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rs_integral_def[T:TYPE+ from real]: THEORY
%----------------------------------------------------------------------------
% Definition of Riemann-Stieltjes Integral
%----------------------------------------------------------------------------
BEGIN
ASSUMING
IMPORTING deriv_domain_def[T]
connected_domain : ASSUMPTION connected?[T]
not_one_element : ASSUMPTION not_one_element?[T]
ENDASSUMING
IMPORTING rs_partition[T],
reals@real_fun_preds
% AUTO_REWRITE+ finseq_appl
a,b,c,x: VAR T
S: VAR real
f,g,h: VAR [T -> real]
IMPORTING reals@intervals_real[T]
xis_pred?(a:T,(b|a<b),P:partition(a,b))(fs:fseq): MACRO bool = (fs`length = P`length-1 AND (FORALL (ii: below(P`length-1)):
P`seq(ii) <= fs`seq(ii) AND fs`seq(ii) <= P`seq(ii+1)))
xis?(a:T,(b|a<b),P:partition(a,b)): TYPE = (xis_pred?(a,b,P))
xis_join(a:T,(b | a<b), (c| b<c))(Pab: partition(a,b),Pbc: partition(b,c))(xab: xis?(a,b,Pab),xbc:xis?(b,c,Pbc)):
xis?(a,c,partjoin(a,b,c)(Pab,Pbc)) =
xab o xbc
xis_lem: LEMMA FORALL (P: partition(a,b), (xis: xis?(a,b,P)),
(ii: below(P`length-1))):
P(ii) <= xis(ii) AND xis(ii) <= P(ii+1)
xis_bounded: LEMMA FORALL (P: partition(a,b), (xis: xis?(a,b,P)),
(ii: below(P`length-1))):
a <= xis(ii) AND xis(ii) <= b
% AUTO_REWRITE+ xis_lem
Rie_sum(a:T,b:{x:T|a<x},g:[T->real],P:partition(a,b),
xis: xis?(a,b,P),f:[T->real]): real =
LET N = P`length-1 IN
sigma[below(N)](0,N-1,(LAMBDA (n: below(N)):
(g(P`seq(n+1)) - g(P`seq(n))) * f(xis(n))))
Rie_sec(a:T,b:{x:T|a<x},g:[T->real],P:partition(a,b), xis: xis?(a,b,P), f:[T->real],
n: upto(P`length-1)): real =
IF n = 0 THEN 0
ELSE (g(P`seq(n)) - g(P`seq(n - 1))) * f(xis(n-1))
ENDIF
Rie_sum_alt(a:T,b:{x:T|a<x},g:[T->real],P:partition(a,b),
xis: xis?(a,b,P),f:[T->real]): real =
LET N = P`length-1 IN
sigma[upto(N)](1,N,(LAMBDA (n: upto(N)):
Rie_sec(a,b,g,P,xis,f,n)))
Rie_sum_alt_lem: LEMMA a < b IMPLIES
FORALL (P: partition(a,b), (xis: xis?(a,b,P))):
Rie_sum(a,b,g,P,xis,f) = Rie_sum_alt(a,b,g,P,xis,f)
Riemann_sum?(a:T,b:{x:T|a<x},P:partition(a,b),g,f:[T->real])(S:real): bool =
(EXISTS (xis: xis?(a,b,P)): LET N = P`length-1 IN
S = Rie_sum(a,b,g,P,xis,f))
Riemann_sum_strictly_sort: LEMMA a<b IMPLIES FORALL (P:partition(a,b),RS:real):
(EXISTS (xis:xis?(a,b,P)): RS = Rie_sum(a,b,g,P,xis,f))
IFF
(EXISTS (xis:xis?(a,b,strictly_sort(P))): RS = Rie_sum(a,b,g,strictly_sort(P),xis,f))
integral?(a:T,b:{x:T|a<x},g,f:[T->real],S:real): bool =
(FORALL (epsi: posreal): (EXISTS (delta: posreal):
(FORALL (P: partition(a,b)):
width(a,b,P) < delta IMPLIES
(FORALL (R: (Riemann_sum?(a,b,P,g,f))):
abs(S - R) < epsi))))
x_in(aa:T,bb:{x:T|aa<=x}): {t:T | aa <= t AND t <= bb}
pick_one(a:T,b:{x:T|a<x},P: partition(a,b))(ii: nat): T =
IF ii>= P`length-1 THEN default
ELSE x_in(P(ii), P(ii+1)) ENDIF
gen_xis(a:T,b:{x:T|a<x},P: partition(a,b)): xis?(a,b,P) =
(# length := P`length - 1,
seq := pick_one(a,b,P)
#)
Riemann?_Rie: LEMMA a < b IMPLIES FORALL (P: partition(a,b),
xis: xis?(a,b,P)):
Riemann_sum?(a,b,P,g,f)(Rie_sum(a,b,g,P,xis,f))
% AUTO_REWRITE+ Riemann?_Rie
A1, A2: VAR real
integral_unique: LEMMA a < b AND integral?(a,b,g,f,A1) AND
integral?(a,b,g,f,A2) IMPLIES A1 = A2
integrable?(a:T,b:{x:T|a<x},g,f:[T->real]): bool =
(EXISTS (S:real): integral?(a,b,g,f,S))
integral(a:T,b:{x:T|a<x}, gg:[T->real],ff: { f | integrable?(a,b,gg,f)} ):
{S:real | integral?(a,b,gg,ff,S)}
s: VAR real
integral_def: LEMMA a < b IMPLIES
( (integrable?(a,b,g,f) AND integral(a,b,g,f) = s)
IFF integral?(a,b,g,f,s) )
integral_restrict_eq: LEMMA a < b AND
(FORALL x: a <= x AND x <= b IMPLIES
f(x) = h(x)) AND
integrable?(a,b,g,f)
IMPLIES integrable?(a,b,g,h) AND
integral(a,b,g,h) = integral(a,b,g,f)
Integrable?(a:T,b:T,g,f:[T->real]): bool = (a = b) OR
(a < b AND integrable?(a,b,g,f)) OR
(b < a AND integrable?(b,a,g,f))
Integrable_funs(a,b,g): TYPE = {f | Integrable?(a,b,g,f)}
Integral?(a:T,b:T,g,f:[T->real],S:real): bool = (a = b AND S = 0) OR
(a < b AND integral?(a,b,g,f,S))
Integral(a:T,b:T,g:[T->real],f:Integrable_funs(a,b,g)): real =
IF a = b THEN 0
ELSIF a < b THEN integral(a,b,g,f)
ELSE -integral(b,a,g,f)
ENDIF
% -----------------------------%
% Extending a riemann sum from a partition to a refinement and vice-versa
Rie_sum_extend_union : LEMMA a<b IMPLIES
FORALL (P,Q: partition(a,b)): LET PQ = partition_union(a,b)(P,Q) IN
strictly_increasing?(P) IMPLIES FORALL (xis:xis?(a,b,P)): Rie_sum(a,b,g,P,xis,f) =
sigma[below(PQ`length-1)](0,PQ`length-2,LAMBDA (n:below(PQ`length-1)):
f(xis`seq(partition_union_map_inv(a,b,P,Q)(n)))*(g(PQ`seq(1+n))-g(PQ`seq(n))))
Rie_sum_restrict_union: LEMMA a<b IMPLIES FORALL (P,Q: partition(a,b)):
LET
PQ = partition_union(a,b)(P,Q),
pum = partition_union_map(a,b,P,Q)
IN FORALL (xispq:xis?(a,b,PQ)):
strictly_increasing?(P) AND
(FORALL (ii:below(P`length),jj,kk:below(PQ`length)):
pum(ii)<=jj AND pum(ii)<=kk AND (ii<P`length-1 IMPLIES jj<pum(ii+1) AND kk<pum(ii+1))
IMPLIES xispq`seq(jj) = xispq`seq(kk))
IMPLIES
EXISTS (xis:xis?(a,b,P)):
Rie_sum(a,b,g,PQ,xispq,f) = Rie_sum(a,b,g,P,xis,f)
Rie_sum_diff_extend_union : LEMMA a<b IMPLIES LET CI = closed_intv(a,b) IN
(increasing?[(CI)](g) IMPLIES
FORALL (P,Q: partition(a,b)): LET PQ = partition_union(a,b)(P,Q) IN
FORALL (xispq:xis?(a,b,PQ),xis:xis?(a,b,P)): EXISTS (xis2:xis?(a,b,P)):
abs(Rie_sum(a,b,g,P,xis,f) - Rie_sum(a,b,g,PQ,xispq,f)) <=
abs(Rie_sum(a,b,g,P,xis,f)-Rie_sum(a,b,g,P,xis2,f)))
END rs_integral_def
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