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:: Fix Point Theorem for Compact Spaces | |
:: by Alicia de la Cruz | |
environ | |
vocabularies NUMBERS, XBOOLE_0, METRIC_1, FUNCT_1, REAL_1, CARD_1, ARYTM_3, | |
PRE_TOPC, XXREAL_0, RELAT_1, STRUCT_0, FUNCOP_1, PCOMPS_1, RCOMP_1, | |
SUBSET_1, POWER, SETFAM_1, TARSKI, ARYTM_1, FINSET_1, ORDINAL1, SEQ_1, | |
VALUED_1, ORDINAL2, SEQ_2, ALI2, NAT_1, ASYMPT_1; | |
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, XREAL_0, | |
FINSET_1, SETFAM_1, RELAT_1, FUNCT_1, FUNCT_2, FUNCOP_1, STRUCT_0, | |
METRIC_1, PRE_TOPC, POWER, COMPTS_1, PCOMPS_1, TOPS_2, VALUED_1, SEQ_1, | |
SEQ_2, XXREAL_0, REAL_1, NAT_1; | |
constructors SETFAM_1, FUNCOP_1, FINSET_1, XXREAL_0, REAL_1, NAT_1, SEQ_2, | |
POWER, TOPS_2, COMPTS_1, PCOMPS_1, VALUED_1, PARTFUN1, BINOP_2, RVSUM_1, | |
COMSEQ_2, SEQ_1, RELSET_1; | |
registrations SUBSET_1, ORDINAL1, NUMBERS, XXREAL_0, MEMBERED, STRUCT_0, | |
METRIC_1, PCOMPS_1, VALUED_1, FUNCT_2, XREAL_0, SEQ_1, SEQ_2, RELSET_1, | |
FUNCOP_1; | |
requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; | |
definitions TARSKI, TOPS_2, ORDINAL1, XBOOLE_0, FINSET_1; | |
equalities XBOOLE_0, SUBSET_1, STRUCT_0; | |
theorems METRIC_1, SUBSET_1, PCOMPS_1, COMPTS_1, POWER, SEQ_2, SEQ_4, | |
SERIES_1, SETFAM_1, SEQ_1, PRE_TOPC, TOPS_1, FINSET_1, XREAL_1, XXREAL_0, | |
XBOOLE_0, XREAL_0, FUNCT_1, FUNCT_2; | |
schemes SUBSET_1, SEQ_1, DOMAIN_1, NAT_1; | |
begin | |
definition | |
let M be non empty MetrSpace; | |
let f be Function of M, M; | |
attr f is contraction means | |
:Def1: | |
ex L being Real st 0 < L & L < 1 & | |
for x,y being Point of M holds dist(f.x,f.y) <= L * dist(x,y); | |
end; | |
registration | |
let M be non empty MetrSpace; | |
cluster constant -> contraction for Function of M,M; | |
coherence | |
proof | |
let f be Function of M,M such that | |
A1: f is constant; | |
take 1/2; | |
thus 0<1/2 & 1/2<1; | |
let z,y be Point of M; | |
dom f = the carrier of M by FUNCT_2:def 1; | |
then f.z = f.y by A1,FUNCT_1:def 10; | |
then | |
A2: dist(f.z,f.y) = 0 by METRIC_1:1; | |
dist(z,y)>=0 by METRIC_1:5; | |
hence thesis by A2; | |
end; | |
end; | |
registration | |
let M be non empty MetrSpace; | |
cluster constant for Function of M, M; | |
existence | |
proof | |
M --> the Point of M is constant; | |
hence thesis; | |
end; | |
end; | |
definition | |
let M be non empty MetrSpace; | |
mode Contraction of M is contraction Function of M, M; | |
end; | |
::$N Banach fixed-point theorem | |
theorem | |
for M being non empty MetrSpace | |
for f being Contraction of M st TopSpaceMetr(M) is compact | |
ex c being Point of M st f.c = c & | |
for x being Point of M st f.x = x holds x = c | |
proof | |
let M be non empty MetrSpace; | |
let f be Contraction of M; | |
set x0 = the Point of M; | |
set a=dist(x0,f.x0); | |
consider L being Real such that | |
A1: 0<L and | |
A2: L<1 and | |
A3: for x,y being Point of M holds dist(f.x,f.y)<=L*dist(x,y) by Def1; | |
assume | |
A4: TopSpaceMetr(M) is compact; | |
now | |
deffunc F(Nat) = L to_power ($1+1); | |
defpred P[set] means ex n being Nat st $1 = { x where x is | |
Point of M : dist(x,f.x) <= a*L to_power n}; | |
assume a <> 0; | |
consider F being Subset-Family of TopSpaceMetr(M) such that | |
A5: for B being Subset of TopSpaceMetr(M) holds B in F iff P[B] from | |
SUBSET_1:sch 3; | |
defpred P[Point of M] means dist($1,f.($1)) <= a*L to_power (0+1); | |
A6: F is closed | |
proof | |
let B be Subset of TopSpaceMetr(M); | |
A7: TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) | |
by PCOMPS_1:def 5; | |
then reconsider V = B` as Subset of M; | |
assume B in F; | |
then consider n being Nat such that | |
A8: B= { x where x is Point of M : dist(x,f.x) <= a*L to_power n} by A5; | |
for x being Point of M st x in V | |
ex r being Real st r>0 & Ball(x,r) c= V | |
proof | |
let x be Point of M; | |
assume x in V; | |
then not x in B by XBOOLE_0:def 5; | |
then dist(x,f.x)>a*L to_power n by A8; | |
then | |
A9: dist(x,f.x)-a*L to_power n>0 by XREAL_1:50; | |
take r = (dist(x,f.x)-a*L to_power n)/2; | |
thus r>0 by A9,XREAL_1:215; | |
let z be object; | |
assume | |
A10: z in Ball(x,r); | |
then reconsider y=z as Point of M; | |
dist(x,y)<r by A10,METRIC_1:11; | |
then 2*dist(x,y)<2*r by XREAL_1:68; | |
then | |
A11: dist(y,f.y) + 2*dist(x,y)< dist(y,f.y) + 2*r by XREAL_1:6; | |
dist(x,y) + dist(y,f.y) >= dist(x,f.y) by METRIC_1:4; | |
then | |
A12: (dist(x,y) + dist(y,f.y)) + dist(f.y,f.x) >= | |
dist(x,f.y) + dist(f.y,f.x) by XREAL_1:6; | |
dist(f.y,f.x)<=L*dist(y,x) & L*dist(y,x)<=dist(y,x) | |
by A2,A3,METRIC_1:5,XREAL_1:153; | |
then dist(f.y,f.x)<=dist(y,x) by XXREAL_0:2; | |
then dist(f.y,f.x)+dist(y,x) <= dist(y,x)+dist(y,x) by XREAL_1:6; | |
then | |
A13: dist(y,f.y) + (dist(y,x) + dist(f.y,f.x)) <= dist(y,f.y) + 2*dist | |
(y,x) by XREAL_1:7; | |
dist(x,f.y) + dist(f.y,f.x) >= dist(x,f.x) by METRIC_1:4; | |
then dist(y,f.y)+dist(x,y)+dist(f.y,f.x)>=dist(x,f.x) | |
by A12,XXREAL_0:2; | |
then dist(y,f.y)+2*dist(x,y)>=dist(x,f.x) by A13,XXREAL_0:2; | |
then | |
dist(x,f.x)-2*r = a*L to_power n & dist(y,f.y)+2*r>dist(x,f.x) | |
by A11,XXREAL_0:2; | |
then | |
not ex x being Point of M st y = x & dist(x,f.x)<= a*L to_power n | |
by XREAL_1:19; | |
then not y in B by A8; | |
hence thesis by A7,SUBSET_1:29; | |
end; | |
then B` in Family_open_set(M) by PCOMPS_1:def 4; | |
then B` is open by A7,PRE_TOPC:def 2; | |
hence thesis by TOPS_1:3; | |
end; | |
A14: TopSpaceMetr(M)=TopStruct (#the carrier of M,Family_open_set(M)#) | |
by PCOMPS_1:def 5; | |
A15: {x where x is Point of M:P[x]}is Subset of M from DOMAIN_1:sch 7; | |
F is centered | |
proof | |
thus F <> {} by A5,A14,A15; | |
defpred P[Nat] means | |
{ x where x is Point of M : dist(x,f.x) | |
<= a*L to_power $1}<>{}; | |
let G be set such that | |
A16: G <> {} and | |
A17: G c= F and | |
A18: G is finite; | |
G is c=-linear | |
proof | |
let B,C be set; | |
assume that | |
A19: B in G and | |
A20: C in G; | |
B in F by A17,A19; | |
then consider n being Nat such that | |
A21: B= { x where x is Point of M : dist(x,f.x) <= a*L to_power n} by A5; | |
C in F by A17,A20; | |
then consider m being Nat such that | |
A22: C= { x where x is Point of M : dist(x,f.x) <= a*L to_power m} by A5; | |
A23: for n,m being Nat st n<=m holds L to_power m <= L to_power n | |
proof | |
let n,m be Nat such that | |
A24: n<=m; | |
per cases by A24,XXREAL_0:1; | |
suppose | |
n<m; | |
hence thesis by A1,A2,POWER:40; | |
end; | |
suppose | |
n=m; | |
hence thesis; | |
end; | |
end; | |
A25: for n,m being Nat st n<=m holds a*L to_power m <= a*L | |
to_power n | |
proof | |
A26: a>=0 by METRIC_1:5; | |
let n,m be Nat; | |
assume n<=m; | |
hence thesis by A23,A26,XREAL_1:64; | |
end; | |
now | |
per cases; | |
case | |
A27: n<=m; | |
thus C c= B | |
proof | |
let y be object; | |
assume y in C; | |
then consider x being Point of M such that | |
A28: y = x and | |
A29: dist(x,f.x) <= a*L to_power m by A22; | |
a*L to_power m <= a*L to_power n by A25,A27; | |
then dist(x,f.x) <= a*L to_power n by A29,XXREAL_0:2; | |
hence thesis by A21,A28; | |
end; | |
end; | |
case | |
A30: m<=n; | |
thus B c= C | |
proof | |
let y be object; | |
assume y in B; | |
then consider x being Point of M such that | |
A31: y = x and | |
A32: dist(x,f.x) <= a*L to_power n by A21; | |
a*L to_power n <= a*L to_power m by A25,A30; | |
then dist(x,f.x) <= a*L to_power m by A32,XXREAL_0:2; | |
hence thesis by A22,A31; | |
end; | |
end; | |
end; | |
hence B c= C or C c= B; | |
end; | |
then consider m being set such that | |
A33: m in G and | |
A34: for C being set st C in G holds m c= C by A16,A18,FINSET_1:11; | |
A35: m c= meet G by A16,A34,SETFAM_1:5; | |
A36: for k being Nat st P[k] holds P[k+1] | |
proof | |
let k be Nat; | |
set z = the Element of { x where x is Point of M : | |
dist(x,f.x) <= a*L to_power k}; | |
A37: L*(a*L to_power k)=a*(L*L to_power k) | |
.=a*(L to_power k*L to_power 1) by POWER:25 | |
.= a*L to_power (k+1) by A1,POWER:27; | |
assume { x where x is Point of M : dist(x,f.x) <= a*L to_power k}<> {}; | |
then z in { x where x is Point of M : dist(x,f.x) <= a*L to_power k}; | |
then consider y being Point of M such that | |
y=z and | |
A38: dist(y,f.y)<= a*L to_power k; | |
A39: dist(f.y,f.(f.y)) <= L*dist(y,f.y) by A3; | |
L*dist(y,f.y) <= L*(a*L to_power k) by A1,A38,XREAL_1:64; | |
then dist(f.y,f.(f.y)) <= a*L to_power (k+1) by A37,A39,XXREAL_0:2; | |
then | |
f.y in { x where x is Point of M : dist(x,f.x) <= a*L to_power (k +1)}; | |
hence thesis; | |
end; | |
dist(x0,f.x0) = a*1 .= a*L to_power 0 by POWER:24; | |
then x0 in { x where x is Point of M : dist(x,f.x) <= a*L to_power 0}; | |
then | |
A40: P[0]; | |
A41: for k being Nat holds P[k] from NAT_1:sch 2(A40,A36); | |
m in F by A17,A33; | |
then m <> {} by A5,A41; | |
hence thesis by A35; | |
end; | |
then meet F <> {} by A4,A6,COMPTS_1:4; | |
then consider c9 being Point of TopSpaceMetr(M) such that | |
A42: c9 in meet F by SUBSET_1:4; | |
reconsider c = c9 as Point of M by A14; | |
reconsider dc = dist(c,f.c) as Element of REAL by XREAL_0:def 1; | |
set r = seq_const dist(c,f.c); | |
consider s9 being Real_Sequence such that | |
A43: for n being Nat holds s9.n= F(n) from SEQ_1:sch 1; | |
set s = a (#) s9; | |
lim s9=0 by A1,A2,A43,SERIES_1:1; | |
then | |
A44: lim s = a*0 by A1,A2,A43,SEQ_2:8,SERIES_1:1 | |
.= 0; | |
A45: now | |
let n be Nat; | |
defpred P[Point of M] means dist($1,f.$1) <= a*L to_power (n+1); | |
set B = { x where x is Point of M : P[x]}; | |
B is Subset of M from DOMAIN_1:sch 7; | |
then B in F by A5,A14; | |
then c in B by A42,SETFAM_1:def 1; | |
then | |
A46: ex x being Point of M st c = x & dist(x,f.x) <= a*L to_power ( n+1 ); | |
s.n = a*s9.n by SEQ_1:9 | |
.= a*L to_power (n+1) by A43; | |
hence r.n <= s.n by A46,SEQ_1:57; | |
end; | |
s is convergent by A1,A2,A43,SEQ_2:7,SERIES_1:1; | |
then | |
A47: lim r <= lim s by A45,SEQ_2:18; | |
r.0=dist(c,f.c) by SEQ_1:57; | |
then dist(c,f.c)<=0 by A44,A47,SEQ_4:25; | |
then dist(c,f.c)=0 by METRIC_1:5; | |
hence ex c being Point of M st dist(c,f.c) = 0; | |
end; | |
then consider c being Point of M such that | |
A48: dist(c,f.c) = 0; | |
take c; | |
thus | |
A49: f.c =c by A48,METRIC_1:2; | |
let x be Point of M; | |
assume | |
A50: f.x=x; | |
A51: dist(x,c)>=0 by METRIC_1:5; | |
assume x<>c; | |
then dist(x,c)<>0 by METRIC_1:2; | |
then L*dist(x,c)<dist(x,c) by A2,A51,XREAL_1:157; | |
hence contradiction by A3,A49,A50; | |
end; | |