:: On the {H}ausdorff Distance Between Compact Subsets
:: by Adam Grabowski
::
:: Received January 27, 2003
:: Copyright (c) 2003 Association of Mizar Users

begin

registration

let M be non empty MetrSpace;
cluster TopSpaceMetr M -> T_2 ;
coherence
TopSpaceMetr M is T_2 by PCOMPS_1:38;

end;

theorem Th1: :: HAUSDORF:1

for x, y being real number st x >= 0 & max x,y = 0 holds
x = 0

proof end;

theorem Th2: :: HAUSDORF:2

for M being non empty MetrSpace
for x being Point of M holds (dist x) . x = 0

proof end;

theorem Th3: :: HAUSDORF:3

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M)
for x being Point of M st x in P holds
0 in (dist x) .: P

proof end;

theorem Th4: :: HAUSDORF:4

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M)
for x being Point of M
for y being real number st y in (dist x) .: P holds
y >= 0

proof end;

theorem Th5: :: HAUSDORF:5

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M)
for x being set st x in P holds
(dist_min P) . x = 0

proof end;

theorem :: HAUSDORF:6

canceled;

theorem :: HAUSDORF:7

canceled;

theorem Th8: :: HAUSDORF:8

for M being non empty MetrSpace
for p being Point of M
for A being Subset of (TopSpaceMetr M) holds
( p in Cl A iff for r being real number st r > 0 holds
ex q being Point of M st
( q in A & dist p,q < r ) )

proof end;

theorem Th9: :: HAUSDORF:9

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x being Point of M holds
( (dist_min P) . x = 0 iff for r being real number st r > 0 holds
ex p being Point of M st
( p in P & dist x,p < r ) )

proof end;

theorem Th10: :: HAUSDORF:10

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x being Point of M holds
( x in Cl P iff (dist_min P) . x = 0 )

proof end;

theorem Th11: :: HAUSDORF:11

for M being non empty MetrSpace
for P being non empty closed Subset of (TopSpaceMetr M)
for x being Point of M holds
( x in P iff (dist_min P) . x = 0 )

proof end;

theorem Th12: :: HAUSDORF:12

for A being non empty Subset of R^1 ex X being non empty Subset of REAL st
( A = X & lower_bound A = inf X )

proof end;

theorem Th13: :: HAUSDORF:13

for A being non empty Subset of R^1 ex X being non empty Subset of REAL st
( A = X & upper_bound A = sup X )

proof end;

theorem Th14: :: HAUSDORF:14

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x being Point of M
for X being Subset of REAL st X = (dist x) .: P holds
X is bounded_below

proof end;

theorem Th15: :: HAUSDORF:15

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x, y being Point of M st y in P holds
(dist_min P) . x <= dist x,y

proof end;

theorem Th16: :: HAUSDORF:16

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for r being real number
for x being Point of M st ( for y being Point of M st y in P holds
dist x,y >= r ) holds
(dist_min P) . x >= r

proof end;

theorem Th17: :: HAUSDORF:17

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x, y being Point of M holds (dist_min P) . x <= (dist x,y) + ((dist_min P) . y)

proof end;

theorem Th18: :: HAUSDORF:18

for M being non empty MetrSpace
for P being Subset of (TopSpaceMetr M)
for Q being non empty Subset of M st P = Q holds
(TopSpaceMetr M) | P = TopSpaceMetr (M | Q)

proof end;

theorem Th19: :: HAUSDORF:19

for M being non empty MetrSpace
for A being Subset of M
for B being non empty Subset of M
for C being Subset of (M | B) st A = C & C is bounded holds
A is bounded

proof end;

theorem :: HAUSDORF:20

for M being non empty MetrSpace
for B being Subset of M
for A being Subset of (TopSpaceMetr M) st A = B & A is compact holds
B is bounded

proof end;

theorem Th21: :: HAUSDORF:21

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for z being Point of M ex w being Point of M st
( w in P & (dist_min P) . z <= dist w,z )

proof end;

registration

let M be non empty MetrSpace;
let x be Point of M;
cluster dist x -> continuous ;
coherence
dist x is continuous by WEIERSTR:22;

end;

registration

let M be non empty MetrSpace;
let X be non empty compact Subset of (TopSpaceMetr M);
cluster dist_max X -> continuous ;
coherence
dist_max X is continuous by WEIERSTR:30;
cluster dist_min X -> continuous ;
coherence
dist_min X is continuous by WEIERSTR:33;

end;

Lm1: for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x being Point of M
for X being Subset of REAL st X = (dist x) .: P & P is compact holds
X is bounded_above

proof end;

theorem Th22: :: HAUSDORF:22

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for x, y being Point of M st y in P & P is compact holds
(dist_max P) . x >= dist x,y

proof end;

theorem :: HAUSDORF:23

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for z being Point of M st P is compact holds
ex w being Point of M st
( w in P & (dist_max P) . z >= dist w,z )

proof end;

theorem Th24: :: HAUSDORF:24

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M)
for z being Point of M st P is compact & Q is compact & z in Q holds
(dist_min P) . z <= max_dist_max P,Q

proof end;

theorem Th25: :: HAUSDORF:25

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M)
for z being Point of M st P is compact & Q is compact & z in Q holds
(dist_max P) . z <= max_dist_max P,Q

proof end;

theorem :: HAUSDORF:26

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M)
for X being Subset of REAL st X = (dist_max P) .: Q & P is compact & Q is compact holds
X is bounded_above

proof end;

theorem Th27: :: HAUSDORF:27

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M)
for X being Subset of REAL st X = (dist_min P) .: Q & P is compact & Q is compact holds
X is bounded_above

proof end;

theorem :: HAUSDORF:28

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M)
for z being Point of M st P is compact holds
(dist_min P) . z <= (dist_max P) . z

proof end;

theorem Th29: :: HAUSDORF:29

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M) holds (dist_min P) .: P = {0 }

proof end;

theorem Th30: :: HAUSDORF:30

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

proof end;

theorem Th31: :: HAUSDORF:31

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M) holds max_dist_min P,P = 0

proof end;

theorem :: HAUSDORF:32

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

proof end;

theorem Th33: :: HAUSDORF:33

for M being non empty MetrSpace
for Q, R being non empty Subset of (TopSpaceMetr M)
for y being Point of M st Q is compact & R is compact & y in Q holds
(dist_min R) . y <= max_dist_min R,Q

proof end;

begin

definition

let M be non empty MetrSpace;
let P, Q be Subset of (TopSpaceMetr M);
func HausDist P,Q -> Real equals :: HAUSDORF:def 1
max (max_dist_min P,Q),(max_dist_min Q,P);
coherence
max (max_dist_min P,Q),(max_dist_min Q,P) is Real ;
commutativity
for b₁ being Real
for P, Q being Subset of (TopSpaceMetr M) st b₁ = max (max_dist_min P,Q),(max_dist_min Q,P) holds
b₁ = max (max_dist_min Q,P),(max_dist_min P,Q) ;

end;

:: deftheorem defines HausDist HAUSDORF:def 1 :

for M being non empty MetrSpace
for P, Q being Subset of (TopSpaceMetr M) holds HausDist P,Q = max (max_dist_min P,Q),(max_dist_min Q,P);

theorem Th34: :: HAUSDORF:34

for M being non empty MetrSpace
for Q, R being non empty Subset of (TopSpaceMetr M)
for y being Point of M st Q is compact & R is compact & y in Q holds
(dist_min R) . y <= HausDist Q,R

proof end;

theorem Th35: :: HAUSDORF:35

for M being non empty MetrSpace
for P, Q, R being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact & R is compact holds
max_dist_min P,R <= (HausDist P,Q) + (HausDist Q,R)

proof end;

theorem Th36: :: HAUSDORF:36

for M being non empty MetrSpace
for P, Q, R being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact & R is compact holds
max_dist_min R,P <= (HausDist P,Q) + (HausDist Q,R)

proof end;

theorem Th37: :: HAUSDORF:37

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

proof end;

theorem :: HAUSDORF:38

for M being non empty MetrSpace
for P being non empty Subset of (TopSpaceMetr M) holds HausDist P,P = 0 by Th31;

theorem Th39: :: HAUSDORF:39

for M being non empty MetrSpace
for P, Q being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact & HausDist P,Q = 0 holds
P = Q

proof end;

theorem Th40: :: HAUSDORF:40

for M being non empty MetrSpace
for P, Q, R being non empty Subset of (TopSpaceMetr M) st P is compact & Q is compact & R is compact holds
HausDist P,R <= (HausDist P,Q) + (HausDist Q,R)

proof end;

definition

let n be Element of NAT ;
let P, Q be Subset of (TOP-REAL n);
func max_dist_min P,Q -> Real means :: HAUSDORF:def 2
ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & it = max_dist_min P',Q' );
existence
ex b₁ being Real ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₁ = max_dist_min P',Q' )

proof end;

uniqueness
for b₁, b₂ being Real st ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₁ = max_dist_min P',Q' ) & ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₂ = max_dist_min P',Q' ) holds
b₁ = b₂ ;

end;

:: deftheorem defines max_dist_min HAUSDORF:def 2 :

for n being Element of NAT
for P, Q being Subset of (TOP-REAL n)
for b₄ being Real holds
( b₄ = max_dist_min P,Q iff ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₄ = max_dist_min P',Q' ) );

definition

let n be Element of NAT ;
let P, Q be Subset of (TOP-REAL n);
func HausDist P,Q -> Real means :Def3: :: HAUSDORF:def 3
ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & it = HausDist P',Q' );
existence
ex b₁ being Real ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₁ = HausDist P',Q' )

proof end;

uniqueness
for b₁, b₂ being Real st ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₁ = HausDist P',Q' ) & ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₂ = HausDist P',Q' ) holds
b₁ = b₂ ;
commutativity
for b₁ being Real
for P, Q being Subset of (TOP-REAL n) st ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₁ = HausDist P',Q' ) holds
ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( Q = P' & P = Q' & b₁ = HausDist P',Q' ) ;

end;

:: deftheorem Def3 defines HausDist HAUSDORF:def 3 :

for n being Element of NAT
for P, Q being Subset of (TOP-REAL n)
for b₄ being Real holds
( b₄ = HausDist P,Q iff ex P', Q' being Subset of (TopSpaceMetr (Euclid n)) st
( P = P' & Q = Q' & b₄ = HausDist P',Q' ) );

theorem :: HAUSDORF:41

for n being Element of NAT
for P, Q being non empty Subset of (TOP-REAL n) st P is compact & Q is compact holds
HausDist P,Q >= 0

proof end;

theorem :: HAUSDORF:42

for n being Element of NAT
for P being non empty Subset of (TOP-REAL n) holds HausDist P,P = 0

proof end;

theorem :: HAUSDORF:43

for n being Element of NAT
for P, Q being non empty Subset of (TOP-REAL n) st P is compact & Q is compact & HausDist P,Q = 0 holds
P = Q

proof end;

theorem :: HAUSDORF:44

for n being Element of NAT
for P, Q, R being non empty Subset of (TOP-REAL n) st P is compact & Q is compact & R is compact holds
HausDist P,R <= (HausDist P,Q) + (HausDist Q,R)

proof end;