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integral_def[T: TYPE FROM real]: THEORY
%----------------------------------------------------------------------------
%
% Definition of Riemann Integral
% Author: Rick Butler
% Major Revision: 12/1/03
%
%----------------------------------------------------------------------------
BEGIN
ASSUMING
IMPORTING deriv_domain_def[T]
connected_domain : ASSUMPTION connected?[T]
not_one_element : ASSUMPTION not_one_element?[T]
ENDASSUMING
IMPORTING finite_sets@finite_sets_minmax[real,<=] %, finite_sets_more
AUTO_REWRITE+ finseq_appl
a,b,x: VAR T
f,g: VAR [T -> real]
% open_interval (a:T,b:{x:T|a<x}): TYPE = { x | a < x AND x < b}
% closed_interval(a:T,b:{x:T|a<x}): TYPE = { x | a <= x AND x <= b}
IMPORTING reals@intervals_real[T]
% % Unlike Rosenlicht index runs from 0 to N-1:
% partition(a:T,b:{x:T|a<x}): TYPE =
% {fs: finite_sequence[closed_interval(a,b)] |
% Let N = length(fs), xx = seq(fs) IN
% N > 1 AND xx(0) = a AND xx(N-1) = b AND
% (FORALL (ii: below(N-1)): xx(ii) < xx(ii+1))}
% width(a:T, b:{x:T|a<x}, P: partition(a,b)): posreal =
% LET xx = seq(P), N = length(P) IN
% max({ l: real | EXISTS (ii: below(N-1)):
% l = xx(ii+1) - xx(ii)})
% width_lem : LEMMA (FORALL (a:T), (b:{x:T|a<x}),
% (P: partition(a,b)),(ii: below(length(P)-1)):
% width(a,b,P) >= P(ii+1) - P(ii))
% parts_order : LEMMA FORALL (P: partition(a,b), ii,jj: below(length(P))):
% ii < jj IMPLIES seq(P)(ii) < seq(P)(jj)
% parts_order_le: LEMMA FORALL (P: partition(a,b),ii,jj: below(length(P))):
% ii <= jj IMPLIES seq(P)(ii) <= seq(P)(jj)
% parts_disjoint: LEMMA FORALL (P: partition(a,b), ii,jj: below(length(P)-1)):
% seq(P)(ii) < x AND x < seq(P)(1 + ii) AND
% seq(P)(jj) < x AND x < seq(P)(1 + jj)
% IMPLIES
% jj = ii
% in_part?(a:T, b:{x:T|a<x},P: partition(a,b),xx:T): MACRO bool =
% EXISTS (ii: below(length(P)-1)): xx = seq(P)(ii)
% in_sect?(a:T, b:{x:T|a<x},P: partition(a,b),
% ii: below(length(P)-1),xx:T): MACRO bool =
% seq(P)(ii) < xx AND xx < seq(P)(ii+1)
% part_in : LEMMA FORALL (P: partition(a,b)):
% a < b AND a <= x AND x <= b IMPLIES
% (EXISTS (ii: below(length(P)-1)):
% seq(P)(ii) <= x AND x <= seq(P)(ii+1))
% part_not_in : LEMMA a < b IMPLIES FORALL (P: partition(a,b)):
% FORALL (ii,jj: below(length(P)-1)):
% seq(P)(ii) < x AND x < seq(P)(ii+1)
% IMPLIES x /= seq(P)(jj)
% Prop: VAR [T -> bool]
% part_induction: LEMMA (FORALL (P: partition(a,b)):
% (FORALL ( x: closed_interval(a,b)):
% LET xx = seq(P), N = length(P) IN
% (FORALL (ii : below(N-1)):
% xx(ii) <= x AND x <= xx(ii+1) IMPLIES
% Prop(x))
% IMPLIES Prop(x) ) )
% IMPORTING reals@sigma_below, reals@sigma_upto
% eq_partition(a:T,b:{x:T|a<x},N: above(1)): partition(a,b) =
% (# length := N,
% seq := (LAMBDA (ii: below(N)): a + ii*(b-a)/(N-1)) #)
% xis?(a:T,b:{x:T|a<x},P:partition(a,b))
% (fs: [below(length(P)-1) -> closed_interval(a,b)]): bool =
% (FORALL (ii: below(length(P)-1)):
% P(ii) <= fs(ii) AND fs(ii) <= P(ii+1))
% AUTO_REWRITE+ xis?
% xis_lem: LEMMA FORALL (P: partition(a,b), (xis: (xis?(a,b,P))),
% (ii: below(length(P)-1))):
% P(ii) <= xis(ii) AND xis(ii) <= P(ii+1)
% AUTO_REWRITE+ xis_lem
% Rie_sum(a:T,b:{x:T|a<x},P:partition(a,b),
% xis: (xis?(a,b,P)),f:[T->real]): real =
% LET N = length(P)-1 IN
% sigma[below(N)](0,N-1,(LAMBDA (n: below(N)):
% (P(n+1) - P(n)) * f(xis(n))))
% Rie_sec(a:T,b:{x:T|a<x},P:partition(a,b), xis: (xis?(a,b,P)), f:[T->real],
% n: upto(length(P)-1)): real =
% IF n = 0 THEN 0
% ELSE (seq(P)(n) - seq(P)(n - 1)) * f(xis(n-1))
% ENDIF
% Rie_sum_alt(a:T,b:{x:T|a<x},P:partition(a,b),
% xis: (xis?(a,b,P)),f:[T->real]): real =
% LET N = length(P)-1 IN
% sigma[upto(N)](1,N,(LAMBDA (n: upto(N)):
% Rie_sec(a,b,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,P,xis,f) = Rie_sum_alt(a,b,P,xis,f)
% Riemann_sum?(a:T,b:{x:T|a<x},P:partition(a,b),f:[T->real])(S:real): bool =
% (EXISTS (xis: (xis?(a,b,P))): LET N = length(P)-1 IN
% S = Rie_sum(a,b,P,xis,f))
integral?(a:T,b:{x:T|a<x},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,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: below(length(P)-1)): real =
% x_in(P(ii), P(ii+1))
% gen_xis(a:T,b:{x:T|a<x},P: partition(a,b)): (xis?(a,b,P)) =
% (# length := length(P) - 1,
% seq := pick_one(a,b,P)
% #)
% N: VAR above(1)
% len_eq_part : LEMMA a < b IMPLIES length(eq_partition(a,b,N)) = N
% eq_part_lem_a: LEMMA a < b IMPLIES seq(eq_partition(a,b,N))(0) = a
% eq_part_lem_b: LEMMA a < b IMPLIES seq(eq_partition(a,b,N))(N-1) = b
% width_eq_part: LEMMA a < b IMPLIES
% width(a,b,eq_partition(a,b,N)) = (b-a)/(N-1)
% delta: VAR posreal
% N_from_delta: LEMMA a < b IMPLIES
% LET N = 2 + floor((b - a) / delta),
% EP = eq_partition(a, b, N) IN
% width(a, b, EP) < delta
% Riemann?_Rie: LEMMA a < b IMPLIES FORALL (P: partition(a,b),
% xis: (xis?(a,b,P))):
% Riemann_sum?(a,b,P,f)(Rie_sum(a,b,P,xis,f))
% AUTO_REWRITE+ Riemann?_Rie
% A1, A2: VAR real
% integral_unique: LEMMA a < b AND integral?(a,b,f,A1) AND
% integral?(a,b,f,A2) IMPLIES A1 = A2
integrable?(a:T,b:{x:T|a<x},f:[T->real]): bool =
(EXISTS (S: real): integral?(a,b,f,S))
integral(a:T,b:{x:T|a<x}, ff: { f | integrable?(a,b,f)} ):
{S: real | integral?(a,b,ff,S)}
% s: VAR real
% integral_def: LEMMA a < b IMPLIES
% ( (integrable?(a,b,f) AND integral(a,b,f) = s)
% IFF integral?(a,b,f,s) )
integral_restrict_eq: AXIOM a < b AND
(FORALL x: a <= x AND x <= b IMPLIES
f(x) = g(x)) AND
integrable?(a,b,f)
IMPLIES integrable?(a,b,g) AND
integral(a,b,g) = integral(a,b,f)
Integrable?(a:T,b:T,f:[T->real]): bool % = (a = b) OR
% (a < b AND integrable?(a,b,f)) OR
% (b < a AND integrable?(b,a,f))
Integrable_funs(a,b): TYPE = { f | Integrable?(a,b,f)}
% Integral?(a:T,b:T,f:[T->real],S:real): bool = (a = b AND S = 0) OR
% (a < b AND integral?(a,b,f,S))
Integral(a:T,b:T,f:Integrable_funs(a,b)): real % =
% IF a = b THEN 0
% ELSIF a < b THEN integral(a,b,f)
% ELSE -integral(b,a,f)
% ENDIF
Integrable?_rew : AXIOM a < b AND Integrable?(a,b,f) IMPLIES integrable?(a,b,f)
Integral_rew : AXIOM a < b AND Integrable?(a,b,f) IMPLIES Integral(a,b,f) = integral(a,b,f)
END integral_def
% Note: The definition of Riemann_sum:
%
%
% Rie_sum(a:T,b:{x:T|a<x},P:partition(a,b),
% xis: (xis?(a,b,P)),f:[T->real]): real =
% LET N = length(P)-1 IN
% sigma[below(N)](0,N-1,(LAMBDA (n: below(N)):
% (P(n+1) - P(n)) * f(xis(n))))
%
% uses sigma[below] rather than sigma[nat]. This produces some
% additional tedium, but makes it unnecessary to surround the
% LAMBDA expression with an IF for type-correctness
%
% Rie_sum(a:T,b:{x:T|a<x},P:partition(a,b),
% xis: (xis?(a,b,P)),f:[T->real]): real =
% LET N = length(P)-1 IN
% sigma[nat](0,N-1,(LAMBDA (n: nat): IF n < N THEN
% (P(n+1) - P(n)) * f(xis(n))
% ELSE 0
% ENDIF))
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