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integral_split_scaf[T: TYPE FROM real]: THEORY
%------------------------------------------------------------------------------
%
% Scaffolding : Rick Butler
%
%------------------------------------------------------------------------------
BEGIN
ASSUMING
IMPORTING deriv_domain_def[T]
connected_domain : ASSUMPTION connected?[T]
not_one_element : ASSUMPTION not_one_element?[T]
ENDASSUMING
IMPORTING reals@sigma_below,
step_fun_def[T],
integral_step[T],
%% step_fun_props[T],
integral_bounded[T]
a,b,c,d,x,y,z: VAR T
f,f1,f2,g: VAR [T -> real]
eps, delta: VAR posreal
xv,yv: VAR real
diff_step_is_step_on: LEMMA a < b IMPLIES FORALL (P: partition[T](a,b)):
step_function_on?(a, b, f, P) AND
step_function_on?(a, b, g, P)
IMPLIES
step_function_on?(a, b, f - g, P)
se_not_on: LEMMA FORALL (P: partition[T](a,b)):
FORALL (ii,jj: below(length(P) - 1)):
FORALL (x: open_interval[T](seq(P)(ii), seq(P)(1 + ii))):
x /= seq(P)(jj)
find_P_sec(a:T, b:{x:T|a<x}, P: partition[T](a,b), xx: closed_interval(a,b)):
{ii: below(length(P)-1) |
seq(P)(ii) <= xx AND xx <= seq(P)(ii+1)}
find_P_sec_lem: LEMMA a < b IMPLIES (FORALL (xx: closed_interval(a,b)):
FORALL (P: partition[T](a,b)):
LET JJ = find_P_sec(a,b,P,xx) IN
seq(P)(JJ) <= xx AND xx <= seq(P)(JJ+1) )
find_P_sec_in: LEMMA a < b IMPLIES
FORALL (P: partition[T](a,b)):
FORALL (ii: below(length(P) - 1)):
FORALL (x: open_interval[T](seq(P)(ii),seq(P)(1 + ii))):
find_P_sec(a, b, P, x) = ii
gen3P(a,b,c): finite_sequence[T] =
(# length := 3,
seq := (LAMBDA (ii: below(3)):
IF ii = 0 THEN a
ELSIF ii = 1 then b
ELSE c ENDIF) #)
F1(a:T,b:{x:T|a<x}, P: partition[T](a,b),
f: (bnded_on?(a,b)))(x:T): real =
IF x < a OR x > b THEN 0
ELSE
LET JJ = find_P_sec(a,b,P,x) IN
IF x = P(JJ) OR x = P(JJ+1) THEN MIN_ALL(a,b,P,f)
ELSE MINj(a,b,P,JJ,f)
ENDIF
ENDIF
F2(a:T,b:{x:T|a<x}, P: partition[T](a,b),
f: (bnded_on?(a,b)))(x:T): real =
IF x < a OR x > b THEN 0
ELSE
LET JJ = find_P_sec(a,b,P,x)IN
IF x = P(JJ) OR x = P(JJ+1) THEN MAX_ALL(a,b,P,f)
ELSE MAXj(a,b,P,JJ,f)
ENDIF
ENDIF
F1_F2_lem: LEMMA a < b AND integrable?(a,b,f) IMPLIES
FORALL (P: partition[T](a, b)):
step_function_on?(a,b,F1(a,b,P,f),P) AND
step_function_on?(a,b,F2(a,b,P,f),P) AND
(FORALL (x: closed_interval(a,b)):
F1(a,b,P,f)(x) <= f(x) AND f(x) <= F2(a,b,P,f)(x))
epsilon_is_0: LEMMA (FORALL eps: abs(xv) < eps) IMPLIES xv = 0
F2_F1_step_function_on: LEMMA a < b AND integrable?(a,b,f) IMPLIES
FORALL (P: partition[T](a, b)):
LET N = length(P) - 1 IN
step_function_on?[T](a, b, F2(a, b, P, f) - F1(a, b, P, f),P)
integrable_F2_F1: LEMMA a < b AND integrable?(a,b,f) IMPLIES
FORALL (P: partition[T](a, b)):
LET N = length(P) - 1 IN
integrable?[T](a, b, F2(a, b, P, f) - F1(a, b, P, f))
integral_F2_F1: LEMMA a < b AND integrable?(a,b,f) IMPLIES
FORALL (P: partition[T](a, b)):
LET N = length(P) - 1 IN
integral(a,b,F2(a,b,P,f)-F1(a,b,P,f)) =
sigma(0,N-1, (LAMBDA (j: below(N)):
Let DP = P(j+1)-P(j),
DELj = MAXj(a,b,P,j,f) - MINj(a,b,P,j,f) IN
DELj* DP))
se_prep: LEMMA a < b IMPLIES
FORALL (P: partition[T](a, b),
f: (bounded_on_all?(a, b, P)),
j: below(length(P) - 1),
nu: posreal):
(EXISTS (xp: closed_interval[T](P`seq(j),P`seq(j+1))):
f(xp) < MINj(a,b,P,j,f) + nu) AND
(EXISTS (xpp: closed_interval[T](P`seq(j),P`seq(j+1))):
f(xpp) > MAXj(a,b,P,j,f) - nu)
get_xp(a:T,b:{x:T|a<x}, P: partition[T](a,b), f: (bounded_on_all?(a,b,P)),
j: below(length(P) - 1), nu: posreal):
{xp: closed_interval[T](P`seq(j),P`seq(j+1)) |
f(xp) < MINj(a,b,P,j,f) + nu}
get_xpp(a:T,b:{x:T|a<x}, P: partition[T](a,b), f: (bounded_on_all?(a,b,P)),
j: below(length(P) - 1), nu: posreal):
{xpp: closed_interval[T](P`seq(j),P`seq(j+1)) |
f(xpp) > MAXj(a,b,P,j,f) - nu}
lt_via_epsi: LEMMA FORALL (x,a: real):
(FORALL (epsi: posreal): x < a + epsi)
IMPLIES x <= a
sigma_all_parts: LEMMA a < b IMPLIES
FORALL (P: partition[T](a, b)):
sigma(0,length(P)-2,(LAMBDA (j: below(length(P) - 1)): P(j+1) - P(j)))
= b - a
steps_exist: LEMMA % Proposition of Rosenlicht on pg 120 (reverse dir.)
% See step_to_integrable in integral_prep for other
a < b IMPLIES
(integrable?(a,b,f) IMPLIES
(FORALL eps: EXISTS (f1,f2):
step_function?(a,b,f1) AND
step_function?(a,b,f2) AND
(FORALL (x: closed_interval(a,b)):
f1(x) <= f(x) AND f(x) <= f2(x)) AND
integrable?(a,b,f2-f1) AND
integral(a,b,f2-f1) < eps))
partition_exists: LEMMA a < b IMPLIES
(EXISTS (P: partition[T](a,b)): width(a,b,P) < delta)
END integral_split_scaf
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