:: On Pseudometric Spaces
:: by Adam Lecko and Mariusz Startek
::
:: Received September 28, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users

definition

let M be non empty MetrStruct ;
let x, y be Element of M;

pred x tolerates y means :: METRIC_2:def 1
dist (x,y) = 0 ;

end;

:: deftheorem defines tolerates METRIC_2:def 1 :
for M being non empty MetrStruct
for x, y being Element of M holds
( x tolerates y iff dist (x,y) = 0 );

definition

let M be non empty Reflexive MetrStruct ;
let x, y be Element of M;
:: original: tolerates
redefine pred x tolerates y;
reflexivity
for x being Element of M holds (M,b₁,b₁) by METRIC_1:1;

end;

definition

let M be non empty symmetric MetrStruct ;
let x, y be Element of M;
:: original: tolerates
redefine pred x tolerates y;
symmetry
for x, y being Element of M st (M,b₁,b₂) holds
(M,b₂,b₁)

proof end;

end;

definition

let M be non empty MetrStruct ;
let x be Element of M;

func x -neighbour -> Subset of M equals :: METRIC_2:def 2
{ y where y is Element of M : x tolerates y } ;

coherence
{ y where y is Element of M : x tolerates y } is Subset of M

proof end;

end;

:: deftheorem defines -neighbour METRIC_2:def 2 :
for M being non empty MetrStruct
for x being Element of M holds x -neighbour = { y where y is Element of M : x tolerates y } ;

definition

let M be non empty MetrStruct ;

mode equivalence_class of M -> Subset of M means :Def3: :: METRIC_2:def 3
ex x being Element of M st it = x -neighbour ;

existence
ex b₁ being Subset of M ex x being Element of M st b₁ = x -neighbour

proof end;

end;

:: deftheorem Def3 defines equivalence_class METRIC_2:def 3 :
for M being non empty MetrStruct
for b₂ being Subset of M holds
( b₂ is equivalence_class of M iff ex x being Element of M st b₂ = x -neighbour );

Lm1: for M being non empty Reflexive MetrStruct
for x being Element of M holds x tolerates x
;

theorem Th1: :: METRIC_2:1

for M being PseudoMetricSpace
for x, y, z being Element of M st x tolerates y & y tolerates z holds
x tolerates z

proof end;

theorem Th2: :: METRIC_2:2

for M being PseudoMetricSpace
for x, y being Element of M holds
( y in x -neighbour iff y tolerates x )

proof end;

theorem :: METRIC_2:3

for M being PseudoMetricSpace
for x, p, q being Element of M st p in x -neighbour & q in x -neighbour holds
p tolerates q

proof end;

theorem Th4: :: METRIC_2:4

for M being PseudoMetricSpace
for x being Element of M holds x in x -neighbour

proof end;

theorem :: METRIC_2:5

for M being PseudoMetricSpace
for x, y being Element of M st x in y -neighbour holds
y in x -neighbour

proof end;

theorem :: METRIC_2:6

for M being PseudoMetricSpace
for p, x, y being Element of M st p in x -neighbour & x tolerates y holds
p in y -neighbour

proof end;

theorem Th7: :: METRIC_2:7

for M being PseudoMetricSpace
for x, y being Element of M st y in x -neighbour holds
x -neighbour = y -neighbour

proof end;

theorem Th8: :: METRIC_2:8

for M being PseudoMetricSpace
for x, y being Element of M holds
( x -neighbour = y -neighbour iff x tolerates y ) by Th2, Th4, Th7;

theorem :: METRIC_2:9

for M being PseudoMetricSpace
for x, y being Element of M holds
( x -neighbour meets y -neighbour iff x tolerates y )

proof end;

Lm2: for M being PseudoMetricSpace
for V being equivalence_class of M holds V is non empty set

proof end;

registration

let M be PseudoMetricSpace;

cluster -> non empty for equivalence_class of M;

coherence
for b₁ being equivalence_class of M holds not b₁ is empty by Lm2;

end;

theorem Th10: :: METRIC_2:10

for M being PseudoMetricSpace
for x, p, q being Element of M st p in x -neighbour & q in x -neighbour holds
dist (p,q) = 0

proof end;

theorem Th11: :: METRIC_2:11

for M being non empty Reflexive discerning MetrStruct
for x, y being Element of M holds
( x tolerates y iff x = y ) by METRIC_1:2;

theorem Th12: :: METRIC_2:12

for M being non empty MetrSpace
for x, y being Element of M holds
( y in x -neighbour iff y = x )

proof end;

theorem Th13: :: METRIC_2:13

for M being non empty MetrSpace
for x being Element of M holds x -neighbour = {x}

proof end;

theorem :: METRIC_2:14

for M being non empty MetrSpace
for V being Subset of M holds
( V is equivalence_class of M iff ex x being Element of M st V = {x} )

proof end;

:: Set of the equivalence classes

definition

let M be non empty MetrStruct ;

func M -neighbour -> set equals :: METRIC_2:def 4
{ s where s is Subset of M : ex x being Element of M st x -neighbour = s } ;

coherence
{ s where s is Subset of M : ex x being Element of M st x -neighbour = s } is set ;

end;

:: deftheorem defines -neighbour METRIC_2:def 4 :
for M being non empty MetrStruct holds M -neighbour = { s where s is Subset of M : ex x being Element of M st x -neighbour = s } ;

registration

let M be non empty MetrStruct ;

cluster M -neighbour -> non empty ;

coherence
not M -neighbour is empty

proof end;

end;

theorem Th15: :: METRIC_2:15

for V being set
for M being non empty MetrStruct holds
( V in M -neighbour iff ex x being Element of M st V = x -neighbour )

proof end;

theorem :: METRIC_2:16

for M being non empty MetrStruct
for x being Element of M holds x -neighbour in M -neighbour ;

theorem Th17: :: METRIC_2:17

for V being set
for M being non empty MetrStruct holds
( V in M -neighbour iff V is equivalence_class of M )

proof end;

theorem :: METRIC_2:18

for M being non empty MetrSpace
for x being Element of M holds {x} in M -neighbour

proof end;

theorem :: METRIC_2:19

for V being set
for M being non empty MetrSpace holds
( V in M -neighbour iff ex x being Element of M st V = {x} )

proof end;

theorem Th20: :: METRIC_2:20

for M being PseudoMetricSpace
for V, Q being Element of M -neighbour
for p1, p2, q1, q2 being Element of M st p1 in V & q1 in Q & p2 in V & q2 in Q holds
dist (p1,q1) = dist (p2,q2)

proof end;

definition

let M be non empty MetrStruct ;
let V, Q be Element of M -neighbour ;
let v be Real;

pred V,Q is_dst v means :: METRIC_2:def 5
for p, q being Element of M st p in V & q in Q holds
dist (p,q) = v;

end;

:: deftheorem defines is_dst METRIC_2:def 5 :
for M being non empty MetrStruct
for V, Q being Element of M -neighbour
for v being Real holds
( V,Q is_dst v iff for p, q being Element of M st p in V & q in Q holds
dist (p,q) = v );

theorem Th21: :: METRIC_2:21

for M being PseudoMetricSpace
for V, Q being Element of M -neighbour
for v being Real holds
( V,Q is_dst v iff ex p, q being Element of M st
( p in V & q in Q & dist (p,q) = v ) )

proof end;

theorem Th22: :: METRIC_2:22

for M being PseudoMetricSpace
for V, Q being Element of M -neighbour
for v being Element of REAL st V,Q is_dst v holds
Q,V is_dst v

proof end;

definition

let M be non empty MetrStruct ;
let V, Q be Element of M -neighbour ;

func ev_eq_1 (V,Q) -> Subset of REAL equals :: METRIC_2:def 6
{ v where v is Element of REAL : V,Q is_dst v } ;

coherence
{ v where v is Element of REAL : V,Q is_dst v } is Subset of REAL

proof end;

end;

:: deftheorem defines ev_eq_1 METRIC_2:def 6 :
for M being non empty MetrStruct
for V, Q being Element of M -neighbour holds ev_eq_1 (V,Q) = { v where v is Element of REAL : V,Q is_dst v } ;

theorem :: METRIC_2:23

for M being PseudoMetricSpace
for V, Q being Element of M -neighbour
for v being Element of REAL holds
( v in ev_eq_1 (V,Q) iff V,Q is_dst v )

proof end;

definition

let M be non empty MetrStruct ;
let v be Element of REAL ;

func ev_eq_2 (v,M) -> Subset of [:(M -neighbour),(M -neighbour):] equals :: METRIC_2:def 7
{ W where W is Element of [:(M -neighbour),(M -neighbour):] : ex V, Q being Element of M -neighbour st
( W = [V,Q] & V,Q is_dst v ) } ;

coherence
{ W where W is Element of [:(M -neighbour),(M -neighbour):] : ex V, Q being Element of M -neighbour st
( W = [V,Q] & V,Q is_dst v ) } is Subset of [:(M -neighbour),(M -neighbour):]

proof end;

end;

:: deftheorem defines ev_eq_2 METRIC_2:def 7 :
for M being non empty MetrStruct
for v being Element of REAL holds ev_eq_2 (v,M) = { W where W is Element of [:(M -neighbour),(M -neighbour):] : ex V, Q being Element of M -neighbour st
( W = [V,Q] & V,Q is_dst v ) } ;

theorem :: METRIC_2:24

for M being PseudoMetricSpace
for v being Element of REAL
for W being Element of [:(M -neighbour),(M -neighbour):] holds
( W in ev_eq_2 (v,M) iff ex V, Q being Element of M -neighbour st
( W = [V,Q] & V,Q is_dst v ) )

proof end;

definition

let M be non empty MetrStruct ;

func real_in_rel M -> Subset of REAL equals :: METRIC_2:def 8
{ v where v is Element of REAL : ex V, Q being Element of M -neighbour st V,Q is_dst v } ;

coherence
{ v where v is Element of REAL : ex V, Q being Element of M -neighbour st V,Q is_dst v } is Subset of REAL

proof end;

end;

:: deftheorem defines real_in_rel METRIC_2:def 8 :
for M being non empty MetrStruct holds real_in_rel M = { v where v is Element of REAL : ex V, Q being Element of M -neighbour st V,Q is_dst v } ;

theorem :: METRIC_2:25

for M being PseudoMetricSpace
for v being Element of REAL holds
( v in real_in_rel M iff ex V, Q being Element of M -neighbour st V,Q is_dst v )

proof end;

definition

let M be non empty MetrStruct ;

func elem_in_rel_1 M -> Subset of (M -neighbour) equals :: METRIC_2:def 9
{ V where V is Element of M -neighbour : ex Q being Element of M -neighbour ex v being Element of REAL st V,Q is_dst v } ;

coherence
{ V where V is Element of M -neighbour : ex Q being Element of M -neighbour ex v being Element of REAL st V,Q is_dst v } is Subset of (M -neighbour)

proof end;

end;

:: deftheorem defines elem_in_rel_1 METRIC_2:def 9 :
for M being non empty MetrStruct holds elem_in_rel_1 M = { V where V is Element of M -neighbour : ex Q being Element of M -neighbour ex v being Element of REAL st V,Q is_dst v } ;

theorem Th26: :: METRIC_2:26

for M being PseudoMetricSpace
for V being Element of M -neighbour holds
( V in elem_in_rel_1 M iff ex Q being Element of M -neighbour ex v being Element of REAL st V,Q is_dst v )

proof end;

definition

let M be non empty MetrStruct ;

func elem_in_rel_2 M -> Subset of (M -neighbour) equals :: METRIC_2:def 10
{ Q where Q is Element of M -neighbour : ex V being Element of M -neighbour ex v being Element of REAL st V,Q is_dst v } ;

coherence
{ Q where Q is Element of M -neighbour : ex V being Element of M -neighbour ex v being Element of REAL st V,Q is_dst v } is Subset of (M -neighbour)

proof end;

end;

:: deftheorem defines elem_in_rel_2 METRIC_2:def 10 :
for M being non empty MetrStruct holds elem_in_rel_2 M = { Q where Q is Element of M -neighbour : ex V being Element of M -neighbour ex v being Element of REAL st V,Q is_dst v } ;

theorem Th27: :: METRIC_2:27

for M being PseudoMetricSpace
for Q being Element of M -neighbour holds
( Q in elem_in_rel_2 M iff ex V being Element of M -neighbour ex v being Element of REAL st V,Q is_dst v )

proof end;

definition

let M be non empty MetrStruct ;

func elem_in_rel M -> Subset of [:(M -neighbour),(M -neighbour):] equals :: METRIC_2:def 11
{ VQ where VQ is Element of [:(M -neighbour),(M -neighbour):] : ex V, Q being Element of M -neighbour ex v being Element of REAL st
( VQ = [V,Q] & V,Q is_dst v ) } ;

coherence
{ VQ where VQ is Element of [:(M -neighbour),(M -neighbour):] : ex V, Q being Element of M -neighbour ex v being Element of REAL st
( VQ = [V,Q] & V,Q is_dst v ) } is Subset of [:(M -neighbour),(M -neighbour):]

proof end;

end;

:: deftheorem defines elem_in_rel METRIC_2:def 11 :
for M being non empty MetrStruct holds elem_in_rel M = { VQ where VQ is Element of [:(M -neighbour),(M -neighbour):] : ex V, Q being Element of M -neighbour ex v being Element of REAL st
( VQ = [V,Q] & V,Q is_dst v ) } ;

theorem :: METRIC_2:28

for M being PseudoMetricSpace
for VQ being Element of [:(M -neighbour),(M -neighbour):] holds
( VQ in elem_in_rel M iff ex V, Q being Element of M -neighbour ex v being Element of REAL st
( VQ = [V,Q] & V,Q is_dst v ) )

proof end;

definition

let M be non empty MetrStruct ;

func set_in_rel M -> Subset of [:(M -neighbour),(M -neighbour),REAL:] equals :: METRIC_2:def 12
{ VQv where VQv is Element of [:(M -neighbour),(M -neighbour),REAL:] : ex V, Q being Element of M -neighbour ex v being Element of REAL st
( VQv = [V,Q,v] & V,Q is_dst v ) } ;

coherence
{ VQv where VQv is Element of [:(M -neighbour),(M -neighbour),REAL:] : ex V, Q being Element of M -neighbour ex v being Element of REAL st
( VQv = [V,Q,v] & V,Q is_dst v ) } is Subset of [:(M -neighbour),(M -neighbour),REAL:]

proof end;

end;

:: deftheorem defines set_in_rel METRIC_2:def 12 :
for M being non empty MetrStruct holds set_in_rel M = { VQv where VQv is Element of [:(M -neighbour),(M -neighbour),REAL:] : ex V, Q being Element of M -neighbour ex v being Element of REAL st
( VQv = [V,Q,v] & V,Q is_dst v ) } ;

theorem Th29: :: METRIC_2:29

for M being PseudoMetricSpace
for VQv being Element of [:(M -neighbour),(M -neighbour),REAL:] holds
( VQv in set_in_rel M iff ex V, Q being Element of M -neighbour ex v being Element of REAL st
( VQv = [V,Q,v] & V,Q is_dst v ) )

proof end;

theorem :: METRIC_2:30

for M being PseudoMetricSpace holds elem_in_rel_1 M = elem_in_rel_2 M

proof end;

theorem :: METRIC_2:31

for M being PseudoMetricSpace holds set_in_rel M c= [:(elem_in_rel_1 M),(elem_in_rel_2 M),(real_in_rel M):]

proof end;

theorem :: METRIC_2:32

for M being PseudoMetricSpace
for V, Q being Element of M -neighbour
for v1, v2 being Element of REAL st V,Q is_dst v1 & V,Q is_dst v2 holds
v1 = v2

proof end;

theorem Th33: :: METRIC_2:33

for M being PseudoMetricSpace
for V, Q being Element of M -neighbour ex v being Real st V,Q is_dst v

proof end;

definition

let M be PseudoMetricSpace;

func nbourdist M -> Function of [:(M -neighbour),(M -neighbour):],REAL means :Def13: :: METRIC_2:def 13
for V, Q being Element of M -neighbour
for p, q being Element of M st p in V & q in Q holds
it . (V,Q) = dist (p,q);

existence
ex b₁ being Function of [:(M -neighbour),(M -neighbour):],REAL st
for V, Q being Element of M -neighbour
for p, q being Element of M st p in V & q in Q holds
b₁ . (V,Q) = dist (p,q)

proof end;

uniqueness
for b₁, b₂ being Function of [:(M -neighbour),(M -neighbour):],REAL st ( for V, Q being Element of M -neighbour
for p, q being Element of M st p in V & q in Q holds
b₁ . (V,Q) = dist (p,q) ) & ( for V, Q being Element of M -neighbour
for p, q being Element of M st p in V & q in Q holds
b₂ . (V,Q) = dist (p,q) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def13 defines nbourdist METRIC_2:def 13 :
for M being PseudoMetricSpace
for b₂ being Function of [:(M -neighbour),(M -neighbour):],REAL holds
( b₂ = nbourdist M iff for V, Q being Element of M -neighbour
for p, q being Element of M st p in V & q in Q holds
b₂ . (V,Q) = dist (p,q) );

theorem Th34: :: METRIC_2:34

for M being PseudoMetricSpace
for V, Q being Element of M -neighbour holds
( (nbourdist M) . (V,Q) = 0 iff V = Q )

proof end;

theorem Th35: :: METRIC_2:35

for M being PseudoMetricSpace
for V, Q being Element of M -neighbour holds (nbourdist M) . (V,Q) = (nbourdist M) . (Q,V)

proof end;

theorem Th36: :: METRIC_2:36

for M being PseudoMetricSpace
for V, Q, W being Element of M -neighbour holds (nbourdist M) . (V,W) <= ((nbourdist M) . (V,Q)) + ((nbourdist M) . (Q,W))

proof end;

definition

let M be PseudoMetricSpace;

func Eq_classMetricSpace M -> MetrSpace equals :: METRIC_2:def 14
MetrStruct(# (M -neighbour),(nbourdist M) #);

coherence
MetrStruct(# (M -neighbour),(nbourdist M) #) is MetrSpace

proof end;

end;

:: deftheorem defines Eq_classMetricSpace METRIC_2:def 14 :
for M being PseudoMetricSpace holds Eq_classMetricSpace M = MetrStruct(# (M -neighbour),(nbourdist M) #);

registration

let M be PseudoMetricSpace;

cluster Eq_classMetricSpace M -> non empty strict ;

coherence
( Eq_classMetricSpace M is strict & not Eq_classMetricSpace M is empty ) ;

end;