y, the topology of Y #):]

proof end;

registration

let n be Nat;

cluster Eucl_dist n -> continuous ;

coherence
Eucl_dist n is continuous

proof end;

end;

theorem Th7: :: JORDAN_A:7

for n being Nat
for A, B being non empty compact Subset of (TOP-REAL n) st A misses B holds
dist_min (A,B) > 0

proof end;

theorem Th8: :: JORDAN_A:8

for C being Simple_closed_curve
for p, q being Point of (TOP-REAL 2) st LE p,q,C & LE q, E-max C,C & p <> q holds
Segment (p,q,C) = Segment ((Upper_Arc C),(W-min C),(E-max C),p,q)

proof end;

theorem Th9: :: JORDAN_A:9

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st LE E-max C,q,C holds
Segment ((E-max C),q,C) = Segment ((Lower_Arc C),(E-max C),(W-min C),(E-max C),q)

proof end;

theorem Th10: :: JORDAN_A:10

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st LE E-max C,q,C holds
Segment (q,(W-min C),C) = Segment ((Lower_Arc C),(E-max C),(W-min C),q,(W-min C))

proof end;

theorem Th11: :: JORDAN_A:11

for C being Simple_closed_curve
for p, q being Point of (TOP-REAL 2) st LE p,q,C & LE E-max C,p,C holds
Segment (p,q,C) = Segment ((Lower_Arc C),(E-max C),(W-min C),p,q)

proof end;

theorem Th12: :: JORDAN_A:12

for C being Simple_closed_curve
for p, q being Point of (TOP-REAL 2) st LE p, E-max C,C & LE E-max C,q,C holds
Segment (p,q,C) = (R_Segment ((Upper_Arc C),(W-min C),(E-max C),p)) \/ (L_Segment ((Lower_Arc C),(E-max C),(W-min C),q))

proof end;

theorem Th13: :: JORDAN_A:13

for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) st LE p, E-max C,C holds
Segment (p,(W-min C),C) = (R_Segment ((Upper_Arc C),(W-min C),(E-max C),p)) \/ (L_Segment ((Lower_Arc C),(E-max C),(W-min C),(W-min C)))

proof end;

theorem Th14: :: JORDAN_A:14

for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) holds R_Segment ((Upper_Arc C),(W-min C),(E-max C),p) = Segment ((Upper_Arc C),(W-min C),(E-max C),p,(E-max C))

proof end;

theorem Th15: :: JORDAN_A:15

for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) holds L_Segment ((Lower_Arc C),(E-max C),(W-min C),p) = Segment ((Lower_Arc C),(E-max C),(W-min C),(E-max C),p)

proof end;

theorem Th16: :: JORDAN_A:16

for C being Simple_closed_curve
for p being Point of (TOP-REAL 2) st p in C & p <> W-min C holds
Segment (p,(W-min C),C) is_an_arc_of p, W-min C

proof end;

theorem Th17: :: JORDAN_A:17

for C being Simple_closed_curve
for p, q being Point of (TOP-REAL 2) st p <> q & LE p,q,C holds
Segment (p,q,C) is_an_arc_of p,q

proof end;

theorem Th18: :: JORDAN_A:18

for C being Simple_closed_curve holds C = Segment ((W-min C),(W-min C),C)

proof end;

theorem Th19: :: JORDAN_A:19

for C being Simple_closed_curve
for q being Point of (TOP-REAL 2) st q in C holds
Segment (q,(W-min C),C) is compact

proof end;

theorem Th20: :: JORDAN_A:20

for C being Simple_closed_curve
for q1, q2 being Point of (TOP-REAL 2) st LE q1,q2,C holds
Segment (q1,q2,C) is compact

proof end;

definition

let C be Simple_closed_curve;

mode Segmentation of C -> FinSequence of (TOP-REAL 2) means :Def3: :: JORDAN_A:def 3
( it /. 1 = W-min C & it is one-to-one & 8 <= len it & rng it c= C & ( for i being Nat st 1 <= i & i < len it holds
LE it /. i,it /. (i + 1),C ) & ( for i being Nat st 1 <= i & i + 1 < len it holds
(Segment ((it /. i),(it /. (i + 1)),C)) /\ (Segment ((it /. (i + 1)),(it /. (i + 2)),C)) = {(it /. (i + 1))} ) & (Segment ((it /. (len it)),(it /. 1),C)) /\ (Segment ((it /. 1),(it /. 2),C)) = {(it /. 1)} & (Segment ((it /. ((len it) -' 1)),(it /. (len it)),C)) /\ (Segment ((it /. (len it)),(it /. 1),C)) = {(it /. (len it))} & Segment ((it /. ((len it) -' 1)),(it /. (len it)),C) misses Segment ((it /. 1),(it /. 2),C) & ( for i, j being Nat st 1 <= i & i < j & j < len it & i,j aren't_adjacent holds
Segment ((it /. i),(it /. (i + 1)),C) misses Segment ((it /. j),(it /. (j + 1)),C) ) & ( for i being Nat st 1 < i & i + 1 < len it holds
Segment ((it /. (len it)),(it /. 1),C) misses Segment ((it /. i),(it /. (i + 1)),C) ) );

existence
ex b₁ being FinSequence of (TOP-REAL 2) st
( b₁ /. 1 = W-min C & b₁ is one-to-one & 8 <= len b₁ & rng b₁ c= C & ( for i being Nat st 1 <= i & i < len b₁ holds
LE b₁ /. i,b₁ /. (i + 1),C ) & ( for i being Nat st 1 <= i & i + 1 < len b₁ holds
(Segment ((b₁ /. i),(b₁ /. (i + 1)),C)) /\ (Segment ((b₁ /. (i + 1)),(b₁ /. (i + 2)),C)) = {(b₁ /. (i + 1))} ) & (Segment ((b₁ /. (len b₁)),(b₁ /. 1),C)) /\ (Segment ((b₁ /. 1),(b₁ /. 2),C)) = {(b₁ /. 1)} & (Segment ((b₁ /. ((len b₁) -' 1)),(b₁ /. (len b₁)),C)) /\ (Segment ((b₁ /. (len b₁)),(b₁ /. 1),C)) = {(b₁ /. (len b₁))} & Segment ((b₁ /. ((len b₁) -' 1)),(b₁ /. (len b₁)),C) misses Segment ((b₁ /. 1),(b₁ /. 2),C) & ( for i, j being Nat st 1 <= i & i < j & j < len b₁ & i,j aren't_adjacent holds
Segment ((b₁ /. i),(b₁ /. (i + 1)),C) misses Segment ((b₁ /. j),(b₁ /. (j + 1)),C) ) & ( for i being Nat st 1 < i & i + 1 < len b₁ holds
Segment ((b₁ /. (len b₁)),(b₁ /. 1),C) misses Segment ((b₁ /. i),(b₁ /. (i + 1)),C) ) )

proof end;

end;

:: deftheorem Def3 defines Segmentation JORDAN_A:def 3 :
for C being Simple_closed_curve
for b₂ being FinSequence of (TOP-REAL 2) holds
( b₂ is Segmentation of C iff ( b₂ /. 1 = W-min C & b₂ is one-to-one & 8 <= len b₂ & rng b₂ c= C & ( for i being Nat st 1 <= i & i < len b₂ holds
LE b₂ /. i,b₂ /. (i + 1),C ) & ( for i being Nat st 1 <= i & i + 1 < len b₂ holds
(Segment ((b₂ /. i),(b₂ /. (i + 1)),C)) /\ (Segment ((b₂ /. (i + 1)),(b₂ /. (i + 2)),C)) = {(b₂ /. (i + 1))} ) & (Segment ((b₂ /. (len b₂)),(b₂ /. 1),C)) /\ (Segment ((b₂ /. 1),(b₂ /. 2),C)) = {(b₂ /. 1)} & (Segment ((b₂ /. ((len b₂) -' 1)),(b₂ /. (len b₂)),C)) /\ (Segment ((b₂ /. (len b₂)),(b₂ /. 1),C)) = {(b₂ /. (len b₂))} & Segment ((b₂ /. ((len b₂) -' 1)),(b₂ /. (len b₂)),C) misses Segment ((b₂ /. 1),(b₂ /. 2),C) & ( for i, j being Nat st 1 <= i & i < j & j < len b₂ & i,j aren't_adjacent holds
Segment ((b₂ /. i),(b₂ /. (i + 1)),C) misses Segment ((b₂ /. j),(b₂ /. (j + 1)),C) ) & ( for i being Nat st 1 < i & i + 1 < len b₂ holds
Segment ((b₂ /. (len b₂)),(b₂ /. 1),C) misses Segment ((b₂ /. i),(b₂ /. (i + 1)),C) ) ) );

registration

let C be Simple_closed_curve;

cluster -> non trivial for Segmentation of C;

coherence
for b₁ being Segmentation of C holds not b₁ is trivial

proof end;

end;

theorem Th21: :: JORDAN_A:21

for C being Simple_closed_curve
for S being Segmentation of C
for i being Nat st 1 <= i & i <= len S holds
S /. i in C

proof end;

definition

let C be Simple_closed_curve;
let i be Nat;
let S be Segmentation of C;

func Segm (S,i) -> Subset of (TOP-REAL 2) equals :Def4: :: JORDAN_A:def 4
Segment ((S /. i),(S /. (i + 1)),C) if ( 1 <= i & i < len S )
otherwise Segment ((S /. (len S)),(S /. 1),C);

correctness
coherence
( ( 1 <= i & i < len S implies Segment ((S /. i),(S /. (i + 1)),C) is Subset of (TOP-REAL 2) ) & ( ( not 1 <= i or not i < len S ) implies Segment ((S /. (len S)),(S /. 1),C) is Subset of (TOP-REAL 2) ) );
consistency
for b₁ being Subset of (TOP-REAL 2) holds verum;
;

end;

:: deftheorem Def4 defines Segm JORDAN_A:def 4 :
for C being Simple_closed_curve
for i being Nat
for S being Segmentation of C holds
( ( 1 <= i & i < len S implies Segm (S,i) = Segment ((S /. i),(S /. (i + 1)),C) ) & ( ( not 1 <= i or not i < len S ) implies Segm (S,i) = Segment ((S /. (len S)),(S /. 1),C) ) );

theorem :: JORDAN_A:22

for C being Simple_closed_curve
for i being Nat
for S being Segmentation of C st i in dom S holds
Segm (S,i) c= C

proof end;

registration

let C be Simple_closed_curve;
let S be Segmentation of C;
let i be Nat;

cluster Segm (S,i) -> non empty compact ;

coherence
( not Segm (S,i) is empty & Segm (S,i) is compact )

proof end;

end;

theorem :: JORDAN_A:23

for C being Simple_closed_curve
for S being Segmentation of C
for p being Point of (TOP-REAL 2) st p in C holds
ex i being Nat st
( i in dom S & p in Segm (S,i) )

proof end;

theorem Th24: :: JORDAN_A:24

for C being Simple_closed_curve
for S being Segmentation of C
for i, j being Nat st 1 <= i & i < j & j < len S & i,j aren't_adjacent holds
Segm (S,i) misses Segm (S,j)

proof end;

theorem Th25: :: JORDAN_A:25

for C being Simple_closed_curve
for S being Segmentation of C
for j being Nat st 1 < j & j < (len S) -' 1 holds
Segm (S,(len S)) misses Segm (S,j)

proof end;

theorem Th26: :: JORDAN_A:26

for C being Simple_closed_curve
for S being Segmentation of C
for i, j being Nat st 1 <= i & i < j & j < len S & i,j are_adjacent holds
(Segm (S,i)) /\ (Segm (S,j)) = {(S /. (i + 1))}

proof end;

theorem Th27: :: JORDAN_A:27

for C being Simple_closed_curve
for S being Segmentation of C
for i, j being Nat st 1 <= i & i < j & j < len S & i,j are_adjacent holds
Segm (S,i) meets Segm (S,j)

proof end;

theorem Th28: :: JORDAN_A:28

for C being Simple_closed_curve
for S being Segmentation of C holds (Segm (S,(len S))) /\ (Segm (S,1)) = {(S /. 1)}

proof end;

theorem Th29: :: JORDAN_A:29

for C being Simple_closed_curve
for S being Segmentation of C holds Segm (S,(len S)) meets Segm (S,1)

proof end;

theorem Th30: :: JORDAN_A:30

for C being Simple_closed_curve
for S being Segmentation of C holds (Segm (S,(len S))) /\ (Segm (S,((len S) -' 1))) = {(S /. (len S))}

proof end;

theorem Th31: :: JORDAN_A:31

for C being Simple_closed_curve
for S being Segmentation of C holds Segm (S,(len S)) meets Segm (S,((len S) -' 1))

proof end;

definition

let n be Nat;
let C be Subset of (TOP-REAL n);

func diameter C -> Real means :Def5: :: JORDAN_A:def 5
ex W being Subset of (Euclid n) st
( W = C & it = diameter W );

existence
ex b₁ being Real ex W being Subset of (Euclid n) st
( W = C & b₁ = diameter W )

proof end;

correctness
uniqueness
for b₁, b₂ being Real st ex W being Subset of (Euclid n) st
( W = C & b₁ = diameter W ) & ex W being Subset of (Euclid n) st
( W = C & b₂ = diameter W ) holds
b₁ = b₂;
;

end;

:: deftheorem Def5 defines diameter JORDAN_A:def 5 :
for n being Nat
for C being Subset of (TOP-REAL n)
for b₃ being Real holds
( b₃ = diameter C iff ex W being Subset of (Euclid n) st
( W = C & b₃ = diameter W ) );

definition

let C be Simple_closed_curve;
let S be Segmentation of C;

func diameter S -> Real means :Def6: :: JORDAN_A:def 6
ex S1 being non empty finite Subset of REAL st
( S1 = { (diameter (Segm (S,i))) where i is Element of NAT : i in dom S } & it = max S1 );

existence
ex b₁ being Real ex S1 being non empty finite Subset of REAL st
( S1 = { (diameter (Segm (S,i))) where i is Element of NAT : i in dom S } & b₁ = max S1 )

proof end;

correctness
uniqueness
for b₁, b₂ being Real st ex S1 being non empty finite Subset of REAL st
( S1 = { (diameter (Segm (S,i))) where i is Element of NAT : i in dom S } & b₁ = max S1 ) & ex S1 being non empty finite Subset of REAL st
( S1 = { (diameter (Segm (S,i))) where i is Element of NAT : i in dom S } & b₂ = max S1 ) holds
b₁ = b₂;
;

end;

:: deftheorem Def6 defines diameter JORDAN_A:def 6 :
for C being Simple_closed_curve
for S being Segmentation of C
for b₃ being Real holds
( b₃ = diameter S iff ex S1 being non empty finite Subset of REAL st
( S1 = { (diameter (Segm (S,i))) where i is Element of NAT : i in dom S } & b₃ = max S1 ) );

theorem :: JORDAN_A:32

for C being Simple_closed_curve
for S being Segmentation of C
for i being Nat holds diameter (Segm (S,i)) <= diameter S

proof end;

theorem Th33: :: JORDAN_A:33

for C being Simple_closed_curve
for S being Segmentation of C
for e being Real st ( for i being Nat holds diameter (Segm (S,i)) < e ) holds
diameter S < e

proof end;

theorem :: JORDAN_A:34

for C being Simple_closed_curve
for e being Real st e > 0 holds
ex S being Segmentation of C st diameter S < e

proof end;

definition

let C be Simple_closed_curve;
let S be Segmentation of C;

func S-Gap S -> Real means :Def7: :: JORDAN_A:def 7
ex S1, S2 being non empty finite Subset of REAL st
( S1 = { (dist_min ((Segm (S,i)),(Segm (S,j)))) where i, j is Element of NAT : ( 1 <= i & i < j & j < len S & i,j aren't_adjacent ) } & S2 = { (dist_min ((Segm (S,(len S))),(Segm (S,k)))) where k is Element of NAT : ( 1 < k & k < (len S) -' 1 ) } & it = min ((min S1),(min S2)) );

existence
ex b₁ being Real ex S1, S2 being non empty finite Subset of REAL st
( S1 = { (dist_min ((Segm (S,i)),(Segm (S,j)))) where i, j is Element of NAT : ( 1 <= i & i < j & j < len S & i,j aren't_adjacent ) } & S2 = { (dist_min ((Segm (S,(len S))),(Segm (S,k)))) where k is Element of NAT : ( 1 < k & k < (len S) -' 1 ) } & b₁ = min ((min S1),(min S2)) )

proof end;

correctness
uniqueness
for b₁, b₂ being Real st ex S1, S2 being non empty finite Subset of REAL st
( S1 = { (dist_min ((Segm (S,i)),(Segm (S,j)))) where i, j is Element of NAT : ( 1 <= i & i < j & j < len S & i,j aren't_adjacent ) } & S2 = { (dist_min ((Segm (S,(len S))),(Segm (S,k)))) where k is Element of NAT : ( 1 < k & k < (len S) -' 1 ) } & b₁ = min ((min S1),(min S2)) ) & ex S1, S2 being non empty finite Subset of REAL st
( S1 = { (dist_min ((Segm (S,i)),(Segm (S,j)))) where i, j is Element of NAT : ( 1 <= i & i < j & j < len S & i,j aren't_adjacent ) } & S2 = { (dist_min ((Segm (S,(len S))),(Segm (S,k)))) where k is Element of NAT : ( 1 < k & k < (len S) -' 1 ) } & b₂ = min ((min S1),(min S2)) ) holds
b₁ = b₂;
;

end;

:: deftheorem Def7 defines S-Gap JORDAN_A:def 7 :
for C being Simple_closed_curve
for S being Segmentation of C
for b₃ being Real holds
( b₃ = S-Gap S iff ex S1, S2 being non empty finite Subset of REAL st
( S1 = { (dist_min ((Segm (S,i)),(Segm (S,j)))) where i, j is Element of NAT : ( 1 <= i & i < j & j < len S & i,j aren't_adjacent ) } & S2 = { (dist_min ((Segm (S,(len S))),(Segm (S,k)))) where k is Element of NAT : ( 1 < k & k < (len S) -' 1 ) } & b₃ = min ((min S1),(min S2)) ) );

theorem Th35: :: JORDAN_A:35

for C being Simple_closed_curve
for S being Segmentation of C ex F being non empty finite Subset of REAL st
( F = { (dist_min ((Segm (S,i)),(Segm (S,j)))) where i, j is Nat : ( 1 <= i & i < j & j <= len S & Segm (S,i) misses Segm (S,j) ) } & S-Gap S = min F )

proof end;

theorem :: JORDAN_A:36

for C being Simple_closed_curve
for S being Segmentation of C holds S-Gap S > 0

proof end;