:: Metric Spaces
:: by Stanis{\l}awa Kanas, Adam Lecko and Mariusz Startek
::
:: Received May 3, 1990
:: Copyright (c) 1990-2021 Association of Mizar Users


definition
attr c1 is strict ;
struct MetrStruct -> 1-sorted ;
aggr MetrStruct(# carrier, distance #) -> MetrStruct ;
sel distance c1 -> Function of [: the carrier of c1, the carrier of c1:],REAL;
end;

registration
cluster non empty strict for MetrStruct ;
existence
ex b1 being MetrStruct st
( not b1 is empty & b1 is strict )
proof end;
end;

:: registration
:: let A,B be set, f be PartFunc of [:A,B:],REAL;
:: let a be Element of A;
:: let b be Element of B;
:: cluster f.(a,b) -> real;
:: coherence
:: proof
:: per cases;
:: suppose
:: [a,b] in dom f;
:: hence thesis by PARTFUN1:4;
:: end;
:: suppose
:: not [a,b] in dom f;
:: then f.(a,b) = 0 by FUNCT_1:def 2;
:: hence thesis;
:: end;
:: end;
:: end;
definition
let M be MetrStruct ;
let a, b be Element of M;
func dist (a,b) -> Real equals :: METRIC_1:def 1
 the distance of M . (a,b);
coherence
the distance of M . (a,b) is Real ;
end;

:: deftheorem defines dist METRIC_1:def 1 :
for M being MetrStruct
for a, b being Element of M holds dist (a,b) = the distance of M . (a,b);

notation
synonym Empty^2-to-zero for op2 ;
end;

Lm1: 0 in REAL
by XREAL_0:def 1;

definition
:: original: Empty^2-to-zero
redefine func Empty^2-to-zero -> Function of [:1,1:],REAL;
coherence
Empty^2-to-zero is Function of [:1,1:],REAL
proof end;
end;

Lm2: op2 . (0,0) = 0
proof end;

Lm3: for x, y being Element of 1 holds
( op2 . (x,y) = 0 iff x = y )
proof end;

Lm4: for x, y being Element of 1 holds op2 . (x,y) = op2 . (y,x)
proof end;

registration
cluster Empty^2-to-zero -> natural-valued for Function;
coherence
for b1 being Function st b1 = op2 holds
b1 is natural-valued ;
end;

registration
let f be natural-valued Function;
let x, y be object ;
cluster f . (x,y) -> natural ;
coherence
f . (x,y) is natural ;
end;

Lm5: for x, y, z being Element of 1 holds op2 . (x,z) <= (op2 . (x,y)) + (op2 . (y,z))
proof end;

definition
let A be set ;
let f be PartFunc of [:A,A:],REAL;
attr f is Reflexive means :: METRIC_1:def 2
 for a being Element of A holds f . (a,a) = 0 ;
attr f is discerning means :: METRIC_1:def 3
 for a, b being Element of A st f . (a,b) = 0 holds
a = b;
attr f is symmetric means :: METRIC_1:def 4
 for a, b being Element of A holds f . (a,b) = f . (b,a);
attr f is triangle means :: METRIC_1:def 5
 for a, b, c being Element of A holds f . (a,c) <= (f . (a,b)) + (f . (b,c));
end;

:: deftheorem defines Reflexive METRIC_1:def 2 :
for A being set
for f being PartFunc of [:A,A:],REAL holds
( f is Reflexive iff for a being Element of A holds f . (a,a) = 0 );

:: deftheorem defines discerning METRIC_1:def 3 :
for A being set
for f being PartFunc of [:A,A:],REAL holds
( f is discerning iff for a, b being Element of A st f . (a,b) = 0 holds
a = b );

:: deftheorem defines symmetric METRIC_1:def 4 :
for A being set
for f being PartFunc of [:A,A:],REAL holds
( f is symmetric iff for a, b being Element of A holds f . (a,b) = f . (b,a) );

:: deftheorem defines triangle METRIC_1:def 5 :
for A being set
for f being PartFunc of [:A,A:],REAL holds
( f is triangle iff for a, b, c being Element of A holds f . (a,c) <= (f . (a,b)) + (f . (b,c)) );

definition
let M be MetrStruct ;
attr M is Reflexive means :Def6: :: METRIC_1:def 6
 the distance of M is Reflexive ;
attr M is discerning means :Def7: :: METRIC_1:def 7
 the distance of M is discerning ;
attr M is symmetric means :Def8: :: METRIC_1:def 8
 the distance of M is symmetric ;
attr M is triangle means :Def9: :: METRIC_1:def 9
 the distance of M is triangle ;
end;

:: deftheorem Def6 defines Reflexive METRIC_1:def 6 :
for M being MetrStruct holds
( M is Reflexive iff the distance of M is Reflexive );

:: deftheorem Def7 defines discerning METRIC_1:def 7 :
for M being MetrStruct holds
( M is discerning iff the distance of M is discerning );

:: deftheorem Def8 defines symmetric METRIC_1:def 8 :
for M being MetrStruct holds
( M is symmetric iff the distance of M is symmetric );

:: deftheorem Def9 defines triangle METRIC_1:def 9 :
for M being MetrStruct holds
( M is triangle iff the distance of M is triangle );

registration
cluster non empty strict Reflexive discerning symmetric triangle for MetrStruct ;
existence
ex b1 being MetrStruct st
( b1 is strict & b1 is Reflexive & b1 is discerning & b1 is symmetric & b1 is triangle & not b1 is empty )
proof end;
end;

definition
mode MetrSpace is Reflexive discerning symmetric triangle MetrStruct ;
end;

theorem Th1: :: METRIC_1:1
for M being MetrStruct holds
( ( for a being Element of M holds dist (a,a) = 0 ) iff M is Reflexive )
proof end;

theorem Th2: :: METRIC_1:2
for M being MetrStruct holds
( ( for a, b being Element of M st dist (a,b) = 0 holds
a = b ) iff M is discerning )
proof end;

theorem Th3: :: METRIC_1:3
for M being MetrStruct st ( for a, b being Element of M holds dist (a,b) = dist (b,a) ) holds
M is symmetric
proof end;

theorem Th4: :: METRIC_1:4
for M being MetrStruct holds
( ( for a, b, c being Element of M holds dist (a,c) <= (dist (a,b)) + (dist (b,c)) ) iff M is triangle )
proof end;

definition
let M be symmetric MetrStruct ;
let a, b be Element of M;
:: original: dist
redefine func dist (a,b) -> Real;
commutativity
for a, b being Element of M holds dist (a,b) = dist (b,a)
proof end;
end;

theorem Th5: :: METRIC_1:5
for M being Reflexive symmetric triangle MetrStruct
for a, b being Element of M holds 0 <= dist (a,b)
proof end;

theorem Th6: :: METRIC_1:6
for M being MetrStruct st ( for a, b, c being Element of M holds
( ( dist (a,b) = 0 implies a = b ) & ( a = b implies dist (a,b) = 0 ) & dist (a,b) = dist (b,a) & dist (a,c) <= (dist (a,b)) + (dist (b,c)) ) ) holds
M is MetrSpace
proof end;

theorem Th7: :: METRIC_1:7
for M being MetrSpace
for x, y being Element of M st x <> y holds
0 < dist (x,y)
proof end;

definition
let A be set ;
func discrete_dist A -> Function of [:A,A:],REAL means :Def10: :: METRIC_1:def 10
 for x, y being Element of A holds
( it . (x,x) = 0 & ( x <> y implies it . (x,y) = 1 ) );
existence
ex b1 being Function of [:A,A:],REAL st
for x, y being Element of A holds
( b1 . (x,x) = 0 & ( x <> y implies b1 . (x,y) = 1 ) )
proof end;
uniqueness
for b1, b2 being Function of [:A,A:],REAL st ( for x, y being Element of A holds
( b1 . (x,x) = 0 & ( x <> y implies b1 . (x,y) = 1 ) ) ) & ( for x, y being Element of A holds
( b2 . (x,x) = 0 & ( x <> y implies b2 . (x,y) = 1 ) ) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def10 defines discrete_dist METRIC_1:def 10 :
for A being set
for b2 being Function of [:A,A:],REAL holds
( b2 = discrete_dist A iff for x, y being Element of A holds
( b2 . (x,x) = 0 & ( x <> y implies b2 . (x,y) = 1 ) ) );

definition
let A be set ;
func DiscreteSpace A -> strict MetrStruct equals :: METRIC_1:def 11
 MetrStruct(# A,(discrete_dist A) #);
coherence
MetrStruct(# A,(discrete_dist A) #) is strict MetrStruct ;
end;

:: deftheorem defines DiscreteSpace METRIC_1:def 11 :
for A being set holds DiscreteSpace A = MetrStruct(# A,(discrete_dist A) #);

registration
let A be non empty set ;
cluster DiscreteSpace A -> non empty strict ;
coherence
not DiscreteSpace A is empty ;
end;

registration
let A be set ;
cluster DiscreteSpace A -> strict Reflexive discerning symmetric triangle ;
coherence
( DiscreteSpace A is Reflexive & DiscreteSpace A is discerning & DiscreteSpace A is symmetric & DiscreteSpace A is triangle )
proof end;
end;

definition
func real_dist -> Function of [:REAL,REAL:],REAL means :Def12: :: METRIC_1:def 12
 for x, y being Real holds it . (x,y) = |.(x - y).|;
existence
ex b1 being Function of [:REAL,REAL:],REAL st
for x, y being Real holds b1 . (x,y) = |.(x - y).|
proof end;
uniqueness
for b1, b2 being Function of [:REAL,REAL:],REAL st ( for x, y being Real holds b1 . (x,y) = |.(x - y).| ) & ( for x, y being Real holds b2 . (x,y) = |.(x - y).| ) holds
b1 = b2
proof end;
end;

:: deftheorem Def12 defines real_dist METRIC_1:def 12 :
for b1 being Function of [:REAL,REAL:],REAL holds
( b1 = real_dist iff for x, y being Real holds b1 . (x,y) = |.(x - y).| );

theorem Th8: :: METRIC_1:8
for x, y being Element of REAL holds
( real_dist . (x,y) = 0 iff x = y )
proof end;

theorem Th9: :: METRIC_1:9
for x, y being Element of REAL holds real_dist . (x,y) = real_dist . (y,x)
proof end;

theorem Th10: :: METRIC_1:10
for x, y, z being Element of REAL holds real_dist . (x,y) <= (real_dist . (x,z)) + (real_dist . (z,y))
proof end;

definition
func RealSpace -> strict MetrStruct equals :: METRIC_1:def 13
 MetrStruct(# REAL,real_dist #);
coherence
MetrStruct(# REAL,real_dist #) is strict MetrStruct ;
end;

:: deftheorem defines RealSpace METRIC_1:def 13 :
RealSpace = MetrStruct(# REAL,real_dist #);

registration
cluster RealSpace -> non empty strict ;
coherence
not RealSpace is empty ;
end;

registration
cluster RealSpace -> strict Reflexive discerning symmetric triangle ;
coherence
( RealSpace is Reflexive & RealSpace is discerning & RealSpace is symmetric & RealSpace is triangle )
proof end;
end;

definition
let M be MetrStruct ;
let p be Element of M;
let r be Real;
func Ball (p,r) -> Subset of M means :Def14: :: METRIC_1:def 14
it = { q where q is Element of M : dist (p,q) < r } if not M is empty
otherwise it is empty ;
existence
( ( not M is empty implies ex b1 being Subset of M st b1 = { q where q is Element of M : dist (p,q) < r } ) & ( M is empty implies ex b1 being Subset of M st b1 is empty ) )
proof end;
correctness
consistency
for b1 being Subset of M holds verum;
uniqueness
for b1, b2 being Subset of M holds
( ( not M is empty & b1 = { q where q is Element of M : dist (p,q) < r } & b2 = { q where q is Element of M : dist (p,q) < r } implies b1 = b2 ) & ( M is empty & b1 is empty & b2 is empty implies b1 = b2 ) );
;
end;

:: deftheorem Def14 defines Ball METRIC_1:def 14 :
for M being MetrStruct
for p being Element of M
for r being Real
for b4 being Subset of M holds
( ( not M is empty implies ( b4 = Ball (p,r) iff b4 = { q where q is Element of M : dist (p,q) < r } ) ) & ( M is empty implies ( b4 = Ball (p,r) iff b4 is empty ) ) );

definition
let M be MetrStruct ;
let p be Element of M;
let r be Real;
func cl_Ball (p,r) -> Subset of M means :Def15: :: METRIC_1:def 15
it = { q where q is Element of M : dist (p,q) <= r } if not M is empty
otherwise it is empty ;
existence
( ( not M is empty implies ex b1 being Subset of M st b1 = { q where q is Element of M : dist (p,q) <= r } ) & ( M is empty implies ex b1 being Subset of M st b1 is empty ) )
proof end;
correctness
consistency
for b1 being Subset of M holds verum;
uniqueness
for b1, b2 being Subset of M holds
( ( not M is empty & b1 = { q where q is Element of M : dist (p,q) <= r } & b2 = { q where q is Element of M : dist (p,q) <= r } implies b1 = b2 ) & ( M is empty & b1 is empty & b2 is empty implies b1 = b2 ) );
;
end;

:: deftheorem Def15 defines cl_Ball METRIC_1:def 15 :
for M being MetrStruct
for p being Element of M
for r being Real
for b4 being Subset of M holds
( ( not M is empty implies ( b4 = cl_Ball (p,r) iff b4 = { q where q is Element of M : dist (p,q) <= r } ) ) & ( M is empty implies ( b4 = cl_Ball (p,r) iff b4 is empty ) ) );

definition
let M be MetrStruct ;
let p be Element of M;
let r be Real;
func Sphere (p,r) -> Subset of M means :Def16: :: METRIC_1:def 16
it = { q where q is Element of M : dist (p,q) = r } if not M is empty
otherwise it is empty ;
existence
( ( not M is empty implies ex b1 being Subset of M st b1 = { q where q is Element of M : dist (p,q) = r } ) & ( M is empty implies ex b1 being Subset of M st b1 is empty ) )
proof end;
correctness
consistency
for b1 being Subset of M holds verum;
uniqueness
for b1, b2 being Subset of M holds
( ( not M is empty & b1 = { q where q is Element of M : dist (p,q) = r } & b2 = { q where q is Element of M : dist (p,q) = r } implies b1 = b2 ) & ( M is empty & b1 is empty & b2 is empty implies b1 = b2 ) );
;
end;

:: deftheorem Def16 defines Sphere METRIC_1:def 16 :
for M being MetrStruct
for p being Element of M
for r being Real
for b4 being Subset of M holds
( ( not M is empty implies ( b4 = Sphere (p,r) iff b4 = { q where q is Element of M : dist (p,q) = r } ) ) & ( M is empty implies ( b4 = Sphere (p,r) iff b4 is empty ) ) );

theorem Th11: :: METRIC_1:11
for r being Real
for M being MetrStruct
for p, x being Element of M holds
( x in Ball (p,r) iff ( not M is empty & dist (p,x) < r ) )
proof end;

theorem Th12: :: METRIC_1:12
for r being Real
for M being MetrStruct
for p, x being Element of M holds
( x in cl_Ball (p,r) iff ( not M is empty & dist (p,x) <= r ) )
proof end;

theorem Th13: :: METRIC_1:13
for r being Real
for M being MetrStruct
for p, x being Element of M holds
( x in Sphere (p,r) iff ( not M is empty & dist (p,x) = r ) )
proof end;

theorem Th14: :: METRIC_1:14
for r being Real
for M being MetrStruct
for p being Element of M holds Ball (p,r) c= cl_Ball (p,r)
proof end;

theorem Th15: :: METRIC_1:15
for r being Real
for M being MetrStruct
for p being Element of M holds Sphere (p,r) c= cl_Ball (p,r)
proof end;

theorem :: METRIC_1:16
for r being Real
for M being MetrStruct
for p being Element of M holds (Sphere (p,r)) \/ (Ball (p,r)) = cl_Ball (p,r)
proof end;

theorem :: METRIC_1:17
for r being Real
for M being non empty MetrStruct
for p being Element of M holds Ball (p,r) = { q where q is Element of M : dist (p,q) < r } by Def14;

theorem :: METRIC_1:18
for r being Real
for M being non empty MetrStruct
for p being Element of M holds cl_Ball (p,r) = { q where q is Element of M : dist (p,q) <= r } by Def15;

theorem :: METRIC_1:19
for r being Real
for M being non empty MetrStruct
for p being Element of M holds Sphere (p,r) = { q where q is Element of M : dist (p,q) = r } by Def16;

theorem :: METRIC_1:20
for x being set holds Empty^2-to-zero . (x,x) = 0
proof end;

theorem Th21: :: METRIC_1:21
for x, y being Element of 1 st x <> y holds
0 < Empty^2-to-zero . (x,y)
proof end;

theorem Th22: :: METRIC_1:22
for x, y being Element of 1 holds Empty^2-to-zero . (x,y) = Empty^2-to-zero . (y,x) by Lm4;

theorem Th23: :: METRIC_1:23
for x, y, z being Element of 1 holds Empty^2-to-zero . (x,z) <= (Empty^2-to-zero . (x,y)) + (Empty^2-to-zero . (y,z)) by Lm5;

theorem Th24: :: METRIC_1:24
for x, y, z being Element of 1 holds Empty^2-to-zero . (x,z) <= max ((Empty^2-to-zero . (x,y)),(Empty^2-to-zero . (y,z)))
proof end;

set M0 = MetrStruct(# 1,Empty^2-to-zero #);

definition
let A be non empty set ;
let f be Function of [:A,A:],REAL;
attr f is Discerning means :: METRIC_1:def 17
 for a, b being Element of A st a <> b holds
0 < f . (a,b);
end;

:: deftheorem defines Discerning METRIC_1:def 17 :
for A being non empty set
for f being Function of [:A,A:],REAL holds
( f is Discerning iff for a, b being Element of A st a <> b holds
0 < f . (a,b) );

definition
let M be non empty MetrStruct ;
attr M is Discerning means :: METRIC_1:def 18
 the distance of M is Discerning ;
end;

:: deftheorem defines Discerning METRIC_1:def 18 :
for M being non empty MetrStruct holds
( M is Discerning iff the distance of M is Discerning );

theorem Th25: :: METRIC_1:25
for M being non empty MetrStruct holds
( ( for a, b being Element of M st a <> b holds
0 < dist (a,b) ) iff M is Discerning )
proof end;

registration
cluster MetrStruct(# 1,Empty^2-to-zero #) -> non empty ;
coherence
not MetrStruct(# 1,Empty^2-to-zero #) is empty ;
end;

registration
cluster MetrStruct(# 1,Empty^2-to-zero #) -> Reflexive symmetric triangle Discerning ;
coherence
( MetrStruct(# 1,Empty^2-to-zero #) is Reflexive & MetrStruct(# 1,Empty^2-to-zero #) is symmetric & MetrStruct(# 1,Empty^2-to-zero #) is Discerning & MetrStruct(# 1,Empty^2-to-zero #) is triangle )
proof end;
end;

definition
let M be non empty MetrStruct ;
attr M is ultra means :Def19: :: METRIC_1:def 19
 for a, b, c being Element of M holds dist (a,c) <= max ((dist (a,b)),(dist (b,c)));
end;

:: deftheorem Def19 defines ultra METRIC_1:def 19 :
for M being non empty MetrStruct holds
( M is ultra iff for a, b, c being Element of M holds dist (a,c) <= max ((dist (a,b)),(dist (b,c))) );

registration
cluster non empty strict Reflexive symmetric triangle Discerning ultra for MetrStruct ;
existence
ex b1 being non empty MetrStruct st
( b1 is strict & b1 is ultra & b1 is Reflexive & b1 is symmetric & b1 is Discerning & b1 is triangle )
proof end;
end;

theorem Th26: :: METRIC_1:26
for M being non empty Reflexive Discerning MetrStruct
for a, b being Element of M holds 0 <= dist (a,b)
proof end;

definition
mode PseudoMetricSpace is non empty Reflexive symmetric triangle MetrStruct ;
mode SemiMetricSpace is non empty Reflexive symmetric Discerning MetrStruct ;
mode NonSymmetricMetricSpace is non empty Reflexive triangle Discerning MetrStruct ;
mode UltraMetricSpace is non empty Reflexive symmetric Discerning ultra MetrStruct ;
end;

registration
cluster non empty Reflexive discerning symmetric triangle -> non empty Discerning for MetrStruct ;
coherence
for b1 being non empty MetrSpace holds b1 is Discerning
proof end;
end;

registration
cluster non empty Reflexive symmetric Discerning ultra -> discerning triangle for MetrStruct ;
coherence
for b1 being UltraMetricSpace holds
( b1 is triangle & b1 is discerning )
proof end;
end;

definition
func Set_to_zero -> Function of [:2,2:],REAL equals :: METRIC_1:def 20
[:2,2:] --> 0;
coherence
[:2,2:] --> 0 is Function of [:2,2:],REAL
proof end;
end;

:: deftheorem defines Set_to_zero METRIC_1:def 20 :
Set_to_zero = [:2,2:] --> 0;

theorem Th27: :: METRIC_1:27
for x, y being Element of 2 holds Set_to_zero . (x,y) = 0 by ZFMISC_1:87, FUNCOP_1:7;

theorem Th28: :: METRIC_1:28
for x, y being Element of 2 holds Set_to_zero . (x,y) = Set_to_zero . (y,x)
proof end;

theorem Th29: :: METRIC_1:29
for x, y, z being Element of 2 holds Set_to_zero . (x,z) <= (Set_to_zero . (x,y)) + (Set_to_zero . (y,z))
proof end;

definition
func ZeroSpace -> MetrStruct equals :: METRIC_1:def 21
 MetrStruct(# 2,Set_to_zero #);
coherence
MetrStruct(# 2,Set_to_zero #) is MetrStruct ;
end;

:: deftheorem defines ZeroSpace METRIC_1:def 21 :
ZeroSpace = MetrStruct(# 2,Set_to_zero #);

registration
cluster ZeroSpace -> non empty strict ;
coherence
( ZeroSpace is strict & not ZeroSpace is empty ) ;
end;

registration
cluster ZeroSpace -> Reflexive symmetric triangle ;
coherence
( ZeroSpace is Reflexive & ZeroSpace is symmetric & ZeroSpace is triangle )
proof end;
end;

definition
let S be MetrStruct ;
let p, q, r be Element of S;
pred q is_between p,r means :: METRIC_1:def 22
( p <> q & p <> r & q <> r & dist (p,r) = (dist (p,q)) + (dist (q,r)) );
end;

:: deftheorem defines is_between METRIC_1:def 22 :
for S being MetrStruct
for p, q, r being Element of S holds
( q is_between p,r iff ( p <> q & p <> r & q <> r & dist (p,r) = (dist (p,q)) + (dist (q,r)) ) );

theorem :: METRIC_1:30
for S being non empty Reflexive symmetric triangle MetrStruct
for p, q, r being Element of S st q is_between p,r holds
q is_between r,p
proof end;

theorem :: METRIC_1:31
for S being MetrSpace
for p, q, r being Element of S st q is_between p,r holds
( not p is_between q,r & not r is_between p,q )
proof end;

theorem :: METRIC_1:32
for S being MetrSpace
for p, q, r, s being Element of S st q is_between p,r & r is_between p,s holds
( q is_between p,s & r is_between q,s )
proof end;

definition
let M be non empty MetrStruct ;
let p, r be Element of M;
func open_dist_Segment (p,r) -> Subset of M equals :: METRIC_1:def 23
 { q where q is Element of M : q is_between p,r } ;
coherence
{ q where q is Element of M : q is_between p,r } is Subset of M
proof end;
end;

:: deftheorem defines open_dist_Segment METRIC_1:def 23 :
for M being non empty MetrStruct
for p, r being Element of M holds open_dist_Segment (p,r) = { q where q is Element of M : q is_between p,r } ;

theorem :: METRIC_1:33
for M being non empty MetrSpace
for p, r, x being Element of M holds
( x in open_dist_Segment (p,r) iff x is_between p,r )
proof end;

definition
let M be non empty MetrStruct ;
let p, r be Element of M;
func close_dist_Segment (p,r) -> Subset of M equals :: METRIC_1:def 24
 { q where q is Element of M : q is_between p,r } \/ {p,r};
coherence
{ q where q is Element of M : q is_between p,r } \/ {p,r} is Subset of M
proof end;
end;

:: deftheorem defines close_dist_Segment METRIC_1:def 24 :
for M being non empty MetrStruct
for p, r being Element of M holds close_dist_Segment (p,r) = { q where q is Element of M : q is_between p,r } \/ {p,r};

theorem :: METRIC_1:34
for M being non empty MetrStruct
for p, r, x being Element of M holds
( x in close_dist_Segment (p,r) iff ( x is_between p,r or x = p or x = r ) )
proof end;