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integral_prep[T: TYPE FROM real]: THEORY
%------------------------------------------------------------------------------
%
% Basic Properties of Integral (Introduction to Analysis (Maxwell Rosenlicht))
%
% Author: Rick Butler NASA Langley Research Center
%------------------------------------------------------------------------------
BEGIN
ASSUMING
IMPORTING deriv_domain_def[T]
connected_domain : ASSUMPTION connected?[T]
not_one_element : ASSUMPTION not_one_element?[T]
ENDASSUMING
IMPORTING integral_def[T], reals@real_fun_ops
a,b,c,x,y,z: VAR T
D,m,M,v1,v2,cc,RS1,RS2: VAR real
f,g,G: VAR [T -> real]
integral_const_fun: LEMMA a < b IMPLIES integrable?(a,b,const_fun(D))
AND integral(a, b, const_fun[T](D)) = D*(b-a)
integral_const_restrict: LEMMA a < b AND
(FORALL (x: closed_interval(a,b)): f(x) = D) IMPLIES
integral?(a,b,f,D*(b-a))
IMPORTING continuous_functions[T], convergence_sequences
% ------------ Linearity Properties of Integral
integral_scal: LEMMA a < b AND integrable?(a,b,f) IMPLIES
integrable?(a,b,D*f) AND
integral(a,b,D*f) = D*integral(a,b,f)
integral_sum: LEMMA a < b AND integrable?(a,b,f) AND integrable?(a,b,g)
IMPLIES
integrable?(a,b,(LAMBDA x: f(x) + g(x))) AND
integral(a,b,(LAMBDA x: f(x) + g(x))) =
integral(a,b,f) + integral(a,b,g)
integral?_sum: LEMMA a < b AND integral?(a,b,f,v1) AND
integral?(a,b,g,v2)
IMPLIES
integral?(a,b,(LAMBDA x: f(x) + g(x)),v1+v2)
integral_diff: LEMMA a < b AND integrable?(a,b,f) AND integrable?(a,b,g)
IMPLIES
integrable?(a,b,(LAMBDA x: f(x) - g(x))) AND
integral(a,b,(LAMBDA x: f(x) - g(x))) =
integral(a,b,f) - integral(a,b,g)
integral_ge_0: LEMMA a < b AND integrable?(a,b,f) AND
(FORALL (x: closed_interval(a,b)): f(x) >= 0) IMPLIES
integral(a,b,f) >= 0
integral_jmp: LEMMA a < b AND a <= z AND z <= b AND f(z) = cc AND % Example2
(FORALL x: x /= z IMPLIES f(x) = 0) IMPLIES
integrable?(a,b,f) AND integral(a,b,f) = 0
yv: VAR real
integral_chg_one_pt: LEMMA a < b IMPLIES
FORALL y: a <= y AND y <= b AND
integrable?(a,b,f)
IMPLIES integrable?(a,b,f WITH [(y) := yv]) AND
integral(a,b,f) = integral(a,b,f WITH [(y) := yv])
integral_restr_eq: LEMMA 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)
integral_bound: LEMMA a < b AND integrable?(a,b,f) AND
(FORALL (x: closed_interval(a,b)):
m <= f(x) AND f(x) <= M )
IMPLIES m*(b-a) <= integral(a,b,f) AND integral(a,b,f) <= M*(b-a)
integral_bound_abs: LEMMA a < b AND integrable?(a,b,f) AND
(FORALL (x: closed_interval(a,b)):
abs(f(x)) <= M )
IMPLIES abs(integral(a,b,f)) <= M*(b-a)
integral_le : LEMMA a < b AND integrable?(a,b,f) AND integrable?(a,b,g)
AND (FORALL (x: closed_interval(a,b)): f(x) <= g(x))
IMPLIES
integral(a,b,f) <= integral(a,b,g)
% One direction of Rosenlicht Lemma 1 on page 118
eps, delta: VAR posreal
Lemma_1: LEMMA a < b IMPLIES
(integrable?(a,b,f) IMPLIES
(FORALL eps: EXISTS delta:
(FORALL (P1,P2: partition[T](a,b),
xis1:(xis?(a,b,P1)),
xis2:(xis?(a,b,P2))):
LET S1 = Rie_sum(a,b,P1,xis1,f),
S2 = Rie_sum(a,b,P2,xis2,f) IN
(width(a,b,P1) < delta AND
width(a,b,P2) < delta)
IMPLIES
abs(S1 - S2) < eps )))
integrable_lem: THEOREM a < b IMPLIES %% Lemma 1 page 118
(integrable?(a,b,f) IFF
(FORALL (epsi: posreal): (EXISTS (delta: posreal):
(FORALL (P1,P2: partition(a,b)):
width(a,b,P1) < delta AND
width(a,b,P2) < delta IMPLIES
(FORALL (RS1: (Riemann_sum?(a,b,P1,f)),
RS2: (Riemann_sum?(a,b,P2,f))):
abs(RS1 - RS2) < epsi )))))
gxis(a:T, b:{x:T|a<x}, P: partition[T](a,b), j: below(length(P)-1),
flag: bool,xx: closed_interval(P(j),P(j+1))):
(xis?(a,b,P)) = (LAMBDA (ii: below(length(P)-1)): P(ii))
WITH [(j) := IF flag then P(j)
ELSE xx
ENDIF ]
END integral_prep
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