section ‹Kurzweil-Henstock Gauge Integration in many dimensions.›
theory Integration
imports
Derivative
Uniform_Limit
"~~/src/HOL/Library/Indicator_Function"
begin
lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one
subsection ‹Sundries›
lemma conjunctD2: assumes "a ∧ b" shows a b using assms by auto
lemma conjunctD3: assumes "a ∧ b ∧ c" shows a b c using assms by auto
lemma conjunctD4: assumes "a ∧ b ∧ c ∧ d" shows a b c d using assms by auto
declare norm_triangle_ineq4[intro]
lemma simple_image: "{f x |x . x ∈ s} = f ` s"
by blast
lemma linear_simps:
assumes "bounded_linear f"
shows
"f (a + b) = f a + f b"
"f (a - b) = f a - f b"
"f 0 = 0"
"f (- a) = - f a"
"f (s *⇩R v) = s *⇩R (f v)"
proof -
interpret f: bounded_linear f by fact
show "f (a + b) = f a + f b" by (rule f.add)
show "f (a - b) = f a - f b" by (rule f.diff)
show "f 0 = 0" by (rule f.zero)
show "f (- a) = - f a" by (rule f.minus)
show "f (s *⇩R v) = s *⇩R (f v)" by (rule f.scaleR)
qed
lemma bounded_linearI:
assumes "⋀x y. f (x + y) = f x + f y"
and "⋀r x. f (r *⇩R x) = r *⇩R f x"
and "⋀x. norm (f x) ≤ norm x * K"
shows "bounded_linear f"
using assms by (rule bounded_linear_intro)
lemma bounded_linear_component [intro]: "bounded_linear (λx::'a::euclidean_space. x ∙ k)"
by (rule bounded_linear_inner_left)
lemma transitive_stepwise_lt_eq:
assumes "(⋀x y z::nat. R x y ⟹ R y z ⟹ R x z)"
shows "((∀m. ∀n>m. R m n) ⟷ (∀n. R n (Suc n)))"
(is "?l = ?r")
proof safe
assume ?r
fix n m :: nat
assume "m < n"
then show "R m n"
proof (induct n arbitrary: m)
case 0
then show ?case by auto
next
case (Suc n)
show ?case
proof (cases "m < n")
case True
show ?thesis
apply (rule assms[OF Suc(1)[OF True]])
using ‹?r›
apply auto
done
next
case False
then have "m = n"
using Suc(2) by auto
then show ?thesis
using ‹?r› by auto
qed
qed
qed auto
lemma transitive_stepwise_gt:
assumes "⋀x y z. R x y ⟹ R y z ⟹ R x z" "⋀n. R n (Suc n)"
shows "∀n>m. R m n"
proof -
have "∀m. ∀n>m. R m n"
apply (subst transitive_stepwise_lt_eq)
apply (blast intro: assms)+
done
then show ?thesis by auto
qed
lemma transitive_stepwise_le_eq:
assumes "⋀x. R x x" "⋀x y z. R x y ⟹ R y z ⟹ R x z"
shows "(∀m. ∀n≥m. R m n) ⟷ (∀n. R n (Suc n))"
(is "?l = ?r")
proof safe
assume ?r
fix m n :: nat
assume "m ≤ n"
then show "R m n"
proof (induct n arbitrary: m)
case 0
with assms show ?case by auto
next
case (Suc n)
show ?case
proof (cases "m ≤ n")
case True
with Suc.hyps ‹∀n. R n (Suc n)› assms show ?thesis
by blast
next
case False
then have "m = Suc n"
using Suc(2) by auto
then show ?thesis
using assms(1) by auto
qed
qed
qed auto
lemma transitive_stepwise_le:
assumes "⋀x. R x x" "⋀x y z. R x y ⟹ R y z ⟹ R x z"
and "⋀n. R n (Suc n)"
shows "∀n≥m. R m n"
proof -
have "∀m. ∀n≥m. R m n"
apply (subst transitive_stepwise_le_eq)
apply (blast intro: assms)+
done
then show ?thesis by auto
qed
subsection ‹Some useful lemmas about intervals.›
lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
using nonempty_Basis
by (fastforce simp add: set_eq_iff mem_box)
lemma interior_subset_union_intervals:
assumes "i = cbox a b"
and "j = cbox c d"
and "interior j ≠ {}"
and "i ⊆ j ∪ s"
and "interior i ∩ interior j = {}"
shows "interior i ⊆ interior s"
proof -
have "box a b ∩ cbox c d = {}"
using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
unfolding assms(1,2) interior_cbox by auto
moreover
have "box a b ⊆ cbox c d ∪ s"
apply (rule order_trans,rule box_subset_cbox)
using assms(4) unfolding assms(1,2)
apply auto
done
ultimately
show ?thesis
unfolding assms interior_cbox
by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI)
qed
lemma inter_interior_unions_intervals:
fixes f::"('a::euclidean_space) set set"
assumes "finite f"
and "open s"
and "∀t∈f. ∃a b. t = cbox a b"
and "∀t∈f. s ∩ (interior t) = {}"
shows "s ∩ interior (⋃f) = {}"
proof (clarsimp simp only: all_not_in_conv [symmetric])
fix x
assume x: "x ∈ s" "x ∈ interior (⋃f)"
have lem1: "⋀x e s U. ball x e ⊆ s ∩ interior U ⟷ ball x e ⊆ s ∩ U"
using interior_subset
by auto (meson Topology_Euclidean_Space.open_ball contra_subsetD interior_maximal mem_ball)
have "∃t∈f. ∃x. ∃e>0. ball x e ⊆ s ∩ t"
if "finite f" and "∀t∈f. ∃a b. t = cbox a b" and "∃x. x ∈ s ∩ interior (⋃f)" for f
using that
proof (induct rule: finite_induct)
case empty
obtain x where "x ∈ s ∩ interior (⋃{})"
using empty(2) ..
then have False
unfolding Union_empty interior_empty by auto
then show ?case by auto
next
case (insert i f)
obtain x where x: "x ∈ s ∩ interior (⋃insert i f)"
using insert(5) ..
then obtain e where e: "0 < e ∧ ball x e ⊆ s ∩ interior (⋃insert i f)"
unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior], rule_format] ..
obtain a where "∃b. i = cbox a b"
using insert(4)[rule_format,OF insertI1] ..
then obtain b where ab: "i = cbox a b" ..
show ?case
proof (cases "x ∈ i")
case False
then have "x ∈ UNIV - cbox a b"
unfolding ab by auto
then obtain d where "0 < d ∧ ball x d ⊆ UNIV - cbox a b"
unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_cbox],rule_format] ..
then have "0 < d" "ball x (min d e) ⊆ UNIV - i"
unfolding ab ball_min_Int by auto
then have "ball x (min d e) ⊆ s ∩ interior (⋃f)"
using e unfolding lem1 unfolding ball_min_Int by auto
then have "x ∈ s ∩ interior (⋃f)" using ‹d>0› e by auto
then have "∃t∈f. ∃x e. 0 < e ∧ ball x e ⊆ s ∩ t"
using insert.hyps(3) insert.prems(1) by blast
then show ?thesis by auto
next
case True show ?thesis
proof (cases "x∈box a b")
case True
then obtain d where "0 < d ∧ ball x d ⊆ box a b"
unfolding open_contains_ball_eq[OF open_box,rule_format] ..
then show ?thesis
apply (rule_tac x=i in bexI, rule_tac x=x in exI, rule_tac x="min d e" in exI)
unfolding ab
using box_subset_cbox[of a b] and e
apply fastforce+
done
next
case False
then obtain k where "x∙k ≤ a∙k ∨ x∙k ≥ b∙k" and k: "k ∈ Basis"
unfolding mem_box by (auto simp add: not_less)
then have "x∙k = a∙k ∨ x∙k = b∙k"
using True unfolding ab and mem_box
apply (erule_tac x = k in ballE)
apply auto
done
then have "∃x. ball x (e/2) ⊆ s ∩ (⋃f)"
proof (rule disjE)
let ?z = "x - (e/2) *⇩R k"
assume as: "x∙k = a∙k"
have "ball ?z (e / 2) ∩ i = {}"
proof (clarsimp simp only: all_not_in_conv [symmetric])
fix y
assume "y ∈ ball ?z (e / 2)" and yi: "y ∈ i"
then have "dist ?z y < e/2" by auto
then have "¦(?z - y) ∙ k¦ < e/2"
using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
then have "y∙k < a∙k"
using e k
by (auto simp add: field_simps abs_less_iff as inner_simps)
then have "y ∉ i"
unfolding ab mem_box by (auto intro!: bexI[OF _ k])
then show False using yi by auto
qed
moreover
have "ball ?z (e/2) ⊆ s ∩ (⋃insert i f)"
apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
proof
fix y
assume as: "y ∈ ball ?z (e/2)"
have "norm (x - y) ≤ ¦e¦ / 2 + norm (x - y - (e / 2) *⇩R k)"
apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *⇩R k"])
unfolding norm_scaleR norm_Basis[OF k]
apply auto
done
also have "… < ¦e¦ / 2 + ¦e¦ / 2"
apply (rule add_strict_left_mono)
using as e
apply (auto simp add: field_simps dist_norm)
done
finally show "y ∈ ball x e"
unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
qed
ultimately show ?thesis
apply (rule_tac x="?z" in exI)
unfolding Union_insert
apply auto
done
next
let ?z = "x + (e/2) *⇩R k"
assume as: "x∙k = b∙k"
have "ball ?z (e / 2) ∩ i = {}"
proof (clarsimp simp only: all_not_in_conv [symmetric])
fix y
assume "y ∈ ball ?z (e / 2)" and yi: "y ∈ i"
then have "dist ?z y < e/2"
by auto
then have "¦(?z - y) ∙ k¦ < e/2"
using Basis_le_norm[OF k, of "?z - y"]
unfolding dist_norm by auto
then have "y∙k > b∙k"
using e k
by (auto simp add:field_simps inner_simps inner_Basis as)
then have "y ∉ i"
unfolding ab mem_box by (auto intro!: bexI[OF _ k])
then show False using yi by auto
qed
moreover
have "ball ?z (e/2) ⊆ s ∩ (⋃insert i f)"
apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
proof
fix y
assume as: "y∈ ball ?z (e/2)"
have "norm (x - y) ≤ ¦e¦ / 2 + norm (x - y + (e / 2) *⇩R k)"
apply (rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *⇩R k"])
unfolding norm_scaleR
apply (auto simp: k)
done
also have "… < ¦e¦ / 2 + ¦e¦ / 2"
apply (rule add_strict_left_mono)
using as unfolding mem_ball dist_norm
using e apply (auto simp add: field_simps)
done
finally show "y ∈ ball x e"
unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
qed
ultimately show ?thesis
apply (rule_tac x="?z" in exI)
unfolding Union_insert
apply auto
done
qed
then obtain x where "ball x (e / 2) ⊆ s ∩ ⋃f" ..
then have "x ∈ s ∩ interior (⋃f)"
unfolding lem1[where U="⋃f", symmetric]
using centre_in_ball e by auto
then show ?thesis
using insert.hyps(3) insert.prems(1) by blast
qed
qed
qed
from this[OF assms(1,3)] x
obtain t x e where "t ∈ f" "0 < e" "ball x e ⊆ s ∩ t"
by blast
then have "x ∈ s" "x ∈ interior t"
using open_subset_interior[OF open_ball, of x e t]
by auto
then show False
using ‹t ∈ f› assms(4) by auto
qed
subsection ‹Bounds on intervals where they exist.›
definition interval_upperbound :: "('a::euclidean_space) set ⇒ 'a"
where "interval_upperbound s = (∑i∈Basis. (SUP x:s. x∙i) *⇩R i)"
definition interval_lowerbound :: "('a::euclidean_space) set ⇒ 'a"
where "interval_lowerbound s = (∑i∈Basis. (INF x:s. x∙i) *⇩R i)"
lemma interval_upperbound[simp]:
"∀i∈Basis. a∙i ≤ b∙i ⟹
interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
unfolding interval_upperbound_def euclidean_representation_setsum cbox_def
by (safe intro!: cSup_eq) auto
lemma interval_lowerbound[simp]:
"∀i∈Basis. a∙i ≤ b∙i ⟹
interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
unfolding interval_lowerbound_def euclidean_representation_setsum cbox_def
by (safe intro!: cInf_eq) auto
lemmas interval_bounds = interval_upperbound interval_lowerbound
lemma
fixes X::"real set"
shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
by (auto simp: interval_upperbound_def interval_lowerbound_def)
lemma interval_bounds'[simp]:
assumes "cbox a b ≠ {}"
shows "interval_upperbound (cbox a b) = b"
and "interval_lowerbound (cbox a b) = a"
using assms unfolding box_ne_empty by auto
lemma interval_upperbound_Times:
assumes "A ≠ {}" and "B ≠ {}"
shows "interval_upperbound (A × B) = (interval_upperbound A, interval_upperbound B)"
proof-
from assms have fst_image_times': "A = fst ` (A × B)" by simp
have "(∑i∈Basis. (SUP x:A × B. x ∙ (i, 0)) *⇩R i) = (∑i∈Basis. (SUP x:A. x ∙ i) *⇩R i)"
by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
moreover from assms have snd_image_times': "B = snd ` (A × B)" by simp
have "(∑i∈Basis. (SUP x:A × B. x ∙ (0, i)) *⇩R i) = (∑i∈Basis. (SUP x:B. x ∙ i) *⇩R i)"
by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
ultimately show ?thesis unfolding interval_upperbound_def
by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
qed
lemma interval_lowerbound_Times:
assumes "A ≠ {}" and "B ≠ {}"
shows "interval_lowerbound (A × B) = (interval_lowerbound A, interval_lowerbound B)"
proof-
from assms have fst_image_times': "A = fst ` (A × B)" by simp
have "(∑i∈Basis. (INF x:A × B. x ∙ (i, 0)) *⇩R i) = (∑i∈Basis. (INF x:A. x ∙ i) *⇩R i)"
by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
moreover from assms have snd_image_times': "B = snd ` (A × B)" by simp
have "(∑i∈Basis. (INF x:A × B. x ∙ (0, i)) *⇩R i) = (∑i∈Basis. (INF x:B. x ∙ i) *⇩R i)"
by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
ultimately show ?thesis unfolding interval_lowerbound_def
by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
qed
subsection ‹Content (length, area, volume...) of an interval.›
definition "content (s::('a::euclidean_space) set) =
(if s = {} then 0 else (∏i∈Basis. (interval_upperbound s)∙i - (interval_lowerbound s)∙i))"
lemma interval_not_empty: "∀i∈Basis. a∙i ≤ b∙i ⟹ cbox a b ≠ {}"
unfolding box_eq_empty unfolding not_ex not_less by auto
lemma content_cbox:
fixes a :: "'a::euclidean_space"
assumes "∀i∈Basis. a∙i ≤ b∙i"
shows "content (cbox a b) = (∏i∈Basis. b∙i - a∙i)"
using interval_not_empty[OF assms]
unfolding content_def
by auto
lemma content_cbox':
fixes a :: "'a::euclidean_space"
assumes "cbox a b ≠ {}"
shows "content (cbox a b) = (∏i∈Basis. b∙i - a∙i)"
using assms box_ne_empty(1) content_cbox by blast
lemma content_real: "a ≤ b ⟹ content {a..b} = b - a"
by (auto simp: interval_upperbound_def interval_lowerbound_def content_def)
lemma abs_eq_content: "¦y - x¦ = (if x≤y then content {x .. y} else content {y..x})"
by (auto simp: content_real)
lemma content_singleton[simp]: "content {a} = 0"
proof -
have "content (cbox a a) = 0"
by (subst content_cbox) (auto simp: ex_in_conv)
then show ?thesis by (simp add: cbox_sing)
qed
lemma content_unit[iff]: "content(cbox 0 (One::'a::euclidean_space)) = 1"
proof -
have *: "∀i∈Basis. (0::'a)∙i ≤ (One::'a)∙i"
by auto
have "0 ∈ cbox 0 (One::'a)"
unfolding mem_box by auto
then show ?thesis
unfolding content_def interval_bounds[OF *] using setprod.neutral_const by auto
qed
lemma content_pos_le[intro]:
fixes a::"'a::euclidean_space"
shows "0 ≤ content (cbox a b)"
proof (cases "cbox a b = {}")
case False
then have *: "∀i∈Basis. a ∙ i ≤ b ∙ i"
unfolding box_ne_empty .
have "0 ≤ (∏i∈Basis. interval_upperbound (cbox a b) ∙ i - interval_lowerbound (cbox a b) ∙ i)"
apply (rule setprod_nonneg)
unfolding interval_bounds[OF *]
using *
apply auto
done
also have "… = content (cbox a b)" using False by (simp add: content_def)
finally show ?thesis .
qed (simp add: content_def)
corollary content_nonneg [simp]:
fixes a::"'a::euclidean_space"
shows "~ content (cbox a b) < 0"
using not_le by blast
lemma content_pos_lt:
fixes a :: "'a::euclidean_space"
assumes "∀i∈Basis. a∙i < b∙i"
shows "0 < content (cbox a b)"
using assms
by (auto simp: content_def box_eq_empty intro!: setprod_pos)
lemma content_eq_0:
"content (cbox a b) = 0 ⟷ (∃i∈Basis. b∙i ≤ a∙i)"
by (auto simp: content_def box_eq_empty intro!: setprod_pos bexI)
lemma cond_cases: "(P ⟹ Q x) ⟹ (¬ P ⟹ Q y) ⟹ Q (if P then x else y)"
by auto
lemma content_cbox_cases:
"content (cbox a (b::'a::euclidean_space)) =
(if ∀i∈Basis. a∙i ≤ b∙i then setprod (λi. b∙i - a∙i) Basis else 0)"
by (auto simp: not_le content_eq_0 intro: less_imp_le content_cbox)
lemma content_eq_0_interior: "content (cbox a b) = 0 ⟷ interior(cbox a b) = {}"
unfolding content_eq_0 interior_cbox box_eq_empty
by auto
lemma content_pos_lt_eq:
"0 < content (cbox a (b::'a::euclidean_space)) ⟷ (∀i∈Basis. a∙i < b∙i)"
proof (rule iffI)
assume "0 < content (cbox a b)"
then have "content (cbox a b) ≠ 0" by auto
then show "∀i∈Basis. a∙i < b∙i"
unfolding content_eq_0 not_ex not_le by fastforce
next
assume "∀i∈Basis. a ∙ i < b ∙ i"
then show "0 < content (cbox a b)"
by (metis content_pos_lt)
qed
lemma content_empty [simp]: "content {} = 0"
unfolding content_def by auto
lemma content_real_if [simp]: "content {a..b} = (if a ≤ b then b - a else 0)"
by (simp add: content_real)
lemma content_subset:
assumes "cbox a b ⊆ cbox c d"
shows "content (cbox a b) ≤ content (cbox c d)"
proof (cases "cbox a b = {}")
case True
then show ?thesis
using content_pos_le[of c d] by auto
next
case False
then have ab_ne: "∀i∈Basis. a ∙ i ≤ b ∙ i"
unfolding box_ne_empty by auto
then have ab_ab: "a∈cbox a b" "b∈cbox a b"
unfolding mem_box by auto
have "cbox c d ≠ {}" using assms False by auto
then have cd_ne: "∀i∈Basis. c ∙ i ≤ d ∙ i"
using assms unfolding box_ne_empty by auto
have "⋀i. i ∈ Basis ⟹ 0 ≤ b ∙ i - a ∙ i"
using ab_ne by auto
moreover
have "⋀i. i ∈ Basis ⟹ b ∙ i - a ∙ i ≤ d ∙ i - c ∙ i"
using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2)]
assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(1)]
by (metis diff_mono)
ultimately show ?thesis
unfolding content_def interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
by (simp add: setprod_mono if_not_P[OF False] if_not_P[OF ‹cbox c d ≠ {}›])
qed
lemma content_lt_nz: "0 < content (cbox a b) ⟷ content (cbox a b) ≠ 0"
unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
lemma content_times[simp]: "content (A × B) = content A * content B"
proof (cases "A × B = {}")
let ?ub1 = "interval_upperbound" and ?lb1 = "interval_lowerbound"
let ?ub2 = "interval_upperbound" and ?lb2 = "interval_lowerbound"
assume nonempty: "A × B ≠ {}"
hence "content (A × B) = (∏i∈Basis. (?ub1 A, ?ub2 B) ∙ i - (?lb1 A, ?lb2 B) ∙ i)"
unfolding content_def by (simp add: interval_upperbound_Times interval_lowerbound_Times)
also have "... = content A * content B" unfolding content_def using nonempty
apply (subst Basis_prod_def, subst setprod.union_disjoint, force, force, force, simp)
apply (subst (1 2) setprod.reindex, auto intro: inj_onI)
done
finally show ?thesis .
qed (auto simp: content_def)
lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
by (simp add: cbox_Pair_eq)
lemma content_cbox_pair_eq0_D:
"content (cbox (a,c) (b,d)) = 0 ⟹ content (cbox a b) = 0 ∨ content (cbox c d) = 0"
by (simp add: content_Pair)
lemma content_eq_0_gen:
fixes s :: "'a::euclidean_space set"
assumes "bounded s"
shows "content s = 0 ⟷ (∃i∈Basis. ∃v. ∀x ∈ s. x ∙ i = v)" (is "_ = ?rhs")
proof safe
assume "content s = 0" then show ?rhs
apply (clarsimp simp: ex_in_conv content_def split: if_split_asm)
apply (rule_tac x=a in bexI)
apply (rule_tac x="interval_lowerbound s ∙ a" in exI)
apply (clarsimp simp: interval_upperbound_def interval_lowerbound_def)
apply (drule cSUP_eq_cINF_D)
apply (auto simp: bounded_inner_imp_bdd_above [OF assms] bounded_inner_imp_bdd_below [OF assms])
done
next
fix i a
assume "i ∈ Basis" "∀x∈s. x ∙ i = a"
then show "content s = 0"
apply (clarsimp simp: content_def)
apply (rule_tac x=i in bexI)
apply (auto simp: interval_upperbound_def interval_lowerbound_def)
done
qed
lemma content_0_subset_gen:
fixes a :: "'a::euclidean_space"
assumes "content t = 0" "s ⊆ t" "bounded t" shows "content s = 0"
proof -
have "bounded s"
using assms by (metis bounded_subset)
then show ?thesis
using assms
by (auto simp: content_eq_0_gen)
qed
lemma content_0_subset: "⟦content(cbox a b) = 0; s ⊆ cbox a b⟧ ⟹ content s = 0"
by (simp add: content_0_subset_gen bounded_cbox)
subsection ‹The notion of a gauge --- simply an open set containing the point.›
definition "gauge d ⟷ (∀x. x ∈ d x ∧ open (d x))"
lemma gaugeI:
assumes "⋀x. x ∈ g x"
and "⋀x. open (g x)"
shows "gauge g"
using assms unfolding gauge_def by auto
lemma gaugeD[dest]:
assumes "gauge d"
shows "x ∈ d x"
and "open (d x)"
using assms unfolding gauge_def by auto
lemma gauge_ball_dependent: "∀x. 0 < e x ⟹ gauge (λx. ball x (e x))"
unfolding gauge_def by auto
lemma gauge_ball[intro]: "0 < e ⟹ gauge (λx. ball x e)"
unfolding gauge_def by auto
lemma gauge_trivial[intro!]: "gauge (λx. ball x 1)"
by (rule gauge_ball) auto
lemma gauge_inter[intro]: "gauge d1 ⟹ gauge d2 ⟹ gauge (λx. d1 x ∩ d2 x)"
unfolding gauge_def by auto
lemma gauge_inters:
assumes "finite s"
and "∀d∈s. gauge (f d)"
shows "gauge (λx. ⋂{f d x | d. d ∈ s})"
proof -
have *: "⋀x. {f d x |d. d ∈ s} = (λd. f d x) ` s"
by auto
show ?thesis
unfolding gauge_def unfolding *
using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
qed
lemma gauge_existence_lemma:
"(∀x. ∃d :: real. p x ⟶ 0 < d ∧ q d x) ⟷ (∀x. ∃d>0. p x ⟶ q d x)"
by (metis zero_less_one)
subsection ‹Divisions.›
definition division_of (infixl "division'_of" 40)
where
"s division_of i ⟷
finite s ∧
(∀k∈s. k ⊆ i ∧ k ≠ {} ∧ (∃a b. k = cbox a b)) ∧
(∀k1∈s. ∀k2∈s. k1 ≠ k2 ⟶ interior(k1) ∩ interior(k2) = {}) ∧
(⋃s = i)"
lemma division_ofD[dest]:
assumes "s division_of i"
shows "finite s"
and "⋀k. k ∈ s ⟹ k ⊆ i"
and "⋀k. k ∈ s ⟹ k ≠ {}"
and "⋀k. k ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀k1 k2. k1 ∈ s ⟹ k2 ∈ s ⟹ k1 ≠ k2 ⟹ interior(k1) ∩ interior(k2) = {}"
and "⋃s = i"
using assms unfolding division_of_def by auto
lemma division_ofI:
assumes "finite s"
and "⋀k. k ∈ s ⟹ k ⊆ i"
and "⋀k. k ∈ s ⟹ k ≠ {}"
and "⋀k. k ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀k1 k2. k1 ∈ s ⟹ k2 ∈ s ⟹ k1 ≠ k2 ⟹ interior k1 ∩ interior k2 = {}"
and "⋃s = i"
shows "s division_of i"
using assms unfolding division_of_def by auto
lemma division_of_finite: "s division_of i ⟹ finite s"
unfolding division_of_def by auto
lemma division_of_self[intro]: "cbox a b ≠ {} ⟹ {cbox a b} division_of (cbox a b)"
unfolding division_of_def by auto
lemma division_of_trivial[simp]: "s division_of {} ⟷ s = {}"
unfolding division_of_def by auto
lemma division_of_sing[simp]:
"s division_of cbox a (a::'a::euclidean_space) ⟷ s = {cbox a a}"
(is "?l = ?r")
proof
assume ?r
moreover
{ fix k
assume "s = {{a}}" "k∈s"
then have "∃x y. k = cbox x y"
apply (rule_tac x=a in exI)+
apply (force simp: cbox_sing)
done
}
ultimately show ?l
unfolding division_of_def cbox_sing by auto
next
assume ?l
note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
{
fix x
assume x: "x ∈ s" have "x = {a}"
using *(2)[rule_format,OF x] by auto
}
moreover have "s ≠ {}"
using *(4) by auto
ultimately show ?r
unfolding cbox_sing by auto
qed
lemma elementary_empty: obtains p where "p division_of {}"
unfolding division_of_trivial by auto
lemma elementary_interval: obtains p where "p division_of (cbox a b)"
by (metis division_of_trivial division_of_self)
lemma division_contains: "s division_of i ⟹ ∀x∈i. ∃k∈s. x ∈ k"
unfolding division_of_def by auto
lemma forall_in_division:
"d division_of i ⟹ (∀x∈d. P x) ⟷ (∀a b. cbox a b ∈ d ⟶ P (cbox a b))"
unfolding division_of_def by fastforce
lemma division_of_subset:
assumes "p division_of (⋃p)"
and "q ⊆ p"
shows "q division_of (⋃q)"
proof (rule division_ofI)
note * = division_ofD[OF assms(1)]
show "finite q"
using "*"(1) assms(2) infinite_super by auto
{
fix k
assume "k ∈ q"
then have kp: "k ∈ p"
using assms(2) by auto
show "k ⊆ ⋃q"
using ‹k ∈ q› by auto
show "∃a b. k = cbox a b"
using *(4)[OF kp] by auto
show "k ≠ {}"
using *(3)[OF kp] by auto
}
fix k1 k2
assume "k1 ∈ q" "k2 ∈ q" "k1 ≠ k2"
then have **: "k1 ∈ p" "k2 ∈ p" "k1 ≠ k2"
using assms(2) by auto
show "interior k1 ∩ interior k2 = {}"
using *(5)[OF **] by auto
qed auto
lemma division_of_union_self[intro]: "p division_of s ⟹ p division_of (⋃p)"
unfolding division_of_def by auto
lemma division_of_content_0:
assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
shows "∀k∈d. content k = 0"
unfolding forall_in_division[OF assms(2)]
by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
lemma division_inter:
fixes s1 s2 :: "'a::euclidean_space set"
assumes "p1 division_of s1"
and "p2 division_of s2"
shows "{k1 ∩ k2 | k1 k2 .k1 ∈ p1 ∧ k2 ∈ p2 ∧ k1 ∩ k2 ≠ {}} division_of (s1 ∩ s2)"
(is "?A' division_of _")
proof -
let ?A = "{s. s ∈ (λ(k1,k2). k1 ∩ k2) ` (p1 × p2) ∧ s ≠ {}}"
have *: "?A' = ?A" by auto
show ?thesis
unfolding *
proof (rule division_ofI)
have "?A ⊆ (λ(x, y). x ∩ y) ` (p1 × p2)"
by auto
moreover have "finite (p1 × p2)"
using assms unfolding division_of_def by auto
ultimately show "finite ?A" by auto
have *: "⋀s. ⋃{x∈s. x ≠ {}} = ⋃s"
by auto
show "⋃?A = s1 ∩ s2"
apply (rule set_eqI)
unfolding * and UN_iff
using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
apply auto
done
{
fix k
assume "k ∈ ?A"
then obtain k1 k2 where k: "k = k1 ∩ k2" "k1 ∈ p1" "k2 ∈ p2" "k ≠ {}"
by auto
then show "k ≠ {}"
by auto
show "k ⊆ s1 ∩ s2"
using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
unfolding k by auto
obtain a1 b1 where k1: "k1 = cbox a1 b1"
using division_ofD(4)[OF assms(1) k(2)] by blast
obtain a2 b2 where k2: "k2 = cbox a2 b2"
using division_ofD(4)[OF assms(2) k(3)] by blast
show "∃a b. k = cbox a b"
unfolding k k1 k2 unfolding inter_interval by auto
}
fix k1 k2
assume "k1 ∈ ?A"
then obtain x1 y1 where k1: "k1 = x1 ∩ y1" "x1 ∈ p1" "y1 ∈ p2" "k1 ≠ {}"
by auto
assume "k2 ∈ ?A"
then obtain x2 y2 where k2: "k2 = x2 ∩ y2" "x2 ∈ p1" "y2 ∈ p2" "k2 ≠ {}"
by auto
assume "k1 ≠ k2"
then have th: "x1 ≠ x2 ∨ y1 ≠ y2"
unfolding k1 k2 by auto
have *: "interior x1 ∩ interior x2 = {} ∨ interior y1 ∩ interior y2 = {} ⟹
interior (x1 ∩ y1) ⊆ interior x1 ⟹ interior (x1 ∩ y1) ⊆ interior y1 ⟹
interior (x2 ∩ y2) ⊆ interior x2 ⟹ interior (x2 ∩ y2) ⊆ interior y2 ⟹
interior (x1 ∩ y1) ∩ interior (x2 ∩ y2) = {}" by auto
show "interior k1 ∩ interior k2 = {}"
unfolding k1 k2
apply (rule *)
using assms division_ofD(5) k1 k2(2) k2(3) th apply auto
done
qed
qed
lemma division_inter_1:
assumes "d division_of i"
and "cbox a (b::'a::euclidean_space) ⊆ i"
shows "{cbox a b ∩ k | k. k ∈ d ∧ cbox a b ∩ k ≠ {}} division_of (cbox a b)"
proof (cases "cbox a b = {}")
case True
show ?thesis
unfolding True and division_of_trivial by auto
next
case False
have *: "cbox a b ∩ i = cbox a b" using assms(2) by auto
show ?thesis
using division_inter[OF division_of_self[OF False] assms(1)]
unfolding * by auto
qed
lemma elementary_inter:
fixes s t :: "'a::euclidean_space set"
assumes "p1 division_of s"
and "p2 division_of t"
shows "∃p. p division_of (s ∩ t)"
using assms division_inter by blast
lemma elementary_inters:
assumes "finite f"
and "f ≠ {}"
and "∀s∈f. ∃p. p division_of (s::('a::euclidean_space) set)"
shows "∃p. p division_of (⋂f)"
using assms
proof (induct f rule: finite_induct)
case (insert x f)
show ?case
proof (cases "f = {}")
case True
then show ?thesis
unfolding True using insert by auto
next
case False
obtain p where "p division_of ⋂f"
using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
moreover obtain px where "px division_of x"
using insert(5)[rule_format,OF insertI1] ..
ultimately show ?thesis
by (simp add: elementary_inter Inter_insert)
qed
qed auto
lemma division_disjoint_union:
assumes "p1 division_of s1"
and "p2 division_of s2"
and "interior s1 ∩ interior s2 = {}"
shows "(p1 ∪ p2) division_of (s1 ∪ s2)"
proof (rule division_ofI)
note d1 = division_ofD[OF assms(1)]
note d2 = division_ofD[OF assms(2)]
show "finite (p1 ∪ p2)"
using d1(1) d2(1) by auto
show "⋃(p1 ∪ p2) = s1 ∪ s2"
using d1(6) d2(6) by auto
{
fix k1 k2
assume as: "k1 ∈ p1 ∪ p2" "k2 ∈ p1 ∪ p2" "k1 ≠ k2"
moreover
let ?g="interior k1 ∩ interior k2 = {}"
{
assume as: "k1∈p1" "k2∈p2"
have ?g
using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
using assms(3) by blast
}
moreover
{
assume as: "k1∈p2" "k2∈p1"
have ?g
using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
using assms(3) by blast
}
ultimately show ?g
using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
}
fix k
assume k: "k ∈ p1 ∪ p2"
show "k ⊆ s1 ∪ s2"
using k d1(2) d2(2) by auto
show "k ≠ {}"
using k d1(3) d2(3) by auto
show "∃a b. k = cbox a b"
using k d1(4) d2(4) by auto
qed
lemma partial_division_extend_1:
fixes a b c d :: "'a::euclidean_space"
assumes incl: "cbox c d ⊆ cbox a b"
and nonempty: "cbox c d ≠ {}"
obtains p where "p division_of (cbox a b)" "cbox c d ∈ p"
proof
let ?B = "λf::'a⇒'a × 'a.
cbox (∑i∈Basis. (fst (f i) ∙ i) *⇩R i) (∑i∈Basis. (snd (f i) ∙ i) *⇩R i)"
def p ≡ "?B ` (Basis →⇩E {(a, c), (c, d), (d, b)})"
show "cbox c d ∈ p"
unfolding p_def
by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="λ(i::'a)∈Basis. (c, d)"])
{
fix i :: 'a
assume "i ∈ Basis"
with incl nonempty have "a ∙ i ≤ c ∙ i" "c ∙ i ≤ d ∙ i" "d ∙ i ≤ b ∙ i"
unfolding box_eq_empty subset_box by (auto simp: not_le)
}
note ord = this
show "p division_of (cbox a b)"
proof (rule division_ofI)
show "finite p"
unfolding p_def by (auto intro!: finite_PiE)
{
fix k
assume "k ∈ p"
then obtain f where f: "f ∈ Basis →⇩E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
by (auto simp: p_def)
then show "∃a b. k = cbox a b"
by auto
have "k ⊆ cbox a b ∧ k ≠ {}"
proof (simp add: k box_eq_empty subset_box not_less, safe)
fix i :: 'a
assume i: "i ∈ Basis"
with f have "f i = (a, c) ∨ f i = (c, d) ∨ f i = (d, b)"
by (auto simp: PiE_iff)
with i ord[of i]
show "a ∙ i ≤ fst (f i) ∙ i" "snd (f i) ∙ i ≤ b ∙ i" "fst (f i) ∙ i ≤ snd (f i) ∙ i"
by auto
qed
then show "k ≠ {}" "k ⊆ cbox a b"
by auto
{
fix l
assume "l ∈ p"
then obtain g where g: "g ∈ Basis →⇩E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
by (auto simp: p_def)
assume "l ≠ k"
have "∃i∈Basis. f i ≠ g i"
proof (rule ccontr)
assume "¬ ?thesis"
with f g have "f = g"
by (auto simp: PiE_iff extensional_def intro!: ext)
with ‹l ≠ k› show False
by (simp add: l k)
qed
then obtain i where *: "i ∈ Basis" "f i ≠ g i" ..
then have "f i = (a, c) ∨ f i = (c, d) ∨ f i = (d, b)"
"g i = (a, c) ∨ g i = (c, d) ∨ g i = (d, b)"
using f g by (auto simp: PiE_iff)
with * ord[of i] show "interior l ∩ interior k = {}"
by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i])
}
note ‹k ⊆ cbox a b›
}
moreover
{
fix x assume x: "x ∈ cbox a b"
have "∀i∈Basis. ∃l. x ∙ i ∈ {fst l ∙ i .. snd l ∙ i} ∧ l ∈ {(a, c), (c, d), (d, b)}"
proof
fix i :: 'a
assume "i ∈ Basis"
with x ord[of i]
have "(a ∙ i ≤ x ∙ i ∧ x ∙ i ≤ c ∙ i) ∨ (c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i) ∨
(d ∙ i ≤ x ∙ i ∧ x ∙ i ≤ b ∙ i)"
by (auto simp: cbox_def)
then show "∃l. x ∙ i ∈ {fst l ∙ i .. snd l ∙ i} ∧ l ∈ {(a, c), (c, d), (d, b)}"
by auto
qed
then obtain f where
f: "∀i∈Basis. x ∙ i ∈ {fst (f i) ∙ i..snd (f i) ∙ i} ∧ f i ∈ {(a, c), (c, d), (d, b)}"
unfolding bchoice_iff ..
moreover from f have "restrict f Basis ∈ Basis →⇩E {(a, c), (c, d), (d, b)}"
by auto
moreover from f have "x ∈ ?B (restrict f Basis)"
by (auto simp: mem_box)
ultimately have "∃k∈p. x ∈ k"
unfolding p_def by blast
}
ultimately show "⋃p = cbox a b"
by auto
qed
qed
lemma partial_division_extend_interval:
assumes "p division_of (⋃p)" "(⋃p) ⊆ cbox a b"
obtains q where "p ⊆ q" "q division_of cbox a (b::'a::euclidean_space)"
proof (cases "p = {}")
case True
obtain q where "q division_of (cbox a b)"
by (rule elementary_interval)
then show ?thesis
using True that by blast
next
case False
note p = division_ofD[OF assms(1)]
have div_cbox: "∀k∈p. ∃q. q division_of cbox a b ∧ k ∈ q"
proof
fix k
assume kp: "k ∈ p"
obtain c d where k: "k = cbox c d"
using p(4)[OF kp] by blast
have *: "cbox c d ⊆ cbox a b" "cbox c d ≠ {}"
using p(2,3)[OF kp, unfolded k] using assms(2)
by (blast intro: order.trans)+
obtain q where "q division_of cbox a b" "cbox c d ∈ q"
by (rule partial_division_extend_1[OF *])
then show "∃q. q division_of cbox a b ∧ k ∈ q"
unfolding k by auto
qed
obtain q where q: "⋀x. x ∈ p ⟹ q x division_of cbox a b" "⋀x. x ∈ p ⟹ x ∈ q x"
using bchoice[OF div_cbox] by blast
{ fix x
assume x: "x ∈ p"
have "q x division_of ⋃q x"
apply (rule division_ofI)
using division_ofD[OF q(1)[OF x]]
apply auto
done }
then have "⋀x. x ∈ p ⟹ ∃d. d division_of ⋃(q x - {x})"
by (meson Diff_subset division_of_subset)
then have "∃d. d division_of ⋂((λi. ⋃(q i - {i})) ` p)"
apply -
apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
done
then obtain d where d: "d division_of ⋂((λi. ⋃(q i - {i})) ` p)" ..
have "d ∪ p division_of cbox a b"
proof -
have te: "⋀s f t. s ≠ {} ⟹ ∀i∈s. f i ∪ i = t ⟹ t = ⋂(f ` s) ∪ ⋃s" by auto
have cbox_eq: "cbox a b = ⋂((λi. ⋃(q i - {i})) ` p) ∪ ⋃p"
proof (rule te[OF False], clarify)
fix i
assume i: "i ∈ p"
show "⋃(q i - {i}) ∪ i = cbox a b"
using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
qed
{ fix k
assume k: "k ∈ p"
have *: "⋀u t s. t ∩ s = {} ⟹ u ⊆ s ⟹ u ∩ t = {}"
by auto
have "interior (⋂i∈p. ⋃(q i - {i})) ∩ interior k = {}"
proof (rule *[OF inter_interior_unions_intervals])
note qk=division_ofD[OF q(1)[OF k]]
show "finite (q k - {k})" "open (interior k)" "∀t∈q k - {k}. ∃a b. t = cbox a b"
using qk by auto
show "∀t∈q k - {k}. interior k ∩ interior t = {}"
using qk(5) using q(2)[OF k] by auto
show "interior (⋂i∈p. ⋃(q i - {i})) ⊆ interior (⋃(q k - {k}))"
apply (rule interior_mono)+
using k
apply auto
done
qed } note [simp] = this
show "d ∪ p division_of (cbox a b)"
unfolding cbox_eq
apply (rule division_disjoint_union[OF d assms(1)])
apply (rule inter_interior_unions_intervals)
apply (rule p open_interior ballI)+
apply simp_all
done
qed
then show ?thesis
by (meson Un_upper2 that)
qed
lemma elementary_bounded[dest]:
fixes s :: "'a::euclidean_space set"
shows "p division_of s ⟹ bounded s"
unfolding division_of_def by (metis bounded_Union bounded_cbox)
lemma elementary_subset_cbox:
"p division_of s ⟹ ∃a b. s ⊆ cbox a (b::'a::euclidean_space)"
by (meson elementary_bounded bounded_subset_cbox)
lemma division_union_intervals_exists:
fixes a b :: "'a::euclidean_space"
assumes "cbox a b ≠ {}"
obtains p where "(insert (cbox a b) p) division_of (cbox a b ∪ cbox c d)"
proof (cases "cbox c d = {}")
case True
show ?thesis
apply (rule that[of "{}"])
unfolding True
using assms
apply auto
done
next
case False
show ?thesis
proof (cases "cbox a b ∩ cbox c d = {}")
case True
then show ?thesis
by (metis that False assms division_disjoint_union division_of_self insert_is_Un interior_Int interior_empty)
next
case False
obtain u v where uv: "cbox a b ∩ cbox c d = cbox u v"
unfolding inter_interval by auto
have uv_sub: "cbox u v ⊆ cbox c d" using uv by auto
obtain p where "p division_of cbox c d" "cbox u v ∈ p"
by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
note p = this division_ofD[OF this(1)]
have "interior (cbox a b ∩ ⋃(p - {cbox u v})) = interior(cbox u v ∩ ⋃(p - {cbox u v}))"
apply (rule arg_cong[of _ _ interior])
using p(8) uv by auto
also have "… = {}"
unfolding interior_Int
apply (rule inter_interior_unions_intervals)
using p(6) p(7)[OF p(2)] p(3)
apply auto
done
finally have [simp]: "interior (cbox a b) ∩ interior (⋃(p - {cbox u v})) = {}" by simp
have cbe: "cbox a b ∪ cbox c d = cbox a b ∪ ⋃(p - {cbox u v})"
using p(8) unfolding uv[symmetric] by auto
have "insert (cbox a b) (p - {cbox u v}) division_of cbox a b ∪ ⋃(p - {cbox u v})"
proof -
have "{cbox a b} division_of cbox a b"
by (simp add: assms division_of_self)
then show "insert (cbox a b) (p - {cbox u v}) division_of cbox a b ∪ ⋃(p - {cbox u v})"
by (metis (no_types) Diff_subset ‹interior (cbox a b) ∩ interior (⋃(p - {cbox u v})) = {}› division_disjoint_union division_of_subset insert_is_Un p(1) p(8))
qed
with that[of "p - {cbox u v}"] show ?thesis by (simp add: cbe)
qed
qed
lemma division_of_unions:
assumes "finite f"
and "⋀p. p ∈ f ⟹ p division_of (⋃p)"
and "⋀k1 k2. k1 ∈ ⋃f ⟹ k2 ∈ ⋃f ⟹ k1 ≠ k2 ⟹ interior k1 ∩ interior k2 = {}"
shows "⋃f division_of ⋃⋃f"
using assms
by (auto intro!: division_ofI)
lemma elementary_union_interval:
fixes a b :: "'a::euclidean_space"
assumes "p division_of ⋃p"
obtains q where "q division_of (cbox a b ∪ ⋃p)"
proof -
note assm = division_ofD[OF assms]
have lem1: "⋀f s. ⋃⋃(f ` s) = ⋃((λx. ⋃(f x)) ` s)"
by auto
have lem2: "⋀f s. f ≠ {} ⟹ ⋃{s ∪ t |t. t ∈ f} = s ∪ ⋃f"
by auto
{
presume "p = {} ⟹ thesis"
"cbox a b = {} ⟹ thesis"
"cbox a b ≠ {} ⟹ interior (cbox a b) = {} ⟹ thesis"
"p ≠ {} ⟹ interior (cbox a b)≠{} ⟹ cbox a b ≠ {} ⟹ thesis"
then show thesis by auto
next
assume as: "p = {}"
obtain p where "p division_of (cbox a b)"
by (rule elementary_interval)
then show thesis
using as that by auto
next
assume as: "cbox a b = {}"
show thesis
using as assms that by auto
next
assume as: "interior (cbox a b) = {}" "cbox a b ≠ {}"
show thesis
apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
unfolding finite_insert
apply (rule assm(1)) unfolding Union_insert
using assm(2-4) as
apply -
apply (fast dest: assm(5))+
done
next
assume as: "p ≠ {}" "interior (cbox a b) ≠ {}" "cbox a b ≠ {}"
have "∀k∈p. ∃q. (insert (cbox a b) q) division_of (cbox a b ∪ k)"
proof
fix k
assume kp: "k ∈ p"
from assm(4)[OF kp] obtain c d where "k = cbox c d" by blast
then show "∃q. (insert (cbox a b) q) division_of (cbox a b ∪ k)"
by (meson as(3) division_union_intervals_exists)
qed
from bchoice[OF this] obtain q where "∀x∈p. insert (cbox a b) (q x) division_of (cbox a b) ∪ x" ..
note q = division_ofD[OF this[rule_format]]
let ?D = "⋃{insert (cbox a b) (q k) | k. k ∈ p}"
show thesis
proof (rule that[OF division_ofI])
have *: "{insert (cbox a b) (q k) |k. k ∈ p} = (λk. insert (cbox a b) (q k)) ` p"
by auto
show "finite ?D"
using "*" assm(1) q(1) by auto
show "⋃?D = cbox a b ∪ ⋃p"
unfolding * lem1
unfolding lem2[OF as(1), of "cbox a b", symmetric]
using q(6)
by auto
fix k
assume k: "k ∈ ?D"
then show "k ⊆ cbox a b ∪ ⋃p"
using q(2) by auto
show "k ≠ {}"
using q(3) k by auto
show "∃a b. k = cbox a b"
using q(4) k by auto
fix k'
assume k': "k' ∈ ?D" "k ≠ k'"
obtain x where x: "k ∈ insert (cbox a b) (q x)" "x∈p"
using k by auto
obtain x' where x': "k'∈insert (cbox a b) (q x')" "x'∈p"
using k' by auto
show "interior k ∩ interior k' = {}"
proof (cases "x = x'")
case True
show ?thesis
using True k' q(5) x' x by auto
next
case False
{
presume "k = cbox a b ⟹ ?thesis"
and "k' = cbox a b ⟹ ?thesis"
and "k ≠ cbox a b ⟹ k' ≠ cbox a b ⟹ ?thesis"
then show ?thesis by linarith
next
assume as': "k = cbox a b"
show ?thesis
using as' k' q(5) x' by blast
next
assume as': "k' = cbox a b"
show ?thesis
using as' k'(2) q(5) x by blast
}
assume as': "k ≠ cbox a b" "k' ≠ cbox a b"
obtain c d where k: "k = cbox c d"
using q(4)[OF x(2,1)] by blast
have "interior k ∩ interior (cbox a b) = {}"
using as' k'(2) q(5) x by blast
then have "interior k ⊆ interior x"
using interior_subset_union_intervals
by (metis as(2) k q(2) x interior_subset_union_intervals)
moreover
obtain c d where c_d: "k' = cbox c d"
using q(4)[OF x'(2,1)] by blast
have "interior k' ∩ interior (cbox a b) = {}"
using as'(2) q(5) x' by blast
then have "interior k' ⊆ interior x'"
by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
ultimately show ?thesis
using assm(5)[OF x(2) x'(2) False] by auto
qed
qed
}
qed
lemma elementary_unions_intervals:
assumes fin: "finite f"
and "⋀s. s ∈ f ⟹ ∃a b. s = cbox a (b::'a::euclidean_space)"
obtains p where "p division_of (⋃f)"
proof -
have "∃p. p division_of (⋃f)"
proof (induct_tac f rule:finite_subset_induct)
show "∃p. p division_of ⋃{}" using elementary_empty by auto
next
fix x F
assume as: "finite F" "x ∉ F" "∃p. p division_of ⋃F" "x∈f"
from this(3) obtain p where p: "p division_of ⋃F" ..
from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
have *: "⋃F = ⋃p"
using division_ofD[OF p] by auto
show "∃p. p division_of ⋃insert x F"
using elementary_union_interval[OF p[unfolded *], of a b]
unfolding Union_insert x * by metis
qed (insert assms, auto)
then show ?thesis
using that by auto
qed
lemma elementary_union:
fixes s t :: "'a::euclidean_space set"
assumes "ps division_of s" "pt division_of t"
obtains p where "p division_of (s ∪ t)"
proof -
have *: "s ∪ t = ⋃ps ∪ ⋃pt"
using assms unfolding division_of_def by auto
show ?thesis
apply (rule elementary_unions_intervals[of "ps ∪ pt"])
using assms apply auto
by (simp add: * that)
qed
lemma partial_division_extend:
fixes t :: "'a::euclidean_space set"
assumes "p division_of s"
and "q division_of t"
and "s ⊆ t"
obtains r where "p ⊆ r" and "r division_of t"
proof -
note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
obtain a b where ab: "t ⊆ cbox a b"
using elementary_subset_cbox[OF assms(2)] by auto
obtain r1 where "p ⊆ r1" "r1 division_of (cbox a b)"
using assms
by (metis ab dual_order.trans partial_division_extend_interval divp(6))
note r1 = this division_ofD[OF this(2)]
obtain p' where "p' division_of ⋃(r1 - p)"
apply (rule elementary_unions_intervals[of "r1 - p"])
using r1(3,6)
apply auto
done
then obtain r2 where r2: "r2 division_of (⋃(r1 - p)) ∩ (⋃q)"
by (metis assms(2) divq(6) elementary_inter)
{
fix x
assume x: "x ∈ t" "x ∉ s"
then have "x∈⋃r1"
unfolding r1 using ab by auto
then obtain r where r: "r ∈ r1" "x ∈ r"
unfolding Union_iff ..
moreover
have "r ∉ p"
proof
assume "r ∈ p"
then have "x ∈ s" using divp(2) r by auto
then show False using x by auto
qed
ultimately have "x∈⋃(r1 - p)" by auto
}
then have *: "t = ⋃p ∪ (⋃(r1 - p) ∩ ⋃q)"
unfolding divp divq using assms(3) by auto
show ?thesis
apply (rule that[of "p ∪ r2"])
unfolding *
defer
apply (rule division_disjoint_union)
unfolding divp(6)
apply(rule assms r2)+
proof -
have "interior s ∩ interior (⋃(r1-p)) = {}"
proof (rule inter_interior_unions_intervals)
show "finite (r1 - p)" and "open (interior s)" and "∀t∈r1-p. ∃a b. t = cbox a b"
using r1 by auto
have *: "⋀s. (⋀x. x ∈ s ⟹ False) ⟹ s = {}"
by auto
show "∀t∈r1-p. interior s ∩ interior t = {}"
proof
fix m x
assume as: "m ∈ r1 - p"
have "interior m ∩ interior (⋃p) = {}"
proof (rule inter_interior_unions_intervals)
show "finite p" and "open (interior m)" and "∀t∈p. ∃a b. t = cbox a b"
using divp by auto
show "∀t∈p. interior m ∩ interior t = {}"
by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
qed
then show "interior s ∩ interior m = {}"
unfolding divp by auto
qed
qed
then show "interior s ∩ interior (⋃(r1-p) ∩ (⋃q)) = {}"
using interior_subset by auto
qed auto
qed
subsection ‹Tagged (partial) divisions.›
definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
where "s tagged_partial_division_of i ⟷
finite s ∧
(∀x k. (x, k) ∈ s ⟶ x ∈ k ∧ k ⊆ i ∧ (∃a b. k = cbox a b)) ∧
(∀x1 k1 x2 k2. (x1, k1) ∈ s ∧ (x2, k2) ∈ s ∧ (x1, k1) ≠ (x2, k2) ⟶
interior k1 ∩ interior k2 = {})"
lemma tagged_partial_division_ofD[dest]:
assumes "s tagged_partial_division_of i"
shows "finite s"
and "⋀x k. (x,k) ∈ s ⟹ x ∈ k"
and "⋀x k. (x,k) ∈ s ⟹ k ⊆ i"
and "⋀x k. (x,k) ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀x1 k1 x2 k2. (x1,k1) ∈ s ⟹
(x2, k2) ∈ s ⟹ (x1, k1) ≠ (x2, k2) ⟹ interior k1 ∩ interior k2 = {}"
using assms unfolding tagged_partial_division_of_def by blast+
definition tagged_division_of (infixr "tagged'_division'_of" 40)
where "s tagged_division_of i ⟷ s tagged_partial_division_of i ∧ (⋃{k. ∃x. (x,k) ∈ s} = i)"
lemma tagged_division_of_finite: "s tagged_division_of i ⟹ finite s"
unfolding tagged_division_of_def tagged_partial_division_of_def by auto
lemma tagged_division_of:
"s tagged_division_of i ⟷
finite s ∧
(∀x k. (x, k) ∈ s ⟶ x ∈ k ∧ k ⊆ i ∧ (∃a b. k = cbox a b)) ∧
(∀x1 k1 x2 k2. (x1, k1) ∈ s ∧ (x2, k2) ∈ s ∧ (x1, k1) ≠ (x2, k2) ⟶
interior k1 ∩ interior k2 = {}) ∧
(⋃{k. ∃x. (x,k) ∈ s} = i)"
unfolding tagged_division_of_def tagged_partial_division_of_def by auto
lemma tagged_division_ofI:
assumes "finite s"
and "⋀x k. (x,k) ∈ s ⟹ x ∈ k"
and "⋀x k. (x,k) ∈ s ⟹ k ⊆ i"
and "⋀x k. (x,k) ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀x1 k1 x2 k2. (x1,k1) ∈ s ⟹ (x2, k2) ∈ s ⟹ (x1, k1) ≠ (x2, k2) ⟹
interior k1 ∩ interior k2 = {}"
and "(⋃{k. ∃x. (x,k) ∈ s} = i)"
shows "s tagged_division_of i"
unfolding tagged_division_of
using assms
apply auto
apply fastforce+
done
lemma tagged_division_ofD[dest]:
assumes "s tagged_division_of i"
shows "finite s"
and "⋀x k. (x,k) ∈ s ⟹ x ∈ k"
and "⋀x k. (x,k) ∈ s ⟹ k ⊆ i"
and "⋀x k. (x,k) ∈ s ⟹ ∃a b. k = cbox a b"
and "⋀x1 k1 x2 k2. (x1, k1) ∈ s ⟹ (x2, k2) ∈ s ⟹ (x1, k1) ≠ (x2, k2) ⟹
interior k1 ∩ interior k2 = {}"
and "(⋃{k. ∃x. (x,k) ∈ s} = i)"
using assms unfolding tagged_division_of by blast+
lemma division_of_tagged_division:
assumes "s tagged_division_of i"
shows "(snd ` s) division_of i"
proof (rule division_ofI)
note assm = tagged_division_ofD[OF assms]
show "⋃(snd ` s) = i" "finite (snd ` s)"
using assm by auto
fix k
assume k: "k ∈ snd ` s"
then obtain xk where xk: "(xk, k) ∈ s"
by auto
then show "k ⊆ i" "k ≠ {}" "∃a b. k = cbox a b"
using assm by fastforce+
fix k'
assume k': "k' ∈ snd ` s" "k ≠ k'"
from this(1) obtain xk' where xk': "(xk', k') ∈ s"
by auto
then show "interior k ∩ interior k' = {}"
using assm(5) k'(2) xk by blast
qed
lemma partial_division_of_tagged_division:
assumes "s tagged_partial_division_of i"
shows "(snd ` s) division_of ⋃(snd ` s)"
proof (rule division_ofI)
note assm = tagged_partial_division_ofD[OF assms]
show "finite (snd ` s)" "⋃(snd ` s) = ⋃(snd ` s)"
using assm by auto
fix k
assume k: "k ∈ snd ` s"
then obtain xk where xk: "(xk, k) ∈ s"
by auto
then show "k ≠ {}" "∃a b. k = cbox a b" "k ⊆ ⋃(snd ` s)"
using assm by auto
fix k'
assume k': "k' ∈ snd ` s" "k ≠ k'"
from this(1) obtain xk' where xk': "(xk', k') ∈ s"
by auto
then show "interior k ∩ interior k' = {}"
using assm(5) k'(2) xk by auto
qed
lemma tagged_partial_division_subset:
assumes "s tagged_partial_division_of i"
and "t ⊆ s"
shows "t tagged_partial_division_of i"
using assms
unfolding tagged_partial_division_of_def
using finite_subset[OF assms(2)]
by blast
lemma setsum_over_tagged_division_lemma:
assumes "p tagged_division_of i"
and "⋀u v. cbox u v ≠ {} ⟹ content (cbox u v) = 0 ⟹ d (cbox u v) = 0"
shows "setsum (λ(x,k). d k) p = setsum d (snd ` p)"
proof -
have *: "(λ(x,k). d k) = d ∘ snd"
unfolding o_def by (rule ext) auto
note assm = tagged_division_ofD[OF assms(1)]
show ?thesis
unfolding *
proof (rule setsum.reindex_nontrivial[symmetric])
show "finite p"
using assm by auto
fix x y
assume "x∈p" "y∈p" "x≠y" "snd x = snd y"
obtain a b where ab: "snd x = cbox a b"
using assm(4)[of "fst x" "snd x"] ‹x∈p› by auto
have "(fst x, snd y) ∈ p" "(fst x, snd y) ≠ y"
by (metis prod.collapse ‹x∈p› ‹snd x = snd y› ‹x ≠ y›)+
with ‹x∈p› ‹y∈p› have "interior (snd x) ∩ interior (snd y) = {}"
by (intro assm(5)[of "fst x" _ "fst y"]) auto
then have "content (cbox a b) = 0"
unfolding ‹snd x = snd y›[symmetric] ab content_eq_0_interior by auto
then have "d (cbox a b) = 0"
using assm(2)[of "fst x" "snd x"] ‹x∈p› ab[symmetric] by (intro assms(2)) auto
then show "d (snd x) = 0"
unfolding ab by auto
qed
qed
lemma tag_in_interval: "p tagged_division_of i ⟹ (x, k) ∈ p ⟹ x ∈ i"
by auto
lemma tagged_division_of_empty: "{} tagged_division_of {}"
unfolding tagged_division_of by auto
lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} ⟷ p = {}"
unfolding tagged_partial_division_of_def by auto
lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} ⟷ p = {}"
unfolding tagged_division_of by auto
lemma tagged_division_of_self: "x ∈ cbox a b ⟹ {(x,cbox a b)} tagged_division_of (cbox a b)"
by (rule tagged_division_ofI) auto
lemma tagged_division_of_self_real: "x ∈ {a .. b::real} ⟹ {(x,{a .. b})} tagged_division_of {a .. b}"
unfolding box_real[symmetric]
by (rule tagged_division_of_self)
lemma tagged_division_union:
assumes "p1 tagged_division_of s1"
and "p2 tagged_division_of s2"
and "interior s1 ∩ interior s2 = {}"
shows "(p1 ∪ p2) tagged_division_of (s1 ∪ s2)"
proof (rule tagged_division_ofI)
note p1 = tagged_division_ofD[OF assms(1)]
note p2 = tagged_division_ofD[OF assms(2)]
show "finite (p1 ∪ p2)"
using p1(1) p2(1) by auto
show "⋃{k. ∃x. (x, k) ∈ p1 ∪ p2} = s1 ∪ s2"
using p1(6) p2(6) by blast
fix x k
assume xk: "(x, k) ∈ p1 ∪ p2"
show "x ∈ k" "∃a b. k = cbox a b"
using xk p1(2,4) p2(2,4) by auto
show "k ⊆ s1 ∪ s2"
using xk p1(3) p2(3) by blast
fix x' k'
assume xk': "(x', k') ∈ p1 ∪ p2" "(x, k) ≠ (x', k')"
have *: "⋀a b. a ⊆ s1 ⟹ b ⊆ s2 ⟹ interior a ∩ interior b = {}"
using assms(3) interior_mono by blast
show "interior k ∩ interior k' = {}"
apply (cases "(x, k) ∈ p1")
apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2))
by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
qed
lemma tagged_division_unions:
assumes "finite iset"
and "∀i∈iset. pfn i tagged_division_of i"
and "∀i1∈iset. ∀i2∈iset. i1 ≠ i2 ⟶ interior(i1) ∩ interior(i2) = {}"
shows "⋃(pfn ` iset) tagged_division_of (⋃iset)"
proof (rule tagged_division_ofI)
note assm = tagged_division_ofD[OF assms(2)[rule_format]]
show "finite (⋃(pfn ` iset))"
apply (rule finite_Union)
using assms
apply auto
done
have "⋃{k. ∃x. (x, k) ∈ ⋃(pfn ` iset)} = ⋃((λi. ⋃{k. ∃x. (x, k) ∈ pfn i}) ` iset)"
by blast
also have "… = ⋃iset"
using assm(6) by auto
finally show "⋃{k. ∃x. (x, k) ∈ ⋃(pfn ` iset)} = ⋃iset" .
fix x k
assume xk: "(x, k) ∈ ⋃(pfn ` iset)"
then obtain i where i: "i ∈ iset" "(x, k) ∈ pfn i"
by auto
show "x ∈ k" "∃a b. k = cbox a b" "k ⊆ ⋃iset"
using assm(2-4)[OF i] using i(1) by auto
fix x' k'
assume xk': "(x', k') ∈ ⋃(pfn ` iset)" "(x, k) ≠ (x', k')"
then obtain i' where i': "i' ∈ iset" "(x', k') ∈ pfn i'"
by auto
have *: "⋀a b. i ≠ i' ⟹ a ⊆ i ⟹ b ⊆ i' ⟹ interior a ∩ interior b = {}"
using i(1) i'(1)
using assms(3)[rule_format] interior_mono
by blast
show "interior k ∩ interior k' = {}"
apply (cases "i = i'")
using assm(5) i' i(2) xk'(2) apply blast
using "*" assm(3) i' i by auto
qed
lemma tagged_partial_division_of_union_self:
assumes "p tagged_partial_division_of s"
shows "p tagged_division_of (⋃(snd ` p))"
apply (rule tagged_division_ofI)
using tagged_partial_division_ofD[OF assms]
apply auto
done
lemma tagged_division_of_union_self:
assumes "p tagged_division_of s"
shows "p tagged_division_of (⋃(snd ` p))"
apply (rule tagged_division_ofI)
using tagged_division_ofD[OF assms]
apply auto
done
subsection ‹Fine-ness of a partition w.r.t. a gauge.›
definition fine (infixr "fine" 46)
where "d fine s ⟷ (∀(x,k) ∈ s. k ⊆ d x)"
lemma fineI:
assumes "⋀x k. (x, k) ∈ s ⟹ k ⊆ d x"
shows "d fine s"
using assms unfolding fine_def by auto
lemma fineD[dest]:
assumes "d fine s"
shows "⋀x k. (x,k) ∈ s ⟹ k ⊆ d x"
using assms unfolding fine_def by auto
lemma fine_inter: "(λx. d1 x ∩ d2 x) fine p ⟷ d1 fine p ∧ d2 fine p"
unfolding fine_def by auto
lemma fine_inters:
"(λx. ⋂{f d x | d. d ∈ s}) fine p ⟷ (∀d∈s. (f d) fine p)"
unfolding fine_def by blast
lemma fine_union: "d fine p1 ⟹ d fine p2 ⟹ d fine (p1 ∪ p2)"
unfolding fine_def by blast
lemma fine_unions: "(⋀p. p ∈ ps ⟹ d fine p) ⟹ d fine (⋃ps)"
unfolding fine_def by auto
lemma fine_subset: "p ⊆ q ⟹ d fine q ⟹ d fine p"
unfolding fine_def by blast
subsection ‹Gauge integral. Define on compact intervals first, then use a limit.›
definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
where "(f has_integral_compact_interval y) i ⟷
(∀e>0. ∃d. gauge d ∧
(∀p. p tagged_division_of i ∧ d fine p ⟶
norm (setsum (λ(x,k). content k *⇩R f x) p - y) < e))"
definition has_integral ::
"('n::euclidean_space ⇒ 'b::real_normed_vector) ⇒ 'b ⇒ 'n set ⇒ bool"
(infixr "has'_integral" 46)
where "(f has_integral y) i ⟷
(if ∃a b. i = cbox a b
then (f has_integral_compact_interval y) i
else (∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
(∃z. ((λx. if x ∈ i then f x else 0) has_integral_compact_interval z) (cbox a b) ∧
norm (z - y) < e)))"
lemma has_integral:
"(f has_integral y) (cbox a b) ⟷
(∀e>0. ∃d. gauge d ∧
(∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm (setsum (λ(x,k). content(k) *⇩R f x) p - y) < e))"
unfolding has_integral_def has_integral_compact_interval_def
by auto
lemma has_integral_real:
"(f has_integral y) {a .. b::real} ⟷
(∀e>0. ∃d. gauge d ∧
(∀p. p tagged_division_of {a .. b} ∧ d fine p ⟶
norm (setsum (λ(x,k). content(k) *⇩R f x) p - y) < e))"
unfolding box_real[symmetric]
by (rule has_integral)
lemma has_integralD[dest]:
assumes "(f has_integral y) (cbox a b)"
and "e > 0"
obtains d where "gauge d"
and "⋀p. p tagged_division_of (cbox a b) ⟹ d fine p ⟹
norm (setsum (λ(x,k). content(k) *⇩R f(x)) p - y) < e"
using assms unfolding has_integral by auto
lemma has_integral_alt:
"(f has_integral y) i ⟷
(if ∃a b. i = cbox a b
then (f has_integral y) i
else (∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
(∃z. ((λx. if x ∈ i then f(x) else 0) has_integral z) (cbox a b) ∧ norm (z - y) < e)))"
unfolding has_integral
unfolding has_integral_compact_interval_def has_integral_def
by auto
lemma has_integral_altD:
assumes "(f has_integral y) i"
and "¬ (∃a b. i = cbox a b)"
and "e>0"
obtains B where "B > 0"
and "∀a b. ball 0 B ⊆ cbox a b ⟶
(∃z. ((λx. if x ∈ i then f(x) else 0) has_integral z) (cbox a b) ∧ norm(z - y) < e)"
using assms
unfolding has_integral
unfolding has_integral_compact_interval_def has_integral_def
by auto
definition integrable_on (infixr "integrable'_on" 46)
where "f integrable_on i ⟷ (∃y. (f has_integral y) i)"
definition "integral i f = (SOME y. (f has_integral y) i ∨ ~ f integrable_on i ∧ y=0)"
lemma integrable_integral[dest]: "f integrable_on i ⟹ (f has_integral (integral i f)) i"
unfolding integrable_on_def integral_def by (metis (mono_tags, lifting) someI_ex)
lemma not_integrable_integral: "~ f integrable_on i ⟹ integral i f = 0"
unfolding integrable_on_def integral_def by blast
lemma has_integral_integrable[intro]: "(f has_integral i) s ⟹ f integrable_on s"
unfolding integrable_on_def by auto
lemma has_integral_integral: "f integrable_on s ⟷ (f has_integral (integral s f)) s"
by auto
lemma setsum_content_null:
assumes "content (cbox a b) = 0"
and "p tagged_division_of (cbox a b)"
shows "setsum (λ(x,k). content k *⇩R f x) p = (0::'a::real_normed_vector)"
proof (rule setsum.neutral, rule)
fix y
assume y: "y ∈ p"
obtain x k where xk: "y = (x, k)"
using surj_pair[of y] by blast
note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
from this(2) obtain c d where k: "k = cbox c d" by blast
have "(λ(x, k). content k *⇩R f x) y = content k *⇩R f x"
unfolding xk by auto
also have "… = 0"
using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
unfolding assms(1) k
by auto
finally show "(λ(x, k). content k *⇩R f x) y = 0" .
qed
subsection ‹Some basic combining lemmas.›
lemma tagged_division_unions_exists:
assumes "finite iset"
and "∀i∈iset. ∃p. p tagged_division_of i ∧ d fine p"
and "∀i1∈iset. ∀i2∈iset. i1 ≠ i2 ⟶ interior i1 ∩ interior i2 = {}"
and "⋃iset = i"
obtains p where "p tagged_division_of i" and "d fine p"
proof -
obtain pfn where pfn:
"⋀x. x ∈ iset ⟹ pfn x tagged_division_of x"
"⋀x. x ∈ iset ⟹ d fine pfn x"
using bchoice[OF assms(2)] by auto
show thesis
apply (rule_tac p="⋃(pfn ` iset)" in that)
using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
qed
subsection ‹The set we're concerned with must be closed.›
lemma division_of_closed:
fixes i :: "'n::euclidean_space set"
shows "s division_of i ⟹ closed i"
unfolding division_of_def by fastforce
subsection ‹General bisection principle for intervals; might be useful elsewhere.›
lemma interval_bisection_step:
fixes type :: "'a::euclidean_space"
assumes "P {}"
and "∀s t. P s ∧ P t ∧ interior(s) ∩ interior(t) = {} ⟶ P (s ∪ t)"
and "¬ P (cbox a (b::'a))"
obtains c d where "¬ P (cbox c d)"
and "∀i∈Basis. a∙i ≤ c∙i ∧ c∙i ≤ d∙i ∧ d∙i ≤ b∙i ∧ 2 * (d∙i - c∙i) ≤ b∙i - a∙i"
proof -
have "cbox a b ≠ {}"
using assms(1,3) by metis
then have ab: "⋀i. i∈Basis ⟹ a ∙ i ≤ b ∙ i"
by (force simp: mem_box)
{ fix f
have "⟦finite f;
⋀s. s∈f ⟹ P s;
⋀s. s∈f ⟹ ∃a b. s = cbox a b;
⋀s t. s∈f ⟹ t∈f ⟹ s ≠ t ⟹ interior s ∩ interior t = {}⟧ ⟹ P (⋃f)"
proof (induct f rule: finite_induct)
case empty
show ?case
using assms(1) by auto
next
case (insert x f)
show ?case
unfolding Union_insert
apply (rule assms(2)[rule_format])
using inter_interior_unions_intervals [of f "interior x"]
apply (auto simp: insert)
by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
qed
} note UN_cases = this
let ?A = "{cbox c d | c d::'a. ∀i∈Basis. (c∙i = a∙i) ∧ (d∙i = (a∙i + b∙i) / 2) ∨
(c∙i = (a∙i + b∙i) / 2) ∧ (d∙i = b∙i)}"
let ?PP = "λc d. ∀i∈Basis. a∙i ≤ c∙i ∧ c∙i ≤ d∙i ∧ d∙i ≤ b∙i ∧ 2 * (d∙i - c∙i) ≤ b∙i - a∙i"
{
presume "∀c d. ?PP c d ⟶ P (cbox c d) ⟹ False"
then show thesis
unfolding atomize_not not_all
by (blast intro: that)
}
assume as: "∀c d. ?PP c d ⟶ P (cbox c d)"
have "P (⋃?A)"
proof (rule UN_cases)
let ?B = "(λs. cbox (∑i∈Basis. (if i ∈ s then a∙i else (a∙i + b∙i) / 2) *⇩R i::'a)
(∑i∈Basis. (if i ∈ s then (a∙i + b∙i) / 2 else b∙i) *⇩R i)) ` {s. s ⊆ Basis}"
have "?A ⊆ ?B"
proof
fix x
assume "x ∈ ?A"
then obtain c d
where x: "x = cbox c d"
"⋀i. i ∈ Basis ⟹
c ∙ i = a ∙ i ∧ d ∙ i = (a ∙ i + b ∙ i) / 2 ∨
c ∙ i = (a ∙ i + b ∙ i) / 2 ∧ d ∙ i = b ∙ i" by blast
show "x ∈ ?B"
unfolding image_iff x
apply (rule_tac x="{i. i∈Basis ∧ c∙i = a∙i}" in bexI)
apply (rule arg_cong2 [where f = cbox])
using x(2) ab
apply (auto simp add: euclidean_eq_iff[where 'a='a])
by fastforce
qed
then show "finite ?A"
by (rule finite_subset) auto
next
fix s
assume "s ∈ ?A"
then obtain c d
where s: "s = cbox c d"
"⋀i. i ∈ Basis ⟹
c ∙ i = a ∙ i ∧ d ∙ i = (a ∙ i + b ∙ i) / 2 ∨
c ∙ i = (a ∙ i + b ∙ i) / 2 ∧ d ∙ i = b ∙ i"
by blast
show "P s"
unfolding s
apply (rule as[rule_format])
using ab s(2) by force
show "∃a b. s = cbox a b"
unfolding s by auto
fix t
assume "t ∈ ?A"
then obtain e f where t:
"t = cbox e f"
"⋀i. i ∈ Basis ⟹
e ∙ i = a ∙ i ∧ f ∙ i = (a ∙ i + b ∙ i) / 2 ∨
e ∙ i = (a ∙ i + b ∙ i) / 2 ∧ f ∙ i = b ∙ i"
by blast
assume "s ≠ t"
then have "¬ (c = e ∧ d = f)"
unfolding s t by auto
then obtain i where "c∙i ≠ e∙i ∨ d∙i ≠ f∙i" and i': "i ∈ Basis"
unfolding euclidean_eq_iff[where 'a='a] by auto
then have i: "c∙i ≠ e∙i" "d∙i ≠ f∙i"
using s(2) t(2) apply fastforce
using t(2)[OF i'] ‹c ∙ i ≠ e ∙ i ∨ d ∙ i ≠ f ∙ i› i' s(2) t(2) by fastforce
have *: "⋀s t. (⋀a. a ∈ s ⟹ a ∈ t ⟹ False) ⟹ s ∩ t = {}"
by auto
show "interior s ∩ interior t = {}"
unfolding s t interior_cbox
proof (rule *)
fix x
assume "x ∈ box c d" "x ∈ box e f"
then have x: "c∙i < d∙i" "e∙i < f∙i" "c∙i < f∙i" "e∙i < d∙i"
unfolding mem_box using i'
by force+
show False using s(2)[OF i']
proof safe
assume as: "c ∙ i = a ∙ i" "d ∙ i = (a ∙ i + b ∙ i) / 2"
show False
using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
next
assume as: "c ∙ i = (a ∙ i + b ∙ i) / 2" "d ∙ i = b ∙ i"
show False
using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
qed
qed
qed
also have "⋃?A = cbox a b"
proof (rule set_eqI,rule)
fix x
assume "x ∈ ⋃?A"
then obtain c d where x:
"x ∈ cbox c d"
"⋀i. i ∈ Basis ⟹
c ∙ i = a ∙ i ∧ d ∙ i = (a ∙ i + b ∙ i) / 2 ∨
c ∙ i = (a ∙ i + b ∙ i) / 2 ∧ d ∙ i = b ∙ i"
by blast
show "x∈cbox a b"
unfolding mem_box
proof safe
fix i :: 'a
assume i: "i ∈ Basis"
then show "a ∙ i ≤ x ∙ i" "x ∙ i ≤ b ∙ i"
using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
qed
next
fix x
assume x: "x ∈ cbox a b"
have "∀i∈Basis.
∃c d. (c = a∙i ∧ d = (a∙i + b∙i) / 2 ∨ c = (a∙i + b∙i) / 2 ∧ d = b∙i) ∧ c≤x∙i ∧ x∙i ≤ d"
(is "∀i∈Basis. ∃c d. ?P i c d")
unfolding mem_box
proof
fix i :: 'a
assume i: "i ∈ Basis"
have "?P i (a∙i) ((a ∙ i + b ∙ i) / 2) ∨ ?P i ((a ∙ i + b ∙ i) / 2) (b∙i)"
using x[unfolded mem_box,THEN bspec, OF i] by auto
then show "∃c d. ?P i c d"
by blast
qed
then show "x∈⋃?A"
unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
apply auto
apply (rule_tac x="cbox xa xaa" in exI)
unfolding mem_box
apply auto
done
qed
finally show False
using assms by auto
qed
lemma interval_bisection:
fixes type :: "'a::euclidean_space"
assumes "P {}"
and "(∀s t. P s ∧ P t ∧ interior(s) ∩ interior(t) = {} ⟶ P(s ∪ t))"
and "¬ P (cbox a (b::'a))"
obtains x where "x ∈ cbox a b"
and "∀e>0. ∃c d. x ∈ cbox c d ∧ cbox c d ⊆ ball x e ∧ cbox c d ⊆ cbox a b ∧ ¬ P (cbox c d)"
proof -
have "∀x. ∃y. ¬ P (cbox (fst x) (snd x)) ⟶ (¬ P (cbox (fst y) (snd y)) ∧
(∀i∈Basis. fst x∙i ≤ fst y∙i ∧ fst y∙i ≤ snd y∙i ∧ snd y∙i ≤ snd x∙i ∧
2 * (snd y∙i - fst y∙i) ≤ snd x∙i - fst x∙i))" (is "∀x. ?P x")
proof
show "?P x" for x
proof (cases "P (cbox (fst x) (snd x))")
case True
then show ?thesis by auto
next
case as: False
obtain c d where "¬ P (cbox c d)"
"∀i∈Basis.
fst x ∙ i ≤ c ∙ i ∧
c ∙ i ≤ d ∙ i ∧
d ∙ i ≤ snd x ∙ i ∧
2 * (d ∙ i - c ∙ i) ≤ snd x ∙ i - fst x ∙ i"
by (rule interval_bisection_step[of P, OF assms(1-2) as])
then show ?thesis
apply -
apply (rule_tac x="(c,d)" in exI)
apply auto
done
qed
qed
then obtain f where f:
"∀x.
¬ P (cbox (fst x) (snd x)) ⟶
¬ P (cbox (fst (f x)) (snd (f x))) ∧
(∀i∈Basis.
fst x ∙ i ≤ fst (f x) ∙ i ∧
fst (f x) ∙ i ≤ snd (f x) ∙ i ∧
snd (f x) ∙ i ≤ snd x ∙ i ∧
2 * (snd (f x) ∙ i - fst (f x) ∙ i) ≤ snd x ∙ i - fst x ∙ i)"
apply -
apply (drule choice)
apply blast
done
def AB ≡ "λn. (f ^^ n) (a,b)"
def A ≡ "λn. fst(AB n)"
def B ≡ "λn. snd(AB n)"
note ab_def = A_def B_def AB_def
have "A 0 = a" "B 0 = b" "⋀n. ¬ P (cbox (A(Suc n)) (B(Suc n))) ∧
(∀i∈Basis. A(n)∙i ≤ A(Suc n)∙i ∧ A(Suc n)∙i ≤ B(Suc n)∙i ∧ B(Suc n)∙i ≤ B(n)∙i ∧
2 * (B(Suc n)∙i - A(Suc n)∙i) ≤ B(n)∙i - A(n)∙i)" (is "⋀n. ?P n")
proof -
show "A 0 = a" "B 0 = b"
unfolding ab_def by auto
note S = ab_def funpow.simps o_def id_apply
show "?P n" for n
proof (induct n)
case 0
then show ?case
unfolding S
apply (rule f[rule_format]) using assms(3)
apply auto
done
next
case (Suc n)
show ?case
unfolding S
apply (rule f[rule_format])
using Suc
unfolding S
apply auto
done
qed
qed
note AB = this(1-2) conjunctD2[OF this(3),rule_format]
have interv: "∃n. ∀x∈cbox (A n) (B n). ∀y∈cbox (A n) (B n). dist x y < e"
if e: "0 < e" for e
proof -
obtain n where n: "(∑i∈Basis. b ∙ i - a ∙ i) / e < 2 ^ n"
using real_arch_pow[of 2 "(setsum (λi. b∙i - a∙i) Basis) / e"] by auto
show ?thesis
proof (rule exI [where x=n], clarify)
fix x y
assume xy: "x∈cbox (A n) (B n)" "y∈cbox (A n) (B n)"
have "dist x y ≤ setsum (λi. ¦(x - y)∙i¦) Basis"
unfolding dist_norm by(rule norm_le_l1)
also have "… ≤ setsum (λi. B n∙i - A n∙i) Basis"
proof (rule setsum_mono)
fix i :: 'a
assume i: "i ∈ Basis"
show "¦(x - y) ∙ i¦ ≤ B n ∙ i - A n ∙ i"
using xy[unfolded mem_box,THEN bspec, OF i]
by (auto simp: inner_diff_left)
qed
also have "… ≤ setsum (λi. b∙i - a∙i) Basis / 2^n"
unfolding setsum_divide_distrib
proof (rule setsum_mono)
show "B n ∙ i - A n ∙ i ≤ (b ∙ i - a ∙ i) / 2 ^ n" if i: "i ∈ Basis" for i
proof (induct n)
case 0
then show ?case
unfolding AB by auto
next
case (Suc n)
have "B (Suc n) ∙ i - A (Suc n) ∙ i ≤ (B n ∙ i - A n ∙ i) / 2"
using AB(4)[of i n] using i by auto
also have "… ≤ (b ∙ i - a ∙ i) / 2 ^ Suc n"
using Suc by (auto simp add: field_simps)
finally show ?case .
qed
qed
also have "… < e"
using n using e by (auto simp add: field_simps)
finally show "dist x y < e" .
qed
qed
{
fix n m :: nat
assume "m ≤ n" then have "cbox (A n) (B n) ⊆ cbox (A m) (B m)"
proof (induction rule: inc_induct)
case (step i)
show ?case
using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
qed simp
} note ABsubset = this
have "∃a. ∀n. a∈ cbox (A n) (B n)"
by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
(metis nat.exhaust AB(1-3) assms(1,3))
then obtain x0 where x0: "⋀n. x0 ∈ cbox (A n) (B n)"
by blast
show thesis
proof (rule that[rule_format, of x0])
show "x0∈cbox a b"
using x0[of 0] unfolding AB .
fix e :: real
assume "e > 0"
from interv[OF this] obtain n
where n: "∀x∈cbox (A n) (B n). ∀y∈cbox (A n) (B n). dist x y < e" ..
have "¬ P (cbox (A n) (B n))"
apply (cases "0 < n")
using AB(3)[of "n - 1"] assms(3) AB(1-2)
apply auto
done
moreover have "cbox (A n) (B n) ⊆ ball x0 e"
using n using x0[of n] by auto
moreover have "cbox (A n) (B n) ⊆ cbox a b"
unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
ultimately show "∃c d. x0 ∈ cbox c d ∧ cbox c d ⊆ ball x0 e ∧ cbox c d ⊆ cbox a b ∧ ¬ P (cbox c d)"
apply (rule_tac x="A n" in exI)
apply (rule_tac x="B n" in exI)
apply (auto simp: x0)
done
qed
qed
subsection ‹Cousin's lemma.›
lemma fine_division_exists:
fixes a b :: "'a::euclidean_space"
assumes "gauge g"
obtains p where "p tagged_division_of (cbox a b)" "g fine p"
proof -
presume "¬ (∃p. p tagged_division_of (cbox a b) ∧ g fine p) ⟹ False"
then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
by blast
then show thesis ..
next
assume as: "¬ (∃p. p tagged_division_of (cbox a b) ∧ g fine p)"
obtain x where x:
"x ∈ (cbox a b)"
"⋀e. 0 < e ⟹
∃c d.
x ∈ cbox c d ∧
cbox c d ⊆ ball x e ∧
cbox c d ⊆ (cbox a b) ∧
¬ (∃p. p tagged_division_of cbox c d ∧ g fine p)"
apply (rule interval_bisection[of "λs. ∃p. p tagged_division_of s ∧ g fine p", OF _ _ as])
apply (simp add: fine_def)
apply (metis tagged_division_union fine_union)
apply (auto simp: )
done
obtain e where e: "e > 0" "ball x e ⊆ g x"
using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
from x(2)[OF e(1)]
obtain c d where c_d: "x ∈ cbox c d"
"cbox c d ⊆ ball x e"
"cbox c d ⊆ cbox a b"
"¬ (∃p. p tagged_division_of cbox c d ∧ g fine p)"
by blast
have "g fine {(x, cbox c d)}"
unfolding fine_def using e using c_d(2) by auto
then show False
using tagged_division_of_self[OF c_d(1)] using c_d by auto
qed
lemma fine_division_exists_real:
fixes a b :: real
assumes "gauge g"
obtains p where "p tagged_division_of {a .. b}" "g fine p"
by (metis assms box_real(2) fine_division_exists)
subsection ‹Basic theorems about integrals.›
lemma has_integral_unique:
fixes f :: "'n::euclidean_space ⇒ 'a::real_normed_vector"
assumes "(f has_integral k1) i"
and "(f has_integral k2) i"
shows "k1 = k2"
proof (rule ccontr)
let ?e = "norm (k1 - k2) / 2"
assume as: "k1 ≠ k2"
then have e: "?e > 0"
by auto
have lem: False
if f_k1: "(f has_integral k1) (cbox a b)"
and f_k2: "(f has_integral k2) (cbox a b)"
and "k1 ≠ k2"
for f :: "'n ⇒ 'a" and a b k1 k2
proof -
let ?e = "norm (k1 - k2) / 2"
from ‹k1 ≠ k2› have e: "?e > 0" by auto
obtain d1 where d1:
"gauge d1"
"⋀p. p tagged_division_of cbox a b ⟹
d1 fine p ⟹ norm ((∑(x, k)∈p. content k *⇩R f x) - k1) < norm (k1 - k2) / 2"
by (rule has_integralD[OF f_k1 e]) blast
obtain d2 where d2:
"gauge d2"
"⋀p. p tagged_division_of cbox a b ⟹
d2 fine p ⟹ norm ((∑(x, k)∈p. content k *⇩R f x) - k2) < norm (k1 - k2) / 2"
by (rule has_integralD[OF f_k2 e]) blast
obtain p where p:
"p tagged_division_of cbox a b"
"(λx. d1 x ∩ d2 x) fine p"
by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)]])
let ?c = "(∑(x, k)∈p. content k *⇩R f x)"
have "norm (k1 - k2) ≤ norm (?c - k2) + norm (?c - k1)"
using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
by (auto simp add:algebra_simps norm_minus_commute)
also have "… < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
apply (rule add_strict_mono)
apply (rule_tac[!] d2(2) d1(2))
using p unfolding fine_def
apply auto
done
finally show False by auto
qed
{
presume "¬ (∃a b. i = cbox a b) ⟹ False"
then show False
using as assms lem by blast
}
assume as: "¬ (∃a b. i = cbox a b)"
obtain B1 where B1:
"0 < B1"
"⋀a b. ball 0 B1 ⊆ cbox a b ⟹
∃z. ((λx. if x ∈ i then f x else 0) has_integral z) (cbox a b) ∧
norm (z - k1) < norm (k1 - k2) / 2"
by (rule has_integral_altD[OF assms(1) as,OF e]) blast
obtain B2 where B2:
"0 < B2"
"⋀a b. ball 0 B2 ⊆ cbox a b ⟹
∃z. ((λx. if x ∈ i then f x else 0) has_integral z) (cbox a b) ∧
norm (z - k2) < norm (k1 - k2) / 2"
by (rule has_integral_altD[OF assms(2) as,OF e]) blast
have "∃a b::'n. ball 0 B1 ∪ ball 0 B2 ⊆ cbox a b"
apply (rule bounded_subset_cbox)
using bounded_Un bounded_ball
apply auto
done
then obtain a b :: 'n where ab: "ball 0 B1 ⊆ cbox a b" "ball 0 B2 ⊆ cbox a b"
by blast
obtain w where w:
"((λx. if x ∈ i then f x else 0) has_integral w) (cbox a b)"
"norm (w - k1) < norm (k1 - k2) / 2"
using B1(2)[OF ab(1)] by blast
obtain z where z:
"((λx. if x ∈ i then f x else 0) has_integral z) (cbox a b)"
"norm (z - k2) < norm (k1 - k2) / 2"
using B2(2)[OF ab(2)] by blast
have "z = w"
using lem[OF w(1) z(1)] by auto
then have "norm (k1 - k2) ≤ norm (z - k2) + norm (w - k1)"
using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
by (auto simp add: norm_minus_commute)
also have "… < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
apply (rule add_strict_mono)
apply (rule_tac[!] z(2) w(2))
done
finally show False by auto
qed
lemma integral_unique [intro]: "(f has_integral y) k ⟹ integral k f = y"
unfolding integral_def
by (rule some_equality) (auto intro: has_integral_unique)
lemma eq_integralD: "integral k f = y ⟹ (f has_integral y) k ∨ ~ f integrable_on k ∧ y=0"
unfolding integral_def integrable_on_def
apply (erule subst)
apply (rule someI_ex)
by blast
lemma has_integral_is_0:
fixes f :: "'n::euclidean_space ⇒ 'a::real_normed_vector"
assumes "∀x∈s. f x = 0"
shows "(f has_integral 0) s"
proof -
have lem: "⋀a b. ⋀f::'n ⇒ 'a.
(∀x∈cbox a b. f(x) = 0) ⟹ (f has_integral 0) (cbox a b)"
unfolding has_integral
proof clarify
fix a b e
fix f :: "'n ⇒ 'a"
assume as: "∀x∈cbox a b. f x = 0" "0 < (e::real)"
have "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e"
if p: "p tagged_division_of cbox a b" for p
proof -
have "(∑(x, k)∈p. content k *⇩R f x) = 0"
proof (rule setsum.neutral, rule)
fix x
assume x: "x ∈ p"
have "f (fst x) = 0"
using tagged_division_ofD(2-3)[OF p, of "fst x" "snd x"] using as x by auto
then show "(λ(x, k). content k *⇩R f x) x = 0"
apply (subst surjective_pairing[of x])
unfolding split_conv
apply auto
done
qed
then show ?thesis
using as by auto
qed
then show "∃d. gauge d ∧
(∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶ norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e)"
by auto
qed
{
presume "¬ (∃a b. s = cbox a b) ⟹ ?thesis"
with assms lem show ?thesis
by blast
}
have *: "(λx. if x ∈ s then f x else 0) = (λx. 0)"
apply (rule ext)
using assms
apply auto
done
assume "¬ (∃a b. s = cbox a b)"
then show ?thesis
using lem
by (subst has_integral_alt) (force simp add: *)
qed
lemma has_integral_0[simp]: "((λx::'n::euclidean_space. 0) has_integral 0) s"
by (rule has_integral_is_0) auto
lemma has_integral_0_eq[simp]: "((λx. 0) has_integral i) s ⟷ i = 0"
using has_integral_unique[OF has_integral_0] by auto
lemma has_integral_linear:
fixes f :: "'n::euclidean_space ⇒ 'a::real_normed_vector"
assumes "(f has_integral y) s"
and "bounded_linear h"
shows "((h ∘ f) has_integral ((h y))) s"
proof -
interpret bounded_linear h
using assms(2) .
from pos_bounded obtain B where B: "0 < B" "⋀x. norm (h x) ≤ norm x * B"
by blast
have lem: "⋀(f :: 'n ⇒ 'a) y a b.
(f has_integral y) (cbox a b) ⟹ ((h ∘ f) has_integral h y) (cbox a b)"
unfolding has_integral
proof (clarify, goal_cases)
case prems: (1 f y a b e)
from pos_bounded
obtain B where B: "0 < B" "⋀x. norm (h x) ≤ norm x * B"
by blast
have "e / B > 0" using prems(2) B by simp
then obtain g
where g: "gauge g"
"⋀p. p tagged_division_of (cbox a b) ⟹ g fine p ⟹
norm ((∑(x, k)∈p. content k *⇩R f x) - y) < e / B"
using prems(1) by auto
{
fix p
assume as: "p tagged_division_of (cbox a b)" "g fine p"
have hc: "⋀x k. h ((λ(x, k). content k *⇩R f x) x) = (λ(x, k). h (content k *⇩R f x)) x"
by auto
then have "(∑(x, k)∈p. content k *⇩R (h ∘ f) x) = setsum (h ∘ (λ(x, k). content k *⇩R f x)) p"
unfolding o_def unfolding scaleR[symmetric] hc by simp
also have "… = h (∑(x, k)∈p. content k *⇩R f x)"
using setsum[of "λ(x,k). content k *⇩R f x" p] using as by auto
finally have "(∑(x, k)∈p. content k *⇩R (h ∘ f) x) = h (∑(x, k)∈p. content k *⇩R f x)" .
then have "norm ((∑(x, k)∈p. content k *⇩R (h ∘ f) x) - h y) < e"
apply (simp add: diff[symmetric])
apply (rule le_less_trans[OF B(2)])
using g(2)[OF as] B(1)
apply (auto simp add: field_simps)
done
}
with g show ?case
by (rule_tac x=g in exI) auto
qed
{
presume "¬ (∃a b. s = cbox a b) ⟹ ?thesis"
then show ?thesis
using assms(1) lem by blast
}
assume as: "¬ (∃a b. s = cbox a b)"
then show ?thesis
proof (subst has_integral_alt, clarsimp)
fix e :: real
assume e: "e > 0"
have *: "0 < e/B" using e B(1) by simp
obtain M where M:
"M > 0"
"⋀a b. ball 0 M ⊆ cbox a b ⟹
∃z. ((λx. if x ∈ s then f x else 0) has_integral z) (cbox a b) ∧ norm (z - y) < e / B"
using has_integral_altD[OF assms(1) as *] by blast
show "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
(∃z. ((λx. if x ∈ s then (h ∘ f) x else 0) has_integral z) (cbox a b) ∧ norm (z - h y) < e)"
proof (rule_tac x=M in exI, clarsimp simp add: M, goal_cases)
case prems: (1 a b)
obtain z where z:
"((λx. if x ∈ s then f x else 0) has_integral z) (cbox a b)"
"norm (z - y) < e / B"
using M(2)[OF prems(1)] by blast
have *: "(λx. if x ∈ s then (h ∘ f) x else 0) = h ∘ (λx. if x ∈ s then f x else 0)"
using zero by auto
show ?case
apply (rule_tac x="h z" in exI)
apply (simp add: * lem z(1))
apply (metis B diff le_less_trans pos_less_divide_eq z(2))
done
qed
qed
qed
lemma has_integral_scaleR_left:
"(f has_integral y) s ⟹ ((λx. f x *⇩R c) has_integral (y *⇩R c)) s"
using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
lemma has_integral_mult_left:
fixes c :: "_ :: real_normed_algebra"
shows "(f has_integral y) s ⟹ ((λx. f x * c) has_integral (y * c)) s"
using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
text‹The case analysis eliminates the condition @{term "f integrable_on s"} at the cost
of the type class constraint ‹division_ring››
corollary integral_mult_left [simp]:
fixes c:: "'a::{real_normed_algebra,division_ring}"
shows "integral s (λx. f x * c) = integral s f * c"
proof (cases "f integrable_on s ∨ c = 0")
case True then show ?thesis
by (force intro: has_integral_mult_left)
next
case False then have "~ (λx. f x * c) integrable_on s"
using has_integral_mult_left [of "(λx. f x * c)" _ s "inverse c"]
by (force simp add: mult.assoc)
with False show ?thesis by (simp add: not_integrable_integral)
qed
corollary integral_mult_right [simp]:
fixes c:: "'a::{real_normed_field}"
shows "integral s (λx. c * f x) = c * integral s f"
by (simp add: mult.commute [of c])
corollary integral_divide [simp]:
fixes z :: "'a::real_normed_field"
shows "integral S (λx. f x / z) = integral S (λx. f x) / z"
using integral_mult_left [of S f "inverse z"]
by (simp add: divide_inverse_commute)
lemma has_integral_mult_right:
fixes c :: "'a :: real_normed_algebra"
shows "(f has_integral y) i ⟹ ((λx. c * f x) has_integral (c * y)) i"
using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def)
lemma has_integral_cmul: "(f has_integral k) s ⟹ ((λx. c *⇩R f x) has_integral (c *⇩R k)) s"
unfolding o_def[symmetric]
by (metis has_integral_linear bounded_linear_scaleR_right)
lemma has_integral_cmult_real:
fixes c :: real
assumes "c ≠ 0 ⟹ (f has_integral x) A"
shows "((λx. c * f x) has_integral c * x) A"
proof (cases "c = 0")
case True
then show ?thesis by simp
next
case False
from has_integral_cmul[OF assms[OF this], of c] show ?thesis
unfolding real_scaleR_def .
qed
lemma has_integral_neg: "(f has_integral k) s ⟹ ((λx. -(f x)) has_integral -k) s"
by (drule_tac c="-1" in has_integral_cmul) auto
lemma has_integral_add:
fixes f :: "'n::euclidean_space ⇒ 'a::real_normed_vector"
assumes "(f has_integral k) s"
and "(g has_integral l) s"
shows "((λx. f x + g x) has_integral (k + l)) s"
proof -
have lem: "((λx. f x + g x) has_integral (k + l)) (cbox a b)"
if f_k: "(f has_integral k) (cbox a b)"
and g_l: "(g has_integral l) (cbox a b)"
for f :: "'n ⇒ 'a" and g a b k l
unfolding has_integral
proof clarify
fix e :: real
assume e: "e > 0"
then have *: "e / 2 > 0"
by auto
obtain d1 where d1:
"gauge d1"
"⋀p. p tagged_division_of (cbox a b) ⟹ d1 fine p ⟹
norm ((∑(x, k)∈p. content k *⇩R f x) - k) < e / 2"
using has_integralD[OF f_k *] by blast
obtain d2 where d2:
"gauge d2"
"⋀p. p tagged_division_of (cbox a b) ⟹ d2 fine p ⟹
norm ((∑(x, k)∈p. content k *⇩R g x) - l) < e / 2"
using has_integralD[OF g_l *] by blast
show "∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm ((∑(x, k)∈p. content k *⇩R (f x + g x)) - (k + l)) < e)"
proof (rule exI [where x="λx. (d1 x) ∩ (d2 x)"], clarsimp simp add: gauge_inter[OF d1(1) d2(1)])
fix p
assume as: "p tagged_division_of (cbox a b)" "(λx. d1 x ∩ d2 x) fine p"
have *: "(∑(x, k)∈p. content k *⇩R (f x + g x)) =
(∑(x, k)∈p. content k *⇩R f x) + (∑(x, k)∈p. content k *⇩R g x)"
unfolding scaleR_right_distrib setsum.distrib[of "λ(x,k). content k *⇩R f x" "λ(x,k). content k *⇩R g x" p,symmetric]
by (rule setsum.cong) auto
from as have fine: "d1 fine p" "d2 fine p"
unfolding fine_inter by auto
have "norm ((∑(x, k)∈p. content k *⇩R (f x + g x)) - (k + l)) =
norm (((∑(x, k)∈p. content k *⇩R f x) - k) + ((∑(x, k)∈p. content k *⇩R g x) - l))"
unfolding * by (auto simp add: algebra_simps)
also have "… < e/2 + e/2"
apply (rule le_less_trans[OF norm_triangle_ineq])
using as d1 d2 fine
apply (blast intro: add_strict_mono)
done
finally show "norm ((∑(x, k)∈p. content k *⇩R (f x + g x)) - (k + l)) < e"
by auto
qed
qed
{
presume "¬ (∃a b. s = cbox a b) ⟹ ?thesis"
then show ?thesis
using assms lem by force
}
assume as: "¬ (∃a b. s = cbox a b)"
then show ?thesis
proof (subst has_integral_alt, clarsimp, goal_cases)
case (1 e)
then have *: "e / 2 > 0"
by auto
from has_integral_altD[OF assms(1) as *]
obtain B1 where B1:
"0 < B1"
"⋀a b. ball 0 B1 ⊆ cbox a b ⟹
∃z. ((λx. if x ∈ s then f x else 0) has_integral z) (cbox a b) ∧ norm (z - k) < e / 2"
by blast
from has_integral_altD[OF assms(2) as *]
obtain B2 where B2:
"0 < B2"
"⋀a b. ball 0 B2 ⊆ (cbox a b) ⟹
∃z. ((λx. if x ∈ s then g x else 0) has_integral z) (cbox a b) ∧ norm (z - l) < e / 2"
by blast
show ?case
proof (rule_tac x="max B1 B2" in exI, clarsimp simp add: max.strict_coboundedI1 B1)
fix a b
assume "ball 0 (max B1 B2) ⊆ cbox a (b::'n)"
then have *: "ball 0 B1 ⊆ cbox a (b::'n)" "ball 0 B2 ⊆ cbox a (b::'n)"
by auto
obtain w where w:
"((λx. if x ∈ s then f x else 0) has_integral w) (cbox a b)"
"norm (w - k) < e / 2"
using B1(2)[OF *(1)] by blast
obtain z where z:
"((λx. if x ∈ s then g x else 0) has_integral z) (cbox a b)"
"norm (z - l) < e / 2"
using B2(2)[OF *(2)] by blast
have *: "⋀x. (if x ∈ s then f x + g x else 0) =
(if x ∈ s then f x else 0) + (if x ∈ s then g x else 0)"
by auto
show "∃z. ((λx. if x ∈ s then f x + g x else 0) has_integral z) (cbox a b) ∧ norm (z - (k + l)) < e"
apply (rule_tac x="w + z" in exI)
apply (simp add: lem[OF w(1) z(1), unfolded *[symmetric]])
using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2)
apply (auto simp add: field_simps)
done
qed
qed
qed
lemma has_integral_sub:
"(f has_integral k) s ⟹ (g has_integral l) s ⟹
((λx. f x - g x) has_integral (k - l)) s"
using has_integral_add[OF _ has_integral_neg, of f k s g l]
unfolding algebra_simps
by auto
lemma integral_0 [simp]:
"integral s (λx::'n::euclidean_space. 0::'m::real_normed_vector) = 0"
by (rule integral_unique has_integral_0)+
lemma integral_add: "f integrable_on s ⟹ g integrable_on s ⟹
integral s (λx. f x + g x) = integral s f + integral s g"
by (rule integral_unique) (metis integrable_integral has_integral_add)
lemma integral_cmul [simp]: "integral s (λx. c *⇩R f x) = c *⇩R integral s f"
proof (cases "f integrable_on s ∨ c = 0")
case True with has_integral_cmul show ?thesis by force
next
case False then have "~ (λx. c *⇩R f x) integrable_on s"
using has_integral_cmul [of "(λx. c *⇩R f x)" _ s "inverse c"]
by force
with False show ?thesis by (simp add: not_integrable_integral)
qed
lemma integral_neg [simp]: "integral s (λx. - f x) = - integral s f"
proof (cases "f integrable_on s")
case True then show ?thesis
by (simp add: has_integral_neg integrable_integral integral_unique)
next
case False then have "~ (λx. - f x) integrable_on s"
using has_integral_neg [of "(λx. - f x)" _ s ]
by force
with False show ?thesis by (simp add: not_integrable_integral)
qed
lemma integral_diff: "f integrable_on s ⟹ g integrable_on s ⟹
integral s (λx. f x - g x) = integral s f - integral s g"
by (rule integral_unique) (metis integrable_integral has_integral_sub)
lemma integrable_0: "(λx. 0) integrable_on s"
unfolding integrable_on_def using has_integral_0 by auto
lemma integrable_add: "f integrable_on s ⟹ g integrable_on s ⟹ (λx. f x + g x) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_add)
lemma integrable_cmul: "f integrable_on s ⟹ (λx. c *⇩R f(x)) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_cmul)
lemma integrable_on_cmult_iff:
fixes c :: real
assumes "c ≠ 0"
shows "(λx. c * f x) integrable_on s ⟷ f integrable_on s"
using integrable_cmul[of "λx. c * f x" s "1 / c"] integrable_cmul[of f s c] ‹c ≠ 0›
by auto
lemma integrable_on_cmult_left:
assumes "f integrable_on s"
shows "(λx. of_real c * f x) integrable_on s"
using integrable_cmul[of f s "of_real c"] assms
by (simp add: scaleR_conv_of_real)
lemma integrable_neg: "f integrable_on s ⟹ (λx. -f(x)) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_neg)
lemma integrable_diff:
"f integrable_on s ⟹ g integrable_on s ⟹ (λx. f x - g x) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_sub)
lemma integrable_linear:
"f integrable_on s ⟹ bounded_linear h ⟹ (h ∘ f) integrable_on s"
unfolding integrable_on_def by(auto intro: has_integral_linear)
lemma integral_linear:
"f integrable_on s ⟹ bounded_linear h ⟹ integral s (h ∘ f) = h (integral s f)"
apply (rule has_integral_unique [where i=s and f = "h ∘ f"])
apply (simp_all add: integrable_integral integrable_linear has_integral_linear )
done
lemma integral_component_eq[simp]:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "f integrable_on s"
shows "integral s (λx. f x ∙ k) = integral s f ∙ k"
unfolding integral_linear[OF assms(1) bounded_linear_component,unfolded o_def] ..
lemma has_integral_setsum:
assumes "finite t"
and "∀a∈t. ((f a) has_integral (i a)) s"
shows "((λx. setsum (λa. f a x) t) has_integral (setsum i t)) s"
using assms(1) subset_refl[of t]
proof (induct rule: finite_subset_induct)
case empty
then show ?case by auto
next
case (insert x F)
with assms show ?case
by (simp add: has_integral_add)
qed
lemma integral_setsum:
"⟦finite t; ∀a∈t. (f a) integrable_on s⟧ ⟹
integral s (λx. setsum (λa. f a x) t) = setsum (λa. integral s (f a)) t"
by (auto intro: has_integral_setsum integrable_integral)
lemma integrable_setsum:
"⟦finite t; ∀a∈t. (f a) integrable_on s⟧ ⟹ (λx. setsum (λa. f a x) t) integrable_on s"
unfolding integrable_on_def
apply (drule bchoice)
using has_integral_setsum[of t]
apply auto
done
lemma has_integral_eq:
assumes "⋀x. x ∈ s ⟹ f x = g x"
and "(f has_integral k) s"
shows "(g has_integral k) s"
using has_integral_sub[OF assms(2), of "λx. f x - g x" 0]
using has_integral_is_0[of s "λx. f x - g x"]
using assms(1)
by auto
lemma integrable_eq: "(⋀x. x ∈ s ⟹ f x = g x) ⟹ f integrable_on s ⟹ g integrable_on s"
unfolding integrable_on_def
using has_integral_eq[of s f g] has_integral_eq by blast
lemma has_integral_cong:
assumes "⋀x. x ∈ s ⟹ f x = g x"
shows "(f has_integral i) s = (g has_integral i) s"
using has_integral_eq[of s f g] has_integral_eq[of s g f] assms
by auto
lemma integral_cong:
assumes "⋀x. x ∈ s ⟹ f x = g x"
shows "integral s f = integral s g"
unfolding integral_def
by (metis (full_types, hide_lams) assms has_integral_cong integrable_eq)
lemma integrable_on_cmult_left_iff [simp]:
assumes "c ≠ 0"
shows "(λx. of_real c * f x) integrable_on s ⟷ f integrable_on s"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have "(λx. of_real (1 / c) * (of_real c * f x)) integrable_on s"
using integrable_cmul[of "λx. of_real c * f x" s "1 / of_real c"]
by (simp add: scaleR_conv_of_real)
then have "(λx. (of_real (1 / c) * of_real c * f x)) integrable_on s"
by (simp add: algebra_simps)
with ‹c ≠ 0› show ?rhs
by (metis (no_types, lifting) integrable_eq mult.left_neutral nonzero_divide_eq_eq of_real_1 of_real_mult)
qed (blast intro: integrable_on_cmult_left)
lemma integrable_on_cmult_right:
fixes f :: "_ ⇒ 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
assumes "f integrable_on s"
shows "(λx. f x * of_real c) integrable_on s"
using integrable_on_cmult_left [OF assms] by (simp add: mult.commute)
lemma integrable_on_cmult_right_iff [simp]:
fixes f :: "_ ⇒ 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
assumes "c ≠ 0"
shows "(λx. f x * of_real c) integrable_on s ⟷ f integrable_on s"
using integrable_on_cmult_left_iff [OF assms] by (simp add: mult.commute)
lemma integrable_on_cdivide:
fixes f :: "_ ⇒ 'b :: real_normed_field"
assumes "f integrable_on s"
shows "(λx. f x / of_real c) integrable_on s"
by (simp add: integrable_on_cmult_right divide_inverse assms of_real_inverse [symmetric] del: of_real_inverse)
lemma integrable_on_cdivide_iff [simp]:
fixes f :: "_ ⇒ 'b :: real_normed_field"
assumes "c ≠ 0"
shows "(λx. f x / of_real c) integrable_on s ⟷ f integrable_on s"
by (simp add: divide_inverse assms of_real_inverse [symmetric] del: of_real_inverse)
lemma has_integral_null [intro]:
assumes "content(cbox a b) = 0"
shows "(f has_integral 0) (cbox a b)"
proof -
have "gauge (λx. ball x 1)"
by auto
moreover
{
fix e :: real
fix p
assume e: "e > 0"
assume p: "p tagged_division_of (cbox a b)"
have "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) = 0"
unfolding norm_eq_zero diff_0_right
using setsum_content_null[OF assms(1) p, of f] .
then have "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e"
using e by auto
}
ultimately show ?thesis
by (auto simp: has_integral)
qed
lemma has_integral_null_real [intro]:
assumes "content {a .. b::real} = 0"
shows "(f has_integral 0) {a .. b}"
by (metis assms box_real(2) has_integral_null)
lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 ⟹ (f has_integral i) (cbox a b) ⟷ i = 0"
by (auto simp add: has_integral_null dest!: integral_unique)
lemma integral_null [simp]: "content (cbox a b) = 0 ⟹ integral (cbox a b) f = 0"
by (metis has_integral_null integral_unique)
lemma integrable_on_null [intro]: "content (cbox a b) = 0 ⟹ f integrable_on (cbox a b)"
by (simp add: has_integral_integrable)
lemma has_integral_empty[intro]: "(f has_integral 0) {}"
by (simp add: has_integral_is_0)
lemma has_integral_empty_eq[simp]: "(f has_integral i) {} ⟷ i = 0"
by (auto simp add: has_integral_empty has_integral_unique)
lemma integrable_on_empty[intro]: "f integrable_on {}"
unfolding integrable_on_def by auto
lemma integral_empty[simp]: "integral {} f = 0"
by (rule integral_unique) (rule has_integral_empty)
lemma has_integral_refl[intro]:
fixes a :: "'a::euclidean_space"
shows "(f has_integral 0) (cbox a a)"
and "(f has_integral 0) {a}"
proof -
have *: "{a} = cbox a a"
apply (rule set_eqI)
unfolding mem_box singleton_iff euclidean_eq_iff[where 'a='a]
apply safe
prefer 3
apply (erule_tac x=b in ballE)
apply (auto simp add: field_simps)
done
show "(f has_integral 0) (cbox a a)" "(f has_integral 0) {a}"
unfolding *
apply (rule_tac[!] has_integral_null)
unfolding content_eq_0_interior
unfolding interior_cbox
using box_sing
apply auto
done
qed
lemma integrable_on_refl[intro]: "f integrable_on cbox a a"
unfolding integrable_on_def by auto
lemma integral_refl [simp]: "integral (cbox a a) f = 0"
by (rule integral_unique) auto
lemma integral_singleton [simp]: "integral {a} f = 0"
by auto
lemma integral_blinfun_apply:
assumes "f integrable_on s"
shows "integral s (λx. blinfun_apply h (f x)) = blinfun_apply h (integral s f)"
by (subst integral_linear[symmetric, OF assms blinfun.bounded_linear_right]) (simp add: o_def)
lemma blinfun_apply_integral:
assumes "f integrable_on s"
shows "blinfun_apply (integral s f) x = integral s (λy. blinfun_apply (f y) x)"
by (metis (no_types, lifting) assms blinfun.prod_left.rep_eq integral_blinfun_apply integral_cong)
subsection ‹Cauchy-type criterion for integrability.›
lemma integrable_cauchy:
fixes f :: "'n::euclidean_space ⇒ 'a::{real_normed_vector,complete_space}"
shows "f integrable_on cbox a b ⟷
(∀e>0.∃d. gauge d ∧
(∀p1 p2. p1 tagged_division_of (cbox a b) ∧ d fine p1 ∧
p2 tagged_division_of (cbox a b) ∧ d fine p2 ⟶
norm (setsum (λ(x,k). content k *⇩R f x) p1 -
setsum (λ(x,k). content k *⇩R f x) p2) < e))"
(is "?l = (∀e>0. ∃d. ?P e d)")
proof
assume ?l
then guess y unfolding integrable_on_def has_integral .. note y=this
show "∀e>0. ∃d. ?P e d"
proof (clarify, goal_cases)
case (1 e)
then have "e/2 > 0" by auto
then guess d
apply -
apply (drule y[rule_format])
apply (elim exE conjE)
done
note d=this[rule_format]
show ?case
proof (rule_tac x=d in exI, clarsimp simp: d)
fix p1 p2
assume as: "p1 tagged_division_of (cbox a b)" "d fine p1"
"p2 tagged_division_of (cbox a b)" "d fine p2"
show "norm ((∑(x, k)∈p1. content k *⇩R f x) - (∑(x, k)∈p2. content k *⇩R f x)) < e"
apply (rule dist_triangle_half_l[where y=y,unfolded dist_norm])
using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .
qed
qed
next
assume "∀e>0. ∃d. ?P e d"
then have "∀n::nat. ∃d. ?P (inverse(of_nat (n + 1))) d"
by auto
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]
have "⋀n. gauge (λx. ⋂{d i x |i. i ∈ {0..n}})"
apply (rule gauge_inters)
using d(1)
apply auto
done
then have "∀n. ∃p. p tagged_division_of (cbox a b) ∧ (λx. ⋂{d i x |i. i ∈ {0..n}}) fine p"
by (meson fine_division_exists)
from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]
have dp: "⋀i n. i≤n ⟹ d i fine p n"
using p(2) unfolding fine_inters by auto
have "Cauchy (λn. setsum (λ(x,k). content k *⇩R (f x)) (p n))"
proof (rule CauchyI, goal_cases)
case (1 e)
then guess N unfolding real_arch_inverse[of e] .. note N=this
show ?case
apply (rule_tac x=N in exI)
proof clarify
fix m n
assume mn: "N ≤ m" "N ≤ n"
have *: "N = (N - 1) + 1" using N by auto
show "norm ((∑(x, k)∈p m. content k *⇩R f x) - (∑(x, k)∈p n. content k *⇩R f x)) < e"
apply (rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]])
apply(subst *)
using dp p(1) mn d(2) by auto
qed
qed
then guess y unfolding convergent_eq_cauchy[symmetric] .. note y=this[THEN LIMSEQ_D]
show ?l
unfolding integrable_on_def has_integral
proof (rule_tac x=y in exI, clarify)
fix e :: real
assume "e>0"
then have *:"e/2 > 0" by auto
then guess N1 unfolding real_arch_inverse[of "e/2"] .. note N1=this
then have N1': "N1 = N1 - 1 + 1"
by auto
guess N2 using y[OF *] .. note N2=this
have "gauge (d (N1 + N2))"
using d by auto
moreover
{
fix q
assume as: "q tagged_division_of (cbox a b)" "d (N1 + N2) fine q"
have *: "inverse (of_nat (N1 + N2 + 1)) < e / 2"
apply (rule less_trans)
using N1
apply auto
done
have "norm ((∑(x, k)∈q. content k *⇩R f x) - y) < e"
apply (rule norm_triangle_half_r)
apply (rule less_trans[OF _ *])
apply (subst N1', rule d(2)[of "p (N1+N2)"])
using N1' as(1) as(2) dp
apply (simp add: ‹∀x. p x tagged_division_of cbox a b ∧ (λxa. ⋂{d i xa |i. i ∈ {0..x}}) fine p x›)
using N2 le_add2 by blast
}
ultimately show "∃d. gauge d ∧
(∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm ((∑(x, k)∈p. content k *⇩R f x) - y) < e)"
by (rule_tac x="d (N1 + N2)" in exI) auto
qed
qed
subsection ‹Additivity of integral on abutting intervals.›
lemma interval_split:
fixes a :: "'a::euclidean_space"
assumes "k ∈ Basis"
shows
"cbox a b ∩ {x. x∙k ≤ c} = cbox a (∑i∈Basis. (if i = k then min (b∙k) c else b∙i) *⇩R i)"
"cbox a b ∩ {x. x∙k ≥ c} = cbox (∑i∈Basis. (if i = k then max (a∙k) c else a∙i) *⇩R i) b"
apply (rule_tac[!] set_eqI)
unfolding Int_iff mem_box mem_Collect_eq
using assms
apply auto
done
lemma content_split:
fixes a :: "'a::euclidean_space"
assumes "k ∈ Basis"
shows "content (cbox a b) = content(cbox a b ∩ {x. x∙k ≤ c}) + content(cbox a b ∩ {x. x∙k ≥ c})"
proof cases
note simps = interval_split[OF assms] content_cbox_cases
have *: "Basis = insert k (Basis - {k})" "⋀x. finite (Basis-{x})" "⋀x. x∉Basis-{x}"
using assms by auto
have *: "⋀X Y Z. (∏i∈Basis. Z i (if i = k then X else Y i)) = Z k X * (∏i∈Basis-{k}. Z i (Y i))"
"(∏i∈Basis. b∙i - a∙i) = (∏i∈Basis-{k}. b∙i - a∙i) * (b∙k - a∙k)"
apply (subst *(1))
defer
apply (subst *(1))
unfolding setprod.insert[OF *(2-)]
apply auto
done
assume as: "∀i∈Basis. a ∙ i ≤ b ∙ i"
moreover
have "⋀x. min (b ∙ k) c = max (a ∙ k) c ⟹
x * (b∙k - a∙k) = x * (max (a ∙ k) c - a ∙ k) + x * (b ∙ k - max (a ∙ k) c)"
by (auto simp add: field_simps)
moreover
have **: "(∏i∈Basis. ((∑i∈Basis. (if i = k then min (b ∙ k) c else b ∙ i) *⇩R i) ∙ i - a ∙ i)) =
(∏i∈Basis. (if i = k then min (b ∙ k) c else b ∙ i) - a ∙ i)"
"(∏i∈Basis. b ∙ i - ((∑i∈Basis. (if i = k then max (a ∙ k) c else a ∙ i) *⇩R i) ∙ i)) =
(∏i∈Basis. b ∙ i - (if i = k then max (a ∙ k) c else a ∙ i))"
by (auto intro!: setprod.cong)
have "¬ a ∙ k ≤ c ⟹ ¬ c ≤ b ∙ k ⟹ False"
unfolding not_le
using as[unfolded ,rule_format,of k] assms
by auto
ultimately show ?thesis
using assms
unfolding simps **
unfolding *(1)[of "λi x. b∙i - x"] *(1)[of "λi x. x - a∙i"]
unfolding *(2)
by auto
next
assume "¬ (∀i∈Basis. a ∙ i ≤ b ∙ i)"
then have "cbox a b = {}"
unfolding box_eq_empty by (auto simp: not_le)
then show ?thesis
by (auto simp: not_le)
qed
lemma division_split_left_inj:
fixes type :: "'a::euclidean_space"
assumes "d division_of i"
and "k1 ∈ d"
and "k2 ∈ d"
and "k1 ≠ k2"
and "k1 ∩ {x::'a. x∙k ≤ c} = k2 ∩ {x. x∙k ≤ c}"
and k: "k∈Basis"
shows "content(k1 ∩ {x. x∙k ≤ c}) = 0"
proof -
note d=division_ofD[OF assms(1)]
have *: "⋀(a::'a) b c. content (cbox a b ∩ {x. x∙k ≤ c}) = 0 ⟷
interior(cbox a b ∩ {x. x∙k ≤ c}) = {}"
unfolding interval_split[OF k] content_eq_0_interior by auto
guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
have **: "⋀s t u. s ∩ t = {} ⟹ u ⊆ s ⟹ u ⊆ t ⟹ u = {}"
by auto
show ?thesis
unfolding uv1 uv2 *
apply (rule **[OF d(5)[OF assms(2-4)]])
apply (simp add: uv1)
using assms(5) uv1 by auto
qed
lemma division_split_right_inj:
fixes type :: "'a::euclidean_space"
assumes "d division_of i"
and "k1 ∈ d"
and "k2 ∈ d"
and "k1 ≠ k2"
and "k1 ∩ {x::'a. x∙k ≥ c} = k2 ∩ {x. x∙k ≥ c}"
and k: "k ∈ Basis"
shows "content (k1 ∩ {x. x∙k ≥ c}) = 0"
proof -
note d=division_ofD[OF assms(1)]
have *: "⋀a b::'a. ⋀c. content(cbox a b ∩ {x. x∙k ≥ c}) = 0 ⟷
interior(cbox a b ∩ {x. x∙k ≥ c}) = {}"
unfolding interval_split[OF k] content_eq_0_interior by auto
guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
have **: "⋀s t u. s ∩ t = {} ⟹ u ⊆ s ⟹ u ⊆ t ⟹ u = {}"
by auto
show ?thesis
unfolding uv1 uv2 *
apply (rule **[OF d(5)[OF assms(2-4)]])
apply (simp add: uv1)
using assms(5) uv1 by auto
qed
lemma tagged_division_split_left_inj:
fixes x1 :: "'a::euclidean_space"
assumes d: "d tagged_division_of i"
and k12: "(x1, k1) ∈ d"
"(x2, k2) ∈ d"
"k1 ≠ k2"
"k1 ∩ {x. x∙k ≤ c} = k2 ∩ {x. x∙k ≤ c}"
"k ∈ Basis"
shows "content (k1 ∩ {x. x∙k ≤ c}) = 0"
proof -
have *: "⋀a b c. (a,b) ∈ c ⟹ b ∈ snd ` c"
by force
show ?thesis
using k12
by (fastforce intro!: division_split_left_inj[OF division_of_tagged_division[OF d]] *)
qed
lemma tagged_division_split_right_inj:
fixes x1 :: "'a::euclidean_space"
assumes d: "d tagged_division_of i"
and k12: "(x1, k1) ∈ d"
"(x2, k2) ∈ d"
"k1 ≠ k2"
"k1 ∩ {x. x∙k ≥ c} = k2 ∩ {x. x∙k ≥ c}"
"k ∈ Basis"
shows "content (k1 ∩ {x. x∙k ≥ c}) = 0"
proof -
have *: "⋀a b c. (a,b) ∈ c ⟹ b ∈ snd ` c"
by force
show ?thesis
using k12
by (fastforce intro!: division_split_right_inj[OF division_of_tagged_division[OF d]] *)
qed
lemma division_split:
fixes a :: "'a::euclidean_space"
assumes "p division_of (cbox a b)"
and k: "k∈Basis"
shows "{l ∩ {x. x∙k ≤ c} | l. l ∈ p ∧ l ∩ {x. x∙k ≤ c} ≠ {}} division_of(cbox a b ∩ {x. x∙k ≤ c})"
(is "?p1 division_of ?I1")
and "{l ∩ {x. x∙k ≥ c} | l. l ∈ p ∧ l ∩ {x. x∙k ≥ c} ≠ {}} division_of (cbox a b ∩ {x. x∙k ≥ c})"
(is "?p2 division_of ?I2")
proof (rule_tac[!] division_ofI)
note p = division_ofD[OF assms(1)]
show "finite ?p1" "finite ?p2"
using p(1) by auto
show "⋃?p1 = ?I1" "⋃?p2 = ?I2"
unfolding p(6)[symmetric] by auto
{
fix k
assume "k ∈ ?p1"
then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
show "k ⊆ ?I1"
using l p(2) uv by force
show "k ≠ {}"
by (simp add: l)
show "∃a b. k = cbox a b"
apply (simp add: l uv p(2-3)[OF l(2)])
apply (subst interval_split[OF k])
apply (auto intro: order.trans)
done
fix k'
assume "k' ∈ ?p1"
then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
assume "k ≠ k'"
then show "interior k ∩ interior k' = {}"
unfolding l l' using p(5)[OF l(2) l'(2)] by auto
}
{
fix k
assume "k ∈ ?p2"
then guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
show "k ⊆ ?I2"
using l p(2) uv by force
show "k ≠ {}"
by (simp add: l)
show "∃a b. k = cbox a b"
apply (simp add: l uv p(2-3)[OF l(2)])
apply (subst interval_split[OF k])
apply (auto intro: order.trans)
done
fix k'
assume "k' ∈ ?p2"
then guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
assume "k ≠ k'"
then show "interior k ∩ interior k' = {}"
unfolding l l' using p(5)[OF l(2) l'(2)] by auto
}
qed
lemma has_integral_split:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes fi: "(f has_integral i) (cbox a b ∩ {x. x∙k ≤ c})"
and fj: "(f has_integral j) (cbox a b ∩ {x. x∙k ≥ c})"
and k: "k ∈ Basis"
shows "(f has_integral (i + j)) (cbox a b)"
proof (unfold has_integral, rule, rule, goal_cases)
case (1 e)
then have e: "e/2 > 0"
by auto
obtain d1
where d1: "gauge d1"
and d1norm:
"⋀p. ⟦p tagged_division_of cbox a b ∩ {x. x ∙ k ≤ c};
d1 fine p⟧ ⟹ norm ((∑(x, k) ∈ p. content k *⇩R f x) - i) < e / 2"
apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
apply (simp add: interval_split[symmetric] k)
done
obtain d2
where d2: "gauge d2"
and d2norm:
"⋀p. ⟦p tagged_division_of cbox a b ∩ {x. c ≤ x ∙ k};
d2 fine p⟧ ⟹ norm ((∑(x, k) ∈ p. content k *⇩R f x) - j) < e / 2"
apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
apply (simp add: interval_split[symmetric] k)
done
let ?d = "λx. if x∙k = c then (d1 x ∩ d2 x) else ball x ¦x∙k - c¦ ∩ d1 x ∩ d2 x"
have "gauge ?d"
using d1 d2 unfolding gauge_def by auto
then show ?case
proof (rule_tac x="?d" in exI, safe)
fix p
assume "p tagged_division_of (cbox a b)" "?d fine p"
note p = this tagged_division_ofD[OF this(1)]
have xk_le_c: "⋀x kk. (x, kk) ∈ p ⟹ kk ∩ {x. x∙k ≤ c} ≠ {} ⟹ x∙k ≤ c"
proof -
fix x kk
assume as: "(x, kk) ∈ p" and kk: "kk ∩ {x. x∙k ≤ c} ≠ {}"
show "x∙k ≤ c"
proof (rule ccontr)
assume **: "¬ ?thesis"
from this[unfolded not_le]
have "kk ⊆ ball x ¦x ∙ k - c¦"
using p(2)[unfolded fine_def, rule_format,OF as] by auto
with kk obtain y where y: "y ∈ ball x ¦x ∙ k - c¦" "y∙k ≤ c"
by blast
then have "¦x ∙ k - y ∙ k¦ < ¦x ∙ k - c¦"
using Basis_le_norm[OF k, of "x - y"]
by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
with y show False
using ** by (auto simp add: field_simps)
qed
qed
have xk_ge_c: "⋀x kk. (x, kk) ∈ p ⟹ kk ∩ {x. x∙k ≥ c} ≠ {} ⟹ x∙k ≥ c"
proof -
fix x kk
assume as: "(x, kk) ∈ p" and kk: "kk ∩ {x. x∙k ≥ c} ≠ {}"
show "x∙k ≥ c"
proof (rule ccontr)
assume **: "¬ ?thesis"
from this[unfolded not_le] have "kk ⊆ ball x ¦x ∙ k - c¦"
using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto
with kk obtain y where y: "y ∈ ball x ¦x ∙ k - c¦" "y∙k ≥ c"
by blast
then have "¦x ∙ k - y ∙ k¦ < ¦x ∙ k - c¦"
using Basis_le_norm[OF k, of "x - y"]
by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
with y show False
using ** by (auto simp add: field_simps)
qed
qed
have lem1: "⋀f P Q. (∀x k. (x, k) ∈ {(x, f k) | x k. P x k} ⟶ Q x k) ⟷
(∀x k. P x k ⟶ Q x (f k))"
by auto
have fin_finite: "finite {(x,f k) | x k. (x,k) ∈ s ∧ P x k}" if "finite s" for f s P
proof -
from that have "finite ((λ(x, k). (x, f k)) ` s)"
by auto
then show ?thesis
by (rule rev_finite_subset) auto
qed
{ fix g :: "'a set ⇒ 'a set"
fix i :: "'a × 'a set"
assume "i ∈ (λ(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) ∈ p ∧ g k ≠ {}}"
then obtain x k where xk:
"i = (x, g k)" "(x, k) ∈ p"
"(x, g k) ∉ {(x, g k) |x k. (x, k) ∈ p ∧ g k ≠ {}}"
by auto
have "content (g k) = 0"
using xk using content_empty by auto
then have "(λ(x, k). content k *⇩R f x) i = 0"
unfolding xk split_conv by auto
} note [simp] = this
have lem3: "⋀g :: 'a set ⇒ 'a set. finite p ⟹
setsum (λ(x, k). content k *⇩R f x) {(x,g k) |x k. (x,k) ∈ p ∧ g k ≠ {}} =
setsum (λ(x, k). content k *⇩R f x) ((λ(x, k). (x, g k)) ` p)"
by (rule setsum.mono_neutral_left) auto
let ?M1 = "{(x, kk ∩ {x. x∙k ≤ c}) |x kk. (x, kk) ∈ p ∧ kk ∩ {x. x∙k ≤ c} ≠ {}}"
have d1_fine: "d1 fine ?M1"
by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
have "norm ((∑(x, k)∈?M1. content k *⇩R f x) - i) < e/2"
proof (rule d1norm [OF tagged_division_ofI d1_fine])
show "finite ?M1"
by (rule fin_finite p(3))+
show "⋃{k. ∃x. (x, k) ∈ ?M1} = cbox a b ∩ {x. x∙k ≤ c}"
unfolding p(8)[symmetric] by auto
fix x l
assume xl: "(x, l) ∈ ?M1"
then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
show "x ∈ l" "l ⊆ cbox a b ∩ {x. x ∙ k ≤ c}"
unfolding xl'
using p(4-6)[OF xl'(3)] using xl'(4)
using xk_le_c[OF xl'(3-4)] by auto
show "∃a b. l = cbox a b"
unfolding xl'
using p(6)[OF xl'(3)]
by (fastforce simp add: interval_split[OF k,where c=c])
fix y r
let ?goal = "interior l ∩ interior r = {}"
assume yr: "(y, r) ∈ ?M1"
then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
assume as: "(x, l) ≠ (y, r)"
show "interior l ∩ interior r = {}"
proof (cases "l' = r' ⟶ x' = y'")
case False
then show ?thesis
using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
next
case True
then have "l' ≠ r'"
using as unfolding xl' yr' by auto
then show ?thesis
using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
qed
qed
moreover
let ?M2 = "{(x,kk ∩ {x. x∙k ≥ c}) |x kk. (x,kk) ∈ p ∧ kk ∩ {x. x∙k ≥ c} ≠ {}}"
have d2_fine: "d2 fine ?M2"
by (force intro: fineI dest: fineD[OF p(2)] simp add: split: if_split_asm)
have "norm ((∑(x, k)∈?M2. content k *⇩R f x) - j) < e/2"
proof (rule d2norm [OF tagged_division_ofI d2_fine])
show "finite ?M2"
by (rule fin_finite p(3))+
show "⋃{k. ∃x. (x, k) ∈ ?M2} = cbox a b ∩ {x. x∙k ≥ c}"
unfolding p(8)[symmetric] by auto
fix x l
assume xl: "(x, l) ∈ ?M2"
then guess x' l' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note xl'=this
show "x ∈ l" "l ⊆ cbox a b ∩ {x. x ∙ k ≥ c}"
unfolding xl'
using p(4-6)[OF xl'(3)] xl'(4) xk_ge_c[OF xl'(3-4)]
by auto
show "∃a b. l = cbox a b"
unfolding xl'
using p(6)[OF xl'(3)]
by (fastforce simp add: interval_split[OF k, where c=c])
fix y r
let ?goal = "interior l ∩ interior r = {}"
assume yr: "(y, r) ∈ ?M2"
then guess y' r' unfolding mem_Collect_eq unfolding prod.inject by (elim exE conjE) note yr'=this
assume as: "(x, l) ≠ (y, r)"
show "interior l ∩ interior r = {}"
proof (cases "l' = r' ⟶ x' = y'")
case False
then show ?thesis
using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
next
case True
then have "l' ≠ r'"
using as unfolding xl' yr' by auto
then show ?thesis
using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto
qed
qed
ultimately
have "norm (((∑(x, k)∈?M1. content k *⇩R f x) - i) + ((∑(x, k)∈?M2. content k *⇩R f x) - j)) < e/2 + e/2"
using norm_add_less by blast
also {
have eq0: "⋀x y. x = (0::real) ⟹ x *⇩R (y::'b) = 0"
using scaleR_zero_left by auto
have cont_eq: "⋀g. (λ(x,l). content l *⇩R f x) ∘ (λ(x,l). (x,g l)) = (λ(x,l). content (g l) *⇩R f x)"
by auto
have "((∑(x, k)∈?M1. content k *⇩R f x) - i) + ((∑(x, k)∈?M2. content k *⇩R f x) - j) =
(∑(x, k)∈?M1. content k *⇩R f x) + (∑(x, k)∈?M2. content k *⇩R f x) - (i + j)"
by auto
also have "… = (∑(x, ka)∈p. content (ka ∩ {x. x ∙ k ≤ c}) *⇩R f x) +
(∑(x, ka)∈p. content (ka ∩ {x. c ≤ x ∙ k}) *⇩R f x) - (i + j)"
unfolding lem3[OF p(3)]
by (subst setsum.reindex_nontrivial[OF p(3)], auto intro!: k eq0 tagged_division_split_left_inj[OF p(1)] tagged_division_split_right_inj[OF p(1)]
simp: cont_eq)+
also note setsum.distrib[symmetric]
also have "⋀x. x ∈ p ⟹
(λ(x,ka). content (ka ∩ {x. x ∙ k ≤ c}) *⇩R f x) x +
(λ(x,ka). content (ka ∩ {x. c ≤ x ∙ k}) *⇩R f x) x =
(λ(x,ka). content ka *⇩R f x) x"
proof clarify
fix a b
assume "(a, b) ∈ p"
from p(6)[OF this] guess u v by (elim exE) note uv=this
then show "content (b ∩ {x. x ∙ k ≤ c}) *⇩R f a + content (b ∩ {x. c ≤ x ∙ k}) *⇩R f a =
content b *⇩R f a"
unfolding scaleR_left_distrib[symmetric]
unfolding uv content_split[OF k,of u v c]
by auto
qed
note setsum.cong [OF _ this]
finally have "(∑(x, k)∈{(x, kk ∩ {x. x ∙ k ≤ c}) |x kk. (x, kk) ∈ p ∧ kk ∩ {x. x ∙ k ≤ c} ≠ {}}. content k *⇩R f x) - i +
((∑(x, k)∈{(x, kk ∩ {x. c ≤ x ∙ k}) |x kk. (x, kk) ∈ p ∧ kk ∩ {x. c ≤ x ∙ k} ≠ {}}. content k *⇩R f x) - j) =
(∑(x, ka)∈p. content ka *⇩R f x) - (i + j)"
by auto
}
finally show "norm ((∑(x, k)∈p. content k *⇩R f x) - (i + j)) < e"
by auto
qed
qed
subsection ‹A sort of converse, integrability on subintervals.›
lemma tagged_division_union_interval:
fixes a :: "'a::euclidean_space"
assumes "p1 tagged_division_of (cbox a b ∩ {x. x∙k ≤ (c::real)})"
and "p2 tagged_division_of (cbox a b ∩ {x. x∙k ≥ c})"
and k: "k ∈ Basis"
shows "(p1 ∪ p2) tagged_division_of (cbox a b)"
proof -
have *: "cbox a b = (cbox a b ∩ {x. x∙k ≤ c}) ∪ (cbox a b ∩ {x. x∙k ≥ c})"
by auto
show ?thesis
apply (subst *)
apply (rule tagged_division_union[OF assms(1-2)])
unfolding interval_split[OF k] interior_cbox
using k
apply (auto simp add: box_def elim!: ballE[where x=k])
done
qed
lemma tagged_division_union_interval_real:
fixes a :: real
assumes "p1 tagged_division_of ({a .. b} ∩ {x. x∙k ≤ (c::real)})"
and "p2 tagged_division_of ({a .. b} ∩ {x. x∙k ≥ c})"
and k: "k ∈ Basis"
shows "(p1 ∪ p2) tagged_division_of {a .. b}"
using assms
unfolding box_real[symmetric]
by (rule tagged_division_union_interval)
lemma has_integral_separate_sides:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "(f has_integral i) (cbox a b)"
and "e > 0"
and k: "k ∈ Basis"
obtains d where "gauge d"
"∀p1 p2. p1 tagged_division_of (cbox a b ∩ {x. x∙k ≤ c}) ∧ d fine p1 ∧
p2 tagged_division_of (cbox a b ∩ {x. x∙k ≥ c}) ∧ d fine p2 ⟶
norm ((setsum (λ(x,k). content k *⇩R f x) p1 + setsum (λ(x,k). content k *⇩R f x) p2) - i) < e"
proof -
guess d using has_integralD[OF assms(1-2)] . note d=this
{ fix p1 p2
assume "p1 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≤ c}" "d fine p1"
note p1=tagged_division_ofD[OF this(1)] this
assume "p2 tagged_division_of (cbox a b) ∩ {x. c ≤ x ∙ k}" "d fine p2"
note p2=tagged_division_ofD[OF this(1)] this
note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this
{ fix a b
assume ab: "(a, b) ∈ p1 ∩ p2"
have "(a, b) ∈ p1"
using ab by auto
with p1 obtain u v where uv: "b = cbox u v" by auto
have "b ⊆ {x. x∙k = c}"
using ab p1(3)[of a b] p2(3)[of a b] by fastforce
moreover
have "interior {x::'a. x ∙ k = c} = {}"
proof (rule ccontr)
assume "¬ ?thesis"
then obtain x where x: "x ∈ interior {x::'a. x∙k = c}"
by auto
then guess e unfolding mem_interior .. note e=this
have x: "x∙k = c"
using x interior_subset by fastforce
have *: "⋀i. i ∈ Basis ⟹ ¦(x - (x + (e / 2) *⇩R k)) ∙ i¦ = (if i = k then e/2 else 0)"
using e k by (auto simp: inner_simps inner_not_same_Basis)
have "(∑i∈Basis. ¦(x - (x + (e / 2 ) *⇩R k)) ∙ i¦) =
(∑i∈Basis. (if i = k then e / 2 else 0))"
using "*" by (blast intro: setsum.cong)
also have "… < e"
apply (subst setsum.delta)
using e
apply auto
done
finally have "x + (e/2) *⇩R k ∈ ball x e"
unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
then have "x + (e/2) *⇩R k ∈ {x. x∙k = c}"
using e by auto
then show False
unfolding mem_Collect_eq using e x k by (auto simp: inner_simps)
qed
ultimately have "content b = 0"
unfolding uv content_eq_0_interior
using interior_mono by blast
then have "content b *⇩R f a = 0"
by auto
}
then have "norm ((∑(x, k)∈p1. content k *⇩R f x) + (∑(x, k)∈p2. content k *⇩R f x) - i) =
norm ((∑(x, k)∈p1 ∪ p2. content k *⇩R f x) - i)"
by (subst setsum.union_inter_neutral) (auto simp: p1 p2)
also have "… < e"
by (rule k d(2) p12 fine_union p1 p2)+
finally have "norm ((∑(x, k)∈p1. content k *⇩R f x) + (∑(x, k)∈p2. content k *⇩R f x) - i) < e" .
}
then show ?thesis
by (auto intro: that[of d] d elim: )
qed
lemma integrable_split[intro]:
fixes f :: "'a::euclidean_space ⇒ 'b::{real_normed_vector,complete_space}"
assumes "f integrable_on cbox a b"
and k: "k ∈ Basis"
shows "f integrable_on (cbox a b ∩ {x. x∙k ≤ c})" (is ?t1)
and "f integrable_on (cbox a b ∩ {x. x∙k ≥ c})" (is ?t2)
proof -
guess y using assms(1) unfolding integrable_on_def .. note y=this
def b' ≡ "∑i∈Basis. (if i = k then min (b∙k) c else b∙i)*⇩R i::'a"
def a' ≡ "∑i∈Basis. (if i = k then max (a∙k) c else a∙i)*⇩R i::'a"
show ?t1 ?t2
unfolding interval_split[OF k] integrable_cauchy
unfolding interval_split[symmetric,OF k]
proof (rule_tac[!] allI impI)+
fix e :: real
assume "e > 0"
then have "e/2>0"
by auto
from has_integral_separate_sides[OF y this k,of c] guess d . note d=this[rule_format]
let ?P = "λA. ∃d. gauge d ∧ (∀p1 p2. p1 tagged_division_of (cbox a b) ∩ A ∧ d fine p1 ∧
p2 tagged_division_of (cbox a b) ∩ A ∧ d fine p2 ⟶
norm ((∑(x, k)∈p1. content k *⇩R f x) - (∑(x, k)∈p2. content k *⇩R f x)) < e)"
show "?P {x. x ∙ k ≤ c}"
proof (rule_tac x=d in exI, clarsimp simp add: d)
fix p1 p2
assume as: "p1 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≤ c}" "d fine p1"
"p2 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≤ c}" "d fine p2"
show "norm ((∑(x, k)∈p1. content k *⇩R f x) - (∑(x, k)∈p2. content k *⇩R f x)) < e"
proof (rule fine_division_exists[OF d(1), of a' b] )
fix p
assume "p tagged_division_of cbox a' b" "d fine p"
then show ?thesis
using as norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]
unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
by (auto simp add: algebra_simps)
qed
qed
show "?P {x. x ∙ k ≥ c}"
proof (rule_tac x=d in exI, clarsimp simp add: d)
fix p1 p2
assume as: "p1 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≥ c}" "d fine p1"
"p2 tagged_division_of (cbox a b) ∩ {x. x ∙ k ≥ c}" "d fine p2"
show "norm ((∑(x, k)∈p1. content k *⇩R f x) - (∑(x, k)∈p2. content k *⇩R f x)) < e"
proof (rule fine_division_exists[OF d(1), of a b'] )
fix p
assume "p tagged_division_of cbox a b'" "d fine p"
then show ?thesis
using as norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]
unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
by (auto simp add: algebra_simps)
qed
qed
qed
qed
subsection ‹Generalized notion of additivity.›
definition "neutral opp = (SOME x. ∀y. opp x y = y ∧ opp y x = y)"
definition operative :: "('a ⇒ 'a ⇒ 'a) ⇒ (('b::euclidean_space) set ⇒ 'a) ⇒ bool"
where "operative opp f ⟷
(∀a b. content (cbox a b) = 0 ⟶ f (cbox a b) = neutral opp) ∧
(∀a b c. ∀k∈Basis. f (cbox a b) = opp (f(cbox a b ∩ {x. x∙k ≤ c})) (f (cbox a b ∩ {x. x∙k ≥ c})))"
lemma operativeD[dest]:
fixes type :: "'a::euclidean_space"
assumes "operative opp f"
shows "⋀a b::'a. content (cbox a b) = 0 ⟹ f (cbox a b) = neutral opp"
and "⋀a b c k. k ∈ Basis ⟹ f (cbox a b) =
opp (f (cbox a b ∩ {x. x∙k ≤ c})) (f (cbox a b ∩ {x. x∙k ≥ c}))"
using assms unfolding operative_def by auto
lemma property_empty_interval: "∀a b. content (cbox a b) = 0 ⟶ P (cbox a b) ⟹ P {}"
using content_empty unfolding empty_as_interval by auto
lemma operative_empty: "operative opp f ⟹ f {} = neutral opp"
unfolding operative_def by (rule property_empty_interval) auto
subsection ‹Using additivity of lifted function to encode definedness.›
fun lifted where
"lifted (opp :: 'a ⇒ 'a ⇒ 'b) (Some x) (Some y) = Some (opp x y)"
| "lifted opp None _ = (None::'b option)"
| "lifted opp _ None = None"
lemma lifted_simp_1[simp]: "lifted opp v None = None"
by (induct v) auto
definition "monoidal opp ⟷
(∀x y. opp x y = opp y x) ∧
(∀x y z. opp x (opp y z) = opp (opp x y) z) ∧
(∀x. opp (neutral opp) x = x)"
lemma monoidalI:
assumes "⋀x y. opp x y = opp y x"
and "⋀x y z. opp x (opp y z) = opp (opp x y) z"
and "⋀x. opp (neutral opp) x = x"
shows "monoidal opp"
unfolding monoidal_def using assms by fastforce
lemma monoidal_ac:
assumes "monoidal opp"
shows [simp]: "opp (neutral opp) a = a"
and [simp]: "opp a (neutral opp) = a"
and "opp a b = opp b a"
and "opp (opp a b) c = opp a (opp b c)"
and "opp a (opp b c) = opp b (opp a c)"
using assms unfolding monoidal_def by metis+
lemma neutral_lifted [cong]:
assumes "monoidal opp"
shows "neutral (lifted opp) = Some (neutral opp)"
proof -
{ fix x
assume "∀y. lifted opp x y = y ∧ lifted opp y x = y"
then have "Some (neutral opp) = x"
apply (induct x)
apply force
by (metis assms lifted.simps(1) monoidal_ac(2) option.inject) }
note [simp] = this
show ?thesis
apply (subst neutral_def)
apply (intro some_equality allI)
apply (induct_tac y)
apply (auto simp add:monoidal_ac[OF assms])
done
qed
lemma monoidal_lifted[intro]:
assumes "monoidal opp"
shows "monoidal (lifted opp)"
unfolding monoidal_def split_option_all neutral_lifted[OF assms]
using monoidal_ac[OF assms]
by auto
definition "support opp f s = {x. x∈s ∧ f x ≠ neutral opp}"
definition "fold' opp e s = (if finite s then Finite_Set.fold opp e s else e)"
definition "iterate opp s f = fold' (λx a. opp (f x) a) (neutral opp) (support opp f s)"
lemma support_subset[intro]: "support opp f s ⊆ s"
unfolding support_def by auto
lemma support_empty[simp]: "support opp f {} = {}"
using support_subset[of opp f "{}"] by auto
lemma comp_fun_commute_monoidal[intro]:
assumes "monoidal opp"
shows "comp_fun_commute opp"
unfolding comp_fun_commute_def
using monoidal_ac[OF assms]
by auto
lemma support_clauses:
"⋀f g s. support opp f {} = {}"
"⋀f g s. support opp f (insert x s) =
(if f(x) = neutral opp then support opp f s else insert x (support opp f s))"
"⋀f g s. support opp f (s - {x}) = (support opp f s) - {x}"
"⋀f g s. support opp f (s ∪ t) = (support opp f s) ∪ (support opp f t)"
"⋀f g s. support opp f (s ∩ t) = (support opp f s) ∩ (support opp f t)"
"⋀f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"
"⋀f g s. support opp g (f ` s) = f ` (support opp (g ∘ f) s)"
unfolding support_def by auto
lemma finite_support[intro]: "finite s ⟹ finite (support opp f s)"
unfolding support_def by auto
lemma iterate_empty[simp]: "iterate opp {} f = neutral opp"
unfolding iterate_def fold'_def by auto
lemma iterate_insert[simp]:
assumes "monoidal opp"
and "finite s"
shows "iterate opp (insert x s) f =
(if x ∈ s then iterate opp s f else opp (f x) (iterate opp s f))"
proof (cases "x ∈ s")
case True
then show ?thesis by (auto simp: insert_absorb iterate_def)
next
case False
note * = comp_fun_commute.comp_comp_fun_commute [OF comp_fun_commute_monoidal[OF assms(1)]]
show ?thesis
proof (cases "f x = neutral opp")
case True
then show ?thesis
using assms ‹x ∉ s›
by (auto simp: iterate_def support_clauses)
next
case False
with ‹x ∉ s› ‹finite s› support_subset show ?thesis
apply (simp add: iterate_def fold'_def support_clauses)
apply (subst comp_fun_commute.fold_insert[OF * finite_support, simplified comp_def])
apply (force simp add: )+
done
qed
qed
lemma iterate_some:
"⟦monoidal opp; finite s⟧ ⟹ iterate (lifted opp) s (λx. Some(f x)) = Some (iterate opp s f)"
by (erule finite_induct) (auto simp: monoidal_lifted)
subsection ‹Two key instances of additivity.›
lemma neutral_add[simp]: "neutral op + = (0::'a::comm_monoid_add)"
unfolding neutral_def
by (force elim: allE [where x=0])
lemma operative_content[intro]: "operative (op +) content"
by (force simp add: operative_def content_split[symmetric])
lemma monoidal_monoid[intro]: "monoidal ((op +)::('a::comm_monoid_add) ⇒ 'a ⇒ 'a)"
unfolding monoidal_def neutral_add
by (auto simp add: algebra_simps)
lemma operative_integral:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
shows "operative (lifted(op +)) (λi. if f integrable_on i then Some(integral i f) else None)"
unfolding operative_def neutral_lifted[OF monoidal_monoid] neutral_add
proof safe
fix a b c
fix k :: 'a
assume k: "k ∈ Basis"
show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
lifted op + (if f integrable_on cbox a b ∩ {x. x ∙ k ≤ c} then Some (integral (cbox a b ∩ {x. x ∙ k ≤ c}) f) else None)
(if f integrable_on cbox a b ∩ {x. c ≤ x ∙ k} then Some (integral (cbox a b ∩ {x. c ≤ x ∙ k}) f) else None)"
proof (cases "f integrable_on cbox a b")
case True
with k show ?thesis
apply (simp add: integrable_split)
apply (rule integral_unique [OF has_integral_split[OF _ _ k]])
apply (auto intro: integrable_integral)
done
next
case False
have "¬ (f integrable_on cbox a b ∩ {x. x ∙ k ≤ c}) ∨ ¬ ( f integrable_on cbox a b ∩ {x. c ≤ x ∙ k})"
proof (rule ccontr)
assume "¬ ?thesis"
then have "f integrable_on cbox a b"
unfolding integrable_on_def
apply (rule_tac x="integral (cbox a b ∩ {x. x ∙ k ≤ c}) f + integral (cbox a b ∩ {x. x ∙ k ≥ c}) f" in exI)
apply (rule has_integral_split[OF _ _ k])
apply (auto intro: integrable_integral)
done
then show False
using False by auto
qed
then show ?thesis
using False by auto
qed
next
fix a b :: 'a
assume "content (cbox a b) = 0"
then show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = Some 0"
using has_integral_null_eq
by (auto simp: integrable_on_null)
qed
subsection ‹Points of division of a partition.›
definition "division_points (k::('a::euclidean_space) set) d =
{(j,x). j ∈ Basis ∧ (interval_lowerbound k)∙j < x ∧ x < (interval_upperbound k)∙j ∧
(∃i∈d. (interval_lowerbound i)∙j = x ∨ (interval_upperbound i)∙j = x)}"
lemma division_points_finite:
fixes i :: "'a::euclidean_space set"
assumes "d division_of i"
shows "finite (division_points i d)"
proof -
note assm = division_ofD[OF assms]
let ?M = "λj. {(j,x)|x. (interval_lowerbound i)∙j < x ∧ x < (interval_upperbound i)∙j ∧
(∃i∈d. (interval_lowerbound i)∙j = x ∨ (interval_upperbound i)∙j = x)}"
have *: "division_points i d = ⋃(?M ` Basis)"
unfolding division_points_def by auto
show ?thesis
unfolding * using assm by auto
qed
lemma division_points_subset:
fixes a :: "'a::euclidean_space"
assumes "d division_of (cbox a b)"
and "∀i∈Basis. a∙i < b∙i" "a∙k < c" "c < b∙k"
and k: "k ∈ Basis"
shows "division_points (cbox a b ∩ {x. x∙k ≤ c}) {l ∩ {x. x∙k ≤ c} | l . l ∈ d ∧ l ∩ {x. x∙k ≤ c} ≠ {}} ⊆
division_points (cbox a b) d" (is ?t1)
and "division_points (cbox a b ∩ {x. x∙k ≥ c}) {l ∩ {x. x∙k ≥ c} | l . l ∈ d ∧ ~(l ∩ {x. x∙k ≥ c} = {})} ⊆
division_points (cbox a b) d" (is ?t2)
proof -
note assm = division_ofD[OF assms(1)]
have *: "∀i∈Basis. a∙i ≤ b∙i"
"∀i∈Basis. a∙i ≤ (∑i∈Basis. (if i = k then min (b ∙ k) c else b ∙ i) *⇩R i) ∙ i"
"∀i∈Basis. (∑i∈Basis. (if i = k then max (a ∙ k) c else a ∙ i) *⇩R i) ∙ i ≤ b∙i"
"min (b ∙ k) c = c" "max (a ∙ k) c = c"
using assms using less_imp_le by auto
show ?t1
unfolding division_points_def interval_split[OF k, of a b]
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
unfolding *
apply (rule subsetI)
unfolding mem_Collect_eq split_beta
apply (erule bexE conjE)+
apply (simp add: )
apply (erule exE conjE)+
proof
fix i l x
assume as:
"a ∙ fst x < snd x" "snd x < (if fst x = k then c else b ∙ fst x)"
"interval_lowerbound i ∙ fst x = snd x ∨ interval_upperbound i ∙ fst x = snd x"
"i = l ∩ {x. x ∙ k ≤ c}" "l ∈ d" "l ∩ {x. x ∙ k ≤ c} ≠ {}"
and fstx: "fst x ∈ Basis"
from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
have *: "∀i∈Basis. u ∙ i ≤ (∑i∈Basis. (if i = k then min (v ∙ k) c else v ∙ i) *⇩R i) ∙ i"
using as(6) unfolding l interval_split[OF k] box_ne_empty as .
have **: "∀i∈Basis. u∙i ≤ v∙i"
using l using as(6) unfolding box_ne_empty[symmetric] by auto
show "∃i∈d. interval_lowerbound i ∙ fst x = snd x ∨ interval_upperbound i ∙ fst x = snd x"
apply (rule bexI[OF _ ‹l ∈ d›])
using as(1-3,5) fstx
unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
apply (auto split: if_split_asm)
done
show "snd x < b ∙ fst x"
using as(2) ‹c < b∙k› by (auto split: if_split_asm)
qed
show ?t2
unfolding division_points_def interval_split[OF k, of a b]
unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
unfolding *
unfolding subset_eq
apply rule
unfolding mem_Collect_eq split_beta
apply (erule bexE conjE)+
apply (simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
apply (erule exE conjE)+
proof
fix i l x
assume as:
"(if fst x = k then c else a ∙ fst x) < snd x" "snd x < b ∙ fst x"
"interval_lowerbound i ∙ fst x = snd x ∨ interval_upperbound i ∙ fst x = snd x"
"i = l ∩ {x. c ≤ x ∙ k}" "l ∈ d" "l ∩ {x. c ≤ x ∙ k} ≠ {}"
and fstx: "fst x ∈ Basis"
from assm(4)[OF this(5)] guess u v by (elim exE) note l=this
have *: "∀i∈Basis. (∑i∈Basis. (if i = k then max (u ∙ k) c else u ∙ i) *⇩R i) ∙ i ≤ v ∙ i"
using as(6) unfolding l interval_split[OF k] box_ne_empty as .
have **: "∀i∈Basis. u∙i ≤ v∙i"
using l using as(6) unfolding box_ne_empty[symmetric] by auto
show "∃i∈d. interval_lowerbound i ∙ fst x = snd x ∨ interval_upperbound i ∙ fst x = snd x"
apply (rule bexI[OF _ ‹l ∈ d›])
using as(1-3,5) fstx
unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
apply (auto split: if_split_asm)
done
show "a ∙ fst x < snd x"
using as(1) ‹a∙k < c› by (auto split: if_split_asm)
qed
qed
lemma division_points_psubset:
fixes a :: "'a::euclidean_space"
assumes "d division_of (cbox a b)"
and "∀i∈Basis. a∙i < b∙i" "a∙k < c" "c < b∙k"
and "l ∈ d"
and "interval_lowerbound l∙k = c ∨ interval_upperbound l∙k = c"
and k: "k ∈ Basis"
shows "division_points (cbox a b ∩ {x. x∙k ≤ c}) {l ∩ {x. x∙k ≤ c} | l. l∈d ∧ l ∩ {x. x∙k ≤ c} ≠ {}} ⊂
division_points (cbox a b) d" (is "?D1 ⊂ ?D")
and "division_points (cbox a b ∩ {x. x∙k ≥ c}) {l ∩ {x. x∙k ≥ c} | l. l∈d ∧ l ∩ {x. x∙k ≥ c} ≠ {}} ⊂
division_points (cbox a b) d" (is "?D2 ⊂ ?D")
proof -
have ab: "∀i∈Basis. a∙i ≤ b∙i"
using assms(2) by (auto intro!:less_imp_le)
guess u v using division_ofD(4)[OF assms(1,5)] by (elim exE) note l=this
have uv: "∀i∈Basis. u∙i ≤ v∙i" "∀i∈Basis. a∙i ≤ u∙i ∧ v∙i ≤ b∙i"
using division_ofD(2,2,3)[OF assms(1,5)] unfolding l box_ne_empty
using subset_box(1)
apply auto
apply blast+
done
have *: "interval_upperbound (cbox a b ∩ {x. x ∙ k ≤ interval_upperbound l ∙ k}) ∙ k = interval_upperbound l ∙ k"
"interval_upperbound (cbox a b ∩ {x. x ∙ k ≤ interval_lowerbound l ∙ k}) ∙ k = interval_lowerbound l ∙ k"
unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
using uv[rule_format, of k] ab k
by auto
have "∃x. x ∈ ?D - ?D1"
using assms(3-)
unfolding division_points_def interval_bounds[OF ab]
apply -
apply (erule disjE)
apply (rule_tac x="(k,(interval_lowerbound l)∙k)" in exI, force simp add: *)
apply (rule_tac x="(k,(interval_upperbound l)∙k)" in exI, force simp add: *)
done
moreover have "?D1 ⊆ ?D"
by (auto simp add: assms division_points_subset)
ultimately show "?D1 ⊂ ?D"
by blast
have *: "interval_lowerbound (cbox a b ∩ {x. x ∙ k ≥ interval_lowerbound l ∙ k}) ∙ k = interval_lowerbound l ∙ k"
"interval_lowerbound (cbox a b ∩ {x. x ∙ k ≥ interval_upperbound l ∙ k}) ∙ k = interval_upperbound l ∙ k"
unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
using uv[rule_format, of k] ab k
by auto
have "∃x. x ∈ ?D - ?D2"
using assms(3-)
unfolding division_points_def interval_bounds[OF ab]
apply -
apply (erule disjE)
apply (rule_tac x="(k,(interval_lowerbound l)∙k)" in exI, force simp add: *)
apply (rule_tac x="(k,(interval_upperbound l)∙k)" in exI, force simp add: *)
done
moreover have "?D2 ⊆ ?D"
by (auto simp add: assms division_points_subset)
ultimately show "?D2 ⊂ ?D"
by blast
qed
subsection ‹Preservation by divisions and tagged divisions.›
lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"
unfolding support_def by auto
lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"
unfolding iterate_def support_support by auto
lemma iterate_expand_cases:
"iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"
by (simp add: iterate_def fold'_def)
lemma iterate_image:
assumes "monoidal opp"
and "inj_on f s"
shows "iterate opp (f ` s) g = iterate opp s (g ∘ f)"
proof -
have *: "iterate opp (f ` s) g = iterate opp s (g ∘ f)"
if "finite s" "∀x∈s. ∀y∈s. f x = f y ⟶ x = y" for s
using that
proof (induct s)
case empty
then show ?case by simp
next
case insert
with assms(1) show ?case by auto
qed
show ?thesis
apply (cases "finite (support opp g (f ` s))")
prefer 2
apply (metis finite_imageI iterate_expand_cases support_clauses(7))
apply (subst (1) iterate_support[symmetric], subst (2) iterate_support[symmetric])
unfolding support_clauses
apply (rule *)
apply (meson assms(2) finite_imageD subset_inj_on support_subset)
apply (meson assms(2) inj_on_contraD rev_subsetD support_subset)
done
qed
lemma iterate_nonzero_image_lemma:
assumes "monoidal opp"
and "finite s" "g(a) = neutral opp"
and "∀x∈s. ∀y∈s. f x = f y ∧ x ≠ y ⟶ g(f x) = neutral opp"
shows "iterate opp {f x | x. x ∈ s ∧ f x ≠ a} g = iterate opp s (g ∘ f)"
proof -
have *: "{f x |x. x ∈ s ∧ f x ≠ a} = f ` {x. x ∈ s ∧ f x ≠ a}"
by auto
have **: "support opp (g ∘ f) {x ∈ s. f x ≠ a} = support opp (g ∘ f) s"
unfolding support_def using assms(3) by auto
have inj: "inj_on f (support opp (g ∘ f) {x ∈ s. f x ≠ a})"
apply (simp add: inj_on_def)
apply (metis (mono_tags, lifting) assms(4) comp_def mem_Collect_eq support_def)
done
show ?thesis
apply (subst iterate_support[symmetric])
apply (simp add: * support_clauses iterate_image[OF assms(1) inj])
apply (simp add: iterate_def **)
done
qed
lemma iterate_eq_neutral:
assumes "monoidal opp"
and "⋀x. x ∈ s ⟹ f x = neutral opp"
shows "iterate opp s f = neutral opp"
proof -
have [simp]: "support opp f s = {}"
unfolding support_def using assms(2) by auto
show ?thesis
by (subst iterate_support[symmetric]) simp
qed
lemma iterate_op:
"⟦monoidal opp; finite s⟧
⟹ iterate opp s (λx. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)"
by (erule finite_induct) (auto simp: monoidal_ac(4) monoidal_ac(5))
lemma iterate_eq:
assumes "monoidal opp"
and "⋀x. x ∈ s ⟹ f x = g x"
shows "iterate opp s f = iterate opp s g"
proof -
have *: "support opp g s = support opp f s"
unfolding support_def using assms(2) by auto
show ?thesis
proof (cases "finite (support opp f s)")
case False
then show ?thesis
by (simp add: "*" iterate_expand_cases)
next
case True
def su ≡ "support opp f s"
have fsu: "finite su"
using True by (simp add: su_def)
moreover
{ assume "finite su" "su ⊆ s"
then have "iterate opp su f = iterate opp su g"
by (induct su) (auto simp: assms)
}
ultimately have "iterate opp (support opp f s) f = iterate opp (support opp g s) g"
by (simp add: "*" su_def support_subset)
then show ?thesis
by simp
qed
qed
lemma nonempty_witness:
assumes "s ≠ {}"
obtains x where "x ∈ s"
using assms by auto
lemma operative_division:
fixes f :: "'a::euclidean_space set ⇒ 'b"
assumes "monoidal opp"
and "operative opp f"
and "d division_of (cbox a b)"
shows "iterate opp d f = f (cbox a b)"
proof -
def C ≡ "card (division_points (cbox a b) d)"
then show ?thesis
using assms
proof (induct C arbitrary: a b d rule: full_nat_induct)
case (1 a b d)
show ?case
proof (cases "content (cbox a b) = 0")
case True
show "iterate opp d f = f (cbox a b)"
unfolding operativeD(1)[OF assms(2) True]
proof (rule iterate_eq_neutral[OF ‹monoidal opp›])
fix x
assume x: "x ∈ d"
then show "f x = neutral opp"
by (metis division_ofD(4) 1(4) division_of_content_0[OF True] operativeD(1)[OF assms(2)] x)
qed
next
case False
note ab = this[unfolded content_lt_nz[symmetric] content_pos_lt_eq]
then have ab': "∀i∈Basis. a∙i ≤ b∙i"
by (auto intro!: less_imp_le)
show "iterate opp d f = f (cbox a b)"
proof (cases "division_points (cbox a b) d = {}")
case True
{ fix u v and j :: 'a
assume j: "j ∈ Basis" and as: "cbox u v ∈ d"
then have "cbox u v ≠ {}"
using "1.prems"(3) by blast
then have uv: "∀i∈Basis. u∙i ≤ v∙i" "u∙j ≤ v∙j"
using j unfolding box_ne_empty by auto
have *: "⋀p r Q. ¬ j∈Basis ∨ p ∨ r ∨ (∀x∈d. Q x) ⟹ p ∨ r ∨ Q (cbox u v)"
using as j by auto
have "(j, u∙j) ∉ division_points (cbox a b) d"
"(j, v∙j) ∉ division_points (cbox a b) d" using True by auto
note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
moreover
have "a∙j ≤ u∙j" "v∙j ≤ b∙j"
using division_ofD(2,2,3)[OF ‹d division_of cbox a b› as]
apply (metis j subset_box(1) uv(1))
by (metis ‹cbox u v ⊆ cbox a b› j subset_box(1) uv(1))
ultimately have "u∙j = a∙j ∧ v∙j = a∙j ∨ u∙j = b∙j ∧ v∙j = b∙j ∨ u∙j = a∙j ∧ v∙j = b∙j"
unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by force }
then have d': "∀i∈d. ∃u v. i = cbox u v ∧
(∀j∈Basis. u∙j = a∙j ∧ v∙j = a∙j ∨ u∙j = b∙j ∧ v∙j = b∙j ∨ u∙j = a∙j ∧ v∙j = b∙j)"
unfolding forall_in_division[OF 1(4)]
by blast
have "(1/2) *⇩R (a+b) ∈ cbox a b"
unfolding mem_box using ab by(auto intro!: less_imp_le simp: inner_simps)
note this[unfolded division_ofD(6)[OF ‹d division_of cbox a b›,symmetric] Union_iff]
then guess i .. note i=this
guess u v using d'[rule_format,OF i(1)] by (elim exE conjE) note uv=this
have "cbox a b ∈ d"
proof -
have "u = a" "v = b"
unfolding euclidean_eq_iff[where 'a='a]
proof safe
fix j :: 'a
assume j: "j ∈ Basis"
note i(2)[unfolded uv mem_box,rule_format,of j]
then show "u ∙ j = a ∙ j" and "v ∙ j = b ∙ j"
using uv(2)[rule_format,of j] j by (auto simp: inner_simps)
qed
then have "i = cbox a b" using uv by auto
then show ?thesis using i by auto
qed
then have deq: "d = insert (cbox a b) (d - {cbox a b})"
by auto
have "iterate opp (d - {cbox a b}) f = neutral opp"
proof (rule iterate_eq_neutral[OF 1(2)])
fix x
assume x: "x ∈ d - {cbox a b}"
then have "x∈d"
by auto note d'[rule_format,OF this]
then guess u v by (elim exE conjE) note uv=this
have "u ≠ a ∨ v ≠ b"
using x[unfolded uv] by auto
then obtain j where "u∙j ≠ a∙j ∨ v∙j ≠ b∙j" and j: "j ∈ Basis"
unfolding euclidean_eq_iff[where 'a='a] by auto
then have "u∙j = v∙j"
using uv(2)[rule_format,OF j] by auto
then have "content (cbox u v) = 0"
unfolding content_eq_0 using j
by force
then show "f x = neutral opp"
unfolding uv(1) by (rule operativeD(1)[OF 1(3)])
qed
then show "iterate opp d f = f (cbox a b)"
apply (subst deq)
apply (subst iterate_insert[OF 1(2)])
using 1
apply auto
done
next
case False
then have "∃x. x ∈ division_points (cbox a b) d"
by auto
then guess k c
unfolding split_paired_Ex division_points_def mem_Collect_eq split_conv
apply (elim exE conjE)
done
note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
from this(3) guess j .. note j=this
def d1 ≡ "{l ∩ {x. x∙k ≤ c} | l. l ∈ d ∧ l ∩ {x. x∙k ≤ c} ≠ {}}"
def d2 ≡ "{l ∩ {x. x∙k ≥ c} | l. l ∈ d ∧ l ∩ {x. x∙k ≥ c} ≠ {}}"
def cb ≡ "(∑i∈Basis. (if i = k then c else b∙i) *⇩R i)::'a"
def ca ≡ "(∑i∈Basis. (if i = k then c else a∙i) *⇩R i)::'a"
note division_points_psubset[OF ‹d division_of cbox a b› ab kc(1-2) j]
note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
then have *: "(iterate opp d1 f) = f (cbox a b ∩ {x. x∙k ≤ c})"
"(iterate opp d2 f) = f (cbox a b ∩ {x. x∙k ≥ c})"
unfolding interval_split[OF kc(4)]
apply (rule_tac[!] "1.hyps"[rule_format])
using division_split[OF ‹d division_of cbox a b›, where k=k and c=c]
apply (simp_all add: interval_split 1 kc d1_def d2_def division_points_finite[OF ‹d division_of cbox a b›])
done
{ fix l y
assume as: "l ∈ d" "y ∈ d" "l ∩ {x. x ∙ k ≤ c} = y ∩ {x. x ∙ k ≤ c}" "l ≠ y"
from division_ofD(4)[OF ‹d division_of cbox a b› this(1)] guess u v by (elim exE) note leq=this
have "f (l ∩ {x. x ∙ k ≤ c}) = neutral opp"
unfolding leq interval_split[OF kc(4)]
apply (rule operativeD(1) 1)+
unfolding interval_split[symmetric,OF kc(4)]
using division_split_left_inj 1 as kc leq by blast
} note fxk_le = this
{ fix l y
assume as: "l ∈ d" "y ∈ d" "l ∩ {x. c ≤ x ∙ k} = y ∩ {x. c ≤ x ∙ k}" "l ≠ y"
from division_ofD(4)[OF ‹d division_of cbox a b› this(1)] guess u v by (elim exE) note leq=this
have "f (l ∩ {x. x ∙ k ≥ c}) = neutral opp"
unfolding leq interval_split[OF kc(4)]
apply (rule operativeD(1) 1)+
unfolding interval_split[symmetric,OF kc(4)]
using division_split_right_inj 1 leq as kc by blast
} note fxk_ge = this
have "f (cbox a b) = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")
unfolding *
using assms(2) kc(4) by blast
also have "iterate opp d1 f = iterate opp d (λl. f(l ∩ {x. x∙k ≤ c}))"
unfolding d1_def empty_as_interval
apply (rule iterate_nonzero_image_lemma[unfolded o_def])
apply (rule 1 division_of_finite operativeD[OF 1(3)])+
apply (force simp add: empty_as_interval[symmetric] fxk_le)+
done
also have "iterate opp d2 f = iterate opp d (λl. f(l ∩ {x. x∙k ≥ c}))"
unfolding d2_def empty_as_interval
apply (rule iterate_nonzero_image_lemma[unfolded o_def])
apply (rule 1 division_of_finite operativeD[OF 1(3)])+
apply (force simp add: empty_as_interval[symmetric] fxk_ge)+
done
also have *: "∀x∈d. f x = opp (f (x ∩ {x. x ∙ k ≤ c})) (f (x ∩ {x. c ≤ x ∙ k}))"
unfolding forall_in_division[OF ‹d division_of cbox a b›]
using assms(2) kc(4) by blast
have "opp (iterate opp d (λl. f (l ∩ {x. x ∙ k ≤ c}))) (iterate opp d (λl. f (l ∩ {x. c ≤ x ∙ k}))) =
iterate opp d f"
apply (subst(3) iterate_eq[OF _ *[rule_format]])
using 1
apply (auto simp: iterate_op[symmetric])
done
finally show ?thesis by auto
qed
qed
qed
qed
lemma iterate_image_nonzero:
assumes "monoidal opp"
and "finite s"
and "⋀x y. ∀x∈s. ∀y∈s. x ≠ y ∧ f x = f y ⟶ g (f x) = neutral opp"
shows "iterate opp (f ` s) g = iterate opp s (g ∘ f)"
using assms
by (induct rule: finite_subset_induct[OF assms(2) subset_refl]) auto
lemma operative_tagged_division:
assumes "monoidal opp"
and "operative opp f"
and "d tagged_division_of (cbox a b)"
shows "iterate opp d (λ(x,l). f l) = f (cbox a b)"
proof -
have *: "(λ(x,l). f l) = f ∘ snd"
unfolding o_def by rule auto note tagged = tagged_division_ofD[OF assms(3)]
{ fix a b a'
assume as: "(a, b) ∈ d" "(a', b) ∈ d" "(a, b) ≠ (a', b)"
have "f b = neutral opp"
using tagged(4)[OF as(1)]
apply clarify
apply (rule operativeD(1)[OF assms(2)])
by (metis content_eq_0_interior inf.idem tagged_division_ofD(5)[OF assms(3) as(1-3)])
}
then have "iterate opp d (λ(x,l). f l) = iterate opp (snd ` d) f"
unfolding *
by (force intro!: assms iterate_image_nonzero[symmetric, OF _ tagged_division_of_finite])
also have "… = f (cbox a b)"
using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .
finally show ?thesis .
qed
subsection ‹Additivity of content.›
lemma setsum_iterate:
assumes "finite s"
shows "setsum f s = iterate op + s f"
proof -
have "setsum f s = setsum f (support op + f s)"
using assms
by (auto simp: support_def intro: setsum.mono_neutral_right)
then show ?thesis unfolding iterate_def fold'_def setsum.eq_fold
by (simp add: comp_def)
qed
lemma additive_content_division:
"d division_of (cbox a b) ⟹ setsum content d = content (cbox a b)"
by (metis division_ofD(1) monoidal_monoid operative_content operative_division setsum_iterate)
lemma additive_content_tagged_division:
"d tagged_division_of (cbox a b) ⟹ setsum (λ(x,l). content l) d = content (cbox a b)"
unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,symmetric]
using setsum_iterate by blast
subsection ‹Finally, the integral of a constant›
lemma has_integral_const [intro]:
fixes a b :: "'a::euclidean_space"
shows "((λx. c) has_integral (content (cbox a b) *⇩R c)) (cbox a b)"
apply (auto intro!: exI [where x="λx. ball x 1"] simp: split_def has_integral)
apply (subst scaleR_left.setsum[symmetric, unfolded o_def])
apply (subst additive_content_tagged_division[unfolded split_def])
apply auto
done
lemma has_integral_const_real [intro]:
fixes a b :: real
shows "((λx. c) has_integral (content {a .. b} *⇩R c)) {a .. b}"
by (metis box_real(2) has_integral_const)
lemma integral_const [simp]:
fixes a b :: "'a::euclidean_space"
shows "integral (cbox a b) (λx. c) = content (cbox a b) *⇩R c"
by (rule integral_unique) (rule has_integral_const)
lemma integral_const_real [simp]:
fixes a b :: real
shows "integral {a .. b} (λx. c) = content {a .. b} *⇩R c"
by (metis box_real(2) integral_const)
subsection ‹Bounds on the norm of Riemann sums and the integral itself.›
lemma dsum_bound:
assumes "p division_of (cbox a b)"
and "norm c ≤ e"
shows "norm (setsum (λl. content l *⇩R c) p) ≤ e * content(cbox a b)"
proof -
have sumeq: "(∑i∈p. ¦content i¦) = setsum content p"
apply (rule setsum.cong)
using assms
apply simp
apply (metis abs_of_nonneg assms(1) content_pos_le division_ofD(4))
done
have e: "0 ≤ e"
using assms(2) norm_ge_zero order_trans by blast
have "norm (setsum (λl. content l *⇩R c) p) ≤ (∑i∈p. norm (content i *⇩R c))"
using norm_setsum by blast
also have "... ≤ e * (∑i∈p. ¦content i¦)"
apply (simp add: setsum_right_distrib[symmetric] mult.commute)
using assms(2) mult_right_mono by blast
also have "... ≤ e * content (cbox a b)"
apply (rule mult_left_mono [OF _ e])
apply (simp add: sumeq)
using additive_content_division assms(1) eq_iff apply blast
done
finally show ?thesis .
qed
lemma rsum_bound:
assumes p: "p tagged_division_of (cbox a b)"
and "∀x∈cbox a b. norm (f x) ≤ e"
shows "norm (setsum (λ(x,k). content k *⇩R f x) p) ≤ e * content (cbox a b)"
proof (cases "cbox a b = {}")
case True show ?thesis
using p unfolding True tagged_division_of_trivial by auto
next
case False
then have e: "e ≥ 0"
by (metis assms(2) norm_ge_zero order_trans nonempty_witness)
have setsum_le: "setsum (content ∘ snd) p ≤ content (cbox a b)"
unfolding additive_content_tagged_division[OF p, symmetric] split_def
by (auto intro: eq_refl)
have con: "⋀xk. xk ∈ p ⟹ 0 ≤ content (snd xk)"
using tagged_division_ofD(4) [OF p] content_pos_le
by force
have norm: "⋀xk. xk ∈ p ⟹ norm (f (fst xk)) ≤ e"
unfolding fst_conv using tagged_division_ofD(2,3)[OF p] assms
by (metis prod.collapse subset_eq)
have "norm (setsum (λ(x,k). content k *⇩R f x) p) ≤ (∑i∈p. norm (case i of (x, k) ⇒ content k *⇩R f x))"
by (rule norm_setsum)
also have "... ≤ e * content (cbox a b)"
unfolding split_def norm_scaleR
apply (rule order_trans[OF setsum_mono])
apply (rule mult_left_mono[OF _ abs_ge_zero, of _ e])
apply (metis norm)
unfolding setsum_left_distrib[symmetric]
using con setsum_le
apply (auto simp: mult.commute intro: mult_left_mono [OF _ e])
done
finally show ?thesis .
qed
lemma rsum_diff_bound:
assumes "p tagged_division_of (cbox a b)"
and "∀x∈cbox a b. norm (f x - g x) ≤ e"
shows "norm (setsum (λ(x,k). content k *⇩R f x) p - setsum (λ(x,k). content k *⇩R g x) p) ≤
e * content (cbox a b)"
apply (rule order_trans[OF _ rsum_bound[OF assms]])
apply (simp add: split_def scaleR_diff_right setsum_subtractf eq_refl)
done
lemma has_integral_bound:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "0 ≤ B"
and "(f has_integral i) (cbox a b)"
and "∀x∈cbox a b. norm (f x) ≤ B"
shows "norm i ≤ B * content (cbox a b)"
proof (rule ccontr)
assume "¬ ?thesis"
then have *: "norm i - B * content (cbox a b) > 0"
by auto
from assms(2)[unfolded has_integral,rule_format,OF *]
guess d by (elim exE conjE) note d=this[rule_format]
from fine_division_exists[OF this(1), of a b] guess p . note p=this
have *: "⋀s B. norm s ≤ B ⟹ ¬ norm (s - i) < norm i - B"
unfolding not_less
by (metis norm_triangle_sub[of i] add.commute le_less_trans less_diff_eq linorder_not_le norm_minus_commute)
show False
using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto
qed
corollary has_integral_bound_real:
fixes f :: "real ⇒ 'b::real_normed_vector"
assumes "0 ≤ B"
and "(f has_integral i) {a .. b}"
and "∀x∈{a .. b}. norm (f x) ≤ B"
shows "norm i ≤ B * content {a .. b}"
by (metis assms box_real(2) has_integral_bound)
corollary integrable_bound:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "0 ≤ B"
and "f integrable_on (cbox a b)"
and "⋀x. x∈cbox a b ⟹ norm (f x) ≤ B"
shows "norm (integral (cbox a b) f) ≤ B * content (cbox a b)"
by (metis integrable_integral has_integral_bound assms)
subsection ‹Similar theorems about relationship among components.›
lemma rsum_component_le:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "p tagged_division_of (cbox a b)"
and "∀x∈cbox a b. (f x)∙i ≤ (g x)∙i"
shows "(setsum (λ(x,k). content k *⇩R f x) p)∙i ≤ (setsum (λ(x,k). content k *⇩R g x) p)∙i"
unfolding inner_setsum_left
proof (rule setsum_mono, clarify)
fix a b
assume ab: "(a, b) ∈ p"
note tagged = tagged_division_ofD(2-4)[OF assms(1) ab]
from this(3) guess u v by (elim exE) note b=this
show "(content b *⇩R f a) ∙ i ≤ (content b *⇩R g a) ∙ i"
unfolding b inner_simps real_scaleR_def
apply (rule mult_left_mono)
using assms(2) tagged
by (auto simp add: content_pos_le)
qed
lemma has_integral_component_le:
fixes f g :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes k: "k ∈ Basis"
assumes "(f has_integral i) s" "(g has_integral j) s"
and "∀x∈s. (f x)∙k ≤ (g x)∙k"
shows "i∙k ≤ j∙k"
proof -
have lem: "i∙k ≤ j∙k"
if f_i: "(f has_integral i) (cbox a b)"
and g_j: "(g has_integral j) (cbox a b)"
and le: "∀x∈cbox a b. (f x)∙k ≤ (g x)∙k"
for a b i and j :: 'b and f g :: "'a ⇒ 'b"
proof (rule ccontr)
assume "¬ ?thesis"
then have *: "0 < (i∙k - j∙k) / 3"
by auto
guess d1 using f_i[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d1=this[rule_format]
guess d2 using g_j[unfolded has_integral,rule_format,OF *] by (elim exE conjE) note d2=this[rule_format]
obtain p where p: "p tagged_division_of cbox a b" "d1 fine p" "d2 fine p"
using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter
by metis
note le_less_trans[OF Basis_le_norm[OF k]]
then have "¦((∑(x, k)∈p. content k *⇩R f x) - i) ∙ k¦ < (i ∙ k - j ∙ k) / 3"
"¦((∑(x, k)∈p. content k *⇩R g x) - j) ∙ k¦ < (i ∙ k - j ∙ k) / 3"
using k norm_bound_Basis_lt d1 d2 p
by blast+
then show False
unfolding inner_simps
using rsum_component_le[OF p(1) le]
by (simp add: abs_real_def split: if_split_asm)
qed
show ?thesis
proof (cases "∃a b. s = cbox a b")
case True
with lem assms show ?thesis
by auto
next
case False
show ?thesis
proof (rule ccontr)
assume "¬ i∙k ≤ j∙k"
then have ij: "(i∙k - j∙k) / 3 > 0"
by auto
note has_integral_altD[OF _ False this]
from this[OF assms(2)] this[OF assms(3)] guess B1 B2 . note B=this[rule_format]
have "bounded (ball 0 B1 ∪ ball (0::'a) B2)"
unfolding bounded_Un by(rule conjI bounded_ball)+
from bounded_subset_cbox[OF this] guess a b by (elim exE)
note ab = conjunctD2[OF this[unfolded Un_subset_iff]]
guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]
guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]
have *: "⋀w1 w2 j i::real .¦w1 - i¦ < (i - j) / 3 ⟹ ¦w2 - j¦ < (i - j) / 3 ⟹ w1 ≤ w2 ⟹ False"
by (simp add: abs_real_def split: if_split_asm)
note le_less_trans[OF Basis_le_norm[OF k]]
note this[OF w1(2)] this[OF w2(2)]
moreover
have "w1∙k ≤ w2∙k"
by (rule lem[OF w1(1) w2(1)]) (simp add: assms(4))
ultimately show False
unfolding inner_simps by(rule *)
qed
qed
qed
lemma integral_component_le:
fixes g f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "k ∈ Basis"
and "f integrable_on s" "g integrable_on s"
and "∀x∈s. (f x)∙k ≤ (g x)∙k"
shows "(integral s f)∙k ≤ (integral s g)∙k"
apply (rule has_integral_component_le)
using integrable_integral assms
apply auto
done
lemma has_integral_component_nonneg:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "k ∈ Basis"
and "(f has_integral i) s"
and "∀x∈s. 0 ≤ (f x)∙k"
shows "0 ≤ i∙k"
using has_integral_component_le[OF assms(1) has_integral_0 assms(2)]
using assms(3-)
by auto
lemma integral_component_nonneg:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "k ∈ Basis"
and "∀x∈s. 0 ≤ (f x)∙k"
shows "0 ≤ (integral s f)∙k"
proof (cases "f integrable_on s")
case True show ?thesis
apply (rule has_integral_component_nonneg)
using assms True
apply auto
done
next
case False then show ?thesis by (simp add: not_integrable_integral)
qed
lemma has_integral_component_neg:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "k ∈ Basis"
and "(f has_integral i) s"
and "∀x∈s. (f x)∙k ≤ 0"
shows "i∙k ≤ 0"
using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-)
by auto
lemma has_integral_component_lbound:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "(f has_integral i) (cbox a b)"
and "∀x∈cbox a b. B ≤ f(x)∙k"
and "k ∈ Basis"
shows "B * content (cbox a b) ≤ i∙k"
using has_integral_component_le[OF assms(3) has_integral_const assms(1),of "(∑i∈Basis. B *⇩R i)::'b"] assms(2-)
by (auto simp add: field_simps)
lemma has_integral_component_ubound:
fixes f::"'a::euclidean_space => 'b::euclidean_space"
assumes "(f has_integral i) (cbox a b)"
and "∀x∈cbox a b. f x∙k ≤ B"
and "k ∈ Basis"
shows "i∙k ≤ B * content (cbox a b)"
using has_integral_component_le[OF assms(3,1) has_integral_const, of "∑i∈Basis. B *⇩R i"] assms(2-)
by (auto simp add: field_simps)
lemma integral_component_lbound:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "f integrable_on cbox a b"
and "∀x∈cbox a b. B ≤ f(x)∙k"
and "k ∈ Basis"
shows "B * content (cbox a b) ≤ (integral(cbox a b) f)∙k"
apply (rule has_integral_component_lbound)
using assms
unfolding has_integral_integral
apply auto
done
lemma integral_component_lbound_real:
assumes "f integrable_on {a ::real .. b}"
and "∀x∈{a .. b}. B ≤ f(x)∙k"
and "k ∈ Basis"
shows "B * content {a .. b} ≤ (integral {a .. b} f)∙k"
using assms
by (metis box_real(2) integral_component_lbound)
lemma integral_component_ubound:
fixes f :: "'a::euclidean_space ⇒ 'b::euclidean_space"
assumes "f integrable_on cbox a b"
and "∀x∈cbox a b. f x∙k ≤ B"
and "k ∈ Basis"
shows "(integral (cbox a b) f)∙k ≤ B * content (cbox a b)"
apply (rule has_integral_component_ubound)
using assms
unfolding has_integral_integral
apply auto
done
lemma integral_component_ubound_real:
fixes f :: "real ⇒ 'a::euclidean_space"
assumes "f integrable_on {a .. b}"
and "∀x∈{a .. b}. f x∙k ≤ B"
and "k ∈ Basis"
shows "(integral {a .. b} f)∙k ≤ B * content {a .. b}"
using assms
by (metis box_real(2) integral_component_ubound)
subsection ‹Uniform limit of integrable functions is integrable.›
lemma real_arch_invD:
"0 < (e::real) ⟹ (∃n::nat. n ≠ 0 ∧ 0 < inverse (real n) ∧ inverse (real n) < e)"
by (subst(asm) real_arch_inverse)
lemma integrable_uniform_limit:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
assumes "∀e>0. ∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
shows "f integrable_on cbox a b"
proof (cases "content (cbox a b) > 0")
case False then show ?thesis
using has_integral_null
by (simp add: content_lt_nz integrable_on_def)
next
case True
have *: "⋀P. ∀e>(0::real). P e ⟹ ∀n::nat. P (inverse (real n + 1))"
by auto
from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]
from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "λx. x"]]
obtain i where i: "⋀x. (g x has_integral i x) (cbox a b)"
by auto
have "Cauchy i"
unfolding Cauchy_def
proof clarify
fix e :: real
assume "e>0"
then have "e / 4 / content (cbox a b) > 0"
using True by (auto simp add: field_simps)
then obtain M :: nat
where M: "M ≠ 0" "0 < inverse (real_of_nat M)" "inverse (of_nat M) < e / 4 / content (cbox a b)"
by (subst (asm) real_arch_inverse) auto
show "∃M. ∀m≥M. ∀n≥M. dist (i m) (i n) < e"
proof (rule exI [where x=M], clarify)
fix m n
assume m: "M ≤ m" and n: "M ≤ n"
have "e/4>0" using ‹e>0› by auto
note * = i[unfolded has_integral,rule_format,OF this]
from *[of m] guess gm by (elim conjE exE) note gm=this[rule_format]
from *[of n] guess gn by (elim conjE exE) note gn=this[rule_format]
from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b]
obtain p where p: "p tagged_division_of cbox a b" "(λx. gm x ∩ gn x) fine p"
by auto
{ fix s1 s2 i1 and i2::'b
assume no: "norm(s2 - s1) ≤ e/2" "norm (s1 - i1) < e/4" "norm (s2 - i2) < e/4"
have "norm (i1 - i2) ≤ norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
using norm_triangle_ineq[of "s1 - s2" "s2 - i2"]
by (auto simp add: algebra_simps)
also have "… < e"
using no
unfolding norm_minus_commute
by (auto simp add: algebra_simps)
finally have "norm (i1 - i2) < e" .
} note triangle3 = this
have finep: "gm fine p" "gn fine p"
using fine_inter p by auto
{ fix x
assume x: "x ∈ cbox a b"
have "norm (f x - g n x) + norm (f x - g m x) ≤ inverse (real n + 1) + inverse (real m + 1)"
using g(1)[OF x, of n] g(1)[OF x, of m] by auto
also have "… ≤ inverse (real M) + inverse (real M)"
apply (rule add_mono)
using M(2) m n by auto
also have "… = 2 / real M"
unfolding divide_inverse by auto
finally have "norm (g n x - g m x) ≤ 2 / real M"
using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
by (auto simp add: algebra_simps simp add: norm_minus_commute)
} note norm_le = this
have le_e2: "norm ((∑(x, k)∈p. content k *⇩R g n x) - (∑(x, k)∈p. content k *⇩R g m x)) ≤ e / 2"
apply (rule order_trans [OF rsum_diff_bound[OF p(1), where e="2 / real M"]])
apply (blast intro: norm_le)
using M True
by (auto simp add: field_simps)
then show "dist (i m) (i n) < e"
unfolding dist_norm
using gm gn p finep
by (auto intro!: triangle3)
qed
qed
then obtain s where s: "i ⇢ s"
using convergent_eq_cauchy[symmetric] by blast
show ?thesis
unfolding integrable_on_def has_integral
proof (rule_tac x=s in exI, clarify)
fix e::real
assume e: "0 < e"
then have *: "e/3 > 0" by auto
then obtain N1 where N1: "∀n≥N1. norm (i n - s) < e / 3"
using LIMSEQ_D [OF s] by metis
from e True have "e / 3 / content (cbox a b) > 0"
by (auto simp add: field_simps)
from real_arch_invD[OF this] guess N2 by (elim exE conjE) note N2=this
from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]
{ fix sf sg i
assume no: "norm (sf - sg) ≤ e / 3"
"norm(i - s) < e / 3"
"norm (sg - i) < e / 3"
have "norm (sf - s) ≤ norm (sf - sg) + norm (sg - i) + norm (i - s)"
using norm_triangle_ineq[of "sf - sg" "sg - s"]
using norm_triangle_ineq[of "sg - i" " i - s"]
by (auto simp add: algebra_simps)
also have "… < e"
using no
unfolding norm_minus_commute
by (auto simp add: algebra_simps)
finally have "norm (sf - s) < e" .
} note lem = this
{ fix p
assume p: "p tagged_division_of (cbox a b) ∧ g' fine p"
then have norm_less: "norm ((∑(x, k)∈p. content k *⇩R g (N1 + N2) x) - i (N1 + N2)) < e / 3"
using g' by blast
have "content (cbox a b) < e / 3 * (of_nat N2)"
using N2 unfolding inverse_eq_divide using True by (auto simp add: field_simps)
moreover have "e / 3 * of_nat N2 ≤ e / 3 * (of_nat (N1 + N2) + 1)"
using ‹e>0› by auto
ultimately have "content (cbox a b) < e / 3 * (of_nat (N1 + N2) + 1)"
by linarith
then have le_e3: "inverse (real (N1 + N2) + 1) * content (cbox a b) ≤ e / 3"
unfolding inverse_eq_divide
by (auto simp add: field_simps)
have ne3: "norm (i (N1 + N2) - s) < e / 3"
using N1 by auto
have "norm ((∑(x, k)∈p. content k *⇩R f x) - s) < e"
apply (rule lem[OF order_trans [OF _ le_e3] ne3 norm_less])
apply (rule rsum_diff_bound[OF p[THEN conjunct1]])
apply (blast intro: g)
done }
then show "∃d. gauge d ∧
(∀p. p tagged_division_of cbox a b ∧ d fine p ⟶ norm ((∑(x, k)∈p. content k *⇩R f x) - s) < e)"
by (blast intro: g')
qed
qed
lemmas integrable_uniform_limit_real = integrable_uniform_limit [where 'a=real, simplified]
subsection ‹Negligible sets.›
definition "negligible (s:: 'a::euclidean_space set) ⟷
(∀a b. ((indicator s :: 'a⇒real) has_integral 0) (cbox a b))"
subsection ‹Negligibility of hyperplane.›
lemma setsum_nonzero_image_lemma:
assumes "finite s"
and "g a = 0"
and "∀x∈s. ∀y∈s. f x = f y ∧ x ≠ y ⟶ g (f x) = 0"
shows "setsum g {f x |x. x ∈ s ∧ f x ≠ a} = setsum (g ∘ f) s"
apply (subst setsum_iterate)
using assms monoidal_monoid
unfolding setsum_iterate[OF assms(1)]
apply (auto intro!: iterate_nonzero_image_lemma)
done
lemma interval_doublesplit:
fixes a :: "'a::euclidean_space"
assumes "k ∈ Basis"
shows "cbox a b ∩ {x . ¦x∙k - c¦ ≤ (e::real)} =
cbox (∑i∈Basis. (if i = k then max (a∙k) (c - e) else a∙i) *⇩R i)
(∑i∈Basis. (if i = k then min (b∙k) (c + e) else b∙i) *⇩R i)"
proof -
have *: "⋀x c e::real. ¦x - c¦ ≤ e ⟷ x ≥ c - e ∧ x ≤ c + e"
by auto
have **: "⋀s P Q. s ∩ {x. P x ∧ Q x} = (s ∩ {x. Q x}) ∩ {x. P x}"
by blast
show ?thesis
unfolding * ** interval_split[OF assms] by (rule refl)
qed
lemma division_doublesplit:
fixes a :: "'a::euclidean_space"
assumes "p division_of (cbox a b)"
and k: "k ∈ Basis"
shows "{l ∩ {x. ¦x∙k - c¦ ≤ e} |l. l ∈ p ∧ l ∩ {x. ¦x∙k - c¦ ≤ e} ≠ {}}
division_of (cbox a b ∩ {x. ¦x∙k - c¦ ≤ e})"
proof -
have *: "⋀x c. ¦x - c¦ ≤ e ⟷ x ≥ c - e ∧ x ≤ c + e"
by auto
have **: "⋀p q p' q'. p division_of q ⟹ p = p' ⟹ q = q' ⟹ p' division_of q'"
by auto
note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
note division_split(2)[OF this, where c="c-e" and k=k,OF k]
then show ?thesis
apply (rule **)
unfolding interval_doublesplit [OF k]
using k
apply (simp_all add: * interval_split)
apply (rule equalityI, blast)
apply clarsimp
apply (rule_tac x="l ∩ {x. c + e ≥ x ∙ k}" in exI)
apply auto
done
qed
lemma content_doublesplit:
fixes a :: "'a::euclidean_space"
assumes "0 < e"
and k: "k ∈ Basis"
obtains d where "0 < d" and "content (cbox a b ∩ {x. ¦x∙k - c¦ ≤ d}) < e"
proof (cases "content (cbox a b) = 0")
case True
then have ce: "content (cbox a b) < e"
by (metis ‹0 < e›)
show ?thesis
apply (rule that[of 1])
apply simp
unfolding interval_doublesplit[OF k]
apply (rule le_less_trans[OF content_subset ce])
apply (auto simp: interval_doublesplit[symmetric] k)
done
next
case False
def d ≡ "e / 3 / setprod (λi. b∙i - a∙i) (Basis - {k})"
note False[unfolded content_eq_0 not_ex not_le, rule_format]
then have "⋀x. x ∈ Basis ⟹ b∙x > a∙x"
by (auto simp add:not_le)
then have prod0: "0 < setprod (λi. b∙i - a∙i) (Basis - {k})"
by (force simp add: setprod_pos field_simps)
then have "d > 0"
using assms
by (auto simp add: d_def field_simps)
then show ?thesis
proof (rule that[of d])
have *: "Basis = insert k (Basis - {k})"
using k by auto
have less_e: "(min (b ∙ k) (c + d) - max (a ∙ k) (c - d)) * (∏i∈Basis - {k}. b ∙ i - a ∙ i) < e"
proof -
have "(min (b ∙ k) (c + d) - max (a ∙ k) (c - d)) ≤ 2 * d"
by auto
also have "… < e / (∏i∈Basis - {k}. b ∙ i - a ∙ i)"
unfolding d_def
using assms prod0
by (auto simp add: field_simps)
finally show "(min (b ∙ k) (c + d) - max (a ∙ k) (c - d)) * (∏i∈Basis - {k}. b ∙ i - a ∙ i) < e"
unfolding pos_less_divide_eq[OF prod0] .
qed
show "content (cbox a b ∩ {x. ¦x ∙ k - c¦ ≤ d}) < e"
proof (cases "cbox a b ∩ {x. ¦x ∙ k - c¦ ≤ d} = {}")
case True
then show ?thesis
using assms by simp
next
case False
then have
"(∏i∈Basis - {k}. interval_upperbound (cbox a b ∩ {x. ¦x ∙ k - c¦ ≤ d}) ∙ i -
interval_lowerbound (cbox a b ∩ {x. ¦x ∙ k - c¦ ≤ d}) ∙ i)
= (∏i∈Basis - {k}. b∙i - a∙i)"
by (simp add: box_eq_empty interval_doublesplit[OF k])
then show "content (cbox a b ∩ {x. ¦x ∙ k - c¦ ≤ d}) < e"
unfolding content_def
using assms False
apply (subst *)
apply (subst setprod.insert)
apply (simp_all add: interval_doublesplit[OF k] box_eq_empty not_less less_e)
done
qed
qed
qed
lemma negligible_standard_hyperplane[intro]:
fixes k :: "'a::euclidean_space"
assumes k: "k ∈ Basis"
shows "negligible {x. x∙k = c}"
unfolding negligible_def has_integral
proof (clarify, goal_cases)
case (1 a b e)
from this and k obtain d where d: "0 < d" "content (cbox a b ∩ {x. ¦x ∙ k - c¦ ≤ d}) < e"
by (rule content_doublesplit)
let ?i = "indicator {x::'a. x∙k = c} :: 'a⇒real"
show ?case
apply (rule_tac x="λx. ball x d" in exI)
apply rule
apply (rule gauge_ball)
apply (rule d)
proof (rule, rule)
fix p
assume p: "p tagged_division_of (cbox a b) ∧ (λx. ball x d) fine p"
have *: "(∑(x, ka)∈p. content ka *⇩R ?i x) =
(∑(x, ka)∈p. content (ka ∩ {x. ¦x∙k - c¦ ≤ d}) *⇩R ?i x)"
apply (rule setsum.cong)
apply (rule refl)
unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv
apply cases
apply (rule disjI1)
apply assumption
apply (rule disjI2)
proof -
fix x l
assume as: "(x, l) ∈ p" "?i x ≠ 0"
then have xk: "x∙k = c"
unfolding indicator_def
apply -
apply (rule ccontr)
apply auto
done
show "content l = content (l ∩ {x. ¦x ∙ k - c¦ ≤ d})"
apply (rule arg_cong[where f=content])
apply (rule set_eqI)
apply rule
apply rule
unfolding mem_Collect_eq
proof -
fix y
assume y: "y ∈ l"
note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]
note this[unfolded subset_eq mem_ball dist_norm,rule_format,OF y]
note le_less_trans[OF Basis_le_norm[OF k] this]
then show "¦y ∙ k - c¦ ≤ d"
unfolding inner_simps xk by auto
qed auto
qed
note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]
show "norm ((∑(x, ka)∈p. content ka *⇩R ?i x) - 0) < e"
unfolding diff_0_right *
unfolding real_scaleR_def real_norm_def
apply (subst abs_of_nonneg)
apply (rule setsum_nonneg)
apply rule
unfolding split_paired_all split_conv
apply (rule mult_nonneg_nonneg)
apply (drule p'(4))
apply (erule exE)+
apply(rule_tac b=b in back_subst)
prefer 2
apply (subst(asm) eq_commute)
apply assumption
apply (subst interval_doublesplit[OF k])
apply (rule content_pos_le)
apply (rule indicator_pos_le)
proof -
have "(∑(x, ka)∈p. content (ka ∩ {x. ¦x ∙ k - c¦ ≤ d}) * ?i x) ≤
(∑(x, ka)∈p. content (ka ∩ {x. ¦x ∙ k - c¦ ≤ d}))"
apply (rule setsum_mono)
unfolding split_paired_all split_conv
apply (rule mult_right_le_one_le)
apply (drule p'(4))
apply (auto simp add:interval_doublesplit[OF k])
done
also have "… < e"
proof (subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]], goal_cases)
case prems: (1 u v)
have "content (cbox u v ∩ {x. ¦x ∙ k - c¦ ≤ d}) ≤ content (cbox u v)"
unfolding interval_doublesplit[OF k]
apply (rule content_subset)
unfolding interval_doublesplit[symmetric,OF k]
apply auto
done
then show ?case
unfolding prems interval_doublesplit[OF k]
by (blast intro: antisym)
next
have *: "setsum content {l ∩ {x. ¦x ∙ k - c¦ ≤ d} |l. l ∈ snd ` p ∧ l ∩ {x. ¦x ∙ k - c¦ ≤ d} ≠ {}} ≥ 0"
apply (rule setsum_nonneg)
apply rule
unfolding mem_Collect_eq image_iff
apply (erule exE bexE conjE)+
unfolding split_paired_all
proof -
fix x l a b
assume as: "x = l ∩ {x. ¦x ∙ k - c¦ ≤ d}" "(a, b) ∈ p" "l = snd (a, b)"
guess u v using p'(4)[OF as(2)] by (elim exE) note * = this
show "content x ≥ 0"
unfolding as snd_conv * interval_doublesplit[OF k]
by (rule content_pos_le)
qed
have **: "norm (1::real) ≤ 1"
by auto
note division_doublesplit[OF p'' k,unfolded interval_doublesplit[OF k]]
note dsum_bound[OF this **,unfolded interval_doublesplit[symmetric,OF k]]
note this[unfolded real_scaleR_def real_norm_def mult_1_right mult_1, of c d]
note le_less_trans[OF this d(2)]
from this[unfolded abs_of_nonneg[OF *]]
show "(∑ka∈snd ` p. content (ka ∩ {x. ¦x ∙ k - c¦ ≤ d})) < e"
apply (subst setsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,symmetric])
apply (rule finite_imageI p' content_empty)+
unfolding forall_in_division[OF p'']
proof (rule,rule,rule,rule,rule,rule,rule,erule conjE)
fix m n u v
assume as:
"cbox m n ∈ snd ` p" "cbox u v ∈ snd ` p"
"cbox m n ≠ cbox u v"
"cbox m n ∩ {x. ¦x ∙ k - c¦ ≤ d} = cbox u v ∩ {x. ¦x ∙ k - c¦ ≤ d}"
have "(cbox m n ∩ {x. ¦x ∙ k - c¦ ≤ d}) ∩ (cbox u v ∩ {x. ¦x ∙ k - c¦ ≤ d}) ⊆ cbox m n ∩ cbox u v"
by blast
note interior_mono[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_Int[of "cbox m n"]]
then have "interior (cbox m n ∩ {x. ¦x ∙ k - c¦ ≤ d}) = {}"
unfolding as Int_absorb by auto
then show "content (cbox m n ∩ {x. ¦x ∙ k - c¦ ≤ d}) = 0"
unfolding interval_doublesplit[OF k] content_eq_0_interior[symmetric] .
qed
qed
finally show "(∑(x, ka)∈p. content (ka ∩ {x. ¦x ∙ k - c¦ ≤ d}) * ?i x) < e" .
qed
qed
qed
subsection ‹A technical lemma about "refinement" of division.›
lemma tagged_division_finer:
fixes p :: "('a::euclidean_space × ('a::euclidean_space set)) set"
assumes "p tagged_division_of (cbox a b)"
and "gauge d"
obtains q where "q tagged_division_of (cbox a b)"
and "d fine q"
and "∀(x,k) ∈ p. k ⊆ d(x) ⟶ (x,k) ∈ q"
proof -
let ?P = "λp. p tagged_partial_division_of (cbox a b) ⟶ gauge d ⟶
(∃q. q tagged_division_of (⋃{k. ∃x. (x,k) ∈ p}) ∧ d fine q ∧
(∀(x,k) ∈ p. k ⊆ d(x) ⟶ (x,k) ∈ q))"
{
have *: "finite p" "p tagged_partial_division_of (cbox a b)"
using assms(1)
unfolding tagged_division_of_def
by auto
presume "⋀p. finite p ⟹ ?P p"
from this[rule_format,OF * assms(2)] guess q .. note q=this
then show ?thesis
apply -
apply (rule that[of q])
unfolding tagged_division_ofD[OF assms(1)]
apply auto
done
}
fix p :: "('a::euclidean_space × ('a::euclidean_space set)) set"
assume as: "finite p"
show "?P p"
apply rule
apply rule
using as
proof (induct p)
case empty
show ?case
apply (rule_tac x="{}" in exI)
unfolding fine_def
apply auto
done
next
case (insert xk p)
guess x k using surj_pair[of xk] by (elim exE) note xk=this
note tagged_partial_division_subset[OF insert(4) subset_insertI]
from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
have *: "⋃{l. ∃y. (y,l) ∈ insert xk p} = k ∪ ⋃{l. ∃y. (y,l) ∈ p}"
unfolding xk by auto
note p = tagged_partial_division_ofD[OF insert(4)]
from p(4)[unfolded xk, OF insertI1] guess u v by (elim exE) note uv=this
have "finite {k. ∃x. (x, k) ∈ p}"
apply (rule finite_subset[of _ "snd ` p"])
using p
apply safe
apply (metis image_iff snd_conv)
apply auto
done
then have int: "interior (cbox u v) ∩ interior (⋃{k. ∃x. (x, k) ∈ p}) = {}"
apply (rule inter_interior_unions_intervals)
apply (rule open_interior)
apply (rule_tac[!] ballI)
unfolding mem_Collect_eq
apply (erule_tac[!] exE)
apply (drule p(4)[OF insertI2])
apply assumption
apply (rule p(5))
unfolding uv xk
apply (rule insertI1)
apply (rule insertI2)
apply assumption
using insert(2)
unfolding uv xk
apply auto
done
show ?case
proof (cases "cbox u v ⊆ d x")
case True
then show ?thesis
apply (rule_tac x="{(x,cbox u v)} ∪ q1" in exI)
apply rule
unfolding * uv
apply (rule tagged_division_union)
apply (rule tagged_division_of_self)
apply (rule p[unfolded xk uv] insertI1)+
apply (rule q1)
apply (rule int)
apply rule
apply (rule fine_union)
apply (subst fine_def)
defer
apply (rule q1)
unfolding Ball_def split_paired_All split_conv
apply rule
apply rule
apply rule
apply rule
apply (erule insertE)
apply (simp add: uv xk)
apply (rule UnI2)
apply (drule q1(3)[rule_format])
unfolding xk uv
apply auto
done
next
case False
from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
show ?thesis
apply (rule_tac x="q2 ∪ q1" in exI)
apply rule
unfolding * uv
apply (rule tagged_division_union q2 q1 int fine_union)+
unfolding Ball_def split_paired_All split_conv
apply rule
apply (rule fine_union)
apply (rule q1 q2)+
apply rule
apply rule
apply rule
apply rule
apply (erule insertE)
apply (rule UnI2)
apply (simp add: False uv xk)
apply (drule q1(3)[rule_format])
using False
unfolding xk uv
apply auto
done
qed
qed
qed
subsection ‹Hence the main theorem about negligible sets.›
lemma finite_product_dependent:
assumes "finite s"
and "⋀x. x ∈ s ⟹ finite (t x)"
shows "finite {(i, j) |i j. i ∈ s ∧ j ∈ t i}"
using assms
proof induct
case (insert x s)
have *: "{(i, j) |i j. i ∈ insert x s ∧ j ∈ t i} =
(λy. (x,y)) ` (t x) ∪ {(i, j) |i j. i ∈ s ∧ j ∈ t i}" by auto
show ?case
unfolding *
apply (rule finite_UnI)
using insert
apply auto
done
qed auto
lemma sum_sum_product:
assumes "finite s"
and "∀i∈s. finite (t i)"
shows "setsum (λi. setsum (x i) (t i)::real) s =
setsum (λ(i,j). x i j) {(i,j) | i j. i ∈ s ∧ j ∈ t i}"
using assms
proof induct
case (insert a s)
have *: "{(i, j) |i j. i ∈ insert a s ∧ j ∈ t i} =
(λy. (a,y)) ` (t a) ∪ {(i, j) |i j. i ∈ s ∧ j ∈ t i}" by auto
show ?case
unfolding *
apply (subst setsum.union_disjoint)
unfolding setsum.insert[OF insert(1-2)]
prefer 4
apply (subst insert(3))
unfolding add_right_cancel
proof -
show "setsum (x a) (t a) = (∑(xa, y)∈ Pair a ` t a. x xa y)"
apply (subst setsum.reindex)
unfolding inj_on_def
apply auto
done
show "finite {(i, j) |i j. i ∈ s ∧ j ∈ t i}"
apply (rule finite_product_dependent)
using insert
apply auto
done
qed (insert insert, auto)
qed auto
lemma has_integral_negligible:
fixes f :: "'b::euclidean_space ⇒ 'a::real_normed_vector"
assumes "negligible s"
and "∀x∈(t - s). f x = 0"
shows "(f has_integral 0) t"
proof -
presume P: "⋀f::'b::euclidean_space ⇒ 'a.
⋀a b. ∀x. x ∉ s ⟶ f x = 0 ⟹ (f has_integral 0) (cbox a b)"
let ?f = "(λx. if x ∈ t then f x else 0)"
show ?thesis
apply (rule_tac f="?f" in has_integral_eq)
unfolding if_P
apply (rule refl)
apply (subst has_integral_alt)
apply cases
apply (subst if_P, assumption)
unfolding if_not_P
proof -
assume "∃a b. t = cbox a b"
then guess a b apply - by (erule exE)+ note t = this
show "(?f has_integral 0) t"
unfolding t
apply (rule P)
using assms(2)
unfolding t
apply auto
done
next
show "∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
(∃z. ((λx. if x ∈ t then ?f x else 0) has_integral z) (cbox a b) ∧ norm (z - 0) < e)"
apply safe
apply (rule_tac x=1 in exI)
apply rule
apply (rule zero_less_one)
apply safe
apply (rule_tac x=0 in exI)
apply rule
apply (rule P)
using assms(2)
apply auto
done
qed
next
fix f :: "'b ⇒ 'a"
fix a b :: 'b
assume assm: "∀x. x ∉ s ⟶ f x = 0"
show "(f has_integral 0) (cbox a b)"
unfolding has_integral
proof (safe, goal_cases)
case prems: (1 e)
then have "⋀n. e / 2 / ((real n+1) * (2 ^ n)) > 0"
apply -
apply (rule divide_pos_pos)
defer
apply (rule mult_pos_pos)
apply (auto simp add:field_simps)
done
note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b]
note allI[OF this,of "λx. x"]
from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]
show ?case
apply (rule_tac x="λx. d (nat ⌊norm (f x)⌋) x" in exI)
proof safe
show "gauge (λx. d (nat ⌊norm (f x)⌋) x)"
using d(1) unfolding gauge_def by auto
fix p
assume as: "p tagged_division_of (cbox a b)" "(λx. d (nat ⌊norm (f x)⌋) x) fine p"
let ?goal = "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) < e"
{
presume "p ≠ {} ⟹ ?goal"
then show ?goal
apply (cases "p = {}")
using prems
apply auto
done
}
assume as': "p ≠ {}"
from real_arch_simple[of "Max((λ(x,k). norm(f x)) ` p)"] guess N ..
then have N: "∀x∈(λ(x, k). norm (f x)) ` p. x ≤ real N"
by (meson Max_ge as(1) dual_order.trans finite_imageI tagged_division_of_finite)
have "∀i. ∃q. q tagged_division_of (cbox a b) ∧ (d i) fine q ∧ (∀(x, k)∈p. k ⊆ (d i) x ⟶ (x, k) ∈ q)"
by (auto intro: tagged_division_finer[OF as(1) d(1)])
from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]
have *: "⋀i. (∑(x, k)∈q i. content k *⇩R indicator s x) ≥ (0::real)"
apply (rule setsum_nonneg)
apply safe
unfolding real_scaleR_def
apply (drule tagged_division_ofD(4)[OF q(1)])
apply (auto intro: mult_nonneg_nonneg)
done
have **: "finite s ⟹ finite t ⟹ (∀(x,y) ∈ t. (0::real) ≤ g(x,y)) ⟹
(∀y∈s. ∃x. (x,y) ∈ t ∧ f(y) ≤ g(x,y)) ⟹ setsum f s ≤ setsum g t" for f g s t
apply (rule setsum_le_included[of s t g snd f])
prefer 4
apply safe
apply (erule_tac x=x in ballE)
apply (erule exE)
apply (rule_tac x="(xa,x)" in bexI)
apply auto
done
have "norm ((∑(x, k)∈p. content k *⇩R f x) - 0) ≤ setsum (λi. (real i + 1) *
norm (setsum (λ(x,k). content k *⇩R indicator s x :: real) (q i))) {..N+1}"
unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right
apply (rule order_trans)
apply (rule norm_setsum)
apply (subst sum_sum_product)
prefer 3
proof (rule **, safe)
show "finite {(i, j) |i j. i ∈ {..N + 1} ∧ j ∈ q i}"
apply (rule finite_product_dependent)
using q
apply auto
done
fix i a b
assume as'': "(a, b) ∈ q i"
show "0 ≤ (real i + 1) * (content b *⇩R indicator s a)"
unfolding real_scaleR_def
using tagged_division_ofD(4)[OF q(1) as'']
by (auto intro!: mult_nonneg_nonneg)
next
fix i :: nat
show "finite (q i)"
using q by auto
next
fix x k
assume xk: "(x, k) ∈ p"
def n ≡ "nat ⌊norm (f x)⌋"
have *: "norm (f x) ∈ (λ(x, k). norm (f x)) ` p"
using xk by auto
have nfx: "real n ≤ norm (f x)" "norm (f x) ≤ real n + 1"
unfolding n_def by auto
then have "n ∈ {0..N + 1}"
using N[rule_format,OF *] by auto
moreover
note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]
note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this]
note this[unfolded n_def[symmetric]]
moreover
have "norm (content k *⇩R f x) ≤ (real n + 1) * (content k * indicator s x)"
proof (cases "x ∈ s")
case False
then show ?thesis
using assm by auto
next
case True
have *: "content k ≥ 0"
using tagged_division_ofD(4)[OF as(1) xk] by auto
moreover
have "content k * norm (f x) ≤ content k * (real n + 1)"
apply (rule mult_mono)
using nfx *
apply auto
done
ultimately
show ?thesis
unfolding abs_mult
using nfx True
by (auto simp add: field_simps)
qed
ultimately show "∃y. (y, x, k) ∈ {(i, j) |i j. i ∈ {..N + 1} ∧ j ∈ q i} ∧ norm (content k *⇩R f x) ≤
(real y + 1) * (content k *⇩R indicator s x)"
apply (rule_tac x=n in exI)
apply safe
apply (rule_tac x=n in exI)
apply (rule_tac x="(x,k)" in exI)
apply safe
apply auto
done
qed (insert as, auto)
also have "… ≤ setsum (λi. e / 2 / 2 ^ i) {..N+1}"
proof (rule setsum_mono, goal_cases)
case (1 i)
then show ?case
apply (subst mult.commute, subst pos_le_divide_eq[symmetric])
using d(2)[rule_format, of "q i" i]
using q[rule_format]
apply (auto simp add: field_simps)
done
qed
also have "… < e * inverse 2 * 2"
unfolding divide_inverse setsum_right_distrib[symmetric]
apply (rule mult_strict_left_mono)
unfolding power_inverse [symmetric] lessThan_Suc_atMost[symmetric]
apply (subst geometric_sum)
using prems
apply auto
done
finally show "?goal" by auto
qed
qed
qed
lemma has_integral_spike:
fixes f :: "'b::euclidean_space ⇒ 'a::real_normed_vector"
assumes "negligible s"
and "(∀x∈(t - s). g x = f x)"
and "(f has_integral y) t"
shows "(g has_integral y) t"
proof -
{
fix a b :: 'b
fix f g :: "'b ⇒ 'a"
fix y :: 'a
assume as: "∀x ∈ cbox a b - s. g x = f x" "(f has_integral y) (cbox a b)"
have "((λx. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)"
apply (rule has_integral_add[OF as(2)])
apply (rule has_integral_negligible[OF assms(1)])
using as
apply auto
done
then have "(g has_integral y) (cbox a b)"
by auto
} note * = this
show ?thesis
apply (subst has_integral_alt)
using assms(2-)
apply -
apply (rule cond_cases)
apply safe
apply (rule *)
apply assumption+
apply (subst(asm) has_integral_alt)
unfolding if_not_P
apply (erule_tac x=e in allE)
apply safe
apply (rule_tac x=B in exI)
apply safe
apply (erule_tac x=a in allE)
apply (erule_tac x=b in allE)
apply safe
apply (rule_tac x=z in exI)
apply safe
apply (rule *[where fa2="λx. if x∈t then f x else 0"])
apply auto
done
qed
lemma has_integral_spike_eq:
assumes "negligible s"
and "∀x∈(t - s). g x = f x"
shows "((f has_integral y) t ⟷ (g has_integral y) t)"
apply rule
apply (rule_tac[!] has_integral_spike[OF assms(1)])
using assms(2)
apply auto
done
lemma integrable_spike:
assumes "negligible s"
and "∀x∈(t - s). g x = f x"
and "f integrable_on t"
shows "g integrable_on t"
using assms
unfolding integrable_on_def
apply -
apply (erule exE)
apply rule
apply (rule has_integral_spike)
apply fastforce+
done
lemma integral_spike:
assumes "negligible s"
and "∀x∈(t - s). g x = f x"
shows "integral t f = integral t g"
using has_integral_spike_eq[OF assms] by (simp add: integral_def integrable_on_def)
subsection ‹Some other trivialities about negligible sets.›
lemma negligible_subset[intro]:
assumes "negligible s"
and "t ⊆ s"
shows "negligible t"
unfolding negligible_def
proof (safe, goal_cases)
case (1 a b)
show ?case
using assms(1)[unfolded negligible_def,rule_format,of a b]
apply -
apply (rule has_integral_spike[OF assms(1)])
defer
apply assumption
using assms(2)
unfolding indicator_def
apply auto
done
qed
lemma negligible_diff[intro?]:
assumes "negligible s"
shows "negligible (s - t)"
using assms by auto
lemma negligible_inter:
assumes "negligible s ∨ negligible t"
shows "negligible (s ∩ t)"
using assms by auto
lemma negligible_union:
assumes "negligible s"
and "negligible t"
shows "negligible (s ∪ t)"
unfolding negligible_def
proof (safe, goal_cases)
case (1 a b)
note assm = assms[unfolded negligible_def,rule_format,of a b]
then show ?case
apply (subst has_integral_spike_eq[OF assms(2)])
defer
apply assumption
unfolding indicator_def
apply auto
done
qed
lemma negligible_union_eq[simp]: "negligible (s ∪ t) ⟷ negligible s ∧ negligible t"
using negligible_union by auto
lemma negligible_sing[intro]: "negligible {a::'a::euclidean_space}"
using negligible_standard_hyperplane[OF SOME_Basis, of "a ∙ (SOME i. i ∈ Basis)"] by auto
lemma negligible_insert[simp]: "negligible (insert a s) ⟷ negligible s"
apply (subst insert_is_Un)
unfolding negligible_union_eq
apply auto
done
lemma negligible_empty[iff]: "negligible {}"
by auto
lemma negligible_finite[intro]:
assumes "finite s"
shows "negligible s"
using assms by (induct s) auto
lemma negligible_unions[intro]:
assumes "finite s"
and "∀t∈s. negligible t"
shows "negligible(⋃s)"
using assms by induct auto
lemma negligible:
"negligible s ⟷ (∀t::('a::euclidean_space) set. ((indicator s::'a⇒real) has_integral 0) t)"
apply safe
defer
apply (subst negligible_def)
proof -
fix t :: "'a set"
assume as: "negligible s"
have *: "(λx. if x ∈ s ∩ t then 1 else 0) = (λx. if x∈t then if x∈s then 1 else 0 else 0)"
by auto
show "((indicator s::'a⇒real) has_integral 0) t"
apply (subst has_integral_alt)
apply cases
apply (subst if_P,assumption)
unfolding if_not_P
apply safe
apply (rule as[unfolded negligible_def,rule_format])
apply (rule_tac x=1 in exI)
apply safe
apply (rule zero_less_one)
apply (rule_tac x=0 in exI)
using negligible_subset[OF as,of "s ∩ t"]
unfolding negligible_def indicator_def [abs_def]
unfolding *
apply auto
done
qed auto
subsection ‹Finite case of the spike theorem is quite commonly needed.›
lemma has_integral_spike_finite:
assumes "finite s"
and "∀x∈t-s. g x = f x"
and "(f has_integral y) t"
shows "(g has_integral y) t"
apply (rule has_integral_spike)
using assms
apply auto
done
lemma has_integral_spike_finite_eq:
assumes "finite s"
and "∀x∈t-s. g x = f x"
shows "((f has_integral y) t ⟷ (g has_integral y) t)"
apply rule
apply (rule_tac[!] has_integral_spike_finite)
using assms
apply auto
done
lemma integrable_spike_finite:
assumes "finite s"
and "∀x∈t-s. g x = f x"
and "f integrable_on t"
shows "g integrable_on t"
using assms
unfolding integrable_on_def
apply safe
apply (rule_tac x=y in exI)
apply (rule has_integral_spike_finite)
apply auto
done
subsection ‹In particular, the boundary of an interval is negligible.›
lemma negligible_frontier_interval: "negligible(cbox (a::'a::euclidean_space) b - box a b)"
proof -
let ?A = "⋃((λk. {x. x∙k = a∙k} ∪ {x::'a. x∙k = b∙k}) ` Basis)"
have "cbox a b - box a b ⊆ ?A"
apply rule unfolding Diff_iff mem_box
apply simp
apply(erule conjE bexE)+
apply(rule_tac x=i in bexI)
apply auto
done
then show ?thesis
apply -
apply (rule negligible_subset[of ?A])
apply (rule negligible_unions[OF finite_imageI])
apply auto
done
qed
lemma has_integral_spike_interior:
assumes "∀x∈box a b. g x = f x"
and "(f has_integral y) (cbox a b)"
shows "(g has_integral y) (cbox a b)"
apply (rule has_integral_spike[OF negligible_frontier_interval _ assms(2)])
using assms(1)
apply auto
done
lemma has_integral_spike_interior_eq:
assumes "∀x∈box a b. g x = f x"
shows "(f has_integral y) (cbox a b) ⟷ (g has_integral y) (cbox a b)"
apply rule
apply (rule_tac[!] has_integral_spike_interior)
using assms
apply auto
done
lemma integrable_spike_interior:
assumes "∀x∈box a b. g x = f x"
and "f integrable_on cbox a b"
shows "g integrable_on cbox a b"
using assms
unfolding integrable_on_def
using has_integral_spike_interior[OF assms(1)]
by auto
subsection ‹Integrability of continuous functions.›
lemma neutral_and[simp]: "neutral op ∧ = True"
unfolding neutral_def by (rule some_equality) auto
lemma monoidal_and[intro]: "monoidal op ∧"
unfolding monoidal_def by auto
lemma iterate_and[simp]:
assumes "finite s"
shows "(iterate op ∧) s p ⟷ (∀x∈s. p x)"
using assms
apply induct
unfolding iterate_insert[OF monoidal_and]
apply auto
done
lemma operative_division_and:
assumes "operative op ∧ P"
and "d division_of (cbox a b)"
shows "(∀i∈d. P i) ⟷ P (cbox a b)"
using operative_division[OF monoidal_and assms] division_of_finite[OF assms(2)]
by auto
lemma operative_approximable:
fixes f :: "'b::euclidean_space ⇒ 'a::banach"
assumes "0 ≤ e"
shows "operative op ∧ (λi. ∃g. (∀x∈i. norm (f x - g (x::'b)) ≤ e) ∧ g integrable_on i)"
unfolding operative_def neutral_and
proof safe
fix a b :: 'b
show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
if "content (cbox a b) = 0"
apply (rule_tac x=f in exI)
using assms that
apply (auto intro!: integrable_on_null)
done
{
fix c g
fix k :: 'b
assume as: "∀x∈cbox a b. norm (f x - g x) ≤ e" "g integrable_on cbox a b"
assume k: "k ∈ Basis"
show "∃g. (∀x∈cbox a b ∩ {x. x ∙ k ≤ c}. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b ∩ {x. x ∙ k ≤ c}"
"∃g. (∀x∈cbox a b ∩ {x. c ≤ x ∙ k}. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b ∩ {x. c ≤ x ∙ k}"
apply (rule_tac[!] x=g in exI)
using as(1) integrable_split[OF as(2) k]
apply auto
done
}
fix c k g1 g2
assume as: "∀x∈cbox a b ∩ {x. x ∙ k ≤ c}. norm (f x - g1 x) ≤ e" "g1 integrable_on cbox a b ∩ {x. x ∙ k ≤ c}"
"∀x∈cbox a b ∩ {x. c ≤ x ∙ k}. norm (f x - g2 x) ≤ e" "g2 integrable_on cbox a b ∩ {x. c ≤ x ∙ k}"
assume k: "k ∈ Basis"
let ?g = "λx. if x∙k = c then f x else if x∙k ≤ c then g1 x else g2 x"
show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
apply (rule_tac x="?g" in exI)
apply safe
proof goal_cases
case (1 x)
then show ?case
apply -
apply (cases "x∙k=c")
apply (case_tac "x∙k < c")
using as assms
apply auto
done
next
case 2
presume "?g integrable_on cbox a b ∩ {x. x ∙ k ≤ c}"
and "?g integrable_on cbox a b ∩ {x. x ∙ k ≥ c}"
then guess h1 h2 unfolding integrable_on_def by auto
from has_integral_split[OF this k] show ?case
unfolding integrable_on_def by auto
next
show "?g integrable_on cbox a b ∩ {x. x ∙ k ≤ c}" "?g integrable_on cbox a b ∩ {x. x ∙ k ≥ c}"
apply(rule_tac[!] integrable_spike[OF negligible_standard_hyperplane[of k c]])
using k as(2,4)
apply auto
done
qed
qed
lemma approximable_on_division:
fixes f :: "'b::euclidean_space ⇒ 'a::banach"
assumes "0 ≤ e"
and "d division_of (cbox a b)"
and "∀i∈d. ∃g. (∀x∈i. norm (f x - g x) ≤ e) ∧ g integrable_on i"
obtains g where "∀x∈cbox a b. norm (f x - g x) ≤ e" "g integrable_on cbox a b"
proof -
note * = operative_division[OF monoidal_and operative_approximable[OF assms(1)] assms(2)]
note this[unfolded iterate_and[OF division_of_finite[OF assms(2)]]]
from assms(3)[unfolded this[of f]] guess g ..
then show thesis
apply -
apply (rule that[of g])
apply auto
done
qed
lemma integrable_continuous:
fixes f :: "'b::euclidean_space ⇒ 'a::banach"
assumes "continuous_on (cbox a b) f"
shows "f integrable_on cbox a b"
proof (rule integrable_uniform_limit, safe)
fix e :: real
assume e: "e > 0"
from compact_uniformly_continuous[OF assms compact_cbox,unfolded uniformly_continuous_on_def,rule_format,OF e] guess d ..
note d=conjunctD2[OF this,rule_format]
from fine_division_exists[OF gauge_ball[OF d(1)], of a b] guess p . note p=this
note p' = tagged_division_ofD[OF p(1)]
have *: "∀i∈snd ` p. ∃g. (∀x∈i. norm (f x - g x) ≤ e) ∧ g integrable_on i"
proof (safe, unfold snd_conv)
fix x l
assume as: "(x, l) ∈ p"
from p'(4)[OF this] guess a b by (elim exE) note l=this
show "∃g. (∀x∈l. norm (f x - g x) ≤ e) ∧ g integrable_on l"
apply (rule_tac x="λy. f x" in exI)
proof safe
show "(λy. f x) integrable_on l"
unfolding integrable_on_def l
apply rule
apply (rule has_integral_const)
done
fix y
assume y: "y ∈ l"
note fineD[OF p(2) as,unfolded subset_eq,rule_format,OF this]
note d(2)[OF _ _ this[unfolded mem_ball]]
then show "norm (f y - f x) ≤ e"
using y p'(2-3)[OF as] unfolding dist_norm l norm_minus_commute by fastforce
qed
qed
from e have "e ≥ 0"
by auto
from approximable_on_division[OF this division_of_tagged_division[OF p(1)] *] guess g .
then show "∃g. (∀x∈cbox a b. norm (f x - g x) ≤ e) ∧ g integrable_on cbox a b"
by auto
qed
lemma integrable_continuous_real:
fixes f :: "real ⇒ 'a::banach"
assumes "continuous_on {a .. b} f"
shows "f integrable_on {a .. b}"
by (metis assms box_real(2) integrable_continuous)
subsection ‹Specialization of additivity to one dimension.›
lemma
shows real_inner_1_left: "inner 1 x = x"
and real_inner_1_right: "inner x 1 = x"
by simp_all
lemma content_real_eq_0: "content {a .. b::real} = 0 ⟷ a ≥ b"
by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)
lemma interval_real_split:
"{a .. b::real} ∩ {x. x ≤ c} = {a .. min b c}"
"{a .. b} ∩ {x. c ≤ x} = {max a c .. b}"
apply (metis Int_atLeastAtMostL1 atMost_def)
apply (metis Int_atLeastAtMostL2 atLeast_def)
done
lemma operative_1_lt:
assumes "monoidal opp"
shows "operative opp f ⟷ ((∀a b. b ≤ a ⟶ f {a .. b::real} = neutral opp) ∧
(∀a b c. a < c ∧ c < b ⟶ opp (f {a .. c}) (f {c .. b}) = f {a .. b}))"
apply (simp add: operative_def content_real_eq_0 del: content_real_if)
proof safe
fix a b c :: real
assume as:
"∀a b c. f {a..b} = opp (f ({a..b} ∩ {x. x ≤ c})) (f ({a..b} ∩ Collect (op ≤ c)))"
"a < c"
"c < b"
from this(2-) have "cbox a b ∩ {x. x ≤ c} = cbox a c" "cbox a b ∩ {x. x ≥ c} = cbox c b"
by (auto simp: mem_box)
then show "opp (f {a..c}) (f {c..b}) = f {a..b}"
unfolding as(1)[rule_format,of a b "c"] by auto
next
fix a b c :: real
assume as: "∀a b. b ≤ a ⟶ f {a..b} = neutral opp"
"∀a b c. a < c ∧ c < b ⟶ opp (f {a..c}) (f {c..b}) = f {a..b}"
show " f {a..b} = opp (f ({a..b} ∩ {x. x ≤ c})) (f ({a..b} ∩ Collect (op ≤ c)))"
proof (cases "c ∈ {a..b}")
case False
then have "c < a ∨ c > b" by auto
then show ?thesis
proof
assume "c < a"
then have *: "{a..b} ∩ {x. x ≤ c} = {1..0}" "{a..b} ∩ {x. c ≤ x} = {a..b}"
by auto
show ?thesis
unfolding *
apply (subst as(1)[rule_format,of 0 1])
using assms
apply auto
done
next
assume "b < c"
then have *: "{a..b} ∩ {x. x ≤ c} = {a..b}" "{a..b} ∩ {x. c ≤ x} = {1 .. 0}"
by auto
show ?thesis
unfolding *
apply (subst as(1)[rule_format,of 0 1])
using assms
apply auto
done
qed
next
case True
then have *: "min (b) c = c" "max a c = c"
by auto
have **: "(1::real) ∈ Basis"
by simp
have ***: "⋀P Q. (∑i∈Basis. (if i = 1 then P i else Q i) *⇩R i) = (P 1::real)"
by simp
show ?thesis
unfolding interval_real_split unfolding *
proof (cases "c = a ∨ c = b")
case False
then show "f {a..b} = opp (f {a..c}) (f {c..b})"
apply -
apply (subst as(2)[rule_format])
using True
apply auto
done
next
case True
then show "f {a..b} = opp (f {a..c}) (f {c..b})"
proof
assume *: "c = a"
then have "f {a .. c} = neutral opp"
apply -
apply (rule as(1)[rule_format])
apply auto
done
then show ?thesis
using assms unfolding * by auto
next
assume *: "c = b"
then have "f {c .. b} = neutral opp"
apply -
apply (rule as(1)[rule_format])
apply auto
done
then show ?thesis
using assms unfolding * by auto
qed
qed
qed
qed
lemma operative_1_le:
assumes "monoidal opp"
shows "operative opp f ⟷ ((∀a b. b ≤ a ⟶ f {a .. b::real} = neutral opp) ∧
(∀a b c. a ≤ c ∧ c ≤ b ⟶ opp (f {a .. c}) (f {c .. b}) = f {a .. b}))"
unfolding operative_1_lt[OF assms]
proof safe
fix a b c :: real
assume as:
"∀a b c. a ≤ c ∧ c ≤ b ⟶ opp (f {a..c}) (f {c..b}) = f {a..b}"
"a < c"
"c < b"
show "opp (f {a..c}) (f {c..b}) = f {a..b}"
apply (rule as(1)[rule_format])
using as(2-)
apply auto
done
next
fix a b c :: real
assume "∀a b. b ≤ a ⟶ f {a .. b} = neutral opp"
and "∀a b c. a < c ∧ c < b ⟶ opp (f {a..c}) (f {c..b}) = f {a..b}"
and "a ≤ c"
and "c ≤ b"
note as = this[rule_format]
show "opp (f {a..c}) (f {c..b}) = f {a..b}"
proof (cases "c = a ∨ c = b")
case False
then show ?thesis
apply -
apply (subst as(2))
using as(3-)
apply auto
done
next
case True
then show ?thesis
proof
assume *: "c = a"
then have "f {a .. c} = neutral opp"
apply -
apply (rule as(1)[rule_format])
apply auto
done
then show ?thesis
using assms unfolding * by auto
next
assume *: "c = b"
then have "f {c .. b} = neutral opp"
apply -
apply (rule as(1)[rule_format])
apply auto
done
then show ?thesis
using assms unfolding * by auto
qed
qed
qed
subsection ‹Special case of additivity we need for the FCT.›
lemma additive_tagged_division_1:
fixes f :: "real ⇒ 'a::real_normed_vector"
assumes "a ≤ b"
and "p tagged_division_of {a..b}"
shows "setsum (λ(x,k). f(Sup k) - f(Inf k)) p = f b - f a"
proof -
let ?f = "(λk::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
have ***: "∀i∈Basis. a ∙ i ≤ b ∙ i"
using assms by auto
have *: "operative op + ?f"
unfolding operative_1_lt[OF monoidal_monoid] box_eq_empty
by auto
have **: "cbox a b ≠ {}"
using assms(1) by auto
note operative_tagged_division[OF monoidal_monoid * assms(2)[simplified box_real[symmetric]]]
note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
show ?thesis
unfolding *
apply (subst setsum_iterate[symmetric])
defer
apply (rule setsum.cong)
unfolding split_paired_all split_conv
using assms(2)
apply auto
done
qed
subsection ‹A useful lemma allowing us to factor out the content size.›
lemma has_integral_factor_content:
"(f has_integral i) (cbox a b) ⟷
(∀e>0. ∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm (setsum (λ(x,k). content k *⇩R f x) p - i) ≤ e * content (cbox a b)))"
proof (cases "content (cbox a b) = 0")
case True
show ?thesis
unfolding has_integral_null_eq[OF True]
apply safe
apply (rule, rule, rule gauge_trivial, safe)
unfolding setsum_content_null[OF True] True
defer
apply (erule_tac x=1 in allE)
apply safe
defer
apply (rule fine_division_exists[of _ a b])
apply assumption
apply (erule_tac x=p in allE)
unfolding setsum_content_null[OF True]
apply auto
done
next
case False
note F = this[unfolded content_lt_nz[symmetric]]
let ?P = "λe opp. ∃d. gauge d ∧
(∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶ opp (norm ((∑(x, k)∈p. content k *⇩R f x) - i)) e)"
show ?thesis
apply (subst has_integral)
proof safe
fix e :: real
assume e: "e > 0"
{
assume "∀e>0. ?P e op <"
then show "?P (e * content (cbox a b)) op ≤"
apply (erule_tac x="e * content (cbox a b)" in allE)
apply (erule impE)
defer
apply (erule exE,rule_tac x=d in exI)
using F e
apply (auto simp add:field_simps)
done
}
{
assume "∀e>0. ?P (e * content (cbox a b)) op ≤"
then show "?P e op <"
apply (erule_tac x="e / 2 / content (cbox a b)" in allE)
apply (erule impE)
defer
apply (erule exE,rule_tac x=d in exI)
using F e
apply (auto simp add: field_simps)
done
}
qed
qed
lemma has_integral_factor_content_real:
"(f has_integral i) {a .. b::real} ⟷
(∀e>0. ∃d. gauge d ∧ (∀p. p tagged_division_of {a .. b} ∧ d fine p ⟶
norm (setsum (λ(x,k). content k *⇩R f x) p - i) ≤ e * content {a .. b} ))"
unfolding box_real[symmetric]
by (rule has_integral_factor_content)
subsection ‹Fundamental theorem of calculus.›
lemma interval_bounds_real:
fixes q b :: real
assumes "a ≤ b"
shows "Sup {a..b} = b"
and "Inf {a..b} = a"
using assms by auto
lemma fundamental_theorem_of_calculus:
fixes f :: "real ⇒ 'a::banach"
assumes "a ≤ b"
and "∀x∈{a .. b}. (f has_vector_derivative f' x) (at x within {a .. b})"
shows "(f' has_integral (f b - f a)) {a .. b}"
unfolding has_integral_factor_content box_real[symmetric]
proof safe
fix e :: real
assume e: "e > 0"
note assm = assms(2)[unfolded has_vector_derivative_def has_derivative_within_alt]
have *: "⋀P Q. ∀x∈{a .. b}. P x ∧ (∀e>0. ∃d>0. Q x e d) ⟹ ∀x. ∃(d::real)>0. x∈{a .. b} ⟶ Q x e d"
using e by blast
note this[OF assm,unfolded gauge_existence_lemma]
from choice[OF this,unfolded Ball_def[symmetric]] guess d ..
note d=conjunctD2[OF this[rule_format],rule_format]
show "∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm ((∑(x, k)∈p. content k *⇩R f' x) - (f b - f a)) ≤ e * content (cbox a b))"
apply (rule_tac x="λx. ball x (d x)" in exI)
apply safe
apply (rule gauge_ball_dependent)
apply rule
apply (rule d(1))
proof -
fix p
assume as: "p tagged_division_of cbox a b" "(λx. ball x (d x)) fine p"
show "norm ((∑(x, k)∈p. content k *⇩R f' x) - (f b - f a)) ≤ e * content (cbox a b)"
unfolding content_real[OF assms(1), simplified box_real[symmetric]] additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of f,symmetric]
unfolding additive_tagged_division_1[OF assms(1) as(1)[simplified box_real],of "λx. x",symmetric]
unfolding setsum_right_distrib
defer
unfolding setsum_subtractf[symmetric]
proof (rule setsum_norm_le,safe)
fix x k
assume "(x, k) ∈ p"
note xk = tagged_division_ofD(2-4)[OF as(1) this]
from this(3) guess u v by (elim exE) note k=this
have *: "u ≤ v"
using xk unfolding k by auto
have ball: "∀xa∈k. xa ∈ ball x (d x)"
using as(2)[unfolded fine_def,rule_format,OF ‹(x,k)∈p›,unfolded split_conv subset_eq] .
have "norm ((v - u) *⇩R f' x - (f v - f u)) ≤
norm (f u - f x - (u - x) *⇩R f' x) + norm (f v - f x - (v - x) *⇩R f' x)"
apply (rule order_trans[OF _ norm_triangle_ineq4])
apply (rule eq_refl)
apply (rule arg_cong[where f=norm])
unfolding scaleR_diff_left
apply (auto simp add:algebra_simps)
done
also have "… ≤ e * norm (u - x) + e * norm (v - x)"
apply (rule add_mono)
apply (rule d(2)[of "x" "u",unfolded o_def])
prefer 4
apply (rule d(2)[of "x" "v",unfolded o_def])
using ball[rule_format,of u] ball[rule_format,of v]
using xk(1-2)
unfolding k subset_eq
apply (auto simp add:dist_real_def)
done
also have "… ≤ e * (Sup k - Inf k)"
unfolding k interval_bounds_real[OF *]
using xk(1)
unfolding k
by (auto simp add: dist_real_def field_simps)
finally show "norm (content k *⇩R f' x - (f (Sup k) - f (Inf k))) ≤
e * (Sup k - Inf k)"
unfolding box_real k interval_bounds_real[OF *] content_real[OF *]
interval_upperbound_real interval_lowerbound_real
.
qed
qed
qed
lemma ident_has_integral:
fixes a::real
assumes "a ≤ b"
shows "((λx. x) has_integral (b⇧2 - a⇧2) / 2) {a..b}"
proof -
have "((λx. x) has_integral inverse 2 * b⇧2 - inverse 2 * a⇧2) {a..b}"
apply (rule fundamental_theorem_of_calculus [OF assms], clarify)
unfolding power2_eq_square
by (rule derivative_eq_intros | simp)+
then show ?thesis
by (simp add: field_simps)
qed
lemma integral_ident [simp]:
fixes a::real
assumes "a ≤ b"
shows "integral {a..b} (λx. x) = (if a ≤ b then (b⇧2 - a⇧2) / 2 else 0)"
using ident_has_integral integral_unique by fastforce
lemma ident_integrable_on:
fixes a::real
shows "(λx. x) integrable_on {a..b}"
by (metis atLeastatMost_empty_iff integrable_on_def has_integral_empty ident_has_integral)
subsection ‹Taylor series expansion›
lemma (in bounded_bilinear) setsum_prod_derivatives_has_vector_derivative:
assumes "p>0"
and f0: "Df 0 = f"
and Df: "⋀m t. m < p ⟹ a ≤ t ⟹ t ≤ b ⟹
(Df m has_vector_derivative Df (Suc m) t) (at t within {a .. b})"
and g0: "Dg 0 = g"
and Dg: "⋀m t. m < p ⟹ a ≤ t ⟹ t ≤ b ⟹
(Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
and ivl: "a ≤ t" "t ≤ b"
shows "((λt. ∑i<p. (-1)^i *⇩R prod (Df i t) (Dg (p - Suc i) t))
has_vector_derivative
prod (f t) (Dg p t) - (-1)^p *⇩R prod (Df p t) (g t))
(at t within {a .. b})"
using assms
proof cases
assume p: "p ≠ 1"
def p' ≡ "p - 2"
from assms p have p': "{..<p} = {..Suc p'}" "p = Suc (Suc p')"
by (auto simp: p'_def)
have *: "⋀i. i ≤ p' ⟹ Suc (Suc p' - i) = (Suc (Suc p') - i)"
by auto
let ?f = "λi. (-1) ^ i *⇩R (prod (Df i t) (Dg ((p - i)) t))"
have "(∑i<p. (-1) ^ i *⇩R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
prod (Df (Suc i) t) (Dg (p - Suc i) t))) =
(∑i≤(Suc p'). ?f i - ?f (Suc i))"
by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost)
also note setsum_telescope
finally
have "(∑i<p. (-1) ^ i *⇩R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
prod (Df (Suc i) t) (Dg (p - Suc i) t)))
= prod (f t) (Dg p t) - (- 1) ^ p *⇩R prod (Df p t) (g t)"
unfolding p'[symmetric]
by (simp add: assms)
thus ?thesis
using assms
by (auto intro!: derivative_eq_intros has_vector_derivative)
qed (auto intro!: derivative_eq_intros has_vector_derivative)
lemma
fixes f::"real⇒'a::banach"
assumes "p>0"
and f0: "Df 0 = f"
and Df: "⋀m t. m < p ⟹ a ≤ t ⟹ t ≤ b ⟹
(Df m has_vector_derivative Df (Suc m) t) (at t within {a .. b})"
and ivl: "a ≤ b"
defines "i ≡ λx. ((b - x) ^ (p - 1) / fact (p - 1)) *⇩R Df p x"
shows taylor_has_integral:
"(i has_integral f b - (∑i<p. ((b - a) ^ i / fact i) *⇩R Df i a)) {a..b}"
and taylor_integral:
"f b = (∑i<p. ((b - a) ^ i / fact i) *⇩R Df i a) + integral {a..b} i"
and taylor_integrable:
"i integrable_on {a .. b}"
proof goal_cases
case 1
interpret bounded_bilinear "scaleR::real⇒'a⇒'a"
by (rule bounded_bilinear_scaleR)
def g ≡ "λs. (b - s)^(p - 1)/fact (p - 1)"
def Dg ≡ "λn s. if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0"
have g0: "Dg 0 = g"
using ‹p > 0›
by (auto simp add: Dg_def divide_simps g_def split: if_split_asm)
{
fix m
assume "p > Suc m"
hence "p - Suc m = Suc (p - Suc (Suc m))"
by auto
hence "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)"
by auto
} note fact_eq = this
have Dg: "⋀m t. m < p ⟹ a ≤ t ⟹ t ≤ b ⟹
(Dg m has_vector_derivative Dg (Suc m) t) (at t within {a .. b})"
unfolding Dg_def
by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq divide_simps)
let ?sum = "λt. ∑i<p. (- 1) ^ i *⇩R Dg i t *⇩R Df (p - Suc i) t"
from setsum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
OF ‹p > 0› g0 Dg f0 Df]
have deriv: "⋀t. a ≤ t ⟹ t ≤ b ⟹
(?sum has_vector_derivative
g t *⇩R Df p t - (- 1) ^ p *⇩R Dg p t *⇩R f t) (at t within {a..b})"
by auto
from fundamental_theorem_of_calculus[rule_format, OF ‹a ≤ b› deriv]
have "(i has_integral ?sum b - ?sum a) {a .. b}"
by (simp add: i_def g_def Dg_def)
also
have one: "(- 1) ^ p' * (- 1) ^ p' = (1::real)"
and "{..<p} ∩ {i. p = Suc i} = {p - 1}"
for p'
using ‹p > 0›
by (auto simp: power_mult_distrib[symmetric])
then have "?sum b = f b"
using Suc_pred'[OF ‹p > 0›]
by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib
cond_application_beta setsum.If_cases f0)
also
have "{..<p} = (λx. p - x - 1) ` {..<p}"
proof safe
fix x
assume "x < p"
thus "x ∈ (λx. p - x - 1) ` {..<p}"
by (auto intro!: image_eqI[where x = "p - x - 1"])
qed simp
from _ this
have "?sum a = (∑i<p. ((b - a) ^ i / fact i) *⇩R Df i a)"
by (rule setsum.reindex_cong) (auto simp add: inj_on_def Dg_def one)
finally show c: ?case .
case 2 show ?case using c integral_unique by force
case 3 show ?case using c by force
qed
subsection ‹Attempt a systematic general set of "offset" results for components.›
lemma gauge_modify:
assumes "(∀s. open s ⟶ open {x. f(x) ∈ s})" "gauge d"
shows "gauge (λx. {y. f y ∈ d (f x)})"
using assms
unfolding gauge_def
apply safe
defer
apply (erule_tac x="f x" in allE)
apply (erule_tac x="d (f x)" in allE)
apply auto
done
subsection ‹Only need trivial subintervals if the interval itself is trivial.›
lemma division_of_nontrivial:
fixes s :: "'a::euclidean_space set set"
assumes "s division_of (cbox a b)"
and "content (cbox a b) ≠ 0"
shows "{k. k ∈ s ∧ content k ≠ 0} division_of (cbox a b)"
using assms(1)
apply -
proof (induct "card s" arbitrary: s rule: nat_less_induct)
fix s::"'a set set"
assume assm: "s division_of (cbox a b)"
"∀m<card s. ∀x. m = card x ⟶
x division_of (cbox a b) ⟶ {k ∈ x. content k ≠ 0} division_of (cbox a b)"
note s = division_ofD[OF assm(1)]
let ?thesis = "{k ∈ s. content k ≠ 0} division_of (cbox a b)"
{
presume *: "{k ∈ s. content k ≠ 0} ≠ s ⟹ ?thesis"
show ?thesis
apply cases
defer
apply (rule *)
apply assumption
using assm(1)
apply auto
done
}
assume noteq: "{k ∈ s. content k ≠ 0} ≠ s"
then obtain k where k: "k ∈ s" "content k = 0"
by auto
from s(4)[OF k(1)] guess c d by (elim exE) note k=k this
from k have "card s > 0"
unfolding card_gt_0_iff using assm(1) by auto
then have card: "card (s - {k}) < card s"
using assm(1) k(1)
apply (subst card_Diff_singleton_if)
apply auto
done
have *: "closed (⋃(s - {k}))"
apply (rule closed_Union)
defer
apply rule
apply (drule DiffD1,drule s(4))
using assm(1)
apply auto
done
have "k ⊆ ⋃(s - {k})"
apply safe
apply (rule *[unfolded closed_limpt,rule_format])
unfolding islimpt_approachable
proof safe
fix x
fix e :: real
assume as: "x ∈ k" "e > 0"
from k(2)[unfolded k content_eq_0] guess i ..
then have i:"c∙i = d∙i" "i∈Basis"
using s(3)[OF k(1),unfolded k] unfolding box_ne_empty by auto
then have xi: "x∙i = d∙i"
using as unfolding k mem_box by (metis antisym)
def y ≡ "∑j∈Basis. (if j = i then if c∙i ≤ (a∙i + b∙i) / 2 then c∙i +
min e (b∙i - c∙i) / 2 else c∙i - min e (c∙i - a∙i) / 2 else x∙j) *⇩R j"
show "∃x'∈⋃(s - {k}). x' ≠ x ∧ dist x' x < e"
apply (rule_tac x=y in bexI)
proof
have "d ∈ cbox c d"
using s(3)[OF k(1)]
unfolding k box_eq_empty mem_box
by (fastforce simp add: not_less)
then have "d ∈ cbox a b"
using s(2)[OF k(1)]
unfolding k
by auto
note di = this[unfolded mem_box,THEN bspec[where x=i]]
then have xyi: "y∙i ≠ x∙i"
unfolding y_def i xi
using as(2) assms(2)[unfolded content_eq_0] i(2)
by (auto elim!: ballE[of _ _ i])
then show "y ≠ x"
unfolding euclidean_eq_iff[where 'a='a] using i by auto
have *: "Basis = insert i (Basis - {i})"
using i by auto
have "norm (y - x) < e + setsum (λi. 0) Basis"
apply (rule le_less_trans[OF norm_le_l1])
apply (subst *)
apply (subst setsum.insert)
prefer 3
apply (rule add_less_le_mono)
proof -
show "¦(y - x) ∙ i¦ < e"
using di as(2) y_def i xi by (auto simp: inner_simps)
show "(∑i∈Basis - {i}. ¦(y - x) ∙ i¦) ≤ (∑i∈Basis. 0)"
unfolding y_def by (auto simp: inner_simps)
qed auto
then show "dist y x < e"
unfolding dist_norm by auto
have "y ∉ k"
unfolding k mem_box
apply rule
apply (erule_tac x=i in ballE)
using xyi k i xi
apply auto
done
moreover
have "y ∈ ⋃s"
using set_rev_mp[OF as(1) s(2)[OF k(1)]] as(2) di i
unfolding s mem_box y_def
by (auto simp: field_simps elim!: ballE[of _ _ i])
ultimately
show "y ∈ ⋃(s - {k})" by auto
qed
qed
then have "⋃(s - {k}) = cbox a b"
unfolding s(6)[symmetric] by auto
then have "{ka ∈ s - {k}. content ka ≠ 0} division_of (cbox a b)"
apply -
apply (rule assm(2)[rule_format,OF card refl])
apply (rule division_ofI)
defer
apply (rule_tac[1-4] s)
using assm(1)
apply auto
done
moreover
have "{ka ∈ s - {k}. content ka ≠ 0} = {k ∈ s. content k ≠ 0}"
using k by auto
ultimately show ?thesis by auto
qed
subsection ‹Integrability on subintervals.›
lemma operative_integrable:
fixes f :: "'b::euclidean_space ⇒ 'a::banach"
shows "operative op ∧ (λi. f integrable_on i)"
unfolding operative_def neutral_and
apply safe
apply (subst integrable_on_def)
unfolding has_integral_null_eq
apply (rule, rule refl)
apply (rule, assumption, assumption)+
unfolding integrable_on_def
by (auto intro!: has_integral_split)
lemma integrable_subinterval:
fixes f :: "'b::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on cbox a b"
and "cbox c d ⊆ cbox a b"
shows "f integrable_on cbox c d"
apply (cases "cbox c d = {}")
defer
apply (rule partial_division_extend_1[OF assms(2)],assumption)
using operative_division_and[OF operative_integrable,symmetric,of _ _ _ f] assms(1)
apply auto
done
lemma integrable_subinterval_real:
fixes f :: "real ⇒ 'a::banach"
assumes "f integrable_on {a .. b}"
and "{c .. d} ⊆ {a .. b}"
shows "f integrable_on {c .. d}"
by (metis assms(1) assms(2) box_real(2) integrable_subinterval)
subsection ‹Combining adjacent intervals in 1 dimension.›
lemma has_integral_combine:
fixes a b c :: real
assumes "a ≤ c"
and "c ≤ b"
and "(f has_integral i) {a .. c}"
and "(f has_integral (j::'a::banach)) {c .. b}"
shows "(f has_integral (i + j)) {a .. b}"
proof -
note operative_integral[of f, unfolded operative_1_le[OF monoidal_lifted[OF monoidal_monoid]]]
note conjunctD2[OF this,rule_format]
note * = this(2)[OF conjI[OF assms(1-2)],unfolded if_P[OF assms(3)]]
then have "f integrable_on cbox a b"
apply -
apply (rule ccontr)
apply (subst(asm) if_P)
defer
apply (subst(asm) if_P)
using assms(3-)
apply auto
done
with *
show ?thesis
apply -
apply (subst(asm) if_P)
defer
apply (subst(asm) if_P)
defer
apply (subst(asm) if_P)
unfolding lifted.simps
using assms(3-)
apply (auto simp add: integrable_on_def integral_unique)
done
qed
lemma integral_combine:
fixes f :: "real ⇒ 'a::banach"
assumes "a ≤ c"
and "c ≤ b"
and "f integrable_on {a .. b}"
shows "integral {a .. c} f + integral {c .. b} f = integral {a .. b} f"
apply (rule integral_unique[symmetric])
apply (rule has_integral_combine[OF assms(1-2)])
apply (metis assms(2) assms(3) atLeastatMost_subset_iff box_real(2) content_pos_le content_real_eq_0 integrable_integral integrable_subinterval le_add_same_cancel2 monoid_add_class.add.left_neutral)
by (metis assms(1) assms(3) atLeastatMost_subset_iff box_real(2) content_pos_le content_real_eq_0 integrable_integral integrable_subinterval le_add_same_cancel1 monoid_add_class.add.right_neutral)
lemma integrable_combine:
fixes f :: "real ⇒ 'a::banach"
assumes "a ≤ c"
and "c ≤ b"
and "f integrable_on {a .. c}"
and "f integrable_on {c .. b}"
shows "f integrable_on {a .. b}"
using assms
unfolding integrable_on_def
by (fastforce intro!:has_integral_combine)
subsection ‹Reduce integrability to "local" integrability.›
lemma integrable_on_little_subintervals:
fixes f :: "'b::euclidean_space ⇒ 'a::banach"
assumes "∀x∈cbox a b. ∃d>0. ∀u v. x ∈ cbox u v ∧ cbox u v ⊆ ball x d ∧ cbox u v ⊆ cbox a b ⟶
f integrable_on cbox u v"
shows "f integrable_on cbox a b"
proof -
have "∀x. ∃d. x∈cbox a b ⟶ d>0 ∧ (∀u v. x ∈ cbox u v ∧ cbox u v ⊆ ball x d ∧ cbox u v ⊆ cbox a b ⟶
f integrable_on cbox u v)"
using assms by auto
note this[unfolded gauge_existence_lemma]
from choice[OF this] guess d .. note d=this[rule_format]
guess p
apply (rule fine_division_exists[OF gauge_ball_dependent,of d a b])
using d
by auto
note p=this(1-2)
note division_of_tagged_division[OF this(1)]
note * = operative_division_and[OF operative_integrable,OF this,symmetric,of f]
show ?thesis
unfolding *
apply safe
unfolding snd_conv
proof -
fix x k
assume "(x, k) ∈ p"
note tagged_division_ofD(2-4)[OF p(1) this] fineD[OF p(2) this]
then show "f integrable_on k"
apply safe
apply (rule d[THEN conjunct2,rule_format,of x])
apply (auto intro: order.trans)
done
qed
qed
subsection ‹Second FCT or existence of antiderivative.›
lemma integrable_const[intro]: "(λx. c) integrable_on cbox a b"
unfolding integrable_on_def
apply rule
apply (rule has_integral_const)
done
lemma integral_has_vector_derivative_continuous_at:
fixes f :: "real ⇒ 'a::banach"
assumes f: "f integrable_on {a..b}"
and x: "x ∈ {a..b}"
and fx: "continuous (at x within {a..b}) f"
shows "((λu. integral {a..u} f) has_vector_derivative f x) (at x within {a..b})"
proof -
let ?I = "λa b. integral {a..b} f"
{ fix e::real
assume "e > 0"
obtain d where "d>0" and d: "⋀x'. ⟦x' ∈ {a..b}; ¦x' - x¦ < d⟧ ⟹ norm(f x' - f x) ≤ e"
using ‹e>0› fx by (auto simp: continuous_within_eps_delta dist_norm less_imp_le)
have "norm (integral {a..y} f - integral {a..x} f - (y - x) *⇩R f x) ≤ e * ¦y - x¦"
if y: "y ∈ {a..b}" and yx: "¦y - x¦ < d" for y
proof (cases "y < x")
case False
have "f integrable_on {a..y}"
using f y by (simp add: integrable_subinterval_real)
then have Idiff: "?I a y - ?I a x = ?I x y"
using False x by (simp add: algebra_simps integral_combine)
have fux_int: "((λu. f u - f x) has_integral integral {x..y} f - (y - x) *⇩R f x) {x..y}"
apply (rule has_integral_sub)
using x y apply (force intro: integrable_integral [OF integrable_subinterval_real [OF f]])
using has_integral_const_real [of "f x" x y] False
apply (simp add: )
done
show ?thesis
using False
apply (simp add: abs_eq_content del: content_real_if)
apply (rule has_integral_bound_real[where f="(λu. f u - f x)"])
using yx False d x y ‹e>0› apply (auto simp add: Idiff fux_int)
done
next
case True
have "f integrable_on {a..x}"
using f x by (simp add: integrable_subinterval_real)
then have Idiff: "?I a x - ?I a y = ?I y x"
using True x y by (simp add: algebra_simps integral_combine)
have fux_int: "((λu. f u - f x) has_integral integral {y..x} f - (x - y) *⇩R f x) {y..x}"
apply (rule has_integral_sub)
using x y apply (force intro: integrable_integral [OF integrable_subinterval_real [OF f]])
using has_integral_const_real [of "f x" y x] True
apply (simp add: )
done
have "norm (integral {a..x} f - integral {a..y} f - (x - y) *⇩R f x) ≤ e * ¦y - x¦"
using True
apply (simp add: abs_eq_content del: content_real_if)
apply (rule has_integral_bound_real[where f="(λu. f u - f x)"])
using yx True d x y ‹e>0› apply (auto simp add: Idiff fux_int)
done
then show ?thesis
by (simp add: algebra_simps norm_minus_commute)
qed
then have "∃d>0. ∀y∈{a..b}. ¦y - x¦ < d ⟶ norm (integral {a..y} f - integral {a..x} f - (y - x) *⇩R f x) ≤ e * ¦y - x¦"
using ‹d>0› by blast
}
then show ?thesis
by (simp add: has_vector_derivative_def has_derivative_within_alt bounded_linear_scaleR_left)
qed
lemma integral_has_vector_derivative:
fixes f :: "real ⇒ 'a::banach"
assumes "continuous_on {a .. b} f"
and "x ∈ {a .. b}"
shows "((λu. integral {a .. u} f) has_vector_derivative f(x)) (at x within {a .. b})"
apply (rule integral_has_vector_derivative_continuous_at [OF integrable_continuous_real])
using assms
apply (auto simp: continuous_on_eq_continuous_within)
done
lemma antiderivative_continuous:
fixes q b :: real
assumes "continuous_on {a .. b} f"
obtains g where "∀x∈{a .. b}. (g has_vector_derivative (f x::_::banach)) (at x within {a .. b})"
apply (rule that)
apply rule
using integral_has_vector_derivative[OF assms]
apply auto
done
subsection ‹Combined fundamental theorem of calculus.›
lemma antiderivative_integral_continuous:
fixes f :: "real ⇒ 'a::banach"
assumes "continuous_on {a .. b} f"
obtains g where "∀u∈{a .. b}. ∀v ∈ {a .. b}. u ≤ v ⟶ (f has_integral (g v - g u)) {u .. v}"
proof -
from antiderivative_continuous[OF assms] guess g . note g=this
show ?thesis
apply (rule that[of g])
apply safe
proof goal_cases
case prems: (1 u v)
have "∀x∈cbox u v. (g has_vector_derivative f x) (at x within cbox u v)"
apply rule
apply (rule has_vector_derivative_within_subset)
apply (rule g[rule_format])
using prems(1,2)
apply auto
done
then show ?case
using fundamental_theorem_of_calculus[OF prems(3), of g f] by auto
qed
qed
subsection ‹General "twiddling" for interval-to-interval function image.›
lemma has_integral_twiddle:
assumes "0 < r"
and "∀x. h(g x) = x"
and "∀x. g(h x) = x"
and "∀x. continuous (at x) g"
and "∀u v. ∃w z. g ` cbox u v = cbox w z"
and "∀u v. ∃w z. h ` cbox u v = cbox w z"
and "∀u v. content(g ` cbox u v) = r * content (cbox u v)"
and "(f has_integral i) (cbox a b)"
shows "((λx. f(g x)) has_integral (1 / r) *⇩R i) (h ` cbox a b)"
proof -
show ?thesis when *: "cbox a b ≠ {} ⟹ ?thesis"
apply cases
defer
apply (rule *)
apply assumption
proof goal_cases
case prems: 1
then show ?thesis
unfolding prems assms(8)[unfolded prems has_integral_empty_eq] by auto
qed
assume "cbox a b ≠ {}"
from assms(6)[rule_format,of a b] guess w z by (elim exE) note wz=this
have inj: "inj g" "inj h"
unfolding inj_on_def
apply safe
apply(rule_tac[!] ccontr)
using assms(2)
apply(erule_tac x=x in allE)
using assms(2)
apply(erule_tac x=y in allE)
defer
using assms(3)
apply (erule_tac x=x in allE)
using assms(3)
apply(erule_tac x=y in allE)
apply auto
done
show ?thesis
unfolding has_integral_def has_integral_compact_interval_def
apply (subst if_P)
apply rule
apply rule
apply (rule wz)
proof safe
fix e :: real
assume e: "e > 0"
with assms(1) have "e * r > 0" by simp
from assms(8)[unfolded has_integral,rule_format,OF this] guess d by (elim exE conjE) note d=this[rule_format]
def d' ≡ "λx. {y. g y ∈ d (g x)}"
have d': "⋀x. d' x = {y. g y ∈ (d (g x))}"
unfolding d'_def ..
show "∃d. gauge d ∧ (∀p. p tagged_division_of h ` cbox a b ∧ d fine p ⟶ norm ((∑(x, k)∈p. content k *⇩R f (g x)) - (1 / r) *⇩R i) < e)"
proof (rule_tac x=d' in exI, safe)
show "gauge d'"
using d(1)
unfolding gauge_def d'
using continuous_open_preimage_univ[OF assms(4)]
by auto
fix p
assume as: "p tagged_division_of h ` cbox a b" "d' fine p"
note p = tagged_division_ofD[OF as(1)]
have "(λ(x, k). (g x, g ` k)) ` p tagged_division_of (cbox a b) ∧ d fine (λ(x, k). (g x, g ` k)) ` p"
unfolding tagged_division_of
proof safe
show "finite ((λ(x, k). (g x, g ` k)) ` p)"
using as by auto
show "d fine (λ(x, k). (g x, g ` k)) ` p"
using as(2) unfolding fine_def d' by auto
fix x k
assume xk[intro]: "(x, k) ∈ p"
show "g x ∈ g ` k"
using p(2)[OF xk] by auto
show "∃u v. g ` k = cbox u v"
using p(4)[OF xk] using assms(5-6) by auto
{
fix y
assume "y ∈ k"
then show "g y ∈ cbox a b" "g y ∈ cbox a b"
using p(3)[OF xk,unfolded subset_eq,rule_format,of "h (g y)"]
using assms(2)[rule_format,of y]
unfolding inj_image_mem_iff[OF inj(2)]
by auto
}
fix x' k'
assume xk': "(x', k') ∈ p"
fix z
assume "z ∈ interior (g ` k)" and "z ∈ interior (g ` k')"
then have *: "interior (g ` k) ∩ interior (g ` k') ≠ {}"
by auto
have same: "(x, k) = (x', k')"
apply -
apply (rule ccontr)
apply (drule p(5)[OF xk xk'])
proof -
assume as: "interior k ∩ interior k' = {}"
from nonempty_witness[OF *] guess z .
then have "z ∈ g ` (interior k ∩ interior k')"
using interior_image_subset[OF assms(4) inj(1)]
unfolding image_Int[OF inj(1)]
by auto
then show False
using as by blast
qed
then show "g x = g x'"
by auto
{
fix z
assume "z ∈ k"
then show "g z ∈ g ` k'"
using same by auto
}
{
fix z
assume "z ∈ k'"
then show "g z ∈ g ` k"
using same by auto
}
next
fix x
assume "x ∈ cbox a b"
then have "h x ∈ ⋃{k. ∃x. (x, k) ∈ p}"
using p(6) by auto
then guess X unfolding Union_iff .. note X=this
from this(1) guess y unfolding mem_Collect_eq ..
then show "x ∈ ⋃{k. ∃x. (x, k) ∈ (λ(x, k). (g x, g ` k)) ` p}"
apply -
apply (rule_tac X="g ` X" in UnionI)
defer
apply (rule_tac x="h x" in image_eqI)
using X(2) assms(3)[rule_format,of x]
apply auto
done
qed
note ** = d(2)[OF this]
have *: "inj_on (λ(x, k). (g x, g ` k)) p"
using inj(1) unfolding inj_on_def by fastforce
have "(∑(x, k)∈(λ(x, k). (g x, g ` k)) ` p. content k *⇩R f x) - i = r *⇩R (∑(x, k)∈p. content k *⇩R f (g x)) - i" (is "?l = _")
using assms(7)
unfolding algebra_simps add_left_cancel scaleR_right.setsum
by (subst setsum.reindex_bij_betw[symmetric, where h="λ(x, k). (g x, g ` k)" and S=p])
(auto intro!: * setsum.cong simp: bij_betw_def dest!: p(4))
also have "… = r *⇩R ((∑(x, k)∈p. content k *⇩R f (g x)) - (1 / r) *⇩R i)" (is "_ = ?r")
unfolding scaleR_diff_right scaleR_scaleR
using assms(1)
by auto
finally have *: "?l = ?r" .
show "norm ((∑(x, k)∈p. content k *⇩R f (g x)) - (1 / r) *⇩R i) < e"
using **
unfolding *
unfolding norm_scaleR
using assms(1)
by (auto simp add:field_simps)
qed
qed
qed
subsection ‹Special case of a basic affine transformation.›
lemma interval_image_affinity_interval:
"∃u v. (λx. m *⇩R (x::'a::euclidean_space) + c) ` cbox a b = cbox u v"
unfolding image_affinity_cbox
by auto
lemma content_image_affinity_cbox:
"content((λx::'a::euclidean_space. m *⇩R x + c) ` cbox a b) =
¦m¦ ^ DIM('a) * content (cbox a b)" (is "?l = ?r")
proof (cases "cbox a b = {}")
case True then show ?thesis by simp
next
case False
show ?thesis
proof (cases "m ≥ 0")
case True
with ‹cbox a b ≠ {}› have "cbox (m *⇩R a + c) (m *⇩R b + c) ≠ {}"
unfolding box_ne_empty
apply (intro ballI)
apply (erule_tac x=i in ballE)
apply (auto simp: inner_simps mult_left_mono)
done
moreover from True have *: "⋀i. (m *⇩R b + c) ∙ i - (m *⇩R a + c) ∙ i = m *⇩R (b - a) ∙ i"
by (simp add: inner_simps field_simps)
ultimately show ?thesis
by (simp add: image_affinity_cbox True content_cbox'
setprod.distrib setprod_constant inner_diff_left)
next
case False
with ‹cbox a b ≠ {}› have "cbox (m *⇩R b + c) (m *⇩R a + c) ≠ {}"
unfolding box_ne_empty
apply (intro ballI)
apply (erule_tac x=i in ballE)
apply (auto simp: inner_simps mult_left_mono)
done
moreover from False have *: "⋀i. (m *⇩R a + c) ∙ i - (m *⇩R b + c) ∙ i = (-m) *⇩R (b - a) ∙ i"
by (simp add: inner_simps field_simps)
ultimately show ?thesis using False
by (simp add: image_affinity_cbox content_cbox'
setprod.distrib[symmetric] setprod_constant[symmetric] inner_diff_left)
qed
qed
lemma has_integral_affinity:
fixes a :: "'a::euclidean_space"
assumes "(f has_integral i) (cbox a b)"
and "m ≠ 0"
shows "((λx. f(m *⇩R x + c)) has_integral ((1 / (¦m¦ ^ DIM('a))) *⇩R i)) ((λx. (1 / m) *⇩R x + -((1 / m) *⇩R c)) ` cbox a b)"
apply (rule has_integral_twiddle)
using assms
apply (safe intro!: interval_image_affinity_interval content_image_affinity_cbox)
apply (rule zero_less_power)
unfolding scaleR_right_distrib
apply auto
done
lemma integrable_affinity:
assumes "f integrable_on cbox a b"
and "m ≠ 0"
shows "(λx. f(m *⇩R x + c)) integrable_on ((λx. (1 / m) *⇩R x + -((1/m) *⇩R c)) ` cbox a b)"
using assms
unfolding integrable_on_def
apply safe
apply (drule has_integral_affinity)
apply auto
done
lemmas has_integral_affinity01 = has_integral_affinity [of _ _ 0 "1::real", simplified]
subsection ‹Special case of stretching coordinate axes separately.›
lemma image_stretch_interval:
"(λx. ∑k∈Basis. (m k * (x∙k)) *⇩R k) ` cbox a (b::'a::euclidean_space) =
(if (cbox a b) = {} then {} else
cbox (∑k∈Basis. (min (m k * (a∙k)) (m k * (b∙k))) *⇩R k::'a)
(∑k∈Basis. (max (m k * (a∙k)) (m k * (b∙k))) *⇩R k))"
proof cases
assume *: "cbox a b ≠ {}"
show ?thesis
unfolding box_ne_empty if_not_P[OF *]
apply (simp add: cbox_def image_Collect set_eq_iff euclidean_eq_iff[where 'a='a] ball_conj_distrib[symmetric])
apply (subst choice_Basis_iff[symmetric])
proof (intro allI ball_cong refl)
fix x i :: 'a assume "i ∈ Basis"
with * have a_le_b: "a ∙ i ≤ b ∙ i"
unfolding box_ne_empty by auto
show "(∃xa. x ∙ i = m i * xa ∧ a ∙ i ≤ xa ∧ xa ≤ b ∙ i) ⟷
min (m i * (a ∙ i)) (m i * (b ∙ i)) ≤ x ∙ i ∧ x ∙ i ≤ max (m i * (a ∙ i)) (m i * (b ∙ i))"
proof (cases "m i = 0")
case True
with a_le_b show ?thesis by auto
next
case False
then have *: "⋀a b. a = m i * b ⟷ b = a / m i"
by (auto simp add: field_simps)
from False have
"min (m i * (a ∙ i)) (m i * (b ∙ i)) = (if 0 < m i then m i * (a ∙ i) else m i * (b ∙ i))"
"max (m i * (a ∙ i)) (m i * (b ∙ i)) = (if 0 < m i then m i * (b ∙ i) else m i * (a ∙ i))"
using a_le_b by (auto simp: min_def max_def mult_le_cancel_left)
with False show ?thesis using a_le_b
unfolding * by (auto simp add: le_divide_eq divide_le_eq ac_simps)
qed
qed
qed simp
lemma interval_image_stretch_interval:
"∃u v. (λx. ∑k∈Basis. (m k * (x∙k))*⇩R k) ` cbox a (b::'a::euclidean_space) = cbox u (v::'a::euclidean_space)"
unfolding image_stretch_interval by auto
lemma content_image_stretch_interval:
"content ((λx::'a::euclidean_space. (∑k∈Basis. (m k * (x∙k))*⇩R k)::'a) ` cbox a b) =
¦setprod m Basis¦ * content (cbox a b)"
proof (cases "cbox a b = {}")
case True
then show ?thesis
unfolding content_def image_is_empty image_stretch_interval if_P[OF True] by auto
next
case False
then have "(λx. (∑k∈Basis. (m k * (x∙k))*⇩R k)) ` cbox a b ≠ {}"
by auto
then show ?thesis
using False
unfolding content_def image_stretch_interval
apply -
unfolding interval_bounds' if_not_P
unfolding abs_setprod setprod.distrib[symmetric]
apply (rule setprod.cong)
apply (rule refl)
unfolding lessThan_iff
apply (simp only: inner_setsum_left_Basis)
proof -
fix i :: 'a
assume i: "i ∈ Basis"
have "(m i < 0 ∨ m i > 0) ∨ m i = 0"
by auto
then show "max (m i * (a ∙ i)) (m i * (b ∙ i)) - min (m i * (a ∙ i)) (m i * (b ∙ i)) =
¦m i¦ * (b ∙ i - a ∙ i)"
apply -
apply (erule disjE)+
unfolding min_def max_def
using False[unfolded box_ne_empty,rule_format,of i] i
apply (auto simp add:field_simps not_le mult_le_cancel_left_neg mult_le_cancel_left_pos)
done
qed
qed
lemma has_integral_stretch:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "(f has_integral i) (cbox a b)"
and "∀k∈Basis. m k ≠ 0"
shows "((λx. f (∑k∈Basis. (m k * (x∙k))*⇩R k)) has_integral
((1/ ¦setprod m Basis¦) *⇩R i)) ((λx. (∑k∈Basis. (1 / m k * (x∙k))*⇩R k)) ` cbox a b)"
apply (rule has_integral_twiddle[where f=f])
unfolding zero_less_abs_iff content_image_stretch_interval
unfolding image_stretch_interval empty_as_interval euclidean_eq_iff[where 'a='a]
using assms
proof -
show "∀y::'a. continuous (at y) (λx. (∑k∈Basis. (m k * (x∙k))*⇩R k))"
apply rule
apply (rule linear_continuous_at)
unfolding linear_linear
unfolding linear_iff inner_simps euclidean_eq_iff[where 'a='a]
apply (auto simp add: field_simps)
done
qed auto
lemma integrable_stretch:
fixes f :: "'a::euclidean_space ⇒ 'b::real_normed_vector"
assumes "f integrable_on cbox a b"
and "∀k∈Basis. m k ≠ 0"
shows "(λx::'a. f (∑k∈Basis. (m k * (x∙k))*⇩R k)) integrable_on
((λx. ∑k∈Basis. (1 / m k * (x∙k))*⇩R k) ` cbox a b)"
using assms
unfolding integrable_on_def
apply -
apply (erule exE)
apply (drule has_integral_stretch)
apply assumption
apply auto
done
subsection ‹even more special cases.›
lemma uminus_interval_vector[simp]:
fixes a b :: "'a::euclidean_space"
shows "uminus ` cbox a b = cbox (-b) (-a)"
apply (rule set_eqI)
apply rule
defer
unfolding image_iff
apply (rule_tac x="-x" in bexI)
apply (auto simp add:minus_le_iff le_minus_iff mem_box)
done
lemma has_integral_reflect_lemma[intro]:
assumes "(f has_integral i) (cbox a b)"
shows "((λx. f(-x)) has_integral i) (cbox (-b) (-a))"
using has_integral_affinity[OF assms, of "-1" 0]
by auto
lemma has_integral_reflect_lemma_real[intro]:
assumes "(f has_integral i) {a .. b::real}"
shows "((λx. f(-x)) has_integral i) {-b .. -a}"
using assms
unfolding box_real[symmetric]
by (rule has_integral_reflect_lemma)
lemma has_integral_reflect[simp]:
"((λx. f (-x)) has_integral i) (cbox (-b) (-a)) ⟷ (f has_integral i) (cbox a b)"
apply rule
apply (drule_tac[!] has_integral_reflect_lemma)
apply auto
done
lemma integrable_reflect[simp]: "(λx. f(-x)) integrable_on cbox (-b) (-a) ⟷ f integrable_on cbox a b"
unfolding integrable_on_def by auto
lemma integrable_reflect_real[simp]: "(λx. f(-x)) integrable_on {-b .. -a} ⟷ f integrable_on {a .. b::real}"
unfolding box_real[symmetric]
by (rule integrable_reflect)
lemma integral_reflect[simp]: "integral (cbox (-b) (-a)) (λx. f (-x)) = integral (cbox a b) f"
unfolding integral_def by auto
lemma integral_reflect_real[simp]: "integral {-b .. -a} (λx. f (-x)) = integral {a .. b::real} f"
unfolding box_real[symmetric]
by (rule integral_reflect)
subsection ‹Stronger form of FCT; quite a tedious proof.›
lemma bgauge_existence_lemma: "(∀x∈s. ∃d::real. 0 < d ∧ q d x) ⟷ (∀x. ∃d>0. x∈s ⟶ q d x)"
by (meson zero_less_one)
lemma additive_tagged_division_1':
fixes f :: "real ⇒ 'a::real_normed_vector"
assumes "a ≤ b"
and "p tagged_division_of {a..b}"
shows "setsum (λ(x,k). f (Sup k) - f(Inf k)) p = f b - f a"
using additive_tagged_division_1[OF _ assms(2), of f]
using assms(1)
by auto
lemma split_minus[simp]: "(λ(x, k). f x k) x - (λ(x, k). g x k) x = (λ(x, k). f x k - g x k) x"
by (simp add: split_def)
lemma norm_triangle_le_sub: "norm x + norm y ≤ e ⟹ norm (x - y) ≤ e"
apply (subst(asm)(2) norm_minus_cancel[symmetric])
apply (drule norm_triangle_le)
apply (auto simp add: algebra_simps)
done
lemma fundamental_theorem_of_calculus_interior:
fixes f :: "real ⇒ 'a::real_normed_vector"
assumes "a ≤ b"
and "continuous_on {a .. b} f"
and "∀x∈{a <..< b}. (f has_vector_derivative f'(x)) (at x)"
shows "(f' has_integral (f b - f a)) {a .. b}"
proof -
{
presume *: "a < b ⟹ ?thesis"
show ?thesis
proof (cases "a < b")
case True
then show ?thesis by (rule *)
next
case False
then have "a = b"
using assms(1) by auto
then have *: "cbox a b = {b}" "f b - f a = 0"
by (auto simp add: order_antisym)
show ?thesis
unfolding *(2)
unfolding content_eq_0
using * ‹a = b›
by (auto simp: ex_in_conv)
qed
}
assume ab: "a < b"
let ?P = "λe. ∃d. gauge d ∧ (∀p. p tagged_division_of {a .. b} ∧ d fine p ⟶
norm ((∑(x, k)∈p. content k *⇩R f' x) - (f b - f a)) ≤ e * content {a .. b})"
{ presume "⋀e. e > 0 ⟹ ?P e" then show ?thesis unfolding has_integral_factor_content_real by auto }
fix e :: real
assume e: "e > 0"
note assms(3)[unfolded has_vector_derivative_def has_derivative_at_alt ball_conj_distrib]
note conjunctD2[OF this]
note bounded=this(1) and this(2)
from this(2) have "∀x∈box a b. ∃d>0. ∀y. norm (y - x) < d ⟶
norm (f y - f x - (y - x) *⇩R f' x) ≤ e/2 * norm (y - x)"
apply -
apply safe
apply (erule_tac x=x in ballE)
apply (erule_tac x="e/2" in allE)
using e
apply auto
done
note this[unfolded bgauge_existence_lemma]
from choice[OF this] guess d ..
note conjunctD2[OF this[rule_format]]
note d = this[rule_format]
have "bounded (f ` cbox a b)"
apply (rule compact_imp_bounded compact_continuous_image)+
using compact_cbox assms
apply auto
done
from this[unfolded bounded_pos] guess B .. note B = this[rule_format]
have "∃da. 0 < da ∧ (∀c. a ≤ c ∧ {a .. c} ⊆ {a .. b} ∧ {a .. c} ⊆ ball a da ⟶
norm (content {a .. c} *⇩R f' a - (f c - f a)) ≤ (e * (b - a)) / 4)"
proof -
have "a ∈ {a .. b}"
using ab by auto
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
note * = this[unfolded continuous_within Lim_within,rule_format]
have "(e * (b - a)) / 8 > 0"
using e ab by (auto simp add: field_simps)
from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
have "∃l. 0 < l ∧ norm(l *⇩R f' a) ≤ (e * (b - a)) / 8"
proof (cases "f' a = 0")
case True
thus ?thesis using ab e by auto
next
case False
then show ?thesis
apply (rule_tac x="(e * (b - a)) / 8 / norm (f' a)" in exI)
using ab e
apply (auto simp add: field_simps)
done
qed
then guess l .. note l = conjunctD2[OF this]
show ?thesis
apply (rule_tac x="min k l" in exI)
apply safe
unfolding min_less_iff_conj
apply rule
apply (rule l k)+
proof -
fix c
assume as: "a ≤ c" "{a .. c} ⊆ {a .. b}" "{a .. c} ⊆ ball a (min k l)"
note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_box]
have "norm ((c - a) *⇩R f' a - (f c - f a)) ≤ norm ((c - a) *⇩R f' a) + norm (f c - f a)"
by (rule norm_triangle_ineq4)
also have "… ≤ e * (b - a) / 8 + e * (b - a) / 8"
proof (rule add_mono)
have "¦c - a¦ ≤ ¦l¦"
using as' by auto
then show "norm ((c - a) *⇩R f' a) ≤ e * (b - a) / 8"
apply -
apply (rule order_trans[OF _ l(2)])
unfolding norm_scaleR
apply (rule mult_right_mono)
apply auto
done
next
show "norm (f c - f a) ≤ e * (b - a) / 8"
apply (rule less_imp_le)
apply (cases "a = c")
defer
apply (rule k(2)[unfolded dist_norm])
using as' e ab
apply (auto simp add: field_simps)
done
qed
finally show "norm (content {a .. c} *⇩R f' a - (f c - f a)) ≤ e * (b - a) / 4"
unfolding content_real[OF as(1)] by auto
qed
qed
then guess da .. note da=conjunctD2[OF this,rule_format]
have "∃db>0. ∀c≤b. {c .. b} ⊆ {a .. b} ∧ {c .. b} ⊆ ball b db ⟶
norm (content {c .. b} *⇩R f' b - (f b - f c)) ≤ (e * (b - a)) / 4"
proof -
have "b ∈ {a .. b}"
using ab by auto
note assms(2)[unfolded continuous_on_eq_continuous_within,rule_format,OF this]
note * = this[unfolded continuous_within Lim_within,rule_format] have "(e * (b - a)) / 8 > 0"
using e ab by (auto simp add: field_simps)
from *[OF this] guess k .. note k = conjunctD2[OF this,rule_format]
have "∃l. 0 < l ∧ norm (l *⇩R f' b) ≤ (e * (b - a)) / 8"
proof (cases "f' b = 0")
case True
thus ?thesis using ab e by auto
next
case False
then show ?thesis
apply (rule_tac x="(e * (b - a)) / 8 / norm (f' b)" in exI)
using ab e
apply (auto simp add: field_simps)
done
qed
then guess l .. note l = conjunctD2[OF this]
show ?thesis
apply (rule_tac x="min k l" in exI)
apply safe
unfolding min_less_iff_conj
apply rule
apply (rule l k)+
proof -
fix c
assume as: "c ≤ b" "{c..b} ⊆ {a..b}" "{c..b} ⊆ ball b (min k l)"
note as' = this[unfolded subset_eq Ball_def mem_ball dist_real_def mem_box]
have "norm ((b - c) *⇩R f' b - (f b - f c)) ≤ norm ((b - c) *⇩R f' b) + norm (f b - f c)"
by (rule norm_triangle_ineq4)
also have "… ≤ e * (b - a) / 8 + e * (b - a) / 8"
proof (rule add_mono)
have "¦c - b¦ ≤ ¦l¦"
using as' by auto
then show "norm ((b - c) *⇩R f' b) ≤ e * (b - a) / 8"
apply -
apply (rule order_trans[OF _ l(2)])
unfolding norm_scaleR
apply (rule mult_right_mono)
apply auto
done
next
show "norm (f b - f c) ≤ e * (b - a) / 8"
apply (rule less_imp_le)
apply (cases "b = c")
defer
apply (subst norm_minus_commute)
apply (rule k(2)[unfolded dist_norm])
using as' e ab
apply (auto simp add: field_simps)
done
qed
finally show "norm (content {c .. b} *⇩R f' b - (f b - f c)) ≤ e * (b - a) / 4"
unfolding content_real[OF as(1)] by auto
qed
qed
then guess db .. note db=conjunctD2[OF this,rule_format]
let ?d = "(λx. ball x (if x=a then da else if x=b then db else d x))"
show "?P e"
apply (rule_tac x="?d" in exI)
proof (safe, goal_cases)
case 1
show ?case
apply (rule gauge_ball_dependent)
using ab db(1) da(1) d(1)
apply auto
done
next
case as: (2 p)
let ?A = "{t. fst t ∈ {a, b}}"
note p = tagged_division_ofD[OF as(1)]
have pA: "p = (p ∩ ?A) ∪ (p - ?A)" "finite (p ∩ ?A)" "finite (p - ?A)" "(p ∩ ?A) ∩ (p - ?A) = {}"
using as by auto
note * = additive_tagged_division_1'[OF assms(1) as(1), symmetric]
have **: "⋀n1 s1 n2 s2::real. n2 ≤ s2 / 2 ⟹ n1 - s1 ≤ s2 / 2 ⟹ n1 + n2 ≤ s1 + s2"
by arith
show ?case
unfolding content_real[OF assms(1)] and *[of "λx. x"] *[of f] setsum_subtractf[symmetric] split_minus
unfolding setsum_right_distrib
apply (subst(2) pA)
apply (subst pA)
unfolding setsum.union_disjoint[OF pA(2-)]
proof (rule norm_triangle_le, rule **, goal_cases)
case 1
show ?case
apply (rule order_trans)
apply (rule setsum_norm_le)
defer
apply (subst setsum_divide_distrib)
apply (rule order_refl)
apply safe
apply (unfold not_le o_def split_conv fst_conv)
proof (rule ccontr)
fix x k
assume xk: "(x, k) ∈ p"
"e * (Sup k - Inf k) / 2 <
norm (content k *⇩R f' x - (f (Sup k) - f (Inf k)))"
from p(4)[OF this(1)] guess u v by (elim exE) note k=this
then have "u ≤ v" and uv: "{u, v} ⊆ cbox u v"
using p(2)[OF xk(1)] by auto
note result = xk(2)[unfolded k box_real interval_bounds_real[OF this(1)] content_real[OF this(1)]]
assume as': "x ≠ a" "x ≠ b"
then have "x ∈ box a b"
using p(2-3)[OF xk(1)] by (auto simp: mem_box)
note * = d(2)[OF this]
have "norm ((v - u) *⇩R f' (x) - (f (v) - f (u))) =
norm ((f (u) - f (x) - (u - x) *⇩R f' (x)) - (f (v) - f (x) - (v - x) *⇩R f' (x)))"
apply (rule arg_cong[of _ _ norm])
unfolding scaleR_left.diff
apply auto
done
also have "… ≤ e / 2 * norm (u - x) + e / 2 * norm (v - x)"
apply (rule norm_triangle_le_sub)
apply (rule add_mono)
apply (rule_tac[!] *)
using fineD[OF as(2) xk(1)] as'
unfolding k subset_eq
apply -
apply (erule_tac x=u in ballE)
apply (erule_tac[3] x=v in ballE)
using uv
apply (auto simp:dist_real_def)
done
also have "… ≤ e / 2 * norm (v - u)"
using p(2)[OF xk(1)]
unfolding k
by (auto simp add: field_simps)
finally have "e * (v - u) / 2 < e * (v - u) / 2"
apply -
apply (rule less_le_trans[OF result])
using uv
apply auto
done
then show False by auto
qed
next
have *: "⋀x s1 s2::real. 0 ≤ s1 ⟹ x ≤ (s1 + s2) / 2 ⟹ x - s1 ≤ s2 / 2"
by auto
case 2
show ?case
apply (rule *)
apply (rule setsum_nonneg)
apply rule
apply (unfold split_paired_all split_conv)
defer
unfolding setsum.union_disjoint[OF pA(2-),symmetric] pA(1)[symmetric]
unfolding setsum_right_distrib[symmetric]
apply (subst additive_tagged_division_1[OF _ as(1)])
apply (rule assms)
proof -
fix x k
assume "(x, k) ∈ p ∩ {t. fst t ∈ {a, b}}"
note xk=IntD1[OF this]
from p(4)[OF this] guess u v by (elim exE) note uv=this
with p(2)[OF xk] have "cbox u v ≠ {}"
by auto
then show "0 ≤ e * ((Sup k) - (Inf k))"
unfolding uv using e by (auto simp add: field_simps)
next
have *: "⋀s f t e. setsum f s = setsum f t ⟹ norm (setsum f t) ≤ e ⟹ norm (setsum f s) ≤ e"
by auto
show "norm (∑(x, k)∈p ∩ ?A. content k *⇩R f' x -
(f ((Sup k)) - f ((Inf k)))) ≤ e * (b - a) / 2"
apply (rule *[where t1="p ∩ {t. fst t ∈ {a, b} ∧ content(snd t) ≠ 0}"])
apply (rule setsum.mono_neutral_right[OF pA(2)])
defer
apply rule
unfolding split_paired_all split_conv o_def
proof goal_cases
fix x k
assume "(x, k) ∈ p ∩ {t. fst t ∈ {a, b}} - p ∩ {t. fst t ∈ {a, b} ∧ content (snd t) ≠ 0}"
then have xk: "(x, k) ∈ p" "content k = 0"
by auto
from p(4)[OF xk(1)] guess u v by (elim exE) note uv=this
have "k ≠ {}"
using p(2)[OF xk(1)] by auto
then have *: "u = v"
using xk
unfolding uv content_eq_0 box_eq_empty
by auto
then show "content k *⇩R (f' (x)) - (f ((Sup k)) - f ((Inf k))) = 0"
using xk unfolding uv by auto
next
have *: "p ∩ {t. fst t ∈ {a, b} ∧ content(snd t) ≠ 0} =
{t. t∈p ∧ fst t = a ∧ content(snd t) ≠ 0} ∪ {t. t∈p ∧ fst t = b ∧ content(snd t) ≠ 0}"
by blast
have **: "norm (setsum f s) ≤ e"
if "∀x y. x ∈ s ∧ y ∈ s ⟶ x = y"
and "∀x. x ∈ s ⟶ norm (f x) ≤ e"
and "e > 0"
for s f and e :: real
proof (cases "s = {}")
case True
with that show ?thesis by auto
next
case False
then obtain x where "x ∈ s"
by auto
then have *: "s = {x}"
using that(1) by auto
then show ?thesis
using ‹x ∈ s› that(2) by auto
qed
case 2
show ?case
apply (subst *)
apply (subst setsum.union_disjoint)
prefer 4
apply (rule order_trans[of _ "e * (b - a)/4 + e * (b - a)/4"])
apply (rule norm_triangle_le,rule add_mono)
apply (rule_tac[1-2] **)
proof -
let ?B = "λx. {t ∈ p. fst t = x ∧ content (snd t) ≠ 0}"
have pa: "∃v. k = cbox a v ∧ a ≤ v" if "(a, k) ∈ p" for k
proof -
guess u v using p(4)[OF that] by (elim exE) note uv=this
have *: "u ≤ v"
using p(2)[OF that] unfolding uv by auto
have u: "u = a"
proof (rule ccontr)
have "u ∈ cbox u v"
using p(2-3)[OF that(1)] unfolding uv by auto
have "u ≥ a"
using p(2-3)[OF that(1)] unfolding uv subset_eq by auto
moreover assume "¬ ?thesis"
ultimately have "u > a" by auto
then show False
using p(2)[OF that(1)] unfolding uv by (auto simp add:)
qed
then show ?thesis
apply (rule_tac x=v in exI)
unfolding uv
using *
apply auto
done
qed
have pb: "∃v. k = cbox v b ∧ b ≥ v" if "(b, k) ∈ p" for k
proof -
guess u v using p(4)[OF that] by (elim exE) note uv=this
have *: "u ≤ v"
using p(2)[OF that] unfolding uv by auto
have u: "v = b"
proof (rule ccontr)
have "u ∈ cbox u v"
using p(2-3)[OF that(1)] unfolding uv by auto
have "v ≤ b"
using p(2-3)[OF that(1)] unfolding uv subset_eq by auto
moreover assume "¬ ?thesis"
ultimately have "v < b" by auto
then show False
using p(2)[OF that(1)] unfolding uv by (auto simp add:)
qed
then show ?thesis
apply (rule_tac x=u in exI)
unfolding uv
using *
apply auto
done
qed
show "∀x y. x ∈ ?B a ∧ y ∈ ?B a ⟶ x = y"
apply (rule,rule,rule,unfold split_paired_all)
unfolding mem_Collect_eq fst_conv snd_conv
apply safe
proof -
fix x k k'
assume k: "(a, k) ∈ p" "(a, k') ∈ p" "content k ≠ 0" "content k' ≠ 0"
guess v using pa[OF k(1)] .. note v = conjunctD2[OF this]
guess v' using pa[OF k(2)] .. note v' = conjunctD2[OF this] let ?v = "min v v'"
have "box a ?v ⊆ k ∩ k'"
unfolding v v' by (auto simp add: mem_box)
note interior_mono[OF this,unfolded interior_Int]
moreover have "(a + ?v)/2 ∈ box a ?v"
using k(3-)
unfolding v v' content_eq_0 not_le
by (auto simp add: mem_box)
ultimately have "(a + ?v)/2 ∈ interior k ∩ interior k'"
unfolding interior_open[OF open_box] by auto
then have *: "k = k'"
apply -
apply (rule ccontr)
using p(5)[OF k(1-2)]
apply auto
done
{ assume "x ∈ k" then show "x ∈ k'" unfolding * . }
{ assume "x ∈ k'" then show "x ∈ k" unfolding * . }
qed
show "∀x y. x ∈ ?B b ∧ y ∈ ?B b ⟶ x = y"
apply rule
apply rule
apply rule
apply (unfold split_paired_all)
unfolding mem_Collect_eq fst_conv snd_conv
apply safe
proof -
fix x k k'
assume k: "(b, k) ∈ p" "(b, k') ∈ p" "content k ≠ 0" "content k' ≠ 0"
guess v using pb[OF k(1)] .. note v = conjunctD2[OF this]
guess v' using pb[OF k(2)] .. note v' = conjunctD2[OF this]
let ?v = "max v v'"
have "box ?v b ⊆ k ∩ k'"
unfolding v v' by (auto simp: mem_box)
note interior_mono[OF this,unfolded interior_Int]
moreover have " ((b + ?v)/2) ∈ box ?v b"
using k(3-) unfolding v v' content_eq_0 not_le by (auto simp: mem_box)
ultimately have " ((b + ?v)/2) ∈ interior k ∩ interior k'"
unfolding interior_open[OF open_box] by auto
then have *: "k = k'"
apply -
apply (rule ccontr)
using p(5)[OF k(1-2)]
apply auto
done
{ assume "x ∈ k" then show "x ∈ k'" unfolding * . }
{ assume "x ∈ k'" then show "x∈k" unfolding * . }
qed
let ?a = a and ?b = b
show "∀x. x ∈ ?B a ⟶ norm ((λ(x, k). content k *⇩R f' x - (f (Sup k) -
f (Inf k))) x) ≤ e * (b - a) / 4"
apply rule
apply rule
unfolding mem_Collect_eq
unfolding split_paired_all fst_conv snd_conv
proof (safe, goal_cases)
case prems: 1
guess v using pa[OF prems(1)] .. note v = conjunctD2[OF this]
have "?a ∈ {?a..v}"
using v(2) by auto
then have "v ≤ ?b"
using p(3)[OF prems(1)] unfolding subset_eq v by auto
moreover have "{?a..v} ⊆ ball ?a da"
using fineD[OF as(2) prems(1)]
apply -
apply (subst(asm) if_P)
apply (rule refl)
unfolding subset_eq
apply safe
apply (erule_tac x=" x" in ballE)
apply (auto simp add:subset_eq dist_real_def v)
done
ultimately show ?case
unfolding v interval_bounds_real[OF v(2)] box_real
apply -
apply(rule da(2)[of "v"])
using prems fineD[OF as(2) prems(1)]
unfolding v content_eq_0
apply auto
done
qed
show "∀x. x ∈ ?B b ⟶ norm ((λ(x, k). content k *⇩R f' x -
(f (Sup k) - f (Inf k))) x) ≤ e * (b - a) / 4"
apply rule
apply rule
unfolding mem_Collect_eq
unfolding split_paired_all fst_conv snd_conv
proof (safe, goal_cases)
case prems: 1
guess v using pb[OF prems(1)] .. note v = conjunctD2[OF this]
have "?b ∈ {v.. ?b}"
using v(2) by auto
then have "v ≥ ?a" using p(3)[OF prems(1)]
unfolding subset_eq v by auto
moreover have "{v..?b} ⊆ ball ?b db"
using fineD[OF as(2) prems(1)]
apply -
apply (subst(asm) if_P, rule refl)
unfolding subset_eq
apply safe
apply (erule_tac x=" x" in ballE)
using ab
apply (auto simp add:subset_eq v dist_real_def)
done
ultimately show ?case
unfolding v
unfolding interval_bounds_real[OF v(2)] box_real
apply -
apply(rule db(2)[of "v"])
using prems fineD[OF as(2) prems(1)]
unfolding v content_eq_0
apply auto
done
qed
qed (insert p(1) ab e, auto simp add: field_simps)
qed auto
qed
qed
qed
qed
subsection ‹Stronger form with finite number of exceptional points.›
lemma fundamental_theorem_of_calculus_interior_strong:
fixes f :: "real ⇒ 'a::banach"
assumes "finite s"
and "a ≤ b"
and "continuous_on {a .. b} f"
and "∀x∈{a <..< b} - s. (f has_vector_derivative f'(x)) (at x)"
shows "(f' has_integral (f b - f a)) {a .. b}"
using assms
proof (induct "card s" arbitrary: s a b)
case 0
show ?case
apply (rule fundamental_theorem_of_calculus_interior)
using 0
apply auto
done
next
case (Suc n)
from this(2) guess c s'
apply -
apply (subst(asm) eq_commute)
unfolding card_Suc_eq
apply (subst(asm)(2) eq_commute)
apply (elim exE conjE)
done
note cs = this[rule_format]
show ?case
proof (cases "c ∈ box a b")
case False
then show ?thesis
apply -
apply (rule Suc(1)[OF cs(3) _ Suc(4,5)])
apply safe
defer
apply (rule Suc(6)[rule_format])
using Suc(3)
unfolding cs
apply auto
done
next
have *: "f b - f a = (f c - f a) + (f b - f c)"
by auto
case True
then have "a ≤ c" "c ≤ b"
by (auto simp: mem_box)
then show ?thesis
apply (subst *)
apply (rule has_integral_combine)
apply assumption+
apply (rule_tac[!] Suc(1)[OF cs(3)])
using Suc(3)
unfolding cs
proof -
show "continuous_on {a .. c} f" "continuous_on {c .. b} f"
apply (rule_tac[!] continuous_on_subset[OF Suc(5)])
using True
apply (auto simp: mem_box)
done
let ?P = "λi j. ∀x∈{i <..< j} - s'. (f has_vector_derivative f' x) (at x)"
show "?P a c" "?P c b"
apply safe
apply (rule_tac[!] Suc(6)[rule_format])
using True
unfolding cs
apply (auto simp: mem_box)
done
qed auto
qed
qed
lemma fundamental_theorem_of_calculus_strong:
fixes f :: "real ⇒ 'a::banach"
assumes "finite s"
and "a ≤ b"
and "continuous_on {a .. b} f"
and "∀x∈{a .. b} - s. (f has_vector_derivative f'(x)) (at x)"
shows "(f' has_integral (f b - f a)) {a .. b}"
apply (rule fundamental_theorem_of_calculus_interior_strong[OF assms(1-3), of f'])
using assms(4)
apply (auto simp: mem_box)
done
lemma indefinite_integral_continuous_left:
fixes f:: "real ⇒ 'a::banach"
assumes "f integrable_on {a .. b}"
and "a < c"
and "c ≤ b"
and "e > 0"
obtains d where "d > 0"
and "∀t. c - d < t ∧ t ≤ c ⟶ norm (integral {a .. c} f - integral {a .. t} f) < e"
proof -
have "∃w>0. ∀t. c - w < t ∧ t < c ⟶ norm (f c) * norm(c - t) < e / 3"
proof (cases "f c = 0")
case False
hence "0 < e / 3 / norm (f c)" using ‹e>0› by simp
then show ?thesis
apply -
apply rule
apply rule
apply assumption
apply safe
proof -
fix t
assume as: "t < c" and "c - e / 3 / norm (f c) < t"
then have "c - t < e / 3 / norm (f c)"
by auto
then have "norm (c - t) < e / 3 / norm (f c)"
using as by auto
then show "norm (f c) * norm (c - t) < e / 3"
using False
apply -
apply (subst mult.commute)
apply (subst pos_less_divide_eq[symmetric])
apply auto
done
qed
next
case True
show ?thesis
apply (rule_tac x=1 in exI)
unfolding True
using ‹e > 0›
apply auto
done
qed
then guess w .. note w = conjunctD2[OF this,rule_format]
have *: "e / 3 > 0"
using assms by auto
have "f integrable_on {a .. c}"
apply (rule integrable_subinterval_real[OF assms(1)])
using assms(2-3)
apply auto
done
from integrable_integral[OF this,unfolded has_integral_real,rule_format,OF *] guess d1 ..
note d1 = conjunctD2[OF this,rule_format]
def d ≡ "λx. ball x w ∩ d1 x"
have "gauge d"
unfolding d_def using w(1) d1 by auto
note this[unfolded gauge_def,rule_format,of c]
note conjunctD2[OF this]
from this(2)[unfolded open_contains_ball,rule_format,OF this(1)] guess k ..
note k=conjunctD2[OF this]
let ?d = "min k (c - a) / 2"
show ?thesis
apply (rule that[of ?d])
apply safe
proof -
show "?d > 0"
using k(1) using assms(2) by auto
fix t
assume as: "c - ?d < t" "t ≤ c"
let ?thesis = "norm (integral ({a .. c}) f - integral ({a .. t}) f) < e"
{
presume *: "t < c ⟹ ?thesis"
show ?thesis
apply (cases "t = c")
defer
apply (rule *)
apply (subst less_le)
using ‹e > 0› as(2)
apply auto
done
}
assume "t < c"
have "f integrable_on {a .. t}"
apply (rule integrable_subinterval_real[OF assms(1)])
using assms(2-3) as(2)
apply auto
done
from integrable_integral[OF this,unfolded has_integral_real,rule_format,OF *] guess d2 ..
note d2 = conjunctD2[OF this,rule_format]
def d3 ≡ "λx. if x ≤ t then d1 x ∩ d2 x else d1 x"
have "gauge d3"
using d2(1) d1(1) unfolding d3_def gauge_def by auto
from fine_division_exists_real[OF this, of a t] guess p . note p=this
note p'=tagged_division_ofD[OF this(1)]
have pt: "∀(x,k)∈p. x ≤ t"
proof (safe, goal_cases)
case prems: 1
from p'(2,3)[OF prems] show ?case
by auto
qed
with p(2) have "d2 fine p"
unfolding fine_def d3_def
apply safe
apply (erule_tac x="(a,b)" in ballE)+
apply auto
done
note d2_fin = d2(2)[OF conjI[OF p(1) this]]
have *: "{a .. c} ∩ {x. x ∙ 1 ≤ t} = {a .. t}" "{a .. c} ∩ {x. x ∙ 1 ≥ t} = {t .. c}"
using assms(2-3) as by (auto simp add: field_simps)
have "p ∪ {(c, {t .. c})} tagged_division_of {a .. c} ∧ d1 fine p ∪ {(c, {t .. c})}"
apply rule
apply (rule tagged_division_union_interval_real[of _ _ _ 1 "t"])
unfolding *
apply (rule p)
apply (rule tagged_division_of_self_real)
unfolding fine_def
apply safe
proof -
fix x k y
assume "(x,k) ∈ p" and "y ∈ k"
then show "y ∈ d1 x"
using p(2) pt
unfolding fine_def d3_def
apply -
apply (erule_tac x="(x,k)" in ballE)+
apply auto
done
next
fix x assume "x ∈ {t..c}"
then have "dist c x < k"
unfolding dist_real_def
using as(1)
by (auto simp add: field_simps)
then show "x ∈ d1 c"
using k(2)
unfolding d_def
by auto
qed (insert as(2), auto) note d1_fin = d1(2)[OF this]
have *: "integral {a .. c} f - integral {a .. t} f = -(((c - t) *⇩R f c + (∑(x, k)∈p. content k *⇩R f x)) -
integral {a .. c} f) + ((∑(x, k)∈p. content k *⇩R f x) - integral {a .. t} f) + (c - t) *⇩R f c"
"e = (e/3 + e/3) + e/3"
by auto
have **: "(∑(x, k)∈p ∪ {(c, {t .. c})}. content k *⇩R f x) =
(c - t) *⇩R f c + (∑(x, k)∈p. content k *⇩R f x)"
proof -
have **: "⋀x F. F ∪ {x} = insert x F"
by auto
have "(c, cbox t c) ∉ p"
proof (safe, goal_cases)
case prems: 1
from p'(2-3)[OF prems] have "c ∈ cbox a t"
by auto
then show False using ‹t < c›
by auto
qed
then show ?thesis
unfolding ** box_real
apply -
apply (subst setsum.insert)
apply (rule p')
unfolding split_conv
defer
apply (subst content_real)
using as(2)
apply auto
done
qed
have ***: "c - w < t ∧ t < c"
proof -
have "c - k < t"
using ‹k>0› as(1) by (auto simp add: field_simps)
moreover have "k ≤ w"
apply (rule ccontr)
using k(2)
unfolding subset_eq
apply (erule_tac x="c + ((k + w)/2)" in ballE)
unfolding d_def
using ‹k > 0› ‹w > 0›
apply (auto simp add: field_simps not_le not_less dist_real_def)
done
ultimately show ?thesis using ‹t < c›
by (auto simp add: field_simps)
qed
show ?thesis
unfolding *(1)
apply (subst *(2))
apply (rule norm_triangle_lt add_strict_mono)+
unfolding norm_minus_cancel
apply (rule d1_fin[unfolded **])
apply (rule d2_fin)
using w(2)[OF ***]
unfolding norm_scaleR
apply (auto simp add: field_simps)
done
qed
qed
lemma indefinite_integral_continuous_right:
fixes f :: "real ⇒ 'a::banach"
assumes "f integrable_on {a .. b}"
and "a ≤ c"
and "c < b"
and "e > 0"
obtains d where "0 < d"
and "∀t. c ≤ t ∧ t < c + d ⟶ norm (integral {a .. c} f - integral {a .. t} f) < e"
proof -
have *: "(λx. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c ≤ - a"
using assms by auto
from indefinite_integral_continuous_left[OF * ‹e>0›] guess d . note d = this
let ?d = "min d (b - c)"
show ?thesis
apply (rule that[of "?d"])
apply safe
proof -
show "0 < ?d"
using d(1) assms(3) by auto
fix t :: real
assume as: "c ≤ t" "t < c + ?d"
have *: "integral {a .. c} f = integral {a .. b} f - integral {c .. b} f"
"integral {a .. t} f = integral {a .. b} f - integral {t .. b} f"
unfolding algebra_simps
apply (rule_tac[!] integral_combine)
using assms as
apply auto
done
have "(- c) - d < (- t) ∧ - t ≤ - c"
using as by auto note d(2)[rule_format,OF this]
then show "norm (integral {a .. c} f - integral {a .. t} f) < e"
unfolding *
unfolding integral_reflect
apply (subst norm_minus_commute)
apply (auto simp add: algebra_simps)
done
qed
qed
lemma indefinite_integral_continuous:
fixes f :: "real ⇒ 'a::banach"
assumes "f integrable_on {a .. b}"
shows "continuous_on {a .. b} (λx. integral {a .. x} f)"
proof (unfold continuous_on_iff, safe)
fix x e :: real
assume as: "x ∈ {a .. b}" "e > 0"
let ?thesis = "∃d>0. ∀x'∈{a .. b}. dist x' x < d ⟶ dist (integral {a .. x'} f) (integral {a .. x} f) < e"
{
presume *: "a < b ⟹ ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
proof goal_cases
case 1
then have "cbox a b = {x}"
using as(1)
apply -
apply (rule set_eqI)
apply auto
done
then show ?case using ‹e > 0› by auto
qed
}
assume "a < b"
have "(x = a ∨ x = b) ∨ (a < x ∧ x < b)"
using as(1) by auto
then show ?thesis
apply (elim disjE)
proof -
assume "x = a"
have "a ≤ a" ..
from indefinite_integral_continuous_right[OF assms(1) this ‹a<b› ‹e>0›] guess d . note d=this
show ?thesis
apply rule
apply rule
apply (rule d)
apply safe
apply (subst dist_commute)
unfolding ‹x = a› dist_norm
apply (rule d(2)[rule_format])
apply auto
done
next
assume "x = b"
have "b ≤ b" ..
from indefinite_integral_continuous_left[OF assms(1) ‹a<b› this ‹e>0›] guess d . note d=this
show ?thesis
apply rule
apply rule
apply (rule d)
apply safe
apply (subst dist_commute)
unfolding ‹x = b› dist_norm
apply (rule d(2)[rule_format])
apply auto
done
next
assume "a < x ∧ x < b"
then have xl: "a < x" "x ≤ b" and xr: "a ≤ x" "x < b"
by auto
from indefinite_integral_continuous_left [OF assms(1) xl ‹e>0›] guess d1 . note d1=this
from indefinite_integral_continuous_right[OF assms(1) xr ‹e>0›] guess d2 . note d2=this
show ?thesis
apply (rule_tac x="min d1 d2" in exI)
proof safe
show "0 < min d1 d2"
using d1 d2 by auto
fix y
assume "y ∈ {a .. b}" and "dist y x < min d1 d2"
then show "dist (integral {a .. y} f) (integral {a .. x} f) < e"
apply (subst dist_commute)
apply (cases "y < x")
unfolding dist_norm
apply (rule d1(2)[rule_format])
defer
apply (rule d2(2)[rule_format])
unfolding not_less
apply (auto simp add: field_simps)
done
qed
qed
qed
subsection ‹This doesn't directly involve integration, but that gives an easy proof.›
lemma has_derivative_zero_unique_strong_interval:
fixes f :: "real ⇒ 'a::banach"
assumes "finite k"
and "continuous_on {a .. b} f"
and "f a = y"
and "∀x∈({a .. b} - k). (f has_derivative (λh. 0)) (at x within {a .. b})" "x ∈ {a .. b}"
shows "f x = y"
proof -
have ab: "a ≤ b"
using assms by auto
have *: "a ≤ x"
using assms(5) by auto
have "((λx. 0::'a) has_integral f x - f a) {a .. x}"
apply (rule fundamental_theorem_of_calculus_interior_strong[OF assms(1) *])
apply (rule continuous_on_subset[OF assms(2)])
defer
apply safe
unfolding has_vector_derivative_def
apply (subst has_derivative_within_open[symmetric])
apply assumption
apply (rule open_greaterThanLessThan)
apply (rule has_derivative_within_subset[where s="{a .. b}"])
using assms(4) assms(5)
apply (auto simp: mem_box)
done
note this[unfolded *]
note has_integral_unique[OF has_integral_0 this]
then show ?thesis
unfolding assms by auto
qed
subsection ‹Generalize a bit to any convex set.›
lemma has_derivative_zero_unique_strong_convex:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
assumes "convex s"
and "finite k"
and "continuous_on s f"
and "c ∈ s"
and "f c = y"
and "∀x∈(s - k). (f has_derivative (λh. 0)) (at x within s)"
and "x ∈ s"
shows "f x = y"
proof -
{
presume *: "x ≠ c ⟹ ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
unfolding assms(5)[symmetric]
apply auto
done
}
assume "x ≠ c"
note conv = assms(1)[unfolded convex_alt,rule_format]
have as1: "continuous_on {0 ..1} (f ∘ (λt. (1 - t) *⇩R c + t *⇩R x))"
apply (rule continuous_intros)+
apply (rule continuous_on_subset[OF assms(3)])
apply safe
apply (rule conv)
using assms(4,7)
apply auto
done
have *: "t = xa" if "(1 - t) *⇩R c + t *⇩R x = (1 - xa) *⇩R c + xa *⇩R x" for t xa
proof -
from that have "(t - xa) *⇩R x = (t - xa) *⇩R c"
unfolding scaleR_simps by (auto simp add: algebra_simps)
then show ?thesis
using ‹x ≠ c› by auto
qed
have as2: "finite {t. ((1 - t) *⇩R c + t *⇩R x) ∈ k}"
using assms(2)
apply (rule finite_surj[where f="λz. SOME t. (1-t) *⇩R c + t *⇩R x = z"])
apply safe
unfolding image_iff
apply rule
defer
apply assumption
apply (rule sym)
apply (rule some_equality)
defer
apply (drule *)
apply auto
done
have "(f ∘ (λt. (1 - t) *⇩R c + t *⇩R x)) 1 = y"
apply (rule has_derivative_zero_unique_strong_interval[OF as2 as1, of ])
unfolding o_def
using assms(5)
defer
apply -
apply rule
proof -
fix t
assume as: "t ∈ {0 .. 1} - {t. (1 - t) *⇩R c + t *⇩R x ∈ k}"
have *: "c - t *⇩R c + t *⇩R x ∈ s - k"
apply safe
apply (rule conv[unfolded scaleR_simps])
using ‹x ∈ s› ‹c ∈ s› as
by (auto simp add: algebra_simps)
have "(f ∘ (λt. (1 - t) *⇩R c + t *⇩R x) has_derivative (λx. 0) ∘ (λz. (0 - z *⇩R c) + z *⇩R x))
(at t within {0 .. 1})"
apply (intro derivative_eq_intros)
apply simp_all
apply (simp add: field_simps)
unfolding scaleR_simps
apply (rule has_derivative_within_subset,rule assms(6)[rule_format])
apply (rule *)
apply safe
apply (rule conv[unfolded scaleR_simps])
using ‹x ∈ s› ‹c ∈ s›
apply auto
done
then show "((λxa. f ((1 - xa) *⇩R c + xa *⇩R x)) has_derivative (λh. 0)) (at t within {0 .. 1})"
unfolding o_def .
qed auto
then show ?thesis
by auto
qed
text ‹Also to any open connected set with finite set of exceptions. Could
generalize to locally convex set with limpt-free set of exceptions.›
lemma has_derivative_zero_unique_strong_connected:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
assumes "connected s"
and "open s"
and "finite k"
and "continuous_on s f"
and "c ∈ s"
and "f c = y"
and "∀x∈(s - k). (f has_derivative (λh. 0)) (at x within s)"
and "x∈s"
shows "f x = y"
proof -
have "{x ∈ s. f x ∈ {y}} = {} ∨ {x ∈ s. f x ∈ {y}} = s"
apply (rule assms(1)[unfolded connected_clopen,rule_format])
apply rule
defer
apply (rule continuous_closedin_preimage[OF assms(4) closed_singleton])
apply (rule open_openin_trans[OF assms(2)])
unfolding open_contains_ball
proof safe
fix x
assume "x ∈ s"
from assms(2)[unfolded open_contains_ball,rule_format,OF this] guess e .. note e=conjunctD2[OF this]
show "∃e>0. ball x e ⊆ {xa ∈ s. f xa ∈ {f x}}"
apply rule
apply rule
apply (rule e)
proof safe
fix y
assume y: "y ∈ ball x e"
then show "y ∈ s"
using e by auto
show "f y = f x"
apply (rule has_derivative_zero_unique_strong_convex[OF convex_ball])
apply (rule assms)
apply (rule continuous_on_subset)
apply (rule assms)
apply (rule e)+
apply (subst centre_in_ball)
apply (rule e)
apply rule
apply safe
apply (rule has_derivative_within_subset)
apply (rule assms(7)[rule_format])
using y e
apply auto
done
qed
qed
then show ?thesis
using ‹x ∈ s› ‹f c = y› ‹c ∈ s› by auto
qed
lemma has_derivative_zero_connected_constant:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
assumes "connected s"
and "open s"
and "finite k"
and "continuous_on s f"
and "∀x∈(s - k). (f has_derivative (λh. 0)) (at x within s)"
obtains c where "⋀x. x ∈ s ⟹ f(x) = c"
proof (cases "s = {}")
case True
then show ?thesis
by (metis empty_iff that)
next
case False
then obtain c where "c ∈ s"
by (metis equals0I)
then show ?thesis
by (metis has_derivative_zero_unique_strong_connected assms that)
qed
subsection ‹Integrating characteristic function of an interval›
lemma has_integral_restrict_open_subinterval:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
assumes "(f has_integral i) (cbox c d)"
and "cbox c d ⊆ cbox a b"
shows "((λx. if x ∈ box c d then f x else 0) has_integral i) (cbox a b)"
proof -
def g ≡ "λx. if x ∈box c d then f x else 0"
{
presume *: "cbox c d ≠ {} ⟹ ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
proof goal_cases
case prems: 1
then have *: "box c d = {}"
by (metis bot.extremum_uniqueI box_subset_cbox)
show ?thesis
using assms(1)
unfolding *
using prems
by auto
qed
}
assume "cbox c d ≠ {}"
from partial_division_extend_1[OF assms(2) this] guess p . note p=this
note mon = monoidal_lifted[OF monoidal_monoid]
note operat = operative_division[OF this operative_integral p(1), symmetric]
let ?P = "(if g integrable_on cbox a b then Some (integral (cbox a b) g) else None) = Some i"
{
presume "?P"
then have "g integrable_on cbox a b ∧ integral (cbox a b) g = i"
apply -
apply cases
apply (subst(asm) if_P)
apply assumption
apply auto
done
then show ?thesis
using integrable_integral
unfolding g_def
by auto
}
note iterate_eq_neutral[OF mon,unfolded neutral_lifted[OF monoidal_monoid]]
note * = this[unfolded neutral_add]
have iterate:"iterate (lifted op +) (p - {cbox c d})
(λi. if g integrable_on i then Some (integral i g) else None) = Some 0"
proof (rule *)
fix x
assume x: "x ∈ p - {cbox c d}"
then have "x ∈ p"
by auto
note div = division_ofD(2-5)[OF p(1) this]
from div(3) guess u v by (elim exE) note uv=this
have "interior x ∩ interior (cbox c d) = {}"
using div(4)[OF p(2)] x by auto
then have "(g has_integral 0) x"
unfolding uv
apply -
apply (rule has_integral_spike_interior[where f="λx. 0"])
unfolding g_def interior_cbox
apply auto
done
then show "(if g integrable_on x then Some (integral x g) else None) = Some 0"
by auto
qed
have *: "p = insert (cbox c d) (p - {cbox c d})"
using p by auto
have **: "g integrable_on cbox c d"
apply (rule integrable_spike_interior[where f=f])
unfolding g_def using assms(1)
apply auto
done
moreover
have "integral (cbox c d) g = i"
apply (rule has_integral_unique[OF _ assms(1)])
apply (rule has_integral_spike_interior[where f=g])
defer
apply (rule integrable_integral[OF **])
unfolding g_def
apply auto
done
ultimately show ?P
unfolding operat
apply (subst *)
apply (subst iterate_insert)
apply rule+
unfolding iterate
defer
apply (subst if_not_P)
defer
using p
apply auto
done
qed
lemma has_integral_restrict_closed_subinterval:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
assumes "(f has_integral i) (cbox c d)"
and "cbox c d ⊆ cbox a b"
shows "((λx. if x ∈ cbox c d then f x else 0) has_integral i) (cbox a b)"
proof -
note has_integral_restrict_open_subinterval[OF assms]
note * = has_integral_spike[OF negligible_frontier_interval _ this]
show ?thesis
apply (rule *[of c d])
using box_subset_cbox[of c d]
apply auto
done
qed
lemma has_integral_restrict_closed_subintervals_eq:
fixes f :: "'a::euclidean_space ⇒ 'b::banach"
assumes "cbox c d ⊆ cbox a b"
shows "((λx. if x ∈ cbox c d then f x else 0) has_integral i) (cbox a b) ⟷ (f has_integral i) (cbox c d)"
(is "?l = ?r")
proof (cases "cbox c d = {}")
case False
let ?g = "λx. if x ∈ cbox c d then f x else 0"
show ?thesis
apply rule
defer
apply (rule has_integral_restrict_closed_subinterval[OF _ assms])
apply assumption
proof -
assume ?l
then have "?g integrable_on cbox c d"
using assms has_integral_integrable integrable_subinterval by blast
then have *: "f integrable_on cbox c d"
apply -
apply (rule integrable_eq)
apply auto
done
then have "i = integral (cbox c d) f"
apply -
apply (rule has_integral_unique)
apply (rule ‹?l›)
apply (rule has_integral_restrict_closed_subinterval[OF _ assms])
apply auto
done
then show ?r
using * by auto
qed
qed auto
text ‹Hence we can apply the limit process uniformly to all integrals.›
lemma has_integral':
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
shows "(f has_integral i) s ⟷
(∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
(∃z. ((λx. if x ∈ s then f(x) else 0) has_integral z) (cbox a b) ∧ norm(z - i) < e))"
(is "?l ⟷ (∀e>0. ?r e)")
proof -
{
presume *: "∃a b. s = cbox a b ⟹ ?thesis"
show ?thesis
apply cases
apply (rule *)
apply assumption
apply (subst has_integral_alt)
apply auto
done
}
assume "∃a b. s = cbox a b"
then guess a b by (elim exE) note s=this
from bounded_cbox[of a b, unfolded bounded_pos] guess B ..
note B = conjunctD2[OF this,rule_format] show ?thesis
apply safe
proof -
fix e :: real
assume ?l and "e > 0"
show "?r e"
apply (rule_tac x="B+1" in exI)
apply safe
defer
apply (rule_tac x=i in exI)
proof
fix c d :: 'n
assume as: "ball 0 (B+1) ⊆ cbox c d"
then show "((λx. if x ∈ s then f x else 0) has_integral i) (cbox c d)"
unfolding s
apply -
apply (rule has_integral_restrict_closed_subinterval)
apply (rule ‹?l›[unfolded s])
apply safe
apply (drule B(2)[rule_format])
unfolding subset_eq
apply (erule_tac x=x in ballE)
apply (auto simp add: dist_norm)
done
qed (insert B ‹e>0›, auto)
next
assume as: "∀e>0. ?r e"
from this[rule_format,OF zero_less_one] guess C .. note C=conjunctD2[OF this,rule_format]
def c ≡ "(∑i∈Basis. (- max B C) *⇩R i)::'n"
def d ≡ "(∑i∈Basis. max B C *⇩R i)::'n"
have c_d: "cbox a b ⊆ cbox c d"
apply safe
apply (drule B(2))
unfolding mem_box
proof
fix x i
show "c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i" if "norm x ≤ B" and "i ∈ Basis"
using that and Basis_le_norm[OF ‹i∈Basis›, of x]
unfolding c_def d_def
by (auto simp add: field_simps setsum_negf)
qed
have "ball 0 C ⊆ cbox c d"
apply (rule subsetI)
unfolding mem_box mem_ball dist_norm
proof
fix x i :: 'n
assume x: "norm (0 - x) < C" and i: "i ∈ Basis"
show "c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i"
using Basis_le_norm[OF i, of x] and x i
unfolding c_def d_def
by (auto simp: setsum_negf)
qed
from C(2)[OF this] have "∃y. (f has_integral y) (cbox a b)"
unfolding has_integral_restrict_closed_subintervals_eq[OF c_d,symmetric]
unfolding s
by auto
then guess y .. note y=this
have "y = i"
proof (rule ccontr)
assume "¬ ?thesis"
then have "0 < norm (y - i)"
by auto
from as[rule_format,OF this] guess C .. note C=conjunctD2[OF this,rule_format]
def c ≡ "(∑i∈Basis. (- max B C) *⇩R i)::'n"
def d ≡ "(∑i∈Basis. max B C *⇩R i)::'n"
have c_d: "cbox a b ⊆ cbox c d"
apply safe
apply (drule B(2))
unfolding mem_box
proof
fix x i :: 'n
assume "norm x ≤ B" and "i ∈ Basis"
then show "c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i"
using Basis_le_norm[of i x]
unfolding c_def d_def
by (auto simp add: field_simps setsum_negf)
qed
have "ball 0 C ⊆ cbox c d"
apply (rule subsetI)
unfolding mem_box mem_ball dist_norm
proof
fix x i :: 'n
assume "norm (0 - x) < C" and "i ∈ Basis"
then show "c ∙ i ≤ x ∙ i ∧ x ∙ i ≤ d ∙ i"
using Basis_le_norm[of i x]
unfolding c_def d_def
by (auto simp: setsum_negf)
qed
note C(2)[OF this] then guess z .. note z = conjunctD2[OF this, unfolded s]
note this[unfolded has_integral_restrict_closed_subintervals_eq[OF c_d]]
then have "z = y" and "norm (z - i) < norm (y - i)"
apply -
apply (rule has_integral_unique[OF _ y(1)])
apply assumption
apply assumption
done
then show False
by auto
qed
then show ?l
using y
unfolding s
by auto
qed
qed
lemma has_integral_le:
fixes f :: "'n::euclidean_space ⇒ real"
assumes "(f has_integral i) s"
and "(g has_integral j) s"
and "∀x∈s. f x ≤ g x"
shows "i ≤ j"
using has_integral_component_le[OF _ assms(1-2), of 1]
using assms(3)
by auto
lemma integral_le:
fixes f :: "'n::euclidean_space ⇒ real"
assumes "f integrable_on s"
and "g integrable_on s"
and "∀x∈s. f x ≤ g x"
shows "integral s f ≤ integral s g"
by (rule has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)])
lemma has_integral_nonneg:
fixes f :: "'n::euclidean_space ⇒ real"
assumes "(f has_integral i) s"
and "∀x∈s. 0 ≤ f x"
shows "0 ≤ i"
using has_integral_component_nonneg[of 1 f i s]
unfolding o_def
using assms
by auto
lemma integral_nonneg:
fixes f :: "'n::euclidean_space ⇒ real"
assumes "f integrable_on s"
and "∀x∈s. 0 ≤ f x"
shows "0 ≤ integral s f"
by (rule has_integral_nonneg[OF assms(1)[unfolded has_integral_integral] assms(2)])
text ‹Hence a general restriction property.›
lemma has_integral_restrict[simp]:
assumes "s ⊆ t"
shows "((λx. if x ∈ s then f x else (0::'a::banach)) has_integral i) t ⟷ (f has_integral i) s"
proof -
have *: "⋀x. (if x ∈ t then if x ∈ s then f x else 0 else 0) = (if x∈s then f x else 0)"
using assms by auto
show ?thesis
apply (subst(2) has_integral')
apply (subst has_integral')
unfolding *
apply rule
done
qed
lemma has_integral_restrict_univ:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
shows "((λx. if x ∈ s then f x else 0) has_integral i) UNIV ⟷ (f has_integral i) s"
by auto
lemma has_integral_on_superset:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "∀x. x ∉ s ⟶ f x = 0"
and "s ⊆ t"
and "(f has_integral i) s"
shows "(f has_integral i) t"
proof -
have "(λx. if x ∈ s then f x else 0) = (λx. if x ∈ t then f x else 0)"
apply rule
using assms(1-2)
apply auto
done
then show ?thesis
using assms(3)
apply (subst has_integral_restrict_univ[symmetric])
apply (subst(asm) has_integral_restrict_univ[symmetric])
apply auto
done
qed
lemma integrable_on_superset:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "∀x. x ∉ s ⟶ f x = 0"
and "s ⊆ t"
and "f integrable_on s"
shows "f integrable_on t"
using assms
unfolding integrable_on_def
by (auto intro:has_integral_on_superset)
lemma integral_restrict_univ[intro]:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
shows "f integrable_on s ⟹ integral UNIV (λx. if x ∈ s then f x else 0) = integral s f"
apply (rule integral_unique)
unfolding has_integral_restrict_univ
apply auto
done
lemma integrable_restrict_univ:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
shows "(λx. if x ∈ s then f x else 0) integrable_on UNIV ⟷ f integrable_on s"
unfolding integrable_on_def
by auto
lemma negligible_on_intervals: "negligible s ⟷ (∀a b. negligible(s ∩ cbox a b))" (is "?l ⟷ ?r")
proof
assume ?r
show ?l
unfolding negligible_def
proof safe
fix a b
show "(indicator s has_integral 0) (cbox a b)"
apply (rule has_integral_negligible[OF ‹?r›[rule_format,of a b]])
unfolding indicator_def
apply auto
done
qed
qed auto
lemma has_integral_spike_set_eq:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "negligible ((s - t) ∪ (t - s))"
shows "(f has_integral y) s ⟷ (f has_integral y) t"
unfolding has_integral_restrict_univ[symmetric,of f]
apply (rule has_integral_spike_eq[OF assms])
by (auto split: if_split_asm)
lemma has_integral_spike_set[dest]:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "negligible ((s - t) ∪ (t - s))"
and "(f has_integral y) s"
shows "(f has_integral y) t"
using assms has_integral_spike_set_eq
by auto
lemma integrable_spike_set[dest]:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "negligible ((s - t) ∪ (t - s))"
and "f integrable_on s"
shows "f integrable_on t"
using assms(2)
unfolding integrable_on_def
unfolding has_integral_spike_set_eq[OF assms(1)] .
lemma integrable_spike_set_eq:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "negligible ((s - t) ∪ (t - s))"
shows "f integrable_on s ⟷ f integrable_on t"
apply rule
apply (rule_tac[!] integrable_spike_set)
using assms
apply auto
done
subsection ‹More lemmas that are useful later›
lemma has_integral_subset_component_le:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes k: "k ∈ Basis"
and as: "s ⊆ t" "(f has_integral i) s" "(f has_integral j) t" "∀x∈t. 0 ≤ f(x)∙k"
shows "i∙k ≤ j∙k"
proof -
note has_integral_restrict_univ[symmetric, of f]
note as(2-3)[unfolded this] note * = has_integral_component_le[OF k this]
show ?thesis
apply (rule *)
using as(1,4)
apply auto
done
qed
lemma has_integral_subset_le:
fixes f :: "'n::euclidean_space ⇒ real"
assumes "s ⊆ t"
and "(f has_integral i) s"
and "(f has_integral j) t"
and "∀x∈t. 0 ≤ f x"
shows "i ≤ j"
using has_integral_subset_component_le[OF _ assms(1), of 1 f i j]
using assms
by auto
lemma integral_subset_component_le:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "k ∈ Basis"
and "s ⊆ t"
and "f integrable_on s"
and "f integrable_on t"
and "∀x ∈ t. 0 ≤ f x ∙ k"
shows "(integral s f)∙k ≤ (integral t f)∙k"
apply (rule has_integral_subset_component_le)
using assms
apply auto
done
lemma integral_subset_le:
fixes f :: "'n::euclidean_space ⇒ real"
assumes "s ⊆ t"
and "f integrable_on s"
and "f integrable_on t"
and "∀x ∈ t. 0 ≤ f x"
shows "integral s f ≤ integral t f"
apply (rule has_integral_subset_le)
using assms
apply auto
done
lemma has_integral_alt':
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
shows "(f has_integral i) s ⟷ (∀a b. (λx. if x ∈ s then f x else 0) integrable_on cbox a b) ∧
(∀e>0. ∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) - i) < e)"
(is "?l = ?r")
proof
assume ?r
show ?l
apply (subst has_integral')
apply safe
proof goal_cases
case (1 e)
from ‹?r›[THEN conjunct2,rule_format,OF this] guess B .. note B=conjunctD2[OF this]
show ?case
apply rule
apply rule
apply (rule B)
apply safe
apply (rule_tac x="integral (cbox a b) (λx. if x ∈ s then f x else 0)" in exI)
apply (drule B(2)[rule_format])
using integrable_integral[OF ‹?r›[THEN conjunct1,rule_format]]
apply auto
done
qed
next
assume ?l note as = this[unfolded has_integral'[of f],rule_format]
let ?f = "λx. if x ∈ s then f x else 0"
show ?r
proof safe
fix a b :: 'n
from as[OF zero_less_one] guess B .. note B=conjunctD2[OF this,rule_format]
let ?a = "∑i∈Basis. min (a∙i) (-B) *⇩R i::'n"
let ?b = "∑i∈Basis. max (b∙i) B *⇩R i::'n"
show "?f integrable_on cbox a b"
proof (rule integrable_subinterval[of _ ?a ?b])
have "ball 0 B ⊆ cbox ?a ?b"
apply (rule subsetI)
unfolding mem_ball mem_box dist_norm
proof (rule, goal_cases)
case (1 x i)
then show ?case using Basis_le_norm[of i x]
by (auto simp add:field_simps)
qed
from B(2)[OF this] guess z .. note conjunct1[OF this]
then show "?f integrable_on cbox ?a ?b"
unfolding integrable_on_def by auto
show "cbox a b ⊆ cbox ?a ?b"
apply safe
unfolding mem_box
apply rule
apply (erule_tac x=i in ballE)
apply auto
done
qed
fix e :: real
assume "e > 0"
from as[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
show "∃B>0. ∀a b. ball 0 B ⊆ cbox a b ⟶
norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) - i) < e"
apply rule
apply rule
apply (rule B)
apply safe
proof goal_cases
case 1
from B(2)[OF this] guess z .. note z=conjunctD2[OF this]
from integral_unique[OF this(1)] show ?case
using z(2) by auto
qed
qed
qed
subsection ‹Continuity of the integral (for a 1-dimensional interval).›
lemma integrable_alt:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
shows "f integrable_on s ⟷
(∀a b. (λx. if x ∈ s then f x else 0) integrable_on cbox a b) ∧
(∀e>0. ∃B>0. ∀a b c d. ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d ⟶
norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) -
integral (cbox c d) (λx. if x ∈ s then f x else 0)) < e)"
(is "?l = ?r")
proof
assume ?l
then guess y unfolding integrable_on_def .. note this[unfolded has_integral_alt'[of f]]
note y=conjunctD2[OF this,rule_format]
show ?r
apply safe
apply (rule y)
proof goal_cases
case (1 e)
then have "e/2 > 0"
by auto
from y(2)[OF this] guess B .. note B=conjunctD2[OF this,rule_format]
show ?case
apply rule
apply rule
apply (rule B)
apply safe
proof goal_cases
case prems: (1 a b c d)
show ?case
apply (rule norm_triangle_half_l)
using B(2)[OF prems(1)] B(2)[OF prems(2)]
apply auto
done
qed
qed
next
assume ?r
note as = conjunctD2[OF this,rule_format]
let ?cube = "λn. cbox (∑i∈Basis. - real n *⇩R i::'n) (∑i∈Basis. real n *⇩R i)"
have "Cauchy (λn. integral (?cube n) (λx. if x ∈ s then f x else 0))"
proof (unfold Cauchy_def, safe, goal_cases)
case (1 e)
from as(2)[OF this] guess B .. note B = conjunctD2[OF this,rule_format]
from real_arch_simple[of B] guess N .. note N = this
{
fix n
assume n: "n ≥ N"
have "ball 0 B ⊆ ?cube n"
apply (rule subsetI)
unfolding mem_ball mem_box dist_norm
proof (rule, goal_cases)
case (1 x i)
then show ?case
using Basis_le_norm[of i x] ‹i∈Basis›
using n N
by (auto simp add: field_simps setsum_negf)
qed
}
then show ?case
apply -
apply (rule_tac x=N in exI)
apply safe
unfolding dist_norm
apply (rule B(2))
apply auto
done
qed
from this[unfolded convergent_eq_cauchy[symmetric]] guess i ..
note i = this[THEN LIMSEQ_D]
show ?l unfolding integrable_on_def has_integral_alt'[of f]
apply (rule_tac x=i in exI)
apply safe
apply (rule as(1)[unfolded integrable_on_def])
proof goal_cases
case (1 e)
then have *: "e/2 > 0" by auto
from i[OF this] guess N .. note N =this[rule_format]
from as(2)[OF *] guess B .. note B=conjunctD2[OF this,rule_format]
let ?B = "max (real N) B"
show ?case
apply (rule_tac x="?B" in exI)
proof safe
show "0 < ?B"
using B(1) by auto
fix a b :: 'n
assume ab: "ball 0 ?B ⊆ cbox a b"
from real_arch_simple[of ?B] guess n .. note n=this
show "norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) - i) < e"
apply (rule norm_triangle_half_l)
apply (rule B(2))
defer
apply (subst norm_minus_commute)
apply (rule N[of n])
proof safe
show "N ≤ n"
using n by auto
fix x :: 'n
assume x: "x ∈ ball 0 B"
then have "x ∈ ball 0 ?B"
by auto
then show "x ∈ cbox a b"
using ab by blast
show "x ∈ ?cube n"
using x
unfolding mem_box mem_ball dist_norm
apply -
proof (rule, goal_cases)
case (1 i)
then show ?case
using Basis_le_norm[of i x] ‹i ∈ Basis›
using n
by (auto simp add: field_simps setsum_negf)
qed
qed
qed
qed
qed
lemma integrable_altD:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on s"
shows "⋀a b. (λx. if x ∈ s then f x else 0) integrable_on cbox a b"
and "⋀e. e > 0 ⟹ ∃B>0. ∀a b c d. ball 0 B ⊆ cbox a b ∧ ball 0 B ⊆ cbox c d ⟶
norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) - integral (cbox c d) (λx. if x ∈ s then f x else 0)) < e"
using assms[unfolded integrable_alt[of f]] by auto
lemma integrable_on_subcbox:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on s"
and "cbox a b ⊆ s"
shows "f integrable_on cbox a b"
apply (rule integrable_eq)
defer
apply (rule integrable_altD(1)[OF assms(1)])
using assms(2)
apply auto
done
subsection ‹A straddling criterion for integrability›
lemma integrable_straddle_interval:
fixes f :: "'n::euclidean_space ⇒ real"
assumes "∀e>0. ∃g h i j. (g has_integral i) (cbox a b) ∧ (h has_integral j) (cbox a b) ∧
norm (i - j) < e ∧ (∀x∈cbox a b. (g x) ≤ f x ∧ f x ≤ h x)"
shows "f integrable_on cbox a b"
proof (subst integrable_cauchy, safe, goal_cases)
case (1 e)
then have e: "e/3 > 0"
by auto
note assms[rule_format,OF this]
then guess g h i j by (elim exE conjE) note obt = this
from obt(1)[unfolded has_integral[of g], rule_format, OF e] guess d1 .. note d1=conjunctD2[OF this,rule_format]
from obt(2)[unfolded has_integral[of h], rule_format, OF e] guess d2 .. note d2=conjunctD2[OF this,rule_format]
show ?case
apply (rule_tac x="λx. d1 x ∩ d2 x" in exI)
apply (rule conjI gauge_inter d1 d2)+
unfolding fine_inter
proof (safe, goal_cases)
have **: "⋀i j g1 g2 h1 h2 f1 f2. g1 - h2 ≤ f1 - f2 ⟹ f1 - f2 ≤ h1 - g2 ⟹
¦i - j¦ < e / 3 ⟹ ¦g2 - i¦ < e / 3 ⟹ ¦g1 - i¦ < e / 3 ⟹
¦h2 - j¦ < e / 3 ⟹ ¦h1 - j¦ < e / 3 ⟹ ¦f1 - f2¦ < e"
using ‹e > 0› by arith
case prems: (1 p1 p2)
note tagged_division_ofD(2-4) note * = this[OF prems(1)] this[OF prems(4)]
have "(∑(x, k)∈p1. content k *⇩R f x) - (∑(x, k)∈p1. content k *⇩R g x) ≥ 0"
and "0 ≤ (∑(x, k)∈p2. content k *⇩R h x) - (∑(x, k)∈p2. content k *⇩R f x)"
and "(∑(x, k)∈p2. content k *⇩R f x) - (∑(x, k)∈p2. content k *⇩R g x) ≥ 0"
and "0 ≤ (∑(x, k)∈p1. content k *⇩R h x) - (∑(x, k)∈p1. content k *⇩R f x)"
unfolding setsum_subtractf[symmetric]
apply -
apply (rule_tac[!] setsum_nonneg)
apply safe
unfolding real_scaleR_def right_diff_distrib[symmetric]
apply (rule_tac[!] mult_nonneg_nonneg)
proof -
fix a b
assume ab: "(a, b) ∈ p1"
show "0 ≤ content b"
using *(3)[OF ab]
apply safe
apply (rule content_pos_le)
done
then show "0 ≤ content b" .
show "0 ≤ f a - g a" "0 ≤ h a - f a"
using *(1-2)[OF ab]
using obt(4)[rule_format,of a]
by auto
next
fix a b
assume ab: "(a, b) ∈ p2"
show "0 ≤ content b"
using *(6)[OF ab]
apply safe
apply (rule content_pos_le)
done
then show "0 ≤ content b" .
show "0 ≤ f a - g a" and "0 ≤ h a - f a"
using *(4-5)[OF ab] using obt(4)[rule_format,of a] by auto
qed
then show ?case
apply -
unfolding real_norm_def
apply (rule **)
defer
defer
unfolding real_norm_def[symmetric]
apply (rule obt(3))
apply (rule d1(2)[OF conjI[OF prems(4,5)]])
apply (rule d1(2)[OF conjI[OF prems(1,2)]])
apply (rule d2(2)[OF conjI[OF prems(4,6)]])
apply (rule d2(2)[OF conjI[OF prems(1,3)]])
apply auto
done
qed
qed
lemma integrable_straddle:
fixes f :: "'n::euclidean_space ⇒ real"
assumes "∀e>0. ∃g h i j. (g has_integral i) s ∧ (h has_integral j) s ∧
norm (i - j) < e ∧ (∀x∈s. g x ≤ f x ∧ f x ≤ h x)"
shows "f integrable_on s"
proof -
have "⋀a b. (λx. if x ∈ s then f x else 0) integrable_on cbox a b"
proof (rule integrable_straddle_interval, safe, goal_cases)
case (1 a b e)
then have *: "e/4 > 0"
by auto
from assms[rule_format,OF this] guess g h i j by (elim exE conjE) note obt=this
note obt(1)[unfolded has_integral_alt'[of g]]
note conjunctD2[OF this, rule_format]
note g = this(1) and this(2)[OF *]
from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
note obt(2)[unfolded has_integral_alt'[of h]]
note conjunctD2[OF this, rule_format]
note h = this(1) and this(2)[OF *]
from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
def c ≡ "∑i∈Basis. min (a∙i) (- (max B1 B2)) *⇩R i::'n"
def d ≡ "∑i∈Basis. max (b∙i) (max B1 B2) *⇩R i::'n"
have *: "ball 0 B1 ⊆ cbox c d" "ball 0 B2 ⊆ cbox c d"
apply safe
unfolding mem_ball mem_box dist_norm
apply (rule_tac[!] ballI)
proof goal_cases
case (1 x i)
then show ?case using Basis_le_norm[of i x]
unfolding c_def d_def by auto
next
case (2 x i)
then show ?case using Basis_le_norm[of i x]
unfolding c_def d_def by auto
qed
have **: "⋀ch cg ag ah::real. norm (ah - ag) ≤ norm (ch - cg) ⟹ norm (cg - i) < e / 4 ⟹
norm (ch - j) < e / 4 ⟹ norm (ag - ah) < e"
using obt(3)
unfolding real_norm_def
by arith
show ?case
apply (rule_tac x="λx. if x ∈ s then g x else 0" in exI)
apply (rule_tac x="λx. if x ∈ s then h x else 0" in exI)
apply (rule_tac x="integral (cbox a b) (λx. if x ∈ s then g x else 0)" in exI)
apply (rule_tac x="integral (cbox a b) (λx. if x ∈ s then h x else 0)" in exI)
apply safe
apply (rule_tac[1-2] integrable_integral,rule g)
apply (rule h)
apply (rule **[OF _ B1(2)[OF *(1)] B2(2)[OF *(2)]])
proof -
have *: "⋀x f g. (if x ∈ s then f x else 0) - (if x ∈ s then g x else 0) =
(if x ∈ s then f x - g x else (0::real))"
by auto
note ** = abs_of_nonneg[OF integral_nonneg[OF integrable_diff, OF h g]]
show "norm (integral (cbox a b) (λx. if x ∈ s then h x else 0) -
integral (cbox a b) (λx. if x ∈ s then g x else 0)) ≤
norm (integral (cbox c d) (λx. if x ∈ s then h x else 0) -
integral (cbox c d) (λx. if x ∈ s then g x else 0))"
unfolding integral_diff[OF h g,symmetric] real_norm_def
apply (subst **)
defer
apply (subst **)
defer
apply (rule has_integral_subset_le)
defer
apply (rule integrable_integral integrable_diff h g)+
proof safe
fix x
assume "x ∈ cbox a b"
then show "x ∈ cbox c d"
unfolding mem_box c_def d_def
apply -
apply rule
apply (erule_tac x=i in ballE)
apply auto
done
qed (insert obt(4), auto)
qed (insert obt(4), auto)
qed
note interv = this
show ?thesis
unfolding integrable_alt[of f]
apply safe
apply (rule interv)
proof goal_cases
case (1 e)
then have *: "e/3 > 0"
by auto
from assms[rule_format,OF this] guess g h i j by (elim exE conjE) note obt=this
note obt(1)[unfolded has_integral_alt'[of g]]
note conjunctD2[OF this, rule_format]
note g = this(1) and this(2)[OF *]
from this(2) guess B1 .. note B1 = conjunctD2[OF this,rule_format]
note obt(2)[unfolded has_integral_alt'[of h]]
note conjunctD2[OF this, rule_format]
note h = this(1) and this(2)[OF *]
from this(2) guess B2 .. note B2 = conjunctD2[OF this,rule_format]
show ?case
apply (rule_tac x="max B1 B2" in exI)
apply safe
apply (rule max.strict_coboundedI1)
apply (rule B1)
proof -
fix a b c d :: 'n
assume as: "ball 0 (max B1 B2) ⊆ cbox a b" "ball 0 (max B1 B2) ⊆ cbox c d"
have **: "ball 0 B1 ⊆ ball (0::'n) (max B1 B2)" "ball 0 B2 ⊆ ball (0::'n) (max B1 B2)"
by auto
have *: "⋀ga gc ha hc fa fc::real.
¦ga - i¦ < e / 3 ∧ ¦gc - i¦ < e / 3 ∧ ¦ha - j¦ < e / 3 ∧
¦hc - j¦ < e / 3 ∧ ¦i - j¦ < e / 3 ∧ ga ≤ fa ∧ fa ≤ ha ∧ gc ≤ fc ∧ fc ≤ hc ⟹
¦fa - fc¦ < e"
by (simp add: abs_real_def split: if_split_asm)
show "norm (integral (cbox a b) (λx. if x ∈ s then f x else 0) - integral (cbox c d)
(λx. if x ∈ s then f x else 0)) < e"
unfolding real_norm_def
apply (rule *)
apply safe
unfolding real_norm_def[symmetric]
apply (rule B1(2))
apply (rule order_trans)
apply (rule **)
apply (rule as(1))
apply (rule B1(2))
apply (rule order_trans)
apply (rule **)
apply (rule as(2))
apply (rule B2(2))
apply (rule order_trans)
apply (rule **)
apply (rule as(1))
apply (rule B2(2))
apply (rule order_trans)
apply (rule **)
apply (rule as(2))
apply (rule obt)
apply (rule_tac[!] integral_le)
using obt
apply (auto intro!: h g interv)
done
qed
qed
qed
subsection ‹Adding integrals over several sets›
lemma has_integral_union:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "(f has_integral i) s"
and "(f has_integral j) t"
and "negligible (s ∩ t)"
shows "(f has_integral (i + j)) (s ∪ t)"
proof -
note * = has_integral_restrict_univ[symmetric, of f]
show ?thesis
unfolding *
apply (rule has_integral_spike[OF assms(3)])
defer
apply (rule has_integral_add[OF assms(1-2)[unfolded *]])
apply auto
done
qed
lemma has_integral_unions:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "finite t"
and "∀s∈t. (f has_integral (i s)) s"
and "∀s∈t. ∀s'∈t. s ≠ s' ⟶ negligible (s ∩ s')"
shows "(f has_integral (setsum i t)) (⋃t)"
proof -
note * = has_integral_restrict_univ[symmetric, of f]
have **: "negligible (⋃((λ(a,b). a ∩ b) ` {(a,b). a ∈ t ∧ b ∈ {y. y ∈ t ∧ a ≠ y}}))"
apply (rule negligible_unions)
apply (rule finite_imageI)
apply (rule finite_subset[of _ "t × t"])
defer
apply (rule finite_cartesian_product[OF assms(1,1)])
using assms(3)
apply auto
done
note assms(2)[unfolded *]
note has_integral_setsum[OF assms(1) this]
then show ?thesis
unfolding *
apply -
apply (rule has_integral_spike[OF **])
defer
apply assumption
apply safe
proof goal_cases
case prems: (1 x)
then show ?case
proof (cases "x ∈ ⋃t")
case True
then guess s unfolding Union_iff .. note s=this
then have *: "∀b∈t. x ∈ b ⟷ b = s"
using prems(3) by blast
show ?thesis
unfolding if_P[OF True]
apply (rule trans)
defer
apply (rule setsum.cong)
apply (rule refl)
apply (subst *)
apply assumption
apply (rule refl)
unfolding setsum.delta[OF assms(1)]
using s
apply auto
done
qed auto
qed
qed
text ‹In particular adding integrals over a division, maybe not of an interval.›
lemma has_integral_combine_division:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "d division_of s"
and "∀k∈d. (f has_integral (i k)) k"
shows "(f has_integral (setsum i d)) s"
proof -
note d = division_ofD[OF assms(1)]
show ?thesis
unfolding d(6)[symmetric]
apply (rule has_integral_unions)
apply (rule d assms)+
apply rule
apply rule
apply rule
proof goal_cases
case prems: (1 s s')
from d(4)[OF this(1)] d(4)[OF this(2)] guess a c b d by (elim exE) note obt=this
from d(5)[OF prems] show ?case
unfolding obt interior_cbox
apply -
apply (rule negligible_subset[of "(cbox a b-box a b) ∪ (cbox c d-box c d)"])
apply (rule negligible_union negligible_frontier_interval)+
apply auto
done
qed
qed
lemma integral_combine_division_bottomup:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "d division_of s"
and "∀k∈d. f integrable_on k"
shows "integral s f = setsum (λi. integral i f) d"
apply (rule integral_unique)
apply (rule has_integral_combine_division[OF assms(1)])
using assms(2)
unfolding has_integral_integral
apply assumption
done
lemma has_integral_combine_division_topdown:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on s"
and "d division_of k"
and "k ⊆ s"
shows "(f has_integral (setsum (λi. integral i f) d)) k"
apply (rule has_integral_combine_division[OF assms(2)])
apply safe
unfolding has_integral_integral[symmetric]
proof goal_cases
case (1 k)
from division_ofD(2,4)[OF assms(2) this]
show ?case
apply safe
apply (rule integrable_on_subcbox)
apply (rule assms)
using assms(3)
apply auto
done
qed
lemma integral_combine_division_topdown:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on s"
and "d division_of s"
shows "integral s f = setsum (λi. integral i f) d"
apply (rule integral_unique)
apply (rule has_integral_combine_division_topdown)
using assms
apply auto
done
lemma integrable_combine_division:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "d division_of s"
and "∀i∈d. f integrable_on i"
shows "f integrable_on s"
using assms(2)
unfolding integrable_on_def
by (metis has_integral_combine_division[OF assms(1)])
lemma integrable_on_subdivision:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "d division_of i"
and "f integrable_on s"
and "i ⊆ s"
shows "f integrable_on i"
apply (rule integrable_combine_division assms)+
apply safe
proof goal_cases
case 1
note division_ofD(2,4)[OF assms(1) this]
then show ?case
apply safe
apply (rule integrable_on_subcbox[OF assms(2)])
using assms(3)
apply auto
done
qed
subsection ‹Also tagged divisions›
lemma has_integral_combine_tagged_division:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "p tagged_division_of s"
and "∀(x,k) ∈ p. (f has_integral (i k)) k"
shows "(f has_integral (setsum (λ(x,k). i k) p)) s"
proof -
have *: "(f has_integral (setsum (λk. integral k f) (snd ` p))) s"
apply (rule has_integral_combine_division)
apply (rule division_of_tagged_division[OF assms(1)])
using assms(2)
unfolding has_integral_integral[symmetric]
apply safe
apply auto
done
then show ?thesis
apply -
apply (rule subst[where P="λi. (f has_integral i) s"])
defer
apply assumption
apply (rule trans[of _ "setsum (λ(x,k). integral k f) p"])
apply (subst eq_commute)
apply (rule setsum_over_tagged_division_lemma[OF assms(1)])
apply (rule integral_null)
apply assumption
apply (rule setsum.cong)
using assms(2)
apply auto
done
qed
lemma integral_combine_tagged_division_bottomup:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "p tagged_division_of (cbox a b)"
and "∀(x,k)∈p. f integrable_on k"
shows "integral (cbox a b) f = setsum (λ(x,k). integral k f) p"
apply (rule integral_unique)
apply (rule has_integral_combine_tagged_division[OF assms(1)])
using assms(2)
apply auto
done
lemma has_integral_combine_tagged_division_topdown:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on cbox a b"
and "p tagged_division_of (cbox a b)"
shows "(f has_integral (setsum (λ(x,k). integral k f) p)) (cbox a b)"
apply (rule has_integral_combine_tagged_division[OF assms(2)])
apply safe
proof goal_cases
case 1
note tagged_division_ofD(3-4)[OF assms(2) this]
then show ?case
using integrable_subinterval[OF assms(1)] by blast
qed
lemma integral_combine_tagged_division_topdown:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on cbox a b"
and "p tagged_division_of (cbox a b)"
shows "integral (cbox a b) f = setsum (λ(x,k). integral k f) p"
apply (rule integral_unique)
apply (rule has_integral_combine_tagged_division_topdown)
using assms
apply auto
done
subsection ‹Henstock's lemma›
lemma henstock_lemma_part1:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on cbox a b"
and "e > 0"
and "gauge d"
and "(∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm (setsum (λ(x,k). content k *⇩R f x) p - integral(cbox a b) f) < e)"
and p: "p tagged_partial_division_of (cbox a b)" "d fine p"
shows "norm (setsum (λ(x,k). content k *⇩R f x - integral k f) p) ≤ e"
(is "?x ≤ e")
proof -
{ presume "⋀k. 0<k ⟹ ?x ≤ e + k" then show ?thesis by (blast intro: field_le_epsilon) }
fix k :: real
assume k: "k > 0"
note p' = tagged_partial_division_ofD[OF p(1)]
have "⋃(snd ` p) ⊆ cbox a b"
using p'(3) by fastforce
note partial_division_of_tagged_division[OF p(1)] this
from partial_division_extend_interval[OF this] guess q . note q=this and q' = division_ofD[OF this(2)]
def r ≡ "q - snd ` p"
have "snd ` p ∩ r = {}"
unfolding r_def by auto
have r: "finite r"
using q' unfolding r_def by auto
have "∀i∈r. ∃p. p tagged_division_of i ∧ d fine p ∧
norm (setsum (λ(x,j). content j *⇩R f x) p - integral i f) < k / (real (card r) + 1)"
apply safe
proof goal_cases
case (1 i)
then have i: "i ∈ q"
unfolding r_def by auto
from q'(4)[OF this] guess u v by (elim exE) note uv=this
have *: "k / (real (card r) + 1) > 0" using k by simp
have "f integrable_on cbox u v"
apply (rule integrable_subinterval[OF assms(1)])
using q'(2)[OF i]
unfolding uv
apply auto
done
note integrable_integral[OF this, unfolded has_integral[of f]]
from this[rule_format,OF *] guess dd .. note dd=conjunctD2[OF this,rule_format]
note gauge_inter[OF ‹gauge d› dd(1)]
from fine_division_exists[OF this,of u v] guess qq .
then show ?case
apply (rule_tac x=qq in exI)
using dd(2)[of qq]
unfolding fine_inter uv
apply auto
done
qed
from bchoice[OF this] guess qq .. note qq=this[rule_format]
let ?p = "p ∪ ⋃(qq ` r)"
have "norm ((∑(x, k)∈?p. content k *⇩R f x) - integral (cbox a b) f) < e"
apply (rule assms(4)[rule_format])
proof
show "d fine ?p"
apply (rule fine_union)
apply (rule p)
apply (rule fine_unions)
using qq
apply auto
done
note * = tagged_partial_division_of_union_self[OF p(1)]
have "p ∪ ⋃(qq ` r) tagged_division_of ⋃(snd ` p) ∪ ⋃r"
using r
proof (rule tagged_division_union[OF * tagged_division_unions], goal_cases)
case 1
then show ?case
using qq by auto
next
case 2
then show ?case
apply rule
apply rule
apply rule
apply(rule q'(5))
unfolding r_def
apply auto
done
next
case 3
then show ?case
apply (rule inter_interior_unions_intervals)
apply fact
apply rule
apply rule
apply (rule q')
defer
apply rule
apply (subst Int_commute)
apply (rule inter_interior_unions_intervals)
apply (rule finite_imageI)
apply (rule p')
apply rule
defer
apply rule
apply (rule q')
using q(1) p'
unfolding r_def
apply auto
done
qed
moreover have "⋃(snd ` p) ∪ ⋃r = cbox a b" and "{qq i |i. i ∈ r} = qq ` r"
unfolding Union_Un_distrib[symmetric] r_def
using q
by auto
ultimately show "?p tagged_division_of (cbox a b)"
by fastforce
qed
then have "norm ((∑(x, k)∈p. content k *⇩R f x) + (∑(x, k)∈⋃(qq ` r). content k *⇩R f x) -
integral (cbox a b) f) < e"
apply (subst setsum.union_inter_neutral[symmetric])
apply (rule p')
prefer 3
apply assumption
apply rule
apply (rule r)
apply safe
apply (drule qq)
proof -
fix x l k
assume as: "(x, l) ∈ p" "(x, l) ∈ qq k" "k ∈ r"
note qq[OF this(3)]
note tagged_division_ofD(3,4)[OF conjunct1[OF this] as(2)]
from this(2) guess u v by (elim exE) note uv=this
have "l∈snd ` p" unfolding image_iff apply(rule_tac x="(x,l)" in bexI) using as by auto
then have "l ∈ q" "k ∈ q" "l ≠ k"
using as(1,3) q(1) unfolding r_def by auto
note q'(5)[OF this]
then have "interior l = {}"
using interior_mono[OF ‹l ⊆ k›] by blast
then show "content l *⇩R f x = 0"
unfolding uv content_eq_0_interior[symmetric] by auto
qed auto
then have "norm ((∑(x, k)∈p. content k *⇩R f x) + setsum (setsum (λ(x, k). content k *⇩R f x))
(qq ` r) - integral (cbox a b) f) < e"
apply (subst (asm) setsum.Union_comp)
prefer 2
unfolding split_paired_all split_conv image_iff
apply (erule bexE)+
proof -
fix x m k l T1 T2
assume "(x, m) ∈ T1" "(x, m) ∈ T2" "T1 ≠ T2" "k ∈ r" "l ∈ r" "T1 = qq k" "T2 = qq l"
note as = this(1-5)[unfolded this(6-)]
note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]]
from this(2)[OF as(4,1)] guess u v by (elim exE) note uv=this
have *: "interior (k ∩ l) = {}"
by (metis DiffE ‹T1 = qq k› ‹T1 ≠ T2› ‹T2 = qq l› as(4) as(5) interior_Int q'(5) r_def)
have "interior m = {}"
unfolding subset_empty[symmetric]
unfolding *[symmetric]
apply (rule interior_mono)
using kl(1)[OF as(4,1)] kl(1)[OF as(5,2)]
apply auto
done
then show "content m *⇩R f x = 0"
unfolding uv content_eq_0_interior[symmetric]
by auto
qed (insert qq, auto)
then have **: "norm ((∑(x, k)∈p. content k *⇩R f x) + setsum (setsum (λ(x, k). content k *⇩R f x) ∘ qq) r -
integral (cbox a b) f) < e"
apply (subst (asm) setsum.reindex_nontrivial)
apply fact
apply (rule setsum.neutral)
apply rule
unfolding split_paired_all split_conv
defer
apply assumption
proof -
fix k l x m
assume as: "k ∈ r" "l ∈ r" "k ≠ l" "qq k = qq l" "(x, m) ∈ qq k"
note tagged_division_ofD(6)[OF qq[THEN conjunct1]]
from this[OF as(1)] this[OF as(2)] show "content m *⇩R f x = 0"
using as(3) unfolding as by auto
qed
have *: "norm (cp - ip) ≤ e + k"
if "norm ((cp + cr) - i) < e"
and "norm (cr - ir) < k"
and "ip + ir = i"
for ir ip i cr cp
proof -
from that show ?thesis
using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]
unfolding that(3)[symmetric] norm_minus_cancel
by (auto simp add: algebra_simps)
qed
have "?x = norm ((∑(x, k)∈p. content k *⇩R f x) - (∑(x, k)∈p. integral k f))"
unfolding split_def setsum_subtractf ..
also have "… ≤ e + k"
apply (rule *[OF **, where ir1="setsum (λk. integral k f) r"])
proof goal_cases
case 1
have *: "k * real (card r) / (1 + real (card r)) < k"
using k by (auto simp add: field_simps)
show ?case
apply (rule le_less_trans[of _ "setsum (λx. k / (real (card r) + 1)) r"])
unfolding setsum_subtractf[symmetric]
apply (rule setsum_norm_le)
apply rule
apply (drule qq)
defer
unfolding divide_inverse setsum_left_distrib[symmetric]
unfolding divide_inverse[symmetric]
using * apply (auto simp add: field_simps)
done
next
case 2
have *: "(∑(x, k)∈p. integral k f) = (∑k∈snd ` p. integral k f)"
apply (subst setsum.reindex_nontrivial)
apply fact
unfolding split_paired_all snd_conv split_def o_def
proof -
fix x l y m
assume as: "(x, l) ∈ p" "(y, m) ∈ p" "(x, l) ≠ (y, m)" "l = m"
from p'(4)[OF as(1)] guess u v by (elim exE) note uv=this
show "integral l f = 0"
unfolding uv
apply (rule integral_unique)
apply (rule has_integral_null)
unfolding content_eq_0_interior
using p'(5)[OF as(1-3)]
unfolding uv as(4)[symmetric]
apply auto
done
qed auto
from q(1) have **: "snd ` p ∪ q = q" by auto
show ?case
unfolding integral_combine_division_topdown[OF assms(1) q(2)] * r_def
using ** q'(1) p'(1) setsum.union_disjoint [of "snd ` p" "q - snd ` p" "λk. integral k f", symmetric]
by simp
qed
finally show "?x ≤ e + k" .
qed
lemma henstock_lemma_part2:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "f integrable_on cbox a b"
and "e > 0"
and "gauge d"
and "∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm (setsum (λ(x,k). content k *⇩R f x) p - integral (cbox a b) f) < e"
and "p tagged_partial_division_of (cbox a b)"
and "d fine p"
shows "setsum (λ(x,k). norm (content k *⇩R f x - integral k f)) p ≤ 2 * real (DIM('n)) * e"
unfolding split_def
apply (rule setsum_norm_allsubsets_bound)
defer
apply (rule henstock_lemma_part1[unfolded split_def,OF assms(1-3)])
apply safe
apply (rule assms[rule_format,unfolded split_def])
defer
apply (rule tagged_partial_division_subset)
apply (rule assms)
apply assumption
apply (rule fine_subset)
apply assumption
apply (rule assms)
using assms(5)
apply auto
done
lemma henstock_lemma:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "f integrable_on cbox a b"
and "e > 0"
obtains d where "gauge d"
and "∀p. p tagged_partial_division_of (cbox a b) ∧ d fine p ⟶
setsum (λ(x,k). norm(content k *⇩R f x - integral k f)) p < e"
proof -
have *: "e / (2 * (real DIM('n) + 1)) > 0" using assms(2) by simp
from integrable_integral[OF assms(1),unfolded has_integral[of f],rule_format,OF this]
guess d .. note d = conjunctD2[OF this]
show thesis
apply (rule that)
apply (rule d)
proof (safe, goal_cases)
case (1 p)
note * = henstock_lemma_part2[OF assms(1) * d this]
show ?case
apply (rule le_less_trans[OF *])
using ‹e > 0›
apply (auto simp add: field_simps)
done
qed
qed
subsection ‹Geometric progression›
text ‹FIXME: Should one or more of these theorems be moved to @{file
"~~/src/HOL/Set_Interval.thy"}, alongside ‹geometric_sum›?›
lemma sum_gp_basic:
fixes x :: "'a::ring_1"
shows "(1 - x) * setsum (λi. x^i) {0 .. n} = (1 - x^(Suc n))"
proof -
def y ≡ "1 - x"
have "y * (∑i=0..n. (1 - y) ^ i) = 1 - (1 - y) ^ Suc n"
by (induct n) (simp_all add: algebra_simps)
then show ?thesis
unfolding y_def by simp
qed
lemma sum_gp_multiplied:
assumes mn: "m ≤ n"
shows "((1::'a::{field}) - x) * setsum (op ^ x) {m..n} = x^m - x^ Suc n"
(is "?lhs = ?rhs")
proof -
let ?S = "{0..(n - m)}"
from mn have mn': "n - m ≥ 0"
by arith
let ?f = "op + m"
have i: "inj_on ?f ?S"
unfolding inj_on_def by auto
have f: "?f ` ?S = {m..n}"
using mn
apply (auto simp add: image_iff Bex_def)
apply presburger
done
have th: "op ^ x ∘ op + m = (λi. x^m * x^i)"
by (rule ext) (simp add: power_add power_mult)
from setsum.reindex[OF i, of "op ^ x", unfolded f th setsum_right_distrib[symmetric]]
have "?lhs = x^m * ((1 - x) * setsum (op ^ x) {0..n - m})"
by simp
then show ?thesis
unfolding sum_gp_basic
using mn
by (simp add: field_simps power_add[symmetric])
qed
lemma sum_gp:
"setsum (op ^ (x::'a::{field})) {m .. n} =
(if n < m then 0
else if x = 1 then of_nat ((n + 1) - m)
else (x^ m - x^ (Suc n)) / (1 - x))"
proof -
{
assume nm: "n < m"
then have ?thesis by simp
}
moreover
{
assume "¬ n < m"
then have nm: "m ≤ n"
by arith
{
assume x: "x = 1"
then have ?thesis
by simp
}
moreover
{
assume x: "x ≠ 1"
then have nz: "1 - x ≠ 0"
by simp
from sum_gp_multiplied[OF nm, of x] nz have ?thesis
by (simp add: field_simps)
}
ultimately have ?thesis by blast
}
ultimately show ?thesis by blast
qed
lemma sum_gp_offset:
"setsum (op ^ (x::'a::{field})) {m .. m+n} =
(if x = 1 then of_nat n + 1 else x^m * (1 - x^Suc n) / (1 - x))"
unfolding sum_gp[of x m "m + n"] power_Suc
by (simp add: field_simps power_add)
subsection ‹Monotone convergence (bounded interval first)›
lemma monotone_convergence_interval:
fixes f :: "nat ⇒ 'n::euclidean_space ⇒ real"
assumes "∀k. (f k) integrable_on cbox a b"
and "∀k. ∀x∈cbox a b.(f k x) ≤ f (Suc k) x"
and "∀x∈cbox a b. ((λk. f k x) ⤏ g x) sequentially"
and "bounded {integral (cbox a b) (f k) | k . k ∈ UNIV}"
shows "g integrable_on cbox a b ∧ ((λk. integral (cbox a b) (f k)) ⤏ integral (cbox a b) g) sequentially"
proof (cases "content (cbox a b) = 0")
case True
show ?thesis
using integrable_on_null[OF True]
unfolding integral_null[OF True]
using tendsto_const
by auto
next
case False
have fg: "∀x∈cbox a b. ∀k. (f k x) ∙ 1 ≤ (g x) ∙ 1"
proof safe
fix x k
assume x: "x ∈ cbox a b"
note * = Lim_component_ge[OF assms(3)[rule_format, OF x] trivial_limit_sequentially]
show "f k x ∙ 1 ≤ g x ∙ 1"
apply (rule *)
unfolding eventually_sequentially
apply (rule_tac x=k in exI)
apply -
apply (rule transitive_stepwise_le)
using assms(2)[rule_format, OF x]
apply auto
done
qed
have "∃i. ((λk. integral (cbox a b) (f k)) ⤏ i) sequentially"
apply (rule bounded_increasing_convergent)
defer
apply rule
apply (rule integral_le)
apply safe
apply (rule assms(1-2)[rule_format])+
using assms(4)
apply auto
done
then guess i .. note i=this
have i': "⋀k. (integral(cbox a b) (f k)) ≤ i∙1"
apply (rule Lim_component_ge)
apply (rule i)
apply (rule trivial_limit_sequentially)
unfolding eventually_sequentially
apply (rule_tac x=k in exI)
apply (rule transitive_stepwise_le)
prefer 3
unfolding inner_simps real_inner_1_right
apply (rule integral_le)
apply (rule assms(1-2)[rule_format])+
using assms(2)
apply auto
done
have "(g has_integral i) (cbox a b)"
unfolding has_integral
proof (safe, goal_cases)
case e: (1 e)
then have "∀k. (∃d. gauge d ∧ (∀p. p tagged_division_of (cbox a b) ∧ d fine p ⟶
norm ((∑(x, ka)∈p. content ka *⇩R f k x) - integral (cbox a b) (f k)) < e / 2 ^ (k + 2)))"
apply -
apply rule
apply (rule assms(1)[unfolded has_integral_integral has_integral,rule_format])
apply auto
done
from choice[OF this] guess c .. note c=conjunctD2[OF this[rule_format],rule_format]
have "∃r. ∀k≥r. 0 ≤ i∙1 - (integral (cbox a b) (f k)) ∧ i∙1 - (integral (cbox a b) (f k)) < e / 4"
proof -
have "e/4 > 0"
using e by auto
from LIMSEQ_D [OF i this] guess r ..
then show ?thesis
apply (rule_tac x=r in exI)
apply rule
apply (erule_tac x=k in allE)
subgoal for k using i'[of k] by auto
done
qed
then guess r .. note r=conjunctD2[OF this[rule_format]]
have "∀x∈cbox a b. ∃n≥r. ∀k≥n. 0 ≤ (g x)∙1 - (f k x)∙1 ∧
(g x)∙1 - (f k x)∙1 < e / (4 * content(cbox a b))"
proof (rule, goal_cases)
case prems: (1 x)
have "e / (4 * content (cbox a b)) > 0"
using ‹e>0› False content_pos_le[of a b] by auto
from assms(3)[rule_format, OF prems, THEN LIMSEQ_D, OF this]
guess n .. note n=this
then show ?case
apply (rule_tac x="n + r" in exI)
apply safe
apply (erule_tac[2-3] x=k in allE)
unfolding dist_real_def
using fg[rule_format, OF prems]
apply (auto simp add: field_simps)
done
qed
from bchoice[OF this] guess m .. note m=conjunctD2[OF this[rule_format],rule_format]
def d ≡ "λx. c (m x) x"
show ?case
apply (rule_tac x=d in exI)
proof safe
show "gauge d"
using c(1) unfolding gauge_def d_def by auto
next
fix p
assume p: "p tagged_division_of (cbox a b)" "d fine p"
note p'=tagged_division_ofD[OF p(1)]
have "∃a. ∀x∈p. m (fst x) ≤ a"
by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
then guess s .. note s=this
have *: "∀a b c d. norm(a - b) ≤ e / 4 ∧ norm(b - c) < e / 2 ∧
norm (c - d) < e / 4 ⟶ norm (a - d) < e"
proof (safe, goal_cases)
case (1 a b c d)
then show ?case
using norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
norm_triangle_lt[of "a - b + (b - c)" "c - d" e]
unfolding norm_minus_cancel
by (auto simp add: algebra_simps)
qed
show "norm ((∑(x, k)∈p. content k *⇩R g x) - i) < e"
apply (rule *[rule_format,where
b="∑(x, k)∈p. content k *⇩R f (m x) x" and c="∑(x, k)∈p. integral k (f (m x))"])
proof (safe, goal_cases)
case 1
show ?case
apply (rule order_trans[of _ "∑(x, k)∈p. content k * (e / (4 * content (cbox a b)))"])
unfolding setsum_subtractf[symmetric]
apply (rule order_trans)
apply (rule norm_setsum)
apply (rule setsum_mono)
unfolding split_paired_all split_conv
unfolding split_def setsum_left_distrib[symmetric] scaleR_diff_right[symmetric]
unfolding additive_content_tagged_division[OF p(1), unfolded split_def]
proof -
fix x k
assume xk: "(x, k) ∈ p"
then have x: "x ∈ cbox a b"
using p'(2-3)[OF xk] by auto
from p'(4)[OF xk] guess u v by (elim exE) note uv=this
show "norm (content k *⇩R (g x - f (m x) x)) ≤ content k * (e / (4 * content (cbox a b)))"
unfolding norm_scaleR uv
unfolding abs_of_nonneg[OF content_pos_le]
apply (rule mult_left_mono)
using m(2)[OF x,of "m x"]
apply auto
done
qed (insert False, auto)
next
case 2
show ?case
apply (rule le_less_trans[of _ "norm (∑j = 0..s.
∑(x, k)∈{xk∈p. m (fst xk) = j}. content k *⇩R f (m x) x - integral k (f (m x)))"])
apply (subst setsum_group)
apply fact
apply (rule finite_atLeastAtMost)
defer
apply (subst split_def)+
unfolding setsum_subtractf
apply rule
proof -
show "norm (∑j = 0..s. ∑(x, k)∈{xk ∈ p.
m (fst xk) = j}. content k *⇩R f (m x) x - integral k (f (m x))) < e / 2"
apply (rule le_less_trans[of _ "setsum (λi. e / 2^(i+2)) {0..s}"])
apply (rule setsum_norm_le)
proof
show "(∑i = 0..s. e / 2 ^ (i + 2)) < e / 2"
unfolding power_add divide_inverse inverse_mult_distrib
unfolding setsum_right_distrib[symmetric] setsum_left_distrib[symmetric]
unfolding power_inverse [symmetric] sum_gp
apply(rule mult_strict_left_mono[OF _ e])
unfolding power2_eq_square
apply auto
done
fix t
assume "t ∈ {0..s}"
show "norm (∑(x, k)∈{xk ∈ p. m (fst xk) = t}. content k *⇩R f (m x) x -
integral k (f (m x))) ≤ e / 2 ^ (t + 2)"
apply (rule order_trans
[of _ "norm (setsum (λ(x,k). content k *⇩R f t x - integral k (f t)) {xk ∈ p. m (fst xk) = t})"])
apply (rule eq_refl)
apply (rule arg_cong[where f=norm])
apply (rule setsum.cong)
apply (rule refl)
defer
apply (rule henstock_lemma_part1)
apply (rule assms(1)[rule_format])
apply (simp add: e)
apply safe
apply (rule c)+
apply rule
apply assumption+
apply (rule tagged_partial_division_subset[of p])
apply (rule p(1)[unfolded tagged_division_of_def,THEN conjunct1])
defer
unfolding fine_def
apply safe
apply (drule p(2)[unfolded fine_def,rule_format])
unfolding d_def
apply auto
done
qed
qed (insert s, auto)
next
case 3
note comb = integral_combine_tagged_division_topdown[OF assms(1)[rule_format] p(1)]
have *: "⋀sr sx ss ks kr::real. kr = sr ⟶ ks = ss ⟶
ks ≤ i ∧ sr ≤ sx ∧ sx ≤ ss ∧ 0 ≤ i∙1 - kr∙1 ∧ i∙1 - kr∙1 < e/4 ⟶ ¦sx - i¦ < e/4"
by auto
show ?case
unfolding real_norm_def
apply (rule *[rule_format])
apply safe
apply (rule comb[of r])
apply (rule comb[of s])
apply (rule i'[unfolded real_inner_1_right])
apply (rule_tac[1-2] setsum_mono)
unfolding split_paired_all split_conv
apply (rule_tac[1-2] integral_le[OF ])
proof safe
show "0 ≤ i∙1 - (integral (cbox a b) (f r))∙1"
using r(1) by auto
show "i∙1 - (integral (cbox a b) (f r))∙1 < e / 4"
using r(2) by auto
fix x k
assume xk: "(x, k) ∈ p"
from p'(4)[OF this] guess u v by (elim exE) note uv=this
show "f r integrable_on k"
and "f s integrable_on k"
and "f (m x) integrable_on k"
and "f (m x) integrable_on k"
unfolding uv
apply (rule_tac[!] integrable_on_subcbox[OF assms(1)[rule_format]])
using p'(3)[OF xk]
unfolding uv
apply auto
done
fix y
assume "y ∈ k"
then have "y ∈ cbox a b"
using p'(3)[OF xk] by auto
then have *: "⋀m. ∀n≥m. f m y ≤ f n y"
apply -
apply (rule transitive_stepwise_le)
using assms(2)
apply auto
done
show "f r y ≤ f (m x) y" and "f (m x) y ≤ f s y"
apply (rule_tac[!] *[rule_format])
using s[rule_format,OF xk] m(1)[of x] p'(2-3)[OF xk]
apply auto
done
qed
qed
qed
qed note * = this
have "integral (cbox a b) g = i"
by (rule integral_unique) (rule *)
then show ?thesis
using i * by auto
qed
lemma monotone_convergence_increasing:
fixes f :: "nat ⇒ 'n::euclidean_space ⇒ real"
assumes "∀k. (f k) integrable_on s"
and "∀k. ∀x∈s. (f k x) ≤ (f (Suc k) x)"
and "∀x∈s. ((λk. f k x) ⤏ g x) sequentially"
and "bounded {integral s (f k)| k. True}"
shows "g integrable_on s ∧ ((λk. integral s (f k)) ⤏ integral s g) sequentially"
proof -
have lem: "g integrable_on s ∧ ((λk. integral s (f k)) ⤏ integral s g) sequentially"
if "∀k. ∀x∈s. 0 ≤ f k x"
and "∀k. (f k) integrable_on s"
and "∀k. ∀x∈s. f k x ≤ f (Suc k) x"
and "∀x∈s. ((λk. f k x) ⤏ g x) sequentially"
and "bounded {integral s (f k)| k. True}"
for f :: "nat ⇒ 'n::euclidean_space ⇒ real" and g s
proof -
note assms=that[rule_format]
have "∀x∈s. ∀k. (f k x)∙1 ≤ (g x)∙1"
apply safe
apply (rule Lim_component_ge)
apply (rule that(4)[rule_format])
apply assumption
apply (rule trivial_limit_sequentially)
unfolding eventually_sequentially
apply (rule_tac x=k in exI)
apply (rule transitive_stepwise_le)
using that(3)
apply auto
done
note fg=this[rule_format]
have "∃i. ((λk. integral s (f k)) ⤏ i) sequentially"
apply (rule bounded_increasing_convergent)
apply (rule that(5))
apply rule
apply (rule integral_le)
apply (rule that(2)[rule_format])+
using that(3)
apply auto
done
then guess i .. note i=this
have "⋀k. ∀x∈s. ∀n≥k. f k x ≤ f n x"
apply rule
apply (rule transitive_stepwise_le)
using that(3)
apply auto
done
then have i': "∀k. (integral s (f k))∙1 ≤ i∙1"
apply -
apply rule
apply (rule Lim_component_ge)
apply (rule i)
apply (rule trivial_limit_sequentially)
unfolding eventually_sequentially
apply (rule_tac x=k in exI)
apply safe
apply (rule integral_component_le)
apply simp
apply (rule that(2)[rule_format])+
apply auto
done
note int = assms(2)[unfolded integrable_alt[of _ s],THEN conjunct1,rule_format]
have ifif: "⋀k t. (λx. if x ∈ t then if x ∈ s then f k x else 0 else 0) =
(λx. if x ∈ t ∩ s then f k x else 0)"
by (rule ext) auto
have int': "⋀k a b. f k integrable_on cbox a b ∩ s"
apply (subst integrable_restrict_univ[symmetric])
apply (subst ifif[symmetric])
apply (subst integrable_restrict_univ)
apply (rule int)
done
have "⋀a b. (λx. if x ∈ s then g x else 0) integrable_on cbox a b ∧
((λk. integral (cbox a b) (λx. if x ∈ s then f k x else 0)) ⤏
integral (cbox a b) (λx. if x ∈ s then g x else 0)) sequentially"
proof (rule monotone_convergence_interval, safe, goal_cases)
case 1
show ?case by (rule int)
next
case (2 _ _ _ x)
then show ?case
apply (cases "x ∈ s")
using assms(3)
apply auto
done
next
case (3 _ _ x)
then show ?case
apply (cases "x ∈ s")
using assms(4)
apply auto
done
next
case (4 a b)
note * = integral_nonneg
have "⋀k. norm (integral (cbox a b) (λx. if x ∈ s then f k x else 0)) ≤ norm (integral s (f k))"
unfolding real_norm_def
apply (subst abs_of_nonneg)
apply (rule *[OF int])
apply safe
apply (case_tac "x ∈ s")
apply (drule assms(1))
prefer 3
apply (subst abs_of_nonneg)
apply (rule *[OF assms(2) that(1)[THEN spec]])
apply (subst integral_restrict_univ[symmetric,OF int])
unfolding ifif
unfolding integral_restrict_univ[OF int']
apply (rule integral_subset_le[OF _ int' assms(2)])
using assms(1)
apply auto
done
then show ?case
using assms(5)
unfolding bounded_iff
apply safe
apply (rule_tac x=aa in exI)
apply safe
apply (erule_tac x="integral s (f k)" in ballE)
apply (rule order_trans)
apply assumption
apply auto
done
qed
note g = conjunctD2[OF this]
have "(g has_integral i) s"
unfolding has_integral_alt'
apply safe
apply (rule g(1))
proof goal_cases
case (1 e)
then have "e/4>0"
by auto
from LIMSEQ_D [OF i this] guess N .. note N=this
note assms(2)[of N,unfolded has_integral_integral has_integral_alt'[of "f N"]]
from this[THEN conjunct2,rule_format,OF ‹e/4>0›] guess B .. note B=conjunctD2[OF this]
show ?case
apply rule
apply rule
apply (rule B)
apply safe
proof -
fix a b :: 'n
assume ab: "ball 0 B ⊆ cbox a b"
from ‹e > 0› have "e/2 > 0"
by auto
from LIMSEQ_D [OF g(2)[of a b] this] guess M .. note M=this
have **: "norm (integral (cbox a b) (λx. if x ∈ s then f N x else 0) - i) < e/2"
apply (rule norm_triangle_half_l)
using B(2)[rule_format,OF ab] N[rule_format,of N]
apply -
defer
apply (subst norm_minus_commute)
apply auto
done
have *: "⋀f1 f2 g. ¦f1 - i¦ < e / 2 ⟶ ¦f2 - g¦ < e / 2 ⟶
f1 ≤ f2 ⟶ f2 ≤ i ⟶ ¦g - i¦ < e"
unfolding real_inner_1_right by arith
show "norm (integral (cbox a b) (λx. if x ∈ s then g x else 0) - i) < e"
unfolding real_norm_def
apply (rule *[rule_format])
apply (rule **[unfolded real_norm_def])
apply (rule M[rule_format,of "M + N",unfolded real_norm_def])
apply (rule le_add1)
apply (rule integral_le[OF int int])
defer
apply (rule order_trans[OF _ i'[rule_format,of "M + N",unfolded real_inner_1_right]])
proof (safe, goal_cases)
case (2 x)
have "⋀m. x ∈ s ⟹ ∀n≥m. (f m x)∙1 ≤ (f n x)∙1"
apply (rule transitive_stepwise_le)
using assms(3)
apply auto
done
then show ?case
by auto
next
case 1
show ?case
apply (subst integral_restrict_univ[symmetric,OF int])
unfolding ifif integral_restrict_univ[OF int']
apply (rule integral_subset_le[OF _ int'])
using assms
apply auto
done
qed
qed
qed
then show ?thesis
apply safe
defer
apply (drule integral_unique)
using i
apply auto
done
qed
have sub: "⋀k. integral s (λx. f k x - f 0 x) = integral s (f k) - integral s (f 0)"
apply (subst integral_diff)
apply (rule assms(1)[rule_format])+
apply rule
done
have "⋀x m. x ∈ s ⟹ ∀n≥m. f m x ≤ f n x"
apply (rule transitive_stepwise_le)
using assms(2)
apply auto
done
note * = this[rule_format]
have "(λx. g x - f 0 x) integrable_on s ∧ ((λk. integral s (λx. f (Suc k) x - f 0 x)) ⤏
integral s (λx. g x - f 0 x)) sequentially"
apply (rule lem)
apply safe
proof goal_cases
case (1 k x)
then show ?case
using *[of x 0 "Suc k"] by auto
next
case (2 k)
then show ?case
apply (rule integrable_diff)
using assms(1)
apply auto
done
next
case (3 k x)
then show ?case
using *[of x "Suc k" "Suc (Suc k)"] by auto
next
case (4 x)
then show ?case
apply -
apply (rule tendsto_diff)
using LIMSEQ_ignore_initial_segment[OF assms(3)[rule_format],of x 1]
apply auto
done
next
case 5
then show ?case
using assms(4)
unfolding bounded_iff
apply safe
apply (rule_tac x="a + norm (integral s (λx. f 0 x))" in exI)
apply safe
apply (erule_tac x="integral s (λx. f (Suc k) x)" in ballE)
unfolding sub
apply (rule order_trans[OF norm_triangle_ineq4])
apply auto
done
qed
note conjunctD2[OF this]
note tendsto_add[OF this(2) tendsto_const[of "integral s (f 0)"]]
integrable_add[OF this(1) assms(1)[rule_format,of 0]]
then show ?thesis
unfolding sub
apply -
apply rule
defer
apply (subst(asm) integral_diff)
using assms(1)
apply auto
apply (rule LIMSEQ_imp_Suc)
apply assumption
done
qed
lemma has_integral_monotone_convergence_increasing:
fixes f :: "nat ⇒ 'a::euclidean_space ⇒ real"
assumes f: "⋀k. (f k has_integral x k) s"
assumes "⋀k x. x ∈ s ⟹ f k x ≤ f (Suc k) x"
assumes "⋀x. x ∈ s ⟹ (λk. f k x) ⇢ g x"
assumes "x ⇢ x'"
shows "(g has_integral x') s"
proof -
have x_eq: "x = (λi. integral s (f i))"
by (simp add: integral_unique[OF f])
then have x: "{integral s (f k) |k. True} = range x"
by auto
have "g integrable_on s ∧ (λk. integral s (f k)) ⇢ integral s g"
proof (intro monotone_convergence_increasing allI ballI assms)
show "bounded {integral s (f k) |k. True}"
unfolding x by (rule convergent_imp_bounded) fact
qed (auto intro: f)
moreover then have "integral s g = x'"
by (intro LIMSEQ_unique[OF _ ‹x ⇢ x'›]) (simp add: x_eq)
ultimately show ?thesis
by (simp add: has_integral_integral)
qed
lemma monotone_convergence_decreasing:
fixes f :: "nat ⇒ 'n::euclidean_space ⇒ real"
assumes "∀k. (f k) integrable_on s"
and "∀k. ∀x∈s. f (Suc k) x ≤ f k x"
and "∀x∈s. ((λk. f k x) ⤏ g x) sequentially"
and "bounded {integral s (f k)| k. True}"
shows "g integrable_on s ∧ ((λk. integral s (f k)) ⤏ integral s g) sequentially"
proof -
note assm = assms[rule_format]
have *: "{integral s (λx. - f k x) |k. True} = op *⇩R (- 1) ` {integral s (f k)| k. True}"
apply safe
unfolding image_iff
apply (rule_tac x="integral s (f k)" in bexI)
prefer 3
apply (rule_tac x=k in exI)
apply auto
done
have "(λx. - g x) integrable_on s ∧
((λk. integral s (λx. - f k x)) ⤏ integral s (λx. - g x)) sequentially"
apply (rule monotone_convergence_increasing)
apply safe
apply (rule integrable_neg)
apply (rule assm)
defer
apply (rule tendsto_minus)
apply (rule assm)
apply assumption
unfolding *
apply (rule bounded_scaling)
using assm
apply auto
done
note * = conjunctD2[OF this]
show ?thesis
using integrable_neg[OF *(1)] tendsto_minus[OF *(2)]
by auto
qed
subsection ‹Absolute integrability (this is the same as Lebesgue integrability)›
definition absolutely_integrable_on (infixr "absolutely'_integrable'_on" 46)
where "f absolutely_integrable_on s ⟷ f integrable_on s ∧ (λx. (norm(f x))) integrable_on s"
lemma absolutely_integrable_onI[intro?]:
"f integrable_on s ⟹
(λx. (norm(f x))) integrable_on s ⟹ f absolutely_integrable_on s"
unfolding absolutely_integrable_on_def
by auto
lemma absolutely_integrable_onD[dest]:
assumes "f absolutely_integrable_on s"
shows "f integrable_on s"
and "(λx. norm (f x)) integrable_on s"
using assms
unfolding absolutely_integrable_on_def
by auto
lemma integral_norm_bound_integral:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f integrable_on s"
and "g integrable_on s"
and "∀x∈s. norm (f x) ≤ g x"
shows "norm (integral s f) ≤ integral s g"
proof -
have *: "⋀x y. (∀e::real. 0 < e ⟶ x < y + e) ⟹ x ≤ y"
apply (rule ccontr)
apply (erule_tac x="x - y" in allE)
apply auto
done
have norm: "norm ig < dia + e"
if "norm sg ≤ dsa"
and "¦dsa - dia¦ < e / 2"
and "norm (sg - ig) < e / 2"
for e dsa dia and sg ig :: 'a
apply (rule le_less_trans[OF norm_triangle_sub[of ig sg]])
apply (subst real_sum_of_halves[of e,symmetric])
unfolding add.assoc[symmetric]
apply (rule add_le_less_mono)
defer
apply (subst norm_minus_commute)
apply (rule that(3))
apply (rule order_trans[OF that(1)])
using that(2)
apply arith
done
have lem: "norm (integral(cbox a b) f) ≤ integral (cbox a b) g"
if "f integrable_on cbox a b"
and "g integrable_on cbox a b"
and "∀x∈cbox a b. norm (f x) ≤ g x"
for f :: "'n ⇒ 'a" and g a b
proof (rule *[rule_format])
fix e :: real
assume "e > 0"
then have *: "e/2 > 0"
by auto
from integrable_integral[OF that(1),unfolded has_integral[of f],rule_format,OF *]
guess d1 .. note d1 = conjunctD2[OF this,rule_format]
from integrable_integral[OF that(2),unfolded has_integral[of g],rule_format,OF *]
guess d2 .. note d2 = conjunctD2[OF this,rule_format]
note gauge_inter[OF d1(1) d2(1)]
from fine_division_exists[OF this, of a b] guess p . note p=this
show "norm (integral (cbox a b) f) < integral (cbox a b) g + e"
apply (rule norm)
defer
apply (rule d2(2)[OF conjI[OF p(1)],unfolded real_norm_def])
defer
apply (rule d1(2)[OF conjI[OF p(1)]])
defer
apply (rule setsum_norm_le)
proof safe
fix x k
assume "(x, k) ∈ p"
note as = tagged_division_ofD(2-4)[OF p(1) this]
from this(3) guess u v by (elim exE) note uv=this
show "norm (content k *⇩R f x) ≤ content k *⇩R g x"
unfolding uv norm_scaleR
unfolding abs_of_nonneg[OF content_pos_le] real_scaleR_def
apply (rule mult_left_mono)
using that(3) as
apply auto
done
qed (insert p[unfolded fine_inter], auto)
qed
{ presume "⋀e. 0 < e ⟹ norm (integral s f) < integral s g + e"
then show ?thesis by (rule *[rule_format]) auto }
fix e :: real
assume "e > 0"
then have e: "e/2 > 0"
by auto
note assms(1)[unfolded integrable_alt[of f]] note f=this[THEN conjunct1,rule_format]
note assms(2)[unfolded integrable_alt[of g]] note g=this[THEN conjunct1,rule_format]
from integrable_integral[OF assms(1),unfolded has_integral'[of f],rule_format,OF e]
guess B1 .. note B1=conjunctD2[OF this[rule_format],rule_format]
from integrable_integral[OF assms(2),unfolded has_integral'[of g],rule_format,OF e]
guess B2 .. note B2=conjunctD2[OF this[rule_format],rule_format]
from bounded_subset_cbox[OF bounded_ball, of "0::'n" "max B1 B2"]
guess a b by (elim exE) note ab=this[unfolded ball_max_Un]
have "ball 0 B1 ⊆ cbox a b"
using ab by auto
from B1(2)[OF this] guess z .. note z=conjunctD2[OF this]
have "ball 0 B2 ⊆ cbox a b"
using ab by auto
from B2(2)[OF this] guess w .. note w=conjunctD2[OF this]
show "norm (integral s f) < integral s g + e"
apply (rule norm)
apply (rule lem[OF f g, of a b])
unfolding integral_unique[OF z(1)] integral_unique[OF w(1)]
defer
apply (rule w(2)[unfolded real_norm_def])
apply (rule z(2))
apply safe
apply (case_tac "x ∈ s")
unfolding if_P
apply (rule assms(3)[rule_format])
apply auto
done
qed
lemma integral_norm_bound_integral_component:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
fixes g :: "'n ⇒ 'b::euclidean_space"
assumes "f integrable_on s"
and "g integrable_on s"
and "∀x∈s. norm(f x) ≤ (g x)∙k"
shows "norm (integral s f) ≤ (integral s g)∙k"
proof -
have "norm (integral s f) ≤ integral s ((λx. x ∙ k) ∘ g)"
apply (rule integral_norm_bound_integral[OF assms(1)])
apply (rule integrable_linear[OF assms(2)])
apply rule
unfolding o_def
apply (rule assms)
done
then show ?thesis
unfolding o_def integral_component_eq[OF assms(2)] .
qed
lemma has_integral_norm_bound_integral_component:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
fixes g :: "'n ⇒ 'b::euclidean_space"
assumes "(f has_integral i) s"
and "(g has_integral j) s"
and "∀x∈s. norm (f x) ≤ (g x)∙k"
shows "norm i ≤ j∙k"
using integral_norm_bound_integral_component[of f s g k]
unfolding integral_unique[OF assms(1)] integral_unique[OF assms(2)]
using assms
by auto
lemma absolutely_integrable_le:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f absolutely_integrable_on s"
shows "norm (integral s f) ≤ integral s (λx. norm (f x))"
apply (rule integral_norm_bound_integral)
using assms
apply auto
done
lemma absolutely_integrable_0[intro]:
"(λx. 0) absolutely_integrable_on s"
unfolding absolutely_integrable_on_def
by auto
lemma absolutely_integrable_cmul[intro]:
"f absolutely_integrable_on s ⟹
(λx. c *⇩R f x) absolutely_integrable_on s"
unfolding absolutely_integrable_on_def
using integrable_cmul[of f s c]
using integrable_cmul[of "λx. norm (f x)" s "¦c¦"]
by auto
lemma absolutely_integrable_neg[intro]:
"f absolutely_integrable_on s ⟹
(λx. -f(x)) absolutely_integrable_on s"
apply (drule absolutely_integrable_cmul[where c="-1"])
apply auto
done
lemma absolutely_integrable_norm[intro]:
"f absolutely_integrable_on s ⟹
(λx. norm (f x)) absolutely_integrable_on s"
unfolding absolutely_integrable_on_def
by auto
lemma absolutely_integrable_abs[intro]:
"f absolutely_integrable_on s ⟹
(λx. ¦f x::real¦) absolutely_integrable_on s"
apply (drule absolutely_integrable_norm)
unfolding real_norm_def
apply assumption
done
lemma absolutely_integrable_on_subinterval:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
shows "f absolutely_integrable_on s ⟹
cbox a b ⊆ s ⟹ f absolutely_integrable_on cbox a b"
unfolding absolutely_integrable_on_def
by (metis integrable_on_subcbox)
lemma absolutely_integrable_bounded_variation:
fixes f :: "'n::euclidean_space ⇒ 'a::banach"
assumes "f absolutely_integrable_on UNIV"
obtains B where "∀d. d division_of (⋃d) ⟶ setsum (λk. norm(integral k f)) d ≤ B"
apply (rule that[of "integral UNIV (λx. norm (f x))"])
apply safe
proof goal_cases
case prems: (1 d)
note d = division_ofD[OF prems(2)]
have "(∑k∈d. norm (integral k f)) ≤ (∑i∈d. integral i (λx. norm (f x)))"
apply (rule setsum_mono,rule absolutely_integrable_le)
apply (drule d(4))
apply safe
apply (rule absolutely_integrable_on_subinterval[OF assms])
apply auto
done
also have "… ≤ integral (⋃d) (λx. norm (f x))"
apply (subst integral_combine_division_topdown[OF _ prems(2)])
using integrable_on_subdivision[OF prems(2)]
using assms
apply auto
done
also have "… ≤ integral UNIV (λx. norm (f x))"
apply (rule integral_subset_le)
using integrable_on_subdivision[OF prems(2)]
using assms
apply auto
done
finally show ?case .
qed
lemma helplemma:
assumes "setsum (λx. norm (f x - g x)) s < e"
and "finite s"
shows "¦setsum (λx. norm(f x)) s - setsum (λx. norm(g x)) s¦ < e"
unfolding setsum_subtractf[symmetric]
apply (rule le_less_trans[OF setsum_abs])
apply (rule le_less_trans[OF _ assms(1)])
apply (rule setsum_mono)
apply (rule norm_triangle_ineq3)
done
lemma bounded_variation_absolutely_integrable_interval:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes f: "f integrable_on cbox a b"
and *: "∀d. d division_of (cbox a b) ⟶ setsum (λk. norm(integral k f)) d ≤ B"
shows "f absolutely_integrable_on cbox a b"
proof -
let ?f = "λd. ∑k∈d. norm (integral k f)" and ?D = "{d. d division_of (cbox a b)}"
have D_1: "?D ≠ {}"
by (rule elementary_interval[of a b]) auto
have D_2: "bdd_above (?f`?D)"
by (metis * mem_Collect_eq bdd_aboveI2)
note D = D_1 D_2
let ?S = "SUP x:?D. ?f x"
show ?thesis
apply (rule absolutely_integrable_onI [OF f has_integral_integrable])
apply (subst has_integral[of _ ?S])
apply safe
proof goal_cases
case e: (1 e)
then have "?S - e / 2 < ?S" by simp
then obtain d where d: "d division_of (cbox a b)" "?S - e / 2 < (∑k∈d. norm (integral k f))"
unfolding less_cSUP_iff[OF D] by auto
note d' = division_ofD[OF this(1)]
have "∀x. ∃e>0. ∀i∈d. x ∉ i ⟶ ball x e ∩ i = {}"
proof
fix x
have "∃da>0. ∀xa∈⋃{i ∈ d. x ∉ i}. da ≤ dist x xa"
apply (rule separate_point_closed)
apply (rule closed_Union)
apply (rule finite_subset[OF _ d'(1)])
using d'(4)
apply auto
done
then show "∃e>0. ∀i∈d. x ∉ i ⟶ ball x e ∩ i = {}"
by force
qed
from choice[OF this] guess k .. note k=conjunctD2[OF this[rule_format],rule_format]
have "e/2 > 0"
using e by auto
from henstock_lemma[OF assms(1) this] guess g . note g=this[rule_format]
let ?g = "λx. g x ∩ ball x (k x)"
show ?case
apply (rule_tac x="?g" in exI)
apply safe
proof -
show "gauge ?g"
using g(1) k(1)
unfolding gauge_def
by auto
fix p
assume "p tagged_division_of (cbox a b)" and "?g fine p"
note p = this(1) conjunctD2[OF this(2)[unfolded fine_inter]]
note p' = tagged_division_ofD[OF p(1)]
def p' ≡ "{(x,k) | x k. ∃i l. x ∈ i ∧ i ∈ d ∧ (x,l) ∈ p ∧ k = i ∩ l}"
have gp': "g fine p'"
using p(2)
unfolding p'_def fine_def
by auto
have p'': "p' tagged_division_of (cbox a b)"
apply (rule tagged_division_ofI)
proof -
show "finite p'"
apply (rule finite_subset[of _ "(λ(k,(x,l)). (x,k ∩ l)) `
{(k,xl) | k xl. k ∈ d ∧ xl ∈ p}"])
unfolding p'_def
defer
apply (rule finite_imageI,rule finite_product_dependent[OF d'(1) p'(1)])
apply safe
unfolding image_iff
apply (rule_tac x="(i,x,l)" in bexI)
apply auto
done
fix x k
assume "(x, k) ∈ p'"
then have "∃i l. x ∈ i ∧ i ∈ d ∧ (x, l) ∈ p ∧ k = i ∩ l"
unfolding p'_def by auto
then guess i l by (elim exE) note il=conjunctD4[OF this]
show "x ∈ k" and "k ⊆ cbox a b"
using p'(2-3)[OF il(3)] il by auto
show "∃a b. k = cbox a b"
unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)]
apply safe
unfolding inter_interval
apply auto
done
next
fix x1 k1
assume "(x1, k1) ∈ p'"
then have "∃i l. x1 ∈ i ∧ i ∈ d ∧ (x1, l) ∈ p ∧ k1 = i ∩ l"
unfolding p'_def by auto
then guess i1 l1 by (elim exE) note il1=conjunctD4[OF this]
fix x2 k2
assume "(x2,k2)∈p'"
then have "∃i l. x2 ∈ i ∧ i ∈ d ∧ (x2, l) ∈ p ∧ k2 = i ∩ l"
unfolding p'_def by auto
then guess i2 l2 by (elim exE) note il2=conjunctD4[OF this]
assume "(x1, k1) ≠ (x2, k2)"
then have "interior i1 ∩ interior i2 = {} ∨ interior l1 ∩ interior l2 = {}"
using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)]
unfolding il1 il2
by auto
then show "interior k1 ∩ interior k2 = {}"
unfolding il1 il2 by auto
next
have *: "∀(x, X) ∈ p'. X ⊆ cbox a b"
unfolding p'_def using d' by auto
show "⋃{k. ∃x. (x, k) ∈ p'} = cbox a b"
apply rule
apply (rule Union_least)
unfolding mem_Collect_eq
apply (erule exE)
apply (drule *[rule_format])
apply safe
proof -
fix y
assume y: "y ∈ cbox a b"
then have "∃x l. (x, l) ∈ p ∧ y∈l"
unfolding p'(6)[symmetric] by auto
then guess x l by (elim exE) note xl=conjunctD2[OF this]
then have "∃k. k ∈ d ∧ y ∈ k"
using y unfolding d'(6)[symmetric] by auto
then guess i .. note i = conjunctD2[OF this]
have "x ∈ i"
using fineD[OF p(3) xl(1)]
using k(2)[OF i(1), of x]
using i(2) xl(2)
by auto
then show "y ∈ ⋃{k. ∃x. (x, k) ∈ p'}"
unfolding p'_def Union_iff
apply (rule_tac x="i ∩ l" in bexI)
using i xl
apply auto
done
qed
qed
then have "(∑(x, k)∈p'. norm (content k *⇩R f x - integral k f)) < e / 2"
apply -
apply (rule g(2)[rule_format])
unfolding tagged_division_of_def
apply safe
apply (rule gp')
done
then have **: "¦(∑(x,k)∈p'. norm (content k *⇩R f x)) - (∑(x,k)∈p'. norm (integral k f))¦ < e / 2"
unfolding split_def
using p''
by (force intro!: helplemma)
have p'alt: "p' = {(x,(i ∩ l)) | x i l. (x,l) ∈ p ∧ i ∈ d ∧ i ∩ l ≠ {}}"
proof (safe, goal_cases)
case prems: (2 _ _ x i l)
have "x ∈ i"
using fineD[OF p(3) prems(1)] k(2)[OF prems(2), of x] prems(4-)
by auto
then have "(x, i ∩ l) ∈ p'"
unfolding p'_def
using prems
apply safe
apply (rule_tac x=x in exI)
apply (rule_tac x="i ∩ l" in exI)
apply safe
using prems
apply auto
done
then show ?case
using prems(3) by auto
next
fix x k
assume "(x, k) ∈ p'"
then have "∃i l. x ∈ i ∧ i ∈ d ∧ (x, l) ∈ p ∧ k = i ∩ l"
unfolding p'_def by auto
then guess i l by (elim exE) note il=conjunctD4[OF this]
then show "∃y i l. (x, k) = (y, i ∩ l) ∧ (y, l) ∈ p ∧ i ∈ d ∧ i ∩ l ≠ {}"
apply (rule_tac x=x in exI)
apply (rule_tac x=i in exI)
apply (rule_tac x=l in exI)
using p'(2)[OF il(3)]
apply auto
done
qed
have sum_p': "(∑(x, k)∈p'. norm (integral k f)) = (∑k∈snd ` p'. norm (integral k f))"
apply (subst setsum_over_tagged_division_lemma[OF p'',of "λk. norm (integral k f)"])
unfolding norm_eq_zero
apply (rule integral_null)
apply assumption
apply rule
done
note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]]
have *: "⋀sni sni' sf sf'. ¦sf' - sni'¦ < e / 2 ⟶ ?S - e / 2 < sni ∧ sni' ≤ ?S ∧
sni ≤ sni' ∧ sf' = sf ⟶ ¦sf - ?S¦ < e"
by arith
show "norm ((∑(x, k)∈p. content k *⇩R norm (f x)) - ?S) < e"
unfolding real_norm_def
apply (rule *[rule_format,OF **])
apply safe
apply(rule d(2))
proof goal_cases
case 1
show ?case
by (auto simp: sum_p' division_of_tagged_division[OF p''] D intro!: cSUP_upper)
next
case 2
have *: "{k ∩ l | k l. k ∈ d ∧ l ∈ snd ` p} =
(λ(k,l). k ∩ l) ` {(k,l)|k l. k ∈ d ∧ l ∈ snd ` p}"
by auto
have "(∑k∈d. norm (integral k f)) ≤ (∑i∈d. ∑l∈snd ` p. norm (integral (i ∩ l) f))"
proof (rule setsum_mono, goal_cases)
case k: (1 k)
from d'(4)[OF this] guess u v by (elim exE) note uv=this
def d' ≡ "{cbox u v ∩ l |l. l ∈ snd ` p ∧ cbox u v ∩ l ≠ {}}"
note uvab = d'(2)[OF k[unfolded uv]]
have "d' division_of cbox u v"
apply (subst d'_def)
apply (rule division_inter_1)
apply (rule division_of_tagged_division[OF p(1)])
apply (rule uvab)
done
then have "norm (integral k f) ≤ setsum (λk. norm (integral k f)) d'"
unfolding uv
apply (subst integral_combine_division_topdown[of _ _ d'])
apply (rule integrable_on_subcbox[OF assms(1) uvab])
apply assumption
apply (rule setsum_norm_le)
apply auto
done
also have "… = (∑k∈{k ∩ l |l. l ∈ snd ` p}. norm (integral k f))"
apply (rule setsum.mono_neutral_left)
apply (subst simple_image)
apply (rule finite_imageI)+
apply fact
unfolding d'_def uv
apply blast
proof (rule, goal_cases)
case prems: (1 i)
then have "i ∈ {cbox u v ∩ l |l. l ∈ snd ` p}"
by auto
from this[unfolded mem_Collect_eq] guess l .. note l=this
then have "cbox u v ∩ l = {}"
using prems by auto
then show ?case
using l by auto
qed
also have "… = (∑l∈snd ` p. norm (integral (k ∩ l) f))"
unfolding simple_image
apply (rule setsum.reindex_nontrivial [unfolded o_def])
apply (rule finite_imageI)
apply (rule p')
proof goal_cases
case prems: (1 l y)
have "interior (k ∩ l) ⊆ interior (l ∩ y)"
apply (subst(2) interior_Int)
apply (rule Int_greatest)
defer
apply (subst prems(4))
apply auto
done
then have *: "interior (k ∩ l) = {}"
using snd_p(5)[OF prems(1-3)] by auto
from d'(4)[OF k] snd_p(4)[OF prems(1)] guess u1 v1 u2 v2 by (elim exE) note uv=this
show ?case
using *
unfolding uv inter_interval content_eq_0_interior[symmetric]
by auto
qed
finally show ?case .
qed
also have "… = (∑(i,l)∈{(i, l) |i l. i ∈ d ∧ l ∈ snd ` p}. norm (integral (i∩l) f))"
apply (subst sum_sum_product[symmetric])
apply fact
using p'(1)
apply auto
done
also have "… = (∑x∈{(i, l) |i l. i ∈ d ∧ l ∈ snd ` p}. norm (integral (case_prod op ∩ x) f))"
unfolding split_def ..
also have "… = (∑k∈{i ∩ l |i l. i ∈ d ∧ l ∈ snd ` p}. norm (integral k f))"
unfolding *
apply (rule setsum.reindex_nontrivial [symmetric, unfolded o_def])
apply (rule finite_product_dependent)
apply fact
apply (rule finite_imageI)
apply (rule p')
unfolding split_paired_all mem_Collect_eq split_conv o_def
proof -
note * = division_ofD(4,5)[OF division_of_tagged_division,OF p(1)]
fix l1 l2 k1 k2
assume as:
"(l1, k1) ≠ (l2, k2)"
"l1 ∩ k1 = l2 ∩ k2"
"∃i l. (l1, k1) = (i, l) ∧ i ∈ d ∧ l ∈ snd ` p"
"∃i l. (l2, k2) = (i, l) ∧ i ∈ d ∧ l ∈ snd ` p"
then have "l1 ∈ d" and "k1 ∈ snd ` p"
by auto from d'(4)[OF this(1)] *(1)[OF this(2)]
guess u1 v1 u2 v2 by (elim exE) note uv=this
have "l1 ≠ l2 ∨ k1 ≠ k2"
using as by auto
then have "interior k1 ∩ interior k2 = {} ∨ interior l1 ∩ interior l2 = {}"
apply -
apply (erule disjE)
apply (rule disjI2)
apply (rule d'(5))
prefer 4
apply (rule disjI1)
apply (rule *)
using as
apply auto
done
moreover have "interior (l1 ∩ k1) = interior (l2 ∩ k2)"
using as(2) by auto
ultimately have "interior(l1 ∩ k1) = {}"
by auto
then show "norm (integral (l1 ∩ k1) f) = 0"
unfolding uv inter_interval
unfolding content_eq_0_interior[symmetric]
by auto
qed
also have "… = (∑(x, k)∈p'. norm (integral k f))"
unfolding sum_p'
apply (rule setsum.mono_neutral_right)
apply (subst *)
apply (rule finite_imageI[OF finite_product_dependent])
apply fact
apply (rule finite_imageI[OF p'(1)])
apply safe
proof goal_cases
case (2 i ia l a b)
then have "ia ∩ b = {}"
unfolding p'alt image_iff Bex_def not_ex
apply (erule_tac x="(a, ia ∩ b)" in allE)
apply auto
done
then show ?case
by auto
next
case (1 x a b)
then show ?case
unfolding p'_def
apply safe
apply (rule_tac x=i in exI)
apply (rule_tac x=l in exI)
unfolding snd_conv image_iff
apply safe
apply (rule_tac x="(a,l)" in bexI)
apply auto
done
qed
finally show ?case .
next
case 3
let ?S = "{(x, i ∩ l) |x i l. (x, l) ∈ p ∧ i ∈ d}"
have Sigma_alt: "⋀s t. s × t = {(i, j) |i j. i ∈ s ∧ j ∈ t}"
by auto
have *: "?S = (λ(xl,i). (fst xl, snd xl ∩ i)) ` (p × d)"
apply safe
unfolding image_iff
apply (rule_tac x="((x,l),i)" in bexI)
apply auto
done
note pdfin = finite_cartesian_product[OF p'(1) d'(1)]
have "(∑(x, k)∈p'. norm (content k *⇩R f x)) = (∑(x, k)∈?S. ¦content k¦ * norm (f x))"
unfolding norm_scaleR
apply (rule setsum.mono_neutral_left)
apply (subst *)
apply (rule finite_imageI)
apply fact
unfolding p'alt
apply blast
apply safe
apply (rule_tac x=x in exI)
apply (rule_tac x=i in exI)
apply (rule_tac x=l in exI)
apply auto
done
also have "… = (∑((x,l),i)∈p × d. ¦content (l ∩ i)¦ * norm (f x))"
unfolding *
apply (subst setsum.reindex_nontrivial)
apply fact
unfolding split_paired_all
unfolding o_def split_def snd_conv fst_conv mem_Sigma_iff prod.inject
apply (elim conjE)
proof -
fix x1 l1 k1 x2 l2 k2
assume as: "(x1, l1) ∈ p" "(x2, l2) ∈ p" "k1 ∈ d" "k2 ∈ d"
"x1 = x2" "l1 ∩ k1 = l2 ∩ k2" "¬ ((x1 = x2 ∧ l1 = l2) ∧ k1 = k2)"
from d'(4)[OF as(3)] p'(4)[OF as(1)] guess u1 v1 u2 v2 by (elim exE) note uv=this
from as have "l1 ≠ l2 ∨ k1 ≠ k2"
by auto
then have "interior k1 ∩ interior k2 = {} ∨ interior l1 ∩ interior l2 = {}"
apply -
apply (erule disjE)
apply (rule disjI2)
defer
apply (rule disjI1)
apply (rule d'(5)[OF as(3-4)])
apply assumption
apply (rule p'(5)[OF as(1-2)])
apply auto
done
moreover have "interior (l1 ∩ k1) = interior (l2 ∩ k2)"
unfolding as ..
ultimately have "interior (l1 ∩ k1) = {}"
by auto
then show "¦content (l1 ∩ k1)¦ * norm (f x1) = 0"
unfolding uv inter_interval
unfolding content_eq_0_interior[symmetric]
by auto
qed safe
also have "… = (∑(x, k)∈p. content k *⇩R norm (f x))"
unfolding Sigma_alt
apply (subst sum_sum_product[symmetric])
apply (rule p')
apply rule
apply (rule d')
apply (rule setsum.cong)
apply (rule refl)
unfolding split_paired_all split_conv
proof -
fix x l
assume as: "(x, l) ∈ p"
note xl = p'(2-4)[OF this]
from this(3) guess u v by (elim exE) note uv=this
have "(∑i∈d. ¦content (l ∩ i)¦) = (∑k∈d. content (k ∩ cbox u v))"
apply (rule setsum.cong)
apply (rule refl)
apply (drule d'(4))
apply safe
apply (subst Int_commute)
unfolding inter_interval uv
apply (subst abs_of_nonneg)
apply auto
done
also have "… = setsum content {k ∩ cbox u v| k. k ∈ d}"
unfolding simple_image
apply (rule setsum.reindex_nontrivial [unfolded o_def, symmetric])
apply (rule d')
proof goal_cases
case prems: (1 k y)
from d'(4)[OF this(1)] d'(4)[OF this(2)]
guess u1 v1 u2 v2 by (elim exE) note uv=this
have "{} = interior ((k ∩ y) ∩ cbox u v)"
apply (subst interior_Int)
using d'(5)[OF prems(1-3)]
apply auto
done
also have "… = interior (y ∩ (k ∩ cbox u v))"
by auto
also have "… = interior (k ∩ cbox u v)"
unfolding prems(4) by auto
finally show ?case
unfolding uv inter_interval content_eq_0_interior ..
qed
also have "… = setsum content {cbox u v ∩ k |k. k ∈ d ∧ cbox u v ∩ k ≠ {}}"
apply (rule setsum.mono_neutral_right)
unfolding simple_image
apply (rule finite_imageI)
apply (rule d')
apply blast
apply safe
apply (rule_tac x=k in exI)
proof goal_cases
case prems: (1 i k)
from d'(4)[OF this(1)] guess a b by (elim exE) note ab=this
have "interior (k ∩ cbox u v) ≠ {}"
using prems(2)
unfolding ab inter_interval content_eq_0_interior
by auto
then show ?case
using prems(1)
using interior_subset[of "k ∩ cbox u v"]
by auto
qed
finally show "(∑i∈d. ¦content (l ∩ i)¦ * norm (f x)) = content l *⇩R norm (f x)"
unfolding setsum_left_distrib[symmetric] real_scaleR_def
apply (subst(asm) additive_content_division[OF division_inter_1[OF d(1)]])
using xl(2)[unfolded uv]
unfolding uv
apply auto
done
qed
finally show ?case .
qed
qed
qed
qed
lemma bounded_variation_absolutely_integrable:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "f integrable_on UNIV"
and "∀d. d division_of (⋃d) ⟶ setsum (λk. norm (integral k f)) d ≤ B"
shows "f absolutely_integrable_on UNIV"
proof (rule absolutely_integrable_onI, fact, rule)
let ?f = "λd. ∑k∈d. norm (integral k f)" and ?D = "{d. d division_of (⋃d)}"
have D_1: "?D ≠ {}"
by (rule elementary_interval) auto
have D_2: "bdd_above (?f`?D)"
by (intro bdd_aboveI2[where M=B] assms(2)[rule_format]) simp
note D = D_1 D_2
let ?S = "SUP d:?D. ?f d"
have f_int: "⋀a b. f absolutely_integrable_on cbox a b"
apply (rule bounded_variation_absolutely_integrable_interval[where B=B])
apply (rule integrable_on_subcbox[OF assms(1)])
defer
apply safe
apply (rule assms(2)[rule_format])
apply auto
done
show "((λx. norm (f x)) has_integral ?S) UNIV"
apply (subst has_integral_alt')
apply safe
proof goal_cases
case (1 a b)
show ?case
using f_int[of a b] by auto
next
case prems: (2 e)
have "∃y∈setsum (λk. norm (integral k f)) ` {d. d division_of ⋃d}. ¬ y ≤ ?S - e"
proof (rule ccontr)
assume "¬ ?thesis"
then have "?S ≤ ?S - e"
by (intro cSUP_least[OF D(1)]) auto
then show False
using prems by auto
qed
then obtain K where *: "∃x∈{d. d division_of ⋃d}. K = (∑k∈x. norm (integral k f))"
"SUPREMUM {d. d division_of ⋃d} (setsum (λk. norm (integral k f))) - e < K"
by (auto simp add: image_iff not_le)
from this(1) obtain d where "d division_of ⋃d"
and "K = (∑k∈d. norm (integral k f))"
by auto
note d = this(1) *(2)[unfolded this(2)]
note d'=division_ofD[OF this(1)]
have "bounded (⋃d)"
by (rule elementary_bounded,fact)
from this[unfolded bounded_pos] obtain K where
K: "0 < K" "∀x∈⋃d. norm x ≤ K" by auto
show ?case
apply (rule_tac x="K + 1" in exI)
apply safe
proof -
fix a b :: 'n
assume ab: "ball 0 (K + 1) ⊆ cbox a b"
have *: "∀s s1. ?S - e < s1 ∧ s1 ≤ s ∧ s < ?S + e ⟶ ¦s - ?S¦ < e"
by arith
show "norm (integral (cbox a b) (λx. if x ∈ UNIV then norm (f x) else 0) - ?S) < e"
unfolding real_norm_def
apply (rule *[rule_format])
apply safe
apply (rule d(2))
proof goal_cases
case 1
have "(∑k∈d. norm (integral k f)) ≤ setsum (λk. integral k (λx. norm (f x))) d"
apply (rule setsum_mono)
apply (rule absolutely_integrable_le)
apply (drule d'(4))
apply safe
apply (rule f_int)
done
also have "… = integral (⋃d) (λx. norm (f x))"
apply (rule integral_combine_division_bottomup[symmetric])
apply (rule d)
unfolding forall_in_division[OF d(1)]
using f_int
apply auto
done
also have "… ≤ integral (cbox a b) (λx. if x ∈ UNIV then norm (f x) else 0)"
proof -
have "⋃d ⊆ cbox a b"
apply rule
apply (drule K(2)[rule_format])
apply (rule ab[unfolded subset_eq,rule_format])
apply (auto simp add: dist_norm)
done
then show ?thesis
apply -
apply (subst if_P)
apply rule
apply (rule integral_subset_le)
defer
apply (rule integrable_on_subdivision[of _ _ _ "cbox a b"])
apply (rule d)
using f_int[of a b]
apply auto
done
qed
finally show ?case .
next
note f = absolutely_integrable_onD[OF f_int[of a b]]
note * = this(2)[unfolded has_integral_integral has_integral[of "λx. norm (f x)"],rule_format]
have "e/2>0"
using ‹e > 0› by auto
from * [OF this] obtain d1 where
d1: "gauge d1" "∀p. p tagged_division_of (cbox a b) ∧ d1 fine p ⟶
norm ((∑(x, k)∈p. content k *⇩R norm (f x)) - integral (cbox a b) (λx. norm (f x))) < e / 2"
by auto
from henstock_lemma [OF f(1) ‹e/2>0›] obtain d2 where
d2: "gauge d2" "∀p. p tagged_partial_division_of (cbox a b) ∧ d2 fine p ⟶
(∑(x, k)∈p. norm (content k *⇩R f x - integral k f)) < e / 2" .
obtain p where
p: "p tagged_division_of (cbox a b)" "d1 fine p" "d2 fine p"
by (rule fine_division_exists [OF gauge_inter [OF d1(1) d2(1)], of a b])
(auto simp add: fine_inter)
have *: "⋀sf sf' si di. sf' = sf ⟶ si ≤ ?S ⟶ ¦sf - si¦ < e / 2 ⟶
¦sf' - di¦ < e / 2 ⟶ di < ?S + e"
by arith
show "integral (cbox a b) (λx. if x ∈ UNIV then norm (f x) else 0) < ?S + e"
apply (subst if_P)
apply rule
proof (rule *[rule_format])
show "¦(∑(x,k)∈p. norm (content k *⇩R f x)) - (∑(x,k)∈p. norm (integral k f))¦ < e / 2"
unfolding split_def
apply (rule helplemma)
using d2(2)[rule_format,of p]
using p(1,3)
unfolding tagged_division_of_def split_def
apply auto
done
show "¦(∑(x, k)∈p. content k *⇩R norm (f x)) - integral (cbox a b) (λx. norm(f x))¦ < e / 2"
using d1(2)[rule_format,OF conjI[OF p(1,2)]]
by (simp only: real_norm_def)
show "(∑(x, k)∈p. content k *⇩R norm (f x)) = (∑(x, k)∈p. norm (content k *⇩R f x))"
apply (rule setsum.cong)
apply (rule refl)
unfolding split_paired_all split_conv
apply (drule tagged_division_ofD(4)[OF p(1)])
unfolding norm_scaleR
apply (subst abs_of_nonneg)
apply auto
done
show "(∑(x, k)∈p. norm (integral k f)) ≤ ?S"
using partial_division_of_tagged_division[of p "cbox a b"] p(1)
apply (subst setsum_over_tagged_division_lemma[OF p(1)])
apply (simp add: integral_null)
apply (intro cSUP_upper2[OF D(2), of "snd ` p"])
apply (auto simp: tagged_partial_division_of_def)
done
qed
qed
qed (insert K, auto)
qed
qed
lemma absolutely_integrable_restrict_univ:
"(λx. if x ∈ s then f x else (0::'a::banach)) absolutely_integrable_on UNIV ⟷
f absolutely_integrable_on s"
unfolding absolutely_integrable_on_def if_distrib norm_zero integrable_restrict_univ ..
lemma absolutely_integrable_add[intro]:
fixes f g :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "f absolutely_integrable_on s"
and "g absolutely_integrable_on s"
shows "(λx. f x + g x) absolutely_integrable_on s"
proof -
let ?P = "⋀f g::'n ⇒ 'm. f absolutely_integrable_on UNIV ⟹
g absolutely_integrable_on UNIV ⟹ (λx. f x + g x) absolutely_integrable_on UNIV"
{
presume as: "PROP ?P"
note a = absolutely_integrable_restrict_univ[symmetric]
have *: "⋀x. (if x ∈ s then f x else 0) + (if x ∈ s then g x else 0) =
(if x ∈ s then f x + g x else 0)" by auto
show ?thesis
apply (subst a)
using as[OF assms[unfolded a[of f] a[of g]]]
apply (simp only: *)
done
}
fix f g :: "'n ⇒ 'm"
assume assms: "f absolutely_integrable_on UNIV" "g absolutely_integrable_on UNIV"
note absolutely_integrable_bounded_variation
from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]
show "(λx. f x + g x) absolutely_integrable_on UNIV"
apply (rule bounded_variation_absolutely_integrable[of _ "B1+B2"])
apply (rule integrable_add)
prefer 3
apply safe
proof goal_cases
case prems: (1 d)
have "⋀k. k ∈ d ⟹ f integrable_on k ∧ g integrable_on k"
apply (drule division_ofD(4)[OF prems])
apply safe
apply (rule_tac[!] integrable_on_subcbox[of _ UNIV])
using assms
apply auto
done
then have "(∑k∈d. norm (integral k (λx. f x + g x))) ≤
(∑k∈d. norm (integral k f)) + (∑k∈d. norm (integral k g))"
apply -
unfolding setsum.distrib [symmetric]
apply (rule setsum_mono)
apply (subst integral_add)
prefer 3
apply (rule norm_triangle_ineq)
apply auto
done
also have "… ≤ B1 + B2"
using B(1)[OF prems] B(2)[OF prems] by auto
finally show ?case .
qed (insert assms, auto)
qed
lemma absolutely_integrable_sub[intro]:
fixes f g :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "f absolutely_integrable_on s"
and "g absolutely_integrable_on s"
shows "(λx. f x - g x) absolutely_integrable_on s"
using absolutely_integrable_add[OF assms(1) absolutely_integrable_neg[OF assms(2)]]
by (simp add: algebra_simps)
lemma absolutely_integrable_linear:
fixes f :: "'m::euclidean_space ⇒ 'n::euclidean_space"
and h :: "'n::euclidean_space ⇒ 'p::euclidean_space"
assumes "f absolutely_integrable_on s"
and "bounded_linear h"
shows "(h ∘ f) absolutely_integrable_on s"
proof -
{
presume as: "⋀f::'m ⇒ 'n. ⋀h::'n ⇒ 'p. f absolutely_integrable_on UNIV ⟹
bounded_linear h ⟹ (h ∘ f) absolutely_integrable_on UNIV"
note a = absolutely_integrable_restrict_univ[symmetric]
show ?thesis
apply (subst a)
using as[OF assms[unfolded a[of f] a[of g]]]
apply (simp only: o_def if_distrib linear_simps[OF assms(2)])
done
}
fix f :: "'m ⇒ 'n"
fix h :: "'n ⇒ 'p"
assume assms: "f absolutely_integrable_on UNIV" "bounded_linear h"
from absolutely_integrable_bounded_variation[OF assms(1)] guess B . note B=this
from bounded_linear.pos_bounded[OF assms(2)] guess b .. note b=conjunctD2[OF this]
show "(h ∘ f) absolutely_integrable_on UNIV"
apply (rule bounded_variation_absolutely_integrable[of _ "B * b"])
apply (rule integrable_linear[OF _ assms(2)])
apply safe
proof goal_cases
case prems: (2 d)
have "(∑k∈d. norm (integral k (h ∘ f))) ≤ setsum (λk. norm(integral k f)) d * b"
unfolding setsum_left_distrib
apply (rule setsum_mono)
proof goal_cases
case (1 k)
from division_ofD(4)[OF prems this]
guess u v by (elim exE) note uv=this
have *: "f integrable_on k"
unfolding uv
apply (rule integrable_on_subcbox[of _ UNIV])
using assms
apply auto
done
note this[unfolded has_integral_integral]
note has_integral_linear[OF this assms(2)] integrable_linear[OF * assms(2)]
note * = has_integral_unique[OF this(2)[unfolded has_integral_integral] this(1)]
show ?case
unfolding * using b by auto
qed
also have "… ≤ B * b"
apply (rule mult_right_mono)
using B prems b
apply auto
done
finally show ?case .
qed (insert assms, auto)
qed
lemma absolutely_integrable_setsum:
fixes f :: "'a ⇒ 'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "finite t"
and "⋀a. a ∈ t ⟹ (f a) absolutely_integrable_on s"
shows "(λx. setsum (λa. f a x) t) absolutely_integrable_on s"
using assms(1,2)
by induct auto
lemma absolutely_integrable_vector_abs:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
and T :: "'c::euclidean_space ⇒ 'b"
assumes f: "f absolutely_integrable_on s"
shows "(λx. (∑i∈Basis. ¦f x∙T i¦ *⇩R i)) absolutely_integrable_on s"
(is "?Tf absolutely_integrable_on s")
proof -
have if_distrib: "⋀P A B x. (if P then A else B) *⇩R x = (if P then A *⇩R x else B *⇩R x)"
by simp
have *: "⋀x. ?Tf x = (∑i∈Basis.
((λy. (∑j∈Basis. (if j = i then y else 0) *⇩R j)) o
(λx. (norm (∑j∈Basis. (if j = i then f x∙T i else 0) *⇩R j)))) x)"
by (simp add: comp_def if_distrib setsum.If_cases)
show ?thesis
unfolding *
apply (rule absolutely_integrable_setsum[OF finite_Basis])
apply (rule absolutely_integrable_linear)
apply (rule absolutely_integrable_norm)
apply (rule absolutely_integrable_linear[OF f, unfolded o_def])
apply (auto simp: linear_linear euclidean_eq_iff[where 'a='c] inner_simps intro!: linearI)
done
qed
lemma absolutely_integrable_max:
fixes f g :: "'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "f absolutely_integrable_on s"
and "g absolutely_integrable_on s"
shows "(λx. (∑i∈Basis. max (f(x)∙i) (g(x)∙i) *⇩R i)::'n) absolutely_integrable_on s"
proof -
have *:"⋀x. (1 / 2) *⇩R (((∑i∈Basis. ¦(f x - g x) ∙ i¦ *⇩R i)::'n) + (f x + g x)) =
(∑i∈Basis. max (f(x)∙i) (g(x)∙i) *⇩R i)"
unfolding euclidean_eq_iff[where 'a='n] by (auto simp: inner_simps)
note absolutely_integrable_sub[OF assms] absolutely_integrable_add[OF assms]
note absolutely_integrable_vector_abs[OF this(1), where T="λx. x"] this(2)
note absolutely_integrable_add[OF this]
note absolutely_integrable_cmul[OF this, of "1/2"]
then show ?thesis unfolding * .
qed
lemma absolutely_integrable_min:
fixes f g::"'m::euclidean_space ⇒ 'n::euclidean_space"
assumes "f absolutely_integrable_on s"
and "g absolutely_integrable_on s"
shows "(λx. (∑i∈Basis. min (f(x)∙i) (g(x)∙i) *⇩R i)::'n) absolutely_integrable_on s"
proof -
have *:"⋀x. (1 / 2) *⇩R ((f x + g x) - (∑i∈Basis. ¦(f x - g x) ∙ i¦ *⇩R i::'n)) =
(∑i∈Basis. min (f(x)∙i) (g(x)∙i) *⇩R i)"
unfolding euclidean_eq_iff[where 'a='n] by (auto simp: inner_simps)
note absolutely_integrable_add[OF assms] absolutely_integrable_sub[OF assms]
note this(1) absolutely_integrable_vector_abs[OF this(2), where T="λx. x"]
note absolutely_integrable_sub[OF this]
note absolutely_integrable_cmul[OF this,of "1/2"]
then show ?thesis unfolding * .
qed
lemma absolutely_integrable_abs_eq:
fixes f::"'n::euclidean_space ⇒ 'm::euclidean_space"
shows "f absolutely_integrable_on s ⟷ f integrable_on s ∧
(λx. (∑i∈Basis. ¦f x∙i¦ *⇩R i)::'m) integrable_on s"
(is "?l = ?r")
proof
assume ?l
then show ?r
apply -
apply rule
defer
apply (drule absolutely_integrable_vector_abs)
apply auto
done
next
assume ?r
{
presume lem: "⋀f::'n ⇒ 'm. f integrable_on UNIV ⟹
(λx. (∑i∈Basis. ¦f x∙i¦ *⇩R i)::'m) integrable_on UNIV ⟹
f absolutely_integrable_on UNIV"
have *: "⋀x. (∑i∈Basis. ¦(if x ∈ s then f x else 0) ∙ i¦ *⇩R i) =
(if x ∈ s then (∑i∈Basis. ¦f x ∙ i¦ *⇩R i) else (0::'m))"
unfolding euclidean_eq_iff[where 'a='m]
by auto
show ?l
apply (subst absolutely_integrable_restrict_univ[symmetric])
apply (rule lem)
unfolding integrable_restrict_univ *
using ‹?r›
apply auto
done
}
fix f :: "'n ⇒ 'm"
assume assms: "f integrable_on UNIV" "(λx. (∑i∈Basis. ¦f x∙i¦ *⇩R i)::'m) integrable_on UNIV"
let ?B = "∑i∈Basis. integral UNIV (λx. (∑i∈Basis. ¦f x∙i¦ *⇩R i)::'m) ∙ i"
show "f absolutely_integrable_on UNIV"
apply (rule bounded_variation_absolutely_integrable[OF assms(1), where B="?B"])
apply safe
proof goal_cases
case d: (1 d)
note d'=division_ofD[OF d]
have "(∑k∈d. norm (integral k f)) ≤
(∑k∈d. setsum (op ∙ (integral k (λx. (∑i∈Basis. ¦f x∙i¦ *⇩R i)::'m))) Basis)"
apply (rule setsum_mono)
apply (rule order_trans[OF norm_le_l1])
apply (rule setsum_mono)
unfolding lessThan_iff
proof -
fix k
fix i :: 'm
assume "k ∈ d" and i: "i ∈ Basis"
from d'(4)[OF this(1)] guess a b by (elim exE) note ab=this
show "¦integral k f ∙ i¦ ≤ integral k (λx. (∑i∈Basis. ¦f x∙i¦ *⇩R i)::'m) ∙ i"
apply (rule abs_leI)
unfolding inner_minus_left[symmetric]
defer
apply (subst integral_neg[symmetric])
apply (rule_tac[1-2] integral_component_le[OF i])
using integrable_on_subcbox[OF assms(1),of a b]
integrable_on_subcbox[OF assms(2),of a b] i integrable_neg
unfolding ab
apply auto
done
qed
also have "… ≤ setsum (op ∙ (integral UNIV (λx. (∑i∈Basis. ¦f x∙i¦ *⇩R i)::'m))) Basis"
apply (subst setsum.commute)
apply (rule setsum_mono)
proof goal_cases
case (1 j)
have *: "(λx. ∑i∈Basis. ¦f x∙i¦ *⇩R i::'m) integrable_on ⋃d"
using integrable_on_subdivision[OF d assms(2)] by auto
have "(∑i∈d. integral i (λx. ∑i∈Basis. ¦f x∙i¦ *⇩R i::'m) ∙ j) =
integral (⋃d) (λx. ∑i∈Basis. ¦f x∙i¦ *⇩R i::'m) ∙ j"
unfolding inner_setsum_left[symmetric] integral_combine_division_topdown[OF * d] ..
also have "… ≤ integral UNIV (λx. ∑i∈Basis. ¦f x∙i¦ *⇩R i::'m) ∙ j"
apply (rule integral_subset_component_le)
using assms * ‹j ∈ Basis›
apply auto
done
finally show ?case .
qed
finally show ?case .
qed
qed
lemma nonnegative_absolutely_integrable:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "∀x∈s. ∀i∈Basis. 0 ≤ f x ∙ i"
and "f integrable_on s"
shows "f absolutely_integrable_on s"
unfolding absolutely_integrable_abs_eq
apply rule
apply (rule assms)thm integrable_eq
apply (rule integrable_eq[of _ f])
using assms
apply (auto simp: euclidean_eq_iff[where 'a='m])
done
lemma absolutely_integrable_integrable_bound:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
assumes "∀x∈s. norm (f x) ≤ g x"
and "f integrable_on s"
and "g integrable_on s"
shows "f absolutely_integrable_on s"
proof -
{
presume *: "⋀f::'n ⇒ 'm. ⋀g. ∀x. norm (f x) ≤ g x ⟹ f integrable_on UNIV ⟹
g integrable_on UNIV ⟹ f absolutely_integrable_on UNIV"
show ?thesis
apply (subst absolutely_integrable_restrict_univ[symmetric])
apply (rule *[of _ "λx. if x∈s then g x else 0"])
using assms
unfolding integrable_restrict_univ
apply auto
done
}
fix g
fix f :: "'n ⇒ 'm"
assume assms: "∀x. norm (f x) ≤ g x" "f integrable_on UNIV" "g integrable_on UNIV"
show "f absolutely_integrable_on UNIV"
apply (rule bounded_variation_absolutely_integrable[OF assms(2),where B="integral UNIV g"])
apply safe
proof goal_cases
case d: (1 d)
note d'=division_ofD[OF d]
have "(∑k∈d. norm (integral k f)) ≤ (∑k∈d. integral k g)"
apply (rule setsum_mono)
apply (rule integral_norm_bound_integral)
apply (drule_tac[!] d'(4))
apply safe
apply (rule_tac[1-2] integrable_on_subcbox)
using assms
apply auto
done
also have "… = integral (⋃d) g"
apply (rule integral_combine_division_bottomup[symmetric])
apply (rule d)
apply safe
apply (drule d'(4))
apply safe
apply (rule integrable_on_subcbox[OF assms(3)])
apply auto
done
also have "… ≤ integral UNIV g"
apply (rule integral_subset_le)
defer
apply (rule integrable_on_subdivision[OF d,of _ UNIV])
prefer 4
apply rule
apply (rule_tac y="norm (f x)" in order_trans)
using assms
apply auto
done
finally show ?case .
qed
qed
lemma absolutely_integrable_absolutely_integrable_bound:
fixes f :: "'n::euclidean_space ⇒ 'm::euclidean_space"
and g :: "'n::euclidean_space ⇒ 'p::euclidean_space"
assumes "∀x∈s. norm (f x) ≤ norm (g x)"
and "f integrable_on s"
and "g absolutely_integrable_on s"
shows "f absolutely_integrable_on s"
apply (rule absolutely_integrable_integrable_bound[of s f "λx. norm (g x)"])
using assms
apply auto
done
lemma absolutely_integrable_inf_real:
assumes "finite k"
and "k ≠ {}"
and "∀i∈k. (λx. (fs x i)::real) absolutely_integrable_on s"
shows "(λx. (Inf ((fs x) ` k))) absolutely_integrable_on s"
using assms
proof induct
case (insert a k)
let ?P = "(λx.
if fs x ` k = {} then fs x a
else min (fs x a) (Inf (fs x ` k))) absolutely_integrable_on s"
show ?case
unfolding image_insert
apply (subst Inf_insert_finite)
apply (rule finite_imageI[OF insert(1)])
proof (cases "k = {}")
case True
then show ?P
apply (subst if_P)
defer
apply (rule insert(5)[rule_format])
apply auto
done
next
case False
then show ?P
apply (subst if_not_P)
defer
apply (rule absolutely_integrable_min[where 'n=real, unfolded Basis_real_def, simplified])
defer
apply(rule insert(3)[OF False])
using insert(5)
apply auto
done
qed
next
case empty
then show ?case by auto
qed
lemma absolutely_integrable_sup_real:
assumes "finite k"
and "k ≠ {}"
and "∀i∈k. (λx. (fs x i)::real) absolutely_integrable_on s"
shows "(λx. (Sup ((fs x) ` k))) absolutely_integrable_on s"
using assms
proof induct
case (insert a k)
let ?P = "(λx.
if fs x ` k = {} then fs x a
else max (fs x a) (Sup (fs x ` k))) absolutely_integrable_on s"
show ?case
unfolding image_insert
apply (subst Sup_insert_finite)
apply (rule finite_imageI[OF insert(1)])
proof (cases "k = {}")
case True
then show ?P
apply (subst if_P)
defer
apply (rule insert(5)[rule_format])
apply auto
done
next
case False
then show ?P
apply (subst if_not_P)
defer
apply (rule absolutely_integrable_max[where 'n=real, unfolded Basis_real_def, simplified])
defer
apply (rule insert(3)[OF False])
using insert(5)
apply auto
done
qed
qed auto
subsection ‹differentiation under the integral sign›
lemma tube_lemma:
assumes "compact K"
assumes "open W"
assumes "{x0} × K ⊆ W"
shows "∃X0. x0 ∈ X0 ∧ open X0 ∧ X0 × K ⊆ W"
proof -
{
fix y assume "y ∈ K"
then have "(x0, y) ∈ W" using assms by auto
with ‹open W›
have "∃X0 Y. open X0 ∧ open Y ∧ x0 ∈ X0 ∧ y ∈ Y ∧ X0 × Y ⊆ W"
by (rule open_prod_elim) blast
} then obtain X0 Y where
"∀y ∈ K. open (X0 y) ∧ open (Y y) ∧ x0 ∈ X0 y ∧ y ∈ Y y ∧ X0 y × Y y ⊆ W"
by metis
moreover
then have "∀t∈Y ` K. open t" "K ⊆ ⋃(Y ` K)" by auto
with ‹compact K› obtain CC where "CC ⊆ Y ` K" "finite CC" "K ⊆ ⋃CC"
by (rule compactE)
moreover
then obtain c where c:
"⋀C. C ∈ CC ⟹ c C ∈ K ∧ C = Y (c C)"
by (force intro!: choice)
ultimately show ?thesis
by (force intro!: exI[where x="⋂C∈CC. X0 (c C)"])
qed
lemma continuous_on_prod_compactE:
fixes fx::"'a::topological_space × 'b::topological_space ⇒ 'c::metric_space"
and e::real
assumes cont_fx: "continuous_on (U × C) fx"
assumes "compact C"
assumes [intro]: "x0 ∈ U"
notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
assumes "e > 0"
obtains X0 where "x0 ∈ X0" "open X0"
"∀x∈X0 ∩ U. ∀t ∈ C. dist (fx (x, t)) (fx (x0, t)) ≤ e"
proof -
def psi ≡ "λ(x, t). dist (fx (x, t)) (fx (x0, t))"
def W0 ≡ "{(x, t) ∈ U × C. psi (x, t) < e}"
have W0_eq: "W0 = psi -` {..<e} ∩ U × C"
by (auto simp: vimage_def W0_def)
have "open {..<e}" by simp
have "continuous_on (U × C) psi"
by (auto intro!: continuous_intros simp: psi_def split_beta')
from this[unfolded continuous_on_open_invariant, rule_format, OF ‹open {..<e}›]
obtain W where W: "open W" "W ∩ U × C = W0 ∩ U × C"
unfolding W0_eq by blast
have "{x0} × C ⊆ W ∩ U × C"
unfolding W
by (auto simp: W0_def psi_def ‹0 < e›)
then have "{x0} × C ⊆ W" by blast
from tube_lemma[OF ‹compact C› ‹open W› this]
obtain X0 where X0: "x0 ∈ X0" "open X0" "X0 × C ⊆ W"
by blast
have "∀x∈X0 ∩ U. ∀t ∈ C. dist (fx (x, t)) (fx (x0, t)) ≤ e"
proof safe
fix x assume x: "x ∈ X0" "x ∈ U"
fix t assume t: "t ∈ C"
have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
by (auto simp: psi_def)
also
{
have "(x, t) ∈ X0 × C"
using t x
by auto
also note ‹… ⊆ W›
finally have "(x, t) ∈ W" .
with t x have "(x, t) ∈ W ∩ U × C"
by blast
also note ‹W ∩ U × C = W0 ∩ U × C›
finally have "psi (x, t) < e"
by (auto simp: W0_def)
}
finally show "dist (fx (x, t)) (fx (x0, t)) ≤ e" by simp
qed
from X0(1,2) this show ?thesis ..
qed
lemma integral_continuous_on_param:
fixes f::"'a::topological_space ⇒ 'b::euclidean_space ⇒ 'c::banach"
assumes cont_fx: "continuous_on (U × cbox a b) (λ(x, t). f x t)"
shows "continuous_on U (λx. integral (cbox a b) (f x))"
proof cases
assume "content (cbox a b) ≠ 0"
then have ne: "cbox a b ≠ {}" by auto
note [continuous_intros] =
continuous_on_compose2[OF cont_fx, where f="λy. Pair x y" for x,
unfolded split_beta fst_conv snd_conv]
show ?thesis
unfolding continuous_on_def
proof (safe intro!: tendstoI)
fix e'::real and x
assume "e' > 0"
def e ≡ "e' / (content (cbox a b) + 1)"
have "e > 0" using ‹e' > 0› by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos)
assume "x ∈ U"
from continuous_on_prod_compactE[OF cont_fx compact_cbox ‹x ∈ U› ‹0 < e›]
obtain X0 where X0: "x ∈ X0" "open X0"
and fx_bound: "⋀y t. y ∈ X0 ∩ U ⟹ t ∈ cbox a b ⟹ norm (f y t - f x t) ≤ e"
unfolding split_beta fst_conv snd_conv dist_norm
by metis
have "∀⇩F y in at x within U. y ∈ X0 ∩ U"
using X0(1) X0(2) eventually_at_topological by auto
then show "∀⇩F y in at x within U. dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'"
proof eventually_elim
case (elim y)
have "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) =
norm (integral (cbox a b) (λt. f y t - f x t))"
using elim ‹x ∈ U›
unfolding dist_norm
by (subst integral_diff)
(auto intro!: integrable_continuous continuous_intros)
also have "… ≤ e * content (cbox a b)"
using elim ‹x ∈ U›
by (intro integrable_bound)
(auto intro!: fx_bound ‹x ∈ U › less_imp_le[OF ‹0 < e›]
integrable_continuous continuous_intros)
also have "… < e'"
using ‹0 < e'› ‹e > 0›
by (auto simp: e_def divide_simps)
finally show "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'" .
qed
qed
qed (auto intro!: continuous_on_const)
lemma eventually_closed_segment:
fixes x0::"'a::real_normed_vector"
assumes "open X0" "x0 ∈ X0"
shows "∀⇩F x in at x0 within U. closed_segment x0 x ⊆ X0"
proof -
from openE[OF assms]
obtain e where e: "0 < e" "ball x0 e ⊆ X0" .
then have "∀⇩F x in at x0 within U. x ∈ ball x0 e"
by (auto simp: dist_commute eventually_at)
then show ?thesis
proof eventually_elim
case (elim x)
have "x0 ∈ ball x0 e" using ‹e > 0› by simp
from convex_ball[unfolded convex_contains_segment, rule_format, OF this elim]
have "closed_segment x0 x ⊆ ball x0 e" .
also note ‹… ⊆ X0›
finally show ?case .
qed
qed
lemma leibniz_rule:
fixes f::"'a::banach ⇒ 'b::euclidean_space ⇒ 'c::banach"
assumes fx: "⋀x t. x ∈ U ⟹ t ∈ cbox a b ⟹
((λx. f x t) has_derivative blinfun_apply (fx x t)) (at x within U)"
assumes integrable_f2: "⋀x. x ∈ U ⟹ f x integrable_on cbox a b"
assumes cont_fx: "continuous_on (U × (cbox a b)) (λ(x, t). fx x t)"
assumes [intro]: "x0 ∈ U"
assumes "convex U"
shows
"((λx. integral (cbox a b) (f x)) has_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
(is "(?F has_derivative ?dF) _")
proof cases
assume "content (cbox a b) ≠ 0"
then have ne: "cbox a b ≠ {}" by auto
note [continuous_intros] =
continuous_on_compose2[OF cont_fx, where f="λy. Pair x y" for x,
unfolded split_beta fst_conv snd_conv]
show ?thesis
proof (intro has_derivativeI bounded_linear_scaleR_left tendstoI, fold norm_conv_dist)
have cont_f1: "⋀t. t ∈ cbox a b ⟹ continuous_on U (λx. f x t)"
by (auto simp: continuous_on_eq_continuous_within intro!: has_derivative_continuous fx)
note [continuous_intros] = continuous_on_compose2[OF cont_f1]
fix e'::real
assume "e' > 0"
def e ≡ "e' / (content (cbox a b) + 1)"
have "e > 0" using ‹e' > 0› by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos)
from continuous_on_prod_compactE[OF cont_fx compact_cbox ‹x0 ∈ U› ‹e > 0›]
obtain X0 where X0: "x0 ∈ X0" "open X0"
and fx_bound: "⋀x t. x ∈ X0 ∩ U ⟹ t ∈ cbox a b ⟹ norm (fx x t - fx x0 t) ≤ e"
unfolding split_beta fst_conv snd_conv
by (metis dist_norm)
note eventually_closed_segment[OF ‹open X0› ‹x0 ∈ X0›, of U]
moreover
have "∀⇩F x in at x0 within U. x ∈ X0"
using ‹open X0› ‹x0 ∈ X0› eventually_at_topological by blast
moreover have "∀⇩F x in at x0 within U. x ≠ x0"
by (auto simp: eventually_at_filter)
moreover have "∀⇩F x in at x0 within U. x ∈ U"
by (auto simp: eventually_at_filter)
ultimately
show "∀⇩F x in at x0 within U. norm ((?F x - ?F x0 - ?dF (x - x0)) /⇩R norm (x - x0)) < e'"
proof eventually_elim
case (elim x)
from elim have "0 < norm (x - x0)" by simp
have "closed_segment x0 x ⊆ U"
by (rule ‹convex U›[unfolded convex_contains_segment, rule_format, OF ‹x0 ∈ U› ‹x ∈ U›])
from elim have [intro]: "x ∈ U" by auto
have "?F x - ?F x0 - ?dF (x - x0) =
integral (cbox a b) (λy. f x y - f x0 y - fx x0 y (x - x0))"
(is "_ = ?id")
using ‹x ≠ x0›
by (subst blinfun_apply_integral integral_diff,
auto intro!: integrable_diff integrable_f2 continuous_intros
intro: integrable_continuous)+
also
{
fix t assume t: "t ∈ (cbox a b)"
have seg: "⋀t. t ∈ {0..1} ⟹ x0 + t *⇩R (x - x0) ∈ X0 ∩ U"
using ‹closed_segment x0 x ⊆ U›
‹closed_segment x0 x ⊆ X0›
by (force simp: closed_segment_def algebra_simps)
from t have deriv:
"((λx. f x t) has_derivative (fx y t)) (at y within X0 ∩ U)"
if "y ∈ X0 ∩ U" for y
unfolding has_vector_derivative_def[symmetric]
using that ‹x ∈ X0›
by (intro has_derivative_within_subset[OF fx]) auto
have "∀x ∈ X0 ∩ U. onorm (blinfun_apply (fx x t) - (fx x0 t)) ≤ e"
using fx_bound t
by (auto simp add: norm_blinfun_def fun_diff_def blinfun.bilinear_simps[symmetric])
from differentiable_bound_linearization[OF seg deriv this] X0
have "norm (f x t - f x0 t - fx x0 t (x - x0)) ≤ e * norm (x - x0)"
by (auto simp add: ac_simps)
}
then have "norm ?id ≤ integral (cbox a b) (λ_. e * norm (x - x0))"
by (intro integral_norm_bound_integral)
(auto intro!: continuous_intros integrable_diff integrable_f2
intro: integrable_continuous)
also have "… = content (cbox a b) * e * norm (x - x0)"
by simp
also have "… < e' * norm (x - x0)"
using ‹e' > 0› content_pos_le[of a b]
by (intro mult_strict_right_mono[OF _ ‹0 < norm (x - x0)›])
(auto simp: divide_simps e_def)
finally have "norm (?F x - ?F x0 - ?dF (x - x0)) < e' * norm (x - x0)" .
then show ?case
by (auto simp: divide_simps)
qed
qed (rule blinfun.bounded_linear_right)
qed (auto intro!: derivative_eq_intros simp: blinfun.bilinear_simps)
lemma
has_vector_derivative_eq_has_derivative_blinfun:
"(f has_vector_derivative f') (at x within U) ⟷
(f has_derivative blinfun_scaleR_left f') (at x within U)"
by (simp add: has_vector_derivative_def)
lemma leibniz_rule_vector_derivative:
fixes f::"real ⇒ 'b::euclidean_space ⇒ 'c::banach"
assumes fx: "⋀x t. x ∈ U ⟹ t ∈ cbox a b ⟹
((λx. f x t) has_vector_derivative (fx x t)) (at x within U)"
assumes integrable_f2: "⋀x. x ∈ U ⟹ (f x) integrable_on cbox a b"
assumes cont_fx: "continuous_on (U × cbox a b) (λ(x, t). fx x t)"
assumes U: "x0 ∈ U" "convex U"
shows "((λx. integral (cbox a b) (f x)) has_vector_derivative integral (cbox a b) (fx x0))
(at x0 within U)"
proof -
note [continuous_intros] =
continuous_on_compose2[OF cont_fx, where f="λy. Pair x y" for x,
unfolded split_beta fst_conv snd_conv]
have *: "blinfun_scaleR_left (integral (cbox a b) (fx x0)) =
integral (cbox a b) (λt. blinfun_scaleR_left (fx x0 t))"
by (subst integral_linear[symmetric])
(auto simp: has_vector_derivative_def o_def
intro!: integrable_continuous U continuous_intros bounded_linear_intros)
show ?thesis
unfolding has_vector_derivative_eq_has_derivative_blinfun
apply (rule has_derivative_eq_rhs)
apply (rule leibniz_rule[OF _ integrable_f2 _ U, where fx="λx t. blinfun_scaleR_left (fx x t)"])
using fx cont_fx
apply (auto simp: has_vector_derivative_def * split_beta intro!: continuous_intros)
done
qed
lemma
has_field_derivative_eq_has_derivative_blinfun:
"(f has_field_derivative f') (at x within U) ⟷ (f has_derivative blinfun_mult_right f') (at x within U)"
by (simp add: has_field_derivative_def)
lemma leibniz_rule_field_derivative:
fixes f::"'a::{real_normed_field, banach} ⇒ 'b::euclidean_space ⇒ 'a"
assumes fx: "⋀x t. x ∈ U ⟹ t ∈ cbox a b ⟹ ((λx. f x t) has_field_derivative fx x t) (at x within U)"
assumes integrable_f2: "⋀x. x ∈ U ⟹ (f x) integrable_on cbox a b"
assumes cont_fx: "continuous_on (U × (cbox a b)) (λ(x, t). fx x t)"
assumes U: "x0 ∈ U" "convex U"
shows "((λx. integral (cbox a b) (f x)) has_field_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
proof -
note [continuous_intros] =
continuous_on_compose2[OF cont_fx, where f="λy. Pair x y" for x,
unfolded split_beta fst_conv snd_conv]
have *: "blinfun_mult_right (integral (cbox a b) (fx x0)) =
integral (cbox a b) (λt. blinfun_mult_right (fx x0 t))"
by (subst integral_linear[symmetric])
(auto simp: has_vector_derivative_def o_def
intro!: integrable_continuous U continuous_intros bounded_linear_intros)
show ?thesis
unfolding has_field_derivative_eq_has_derivative_blinfun
apply (rule has_derivative_eq_rhs)
apply (rule leibniz_rule[OF _ integrable_f2 _ U, where fx="λx t. blinfun_mult_right (fx x t)"])
using fx cont_fx
apply (auto simp: has_field_derivative_def * split_beta intro!: continuous_intros)
done
qed
subsection ‹Exchange uniform limit and integral›
lemma
uniform_limit_integral:
fixes f::"'a ⇒ 'b::euclidean_space ⇒ 'c::banach"
assumes u: "uniform_limit (cbox a b) f g F"
assumes c: "⋀n. continuous_on (cbox a b) (f n)"
assumes [simp]: "F ≠ bot"
obtains I J where
"⋀n. (f n has_integral I n) (cbox a b)"
"(g has_integral J) (cbox a b)"
"(I ⤏ J) F"
proof -
have fi[simp]: "f n integrable_on (cbox a b)" for n
by (auto intro!: integrable_continuous assms)
then obtain I where I: "⋀n. (f n has_integral I n) (cbox a b)"
by atomize_elim (auto simp: integrable_on_def intro!: choice)
moreover
have gi[simp]: "g integrable_on (cbox a b)"
by (auto intro!: integrable_continuous uniform_limit_theorem[OF _ u] eventuallyI c)
then obtain J where J: "(g has_integral J) (cbox a b)"
by blast
moreover
have "(I ⤏ J) F"
proof cases
assume "content (cbox a b) = 0"
hence "I = (λ_. 0)" "J = 0"
by (auto intro!: has_integral_unique I J)
thus ?thesis by simp
next
assume content_nonzero: "content (cbox a b) ≠ 0"
show ?thesis
proof (rule tendstoI)
fix e::real
assume "e > 0"
def e' ≡ "e / 2"
with ‹e > 0› have "e' > 0" by simp
then have "∀⇩F n in F. ∀x∈cbox a b. norm (f n x - g x) < e' / content (cbox a b)"
using u content_nonzero content_pos_le[of a b]
by (auto simp: uniform_limit_iff dist_norm)
then show "∀⇩F n in F. dist (I n) J < e"
proof eventually_elim
case (elim n)
have "I n = integral (cbox a b) (f n)"
"J = integral (cbox a b) g"
using I[of n] J by (simp_all add: integral_unique)
then have "dist (I n) J = norm (integral (cbox a b) (λx. f n x - g x))"
by (simp add: integral_diff dist_norm)
also have "… ≤ integral (cbox a b) (λx. (e' / content (cbox a b)))"
using elim
by (intro integral_norm_bound_integral)
(auto intro!: integrable_diff absolutely_integrable_onI)
also have "… < e"
using ‹0 < e›
by (simp add: e'_def)
finally show ?case .
qed
qed
qed
ultimately show ?thesis ..
qed
subsection ‹Dominated convergence›
lemma dominated_convergence:
fixes f :: "nat ⇒ 'n::euclidean_space ⇒ real"
assumes "⋀k. (f k) integrable_on s" "h integrable_on s"
and "⋀k. ∀x ∈ s. norm (f k x) ≤ h x"
and "∀x ∈ s. ((λk. f k x) ⤏ g x) sequentially"
shows "g integrable_on s"
and "((λk. integral s (f k)) ⤏ integral s g) sequentially"
proof -
have bdd_below[simp]: "⋀x P. x ∈ s ⟹ bdd_below {f n x |n. P n}"
proof (safe intro!: bdd_belowI)
fix n x show "x ∈ s ⟹ - h x ≤ f n x"
using assms(3)[rule_format, of x n] by simp
qed
have bdd_above[simp]: "⋀x P. x ∈ s ⟹ bdd_above {f n x |n. P n}"
proof (safe intro!: bdd_aboveI)
fix n x show "x ∈ s ⟹ f n x ≤ h x"
using assms(3)[rule_format, of x n] by simp
qed
have "⋀m. (λx. Inf {f j x |j. m ≤ j}) integrable_on s ∧
((λk. integral s (λx. Inf {f j x |j. j ∈ {m..m + k}})) ⤏
integral s (λx. Inf {f j x |j. m ≤ j}))sequentially"
proof (rule monotone_convergence_decreasing, safe)
fix m :: nat
show "bounded {integral s (λx. Inf {f j x |j. j ∈ {m..m + k}}) |k. True}"
unfolding bounded_iff
apply (rule_tac x="integral s h" in exI)
proof safe
fix k :: nat
show "norm (integral s (λx. Inf {f j x |j. j ∈ {m..m + k}})) ≤ integral s h"
apply (rule integral_norm_bound_integral)
unfolding simple_image
apply (rule absolutely_integrable_onD)
apply (rule absolutely_integrable_inf_real)
prefer 5
unfolding real_norm_def
apply rule
apply (rule cInf_abs_ge)
prefer 5
apply rule
apply (rule_tac g = h in absolutely_integrable_integrable_bound)
using assms
unfolding real_norm_def
apply auto
done
qed
fix k :: nat
show "(λx. Inf {f j x |j. j ∈ {m..m + k}}) integrable_on s"
unfolding simple_image
apply (rule absolutely_integrable_onD)
apply (rule absolutely_integrable_inf_real)
prefer 3
using absolutely_integrable_integrable_bound[OF assms(3,1,2)]
apply auto
done
fix x
assume x: "x ∈ s"
show "Inf {f j x |j. j ∈ {m..m + Suc k}} ≤ Inf {f j x |j. j ∈ {m..m + k}}"
by (rule cInf_superset_mono) auto
let ?S = "{f j x| j. m ≤ j}"
show "((λk. Inf {f j x |j. j ∈ {m..m + k}}) ⤏ Inf ?S) sequentially"
proof (rule LIMSEQ_I, goal_cases)
case r: (1 r)
have "∃y∈?S. y < Inf ?S + r"
by (subst cInf_less_iff[symmetric]) (auto simp: ‹x∈s› r)
then obtain N where N: "f N x < Inf ?S + r" "m ≤ N"
by blast
show ?case
apply (rule_tac x=N in exI)
apply safe
proof goal_cases
case prems: (1 n)
have *: "⋀y ix. y < Inf ?S + r ⟶ Inf ?S ≤ ix ⟶ ix ≤ y ⟶ ¦ix - Inf ?S¦ < r"
by arith
show ?case
unfolding real_norm_def
apply (rule *[rule_format, OF N(1)])
apply (rule cInf_superset_mono, auto simp: ‹x∈s›) []
apply (rule cInf_lower)
using N prems
apply auto []
apply simp
done
qed
qed
qed
note dec1 = conjunctD2[OF this]
have "⋀m. (λx. Sup {f j x |j. m ≤ j}) integrable_on s ∧
((λk. integral s (λx. Sup {f j x |j. j ∈ {m..m + k}})) ⤏
integral s (λx. Sup {f j x |j. m ≤ j})) sequentially"
proof (rule monotone_convergence_increasing,safe)
fix m :: nat
show "bounded {integral s (λx. Sup {f j x |j. j ∈ {m..m + k}}) |k. True}"
unfolding bounded_iff
apply (rule_tac x="integral s h" in exI)
proof safe
fix k :: nat
show "norm (integral s (λx. Sup {f j x |j. j ∈ {m..m + k}})) ≤ integral s h"
apply (rule integral_norm_bound_integral) unfolding simple_image
apply (rule absolutely_integrable_onD)
apply(rule absolutely_integrable_sup_real)
prefer 5 unfolding real_norm_def
apply rule
apply (rule cSup_abs_le)
using assms
apply (force simp add: )
prefer 4
apply rule
apply (rule_tac g=h in absolutely_integrable_integrable_bound)
using assms
unfolding real_norm_def
apply auto
done
qed
fix k :: nat
show "(λx. Sup {f j x |j. j ∈ {m..m + k}}) integrable_on s"
unfolding simple_image
apply (rule absolutely_integrable_onD)
apply (rule absolutely_integrable_sup_real)
prefer 3
using absolutely_integrable_integrable_bound[OF assms(3,1,2)]
apply auto
done
fix x
assume x: "x∈s"
show "Sup {f j x |j. j ∈ {m..m + Suc k}} ≥ Sup {f j x |j. j ∈ {m..m + k}}"
by (rule cSup_subset_mono) auto
let ?S = "{f j x| j. m ≤ j}"
show "((λk. Sup {f j x |j. j ∈ {m..m + k}}) ⤏ Sup ?S) sequentially"
proof (rule LIMSEQ_I, goal_cases)
case r: (1 r)
have "∃y∈?S. Sup ?S - r < y"
by (subst less_cSup_iff[symmetric]) (auto simp: r ‹x∈s›)
then obtain N where N: "Sup ?S - r < f N x" "m ≤ N"
by blast
show ?case
apply (rule_tac x=N in exI)
apply safe
proof goal_cases
case prems: (1 n)
have *: "⋀y ix. Sup ?S - r < y ⟶ ix ≤ Sup ?S ⟶ y ≤ ix ⟶ ¦ix - Sup ?S¦ < r"
by arith
show ?case
apply simp
apply (rule *[rule_format, OF N(1)])
apply (rule cSup_subset_mono, auto simp: ‹x∈s›) []
apply (subst cSup_upper)
using N prems
apply auto
done
qed
qed
qed
note inc1 = conjunctD2[OF this]
have "g integrable_on s ∧
((λk. integral s (λx. Inf {f j x |j. k ≤ j})) ⤏ integral s g) sequentially"
apply (rule monotone_convergence_increasing,safe)
apply fact
proof -
show "bounded {integral s (λx. Inf {f j x |j. k ≤ j}) |k. True}"
unfolding bounded_iff apply(rule_tac x="integral s h" in exI)
proof safe
fix k :: nat
show "norm (integral s (λx. Inf {f j x |j. k ≤ j})) ≤ integral s h"
apply (rule integral_norm_bound_integral)
apply fact+
unfolding real_norm_def
apply rule
apply (rule cInf_abs_ge)
using assms(3)
apply auto
done
qed
fix k :: nat and x
assume x: "x ∈ s"
have *: "⋀x y::real. x ≥ - y ⟹ - x ≤ y" by auto
show "Inf {f j x |j. k ≤ j} ≤ Inf {f j x |j. Suc k ≤ j}"
by (intro cInf_superset_mono) (auto simp: ‹x∈s›)
show "(λk::nat. Inf {f j x |j. k ≤ j}) ⇢ g x"
proof (rule LIMSEQ_I, goal_cases)
case r: (1 r)
then have "0<r/2"
by auto
from assms(4)[THEN bspec, THEN LIMSEQ_D, OF x this] guess N .. note N = this
show ?case
apply (rule_tac x=N in exI)
apply safe
unfolding real_norm_def
apply (rule le_less_trans[of _ "r/2"])
apply (rule cInf_asclose)
apply safe
defer
apply (rule less_imp_le)
using N r
apply auto
done
qed
qed
note inc2 = conjunctD2[OF this]
have "g integrable_on s ∧
((λk. integral s (λx. Sup {f j x |j. k ≤ j})) ⤏ integral s g) sequentially"
apply (rule monotone_convergence_decreasing,safe)
apply fact
proof -
show "bounded {integral s (λx. Sup {f j x |j. k ≤ j}) |k. True}"
unfolding bounded_iff
apply (rule_tac x="integral s h" in exI)
proof safe
fix k :: nat
show "norm (integral s (λx. Sup {f j x |j. k ≤ j})) ≤ integral s h"
apply (rule integral_norm_bound_integral)
apply fact+
unfolding real_norm_def
apply rule
apply (rule cSup_abs_le)
using assms(3)
apply auto
done
qed
fix k :: nat
fix x
assume x: "x ∈ s"
show "Sup {f j x |j. k ≤ j} ≥ Sup {f j x |j. Suc k ≤ j}"
by (rule cSup_subset_mono) (auto simp: ‹x∈s›)
show "((λk. Sup {f j x |j. k ≤ j}) ⤏ g x) sequentially"
proof (rule LIMSEQ_I, goal_cases)
case r: (1 r)
then have "0<r/2"
by auto
from assms(4)[THEN bspec, THEN LIMSEQ_D, OF x this] guess N .. note N=this
show ?case
apply (rule_tac x=N in exI,safe)
unfolding real_norm_def
apply (rule le_less_trans[of _ "r/2"])
apply (rule cSup_asclose)
apply safe
defer
apply (rule less_imp_le)
using N r
apply auto
done
qed
qed
note dec2 = conjunctD2[OF this]
show "g integrable_on s" by fact
show "((λk. integral s (f k)) ⤏ integral s g) sequentially"
proof (rule LIMSEQ_I, goal_cases)
case r: (1 r)
from LIMSEQ_D [OF inc2(2) r] guess N1 .. note N1=this[unfolded real_norm_def]
from LIMSEQ_D [OF dec2(2) r] guess N2 .. note N2=this[unfolded real_norm_def]
show ?case
proof (rule_tac x="N1+N2" in exI, safe)
fix n
assume n: "n ≥ N1 + N2"
have *: "⋀i0 i i1 g. ¦i0 - g¦ < r ⟶ ¦i1 - g¦ < r ⟶ i0 ≤ i ⟶ i ≤ i1 ⟶ ¦i - g¦ < r"
by arith
show "norm (integral s (f n) - integral s g) < r"
unfolding real_norm_def
proof (rule *[rule_format,OF N1[rule_format] N2[rule_format], of n n])
show "integral s (λx. Inf {f j x |j. n ≤ j}) ≤ integral s (f n)"
by (rule integral_le[OF dec1(1) assms(1)]) (auto intro!: cInf_lower)
show "integral s (f n) ≤ integral s (λx. Sup {f j x |j. n ≤ j})"
by (rule integral_le[OF assms(1) inc1(1)]) (auto intro!: cSup_upper)
qed (insert n, auto)
qed
qed
qed
lemma has_integral_dominated_convergence:
fixes f :: "nat ⇒ 'n::euclidean_space ⇒ real"
assumes "⋀k. (f k has_integral y k) s" "h integrable_on s"
"⋀k. ∀x∈s. norm (f k x) ≤ h x" "∀x∈s. (λk. f k x) ⇢ g x"
and x: "y ⇢ x"
shows "(g has_integral x) s"
proof -
have int_f: "⋀k. (f k) integrable_on s"
using assms by (auto simp: integrable_on_def)
have "(g has_integral (integral s g)) s"
by (intro integrable_integral dominated_convergence[OF int_f assms(2)]) fact+
moreover have "integral s g = x"
proof (rule LIMSEQ_unique)
show "(λi. integral s (f i)) ⇢ x"
using integral_unique[OF assms(1)] x by simp
show "(λi. integral s (f i)) ⇢ integral s g"
by (intro dominated_convergence[OF int_f assms(2)]) fact+
qed
ultimately show ?thesis
by simp
qed
subsection‹Compute a double integral using iterated integrals and switching the order of integration›
lemma setcomp_dot1: "{z. P (z ∙ (i,0))} = {(x,y). P(x ∙ i)}"
by auto
lemma setcomp_dot2: "{z. P (z ∙ (0,i))} = {(x,y). P(y ∙ i)}"
by auto
lemma Sigma_Int_Paircomp1: "(Sigma A B) ∩ {(x, y). P x} = Sigma (A ∩ {x. P x}) B"
by blast
lemma Sigma_Int_Paircomp2: "(Sigma A B) ∩ {(x, y). P y} = Sigma A (λz. B z ∩ {y. P y})"
by blast
lemma continuous_on_imp_integrable_on_Pair1:
fixes f :: "_ ⇒ 'b::banach"
assumes con: "continuous_on (cbox (a,c) (b,d)) f" and x: "x ∈ cbox a b"
shows "(λy. f (x, y)) integrable_on (cbox c d)"
proof -
have "f ∘ (λy. (x, y)) integrable_on (cbox c d)"
apply (rule integrable_continuous)
apply (rule continuous_on_compose [OF _ continuous_on_subset [OF con]])
using x
apply (auto intro: continuous_on_Pair continuous_on_const continuous_on_id continuous_on_subset con)
done
then show ?thesis
by (simp add: o_def)
qed
lemma integral_integrable_2dim:
fixes f :: "('a::euclidean_space * 'b::euclidean_space) ⇒ 'c::banach"
assumes "continuous_on (cbox (a,c) (b,d)) f"
shows "(λx. integral (cbox c d) (λy. f (x,y))) integrable_on cbox a b"
proof (cases "content(cbox c d) = 0")
case True
then show ?thesis
by (simp add: True integrable_const)
next
case False
have uc: "uniformly_continuous_on (cbox (a,c) (b,d)) f"
by (simp add: assms compact_cbox compact_uniformly_continuous)
{ fix x::'a and e::real
assume x: "x ∈ cbox a b" and e: "0 < e"
then have e2_gt: "0 < e / 2 / content (cbox c d)" and e2_less: "e / 2 / content (cbox c d) * content (cbox c d) < e"
by (auto simp: False content_lt_nz e)
then obtain dd
where dd: "⋀x x'. ⟦x∈cbox (a, c) (b, d); x'∈cbox (a, c) (b, d); norm (x' - x) < dd⟧
⟹ norm (f x' - f x) ≤ e / (2 * content (cbox c d))" "dd>0"
using uc [unfolded uniformly_continuous_on_def, THEN spec, of "e / (2 * content (cbox c d))"]
by (auto simp: dist_norm intro: less_imp_le)
have "∃delta>0. ∀x'∈cbox a b. norm (x' - x) < delta ⟶ norm (integral (cbox c d) (λu. f (x', u) - f (x, u))) < e"
apply (rule_tac x=dd in exI)
using dd e2_gt assms x
apply clarify
apply (rule le_less_trans [OF _ e2_less])
apply (rule integrable_bound)
apply (auto intro: integrable_diff continuous_on_imp_integrable_on_Pair1)
done
} note * = this
show ?thesis
apply (rule integrable_continuous)
apply (simp add: * continuous_on_iff dist_norm integral_diff [symmetric] continuous_on_imp_integrable_on_Pair1 [OF assms])
done
qed
lemma norm_diff2: "⟦y = y1 + y2; x = x1 + x2; e = e1 + e2; norm(y1 - x1) ≤ e1; norm(y2 - x2) ≤ e2⟧
⟹ norm(y - x) ≤ e"
using norm_triangle_mono [of "y1 - x1" "e1" "y2 - x2" "e2"]
by (simp add: add_diff_add)
lemma integral_split:
fixes f :: "'a::euclidean_space ⇒ 'b::{real_normed_vector,complete_space}"
assumes f: "f integrable_on (cbox a b)"
and k: "k ∈ Basis"
shows "integral (cbox a b) f =
integral (cbox a b ∩ {x. x∙k ≤ c}) f +
integral (cbox a b ∩ {x. x∙k ≥ c}) f"
apply (rule integral_unique [OF has_integral_split [where c=c]])
using k f
apply (auto simp: has_integral_integral [symmetric])
done
lemma integral_swap_operative:
fixes f :: "('a::euclidean_space * 'b::euclidean_space) ⇒ 'c::banach"
assumes f: "continuous_on s f" and e: "0 < e"
shows "operative(op ∧)
(λk. ∀a b c d.
cbox (a,c) (b,d) ⊆ k ∧ cbox (a,c) (b,d) ⊆ s
⟶ norm(integral (cbox (a,c) (b,d)) f -
integral (cbox a b) (λx. integral (cbox c d) (λy. f((x,y)))))
≤ e * content (cbox (a,c) (b,d)))"
proof (auto simp: operative_def)
fix a::'a and c::'b and b::'a and d::'b and u::'a and v::'a and w::'b and z::'b
assume c0: "content (cbox (a, c) (b, d)) = 0"
and cb1: "cbox (u, w) (v, z) ⊆ cbox (a, c) (b, d)"
and cb2: "cbox (u, w) (v, z) ⊆ s"
have c0': "content (cbox (u, w) (v, z)) = 0"
by (fact content_0_subset [OF c0 cb1])
show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (λx. integral (cbox w z) (λy. f (x, y))))
≤ e * content (cbox (u,w) (v,z))"
using content_cbox_pair_eq0_D [OF c0']
by (force simp add: c0')
next
fix a::'a and c::'b and b::'a and d::'b
and M::real and i::'a and j::'b
and u::'a and v::'a and w::'b and z::'b
assume ij: "(i,j) ∈ Basis"
and n1: "∀a' b' c' d'.
cbox (a',c') (b',d') ⊆ cbox (a,c) (b,d) ∧
cbox (a',c') (b',d') ⊆ {x. x ∙ (i,j) ≤ M} ∧ cbox (a',c') (b',d') ⊆ s ⟶
norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (λx. integral (cbox c' d') (λy. f (x,y))))
≤ e * content (cbox (a',c') (b',d'))"
and n2: "∀a' b' c' d'.
cbox (a',c') (b',d') ⊆ cbox (a,c) (b,d) ∧
cbox (a',c') (b',d') ⊆ {x. M ≤ x ∙ (i,j)} ∧ cbox (a',c') (b',d') ⊆ s ⟶
norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (λx. integral (cbox c' d') (λy. f (x,y))))
≤ e * content (cbox (a',c') (b',d'))"
and subs: "cbox (u,w) (v,z) ⊆ cbox (a,c) (b,d)" "cbox (u,w) (v,z) ⊆ s"
have fcont: "continuous_on (cbox (u, w) (v, z)) f"
using assms(1) continuous_on_subset subs(2) by blast
then have fint: "f integrable_on cbox (u, w) (v, z)"
by (metis integrable_continuous)
consider "i ∈ Basis" "j=0" | "j ∈ Basis" "i=0" using ij
by (auto simp: Euclidean_Space.Basis_prod_def)
then show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (λx. integral (cbox w z) (λy. f (x,y))))
≤ e * content (cbox (u,w) (v,z))" (is ?normle)
proof cases
case 1
then have e: "e * content (cbox (u, w) (v, z)) =
e * (content (cbox u v ∩ {x. x ∙ i ≤ M}) * content (cbox w z)) +
e * (content (cbox u v ∩ {x. M ≤ x ∙ i}) * content (cbox w z))"
by (simp add: content_split [where c=M] content_Pair algebra_simps)
have *: "integral (cbox u v) (λx. integral (cbox w z) (λy. f (x, y))) =
integral (cbox u v ∩ {x. x ∙ i ≤ M}) (λx. integral (cbox w z) (λy. f (x, y))) +
integral (cbox u v ∩ {x. M ≤ x ∙ i}) (λx. integral (cbox w z) (λy. f (x, y)))"
using 1 f subs integral_integrable_2dim continuous_on_subset
by (blast intro: integral_split)
show ?normle
apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e])
using 1 subs
apply (simp_all add: cbox_Pair_eq setcomp_dot1 [of "λu. M≤u"] setcomp_dot1 [of "λu. u≤M"] Sigma_Int_Paircomp1)
apply (simp_all add: interval_split ij)
apply (simp_all add: cbox_Pair_eq [symmetric] content_Pair [symmetric])
apply (force simp add: interval_split [symmetric] intro!: n1 [rule_format])
apply (force simp add: interval_split [symmetric] intro!: n2 [rule_format])
done
next
case 2
then have e: "e * content (cbox (u, w) (v, z)) =
e * (content (cbox u v) * content (cbox w z ∩ {x. x ∙ j ≤ M})) +
e * (content (cbox u v) * content (cbox w z ∩ {x. M ≤ x ∙ j}))"
by (simp add: content_split [where c=M] content_Pair algebra_simps)
have "(λx. integral (cbox w z ∩ {x. x ∙ j ≤ M}) (λy. f (x, y))) integrable_on cbox u v"
"(λx. integral (cbox w z ∩ {x. M ≤ x ∙ j}) (λy. f (x, y))) integrable_on cbox u v"
using 2 subs
apply (simp_all add: interval_split)
apply (rule_tac [!] integral_integrable_2dim [OF continuous_on_subset [OF f]])
apply (auto simp: cbox_Pair_eq interval_split [symmetric])
done
with 2 have *: "integral (cbox u v) (λx. integral (cbox w z) (λy. f (x, y))) =
integral (cbox u v) (λx. integral (cbox w z ∩ {x. x ∙ j ≤ M}) (λy. f (x, y))) +
integral (cbox u v) (λx. integral (cbox w z ∩ {x. M ≤ x ∙ j}) (λy. f (x, y)))"
by (simp add: integral_add [symmetric] integral_split [symmetric]
continuous_on_imp_integrable_on_Pair1 [OF fcont] cong: integral_cong)
show ?normle
apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e])
using 2 subs
apply (simp_all add: cbox_Pair_eq setcomp_dot2 [of "λu. M≤u"] setcomp_dot2 [of "λu. u≤M"] Sigma_Int_Paircomp2)
apply (simp_all add: interval_split ij)
apply (simp_all add: cbox_Pair_eq [symmetric] content_Pair [symmetric])
apply (force simp add: interval_split [symmetric] intro!: n1 [rule_format])
apply (force simp add: interval_split [symmetric] intro!: n2 [rule_format])
done
qed
qed
lemma integral_Pair_const:
"integral (cbox (a,c) (b,d)) (λx. k) =
integral (cbox a b) (λx. integral (cbox c d) (λy. k))"
by (simp add: content_Pair)
lemma norm_minus2: "norm (x1-x2, y1-y2) = norm (x2-x1, y2-y1)"
by (simp add: norm_minus_eqI)
lemma integral_prod_continuous:
fixes f :: "('a::euclidean_space * 'b::euclidean_space) ⇒ 'c::banach"
assumes "continuous_on (cbox (a,c) (b,d)) f" (is "continuous_on ?CBOX f")
shows "integral (cbox (a,c) (b,d)) f = integral (cbox a b) (λx. integral (cbox c d) (λy. f(x,y)))"
proof (cases "content ?CBOX = 0")
case True
then show ?thesis
by (auto simp: content_Pair)
next
case False
then have cbp: "content ?CBOX > 0"
using content_lt_nz by blast
have "norm (integral ?CBOX f - integral (cbox a b) (λx. integral (cbox c d) (λy. f (x,y)))) = 0"
proof (rule dense_eq0_I, simp)
fix e::real assume "0 < e"
with cbp have e': "0 < e / content ?CBOX"
by simp
have f_int_acbd: "f integrable_on cbox (a,c) (b,d)"
by (rule integrable_continuous [OF assms])
{ fix p
assume p: "p division_of cbox (a,c) (b,d)"
note opd1 = operative_division_and [OF integral_swap_operative [OF assms e'], THEN iffD1,
THEN spec, THEN spec, THEN spec, THEN spec, of p "(a,c)" "(b,d)" a c b d]
have "(⋀t u v w z.
⟦t ∈ p; cbox (u,w) (v,z) ⊆ t; cbox (u,w) (v,z) ⊆ cbox (a,c) (b,d)⟧ ⟹
norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (λx. integral (cbox w z) (λy. f (x,y))))
≤ e * content (cbox (u,w) (v,z)) / content?CBOX)
⟹
norm (integral ?CBOX f - integral (cbox a b) (λx. integral (cbox c d) (λy. f (x,y)))) ≤ e"
using opd1 [OF p] False by simp
} note op_acbd = this
{ fix k::real and p and u::'a and v w and z::'b and t1 t2 l
assume k: "0 < k"
and nf: "⋀x y u v.
⟦x ∈ cbox a b; y ∈ cbox c d; u ∈ cbox a b; v∈cbox c d; norm (u-x, v-y) < k⟧
⟹ norm (f(u,v) - f(x,y)) < e / (2 * (content ?CBOX))"
and p_acbd: "p tagged_division_of cbox (a,c) (b,d)"
and fine: "(λx. ball x k) fine p" "((t1,t2), l) ∈ p"
and uwvz_sub: "cbox (u,w) (v,z) ⊆ l" "cbox (u,w) (v,z) ⊆ cbox (a,c) (b,d)"
have t: "t1 ∈ cbox a b" "t2 ∈ cbox c d"
by (meson fine p_acbd cbox_Pair_iff tag_in_interval)+
have ls: "l ⊆ ball (t1,t2) k"
using fine by (simp add: fine_def Ball_def)
{ fix x1 x2
assume xuvwz: "x1 ∈ cbox u v" "x2 ∈ cbox w z"
then have x: "x1 ∈ cbox a b" "x2 ∈ cbox c d"
using uwvz_sub by auto
have "norm (x1 - t1, x2 - t2) < k"
using xuvwz ls uwvz_sub unfolding ball_def
by (force simp add: cbox_Pair_eq dist_norm norm_minus2)
then have "norm (f (x1,x2) - f (t1,t2)) ≤ e / content ?CBOX / 2"
using nf [OF t x] by simp
} note nf' = this
have f_int_uwvz: "f integrable_on cbox (u,w) (v,z)"
using f_int_acbd uwvz_sub integrable_on_subcbox by blast
have f_int_uv: "⋀x. x ∈ cbox u v ⟹ (λy. f (x,y)) integrable_on cbox w z"
using assms continuous_on_subset uwvz_sub
by (blast intro: continuous_on_imp_integrable_on_Pair1)
have 1: "norm (integral (cbox (u,w) (v,z)) f - integral (cbox (u,w) (v,z)) (λx. f (t1,t2)))
≤ e * content (cbox (u,w) (v,z)) / content ?CBOX / 2"
apply (simp only: integral_diff [symmetric] f_int_uwvz integrable_const)
apply (rule order_trans [OF integrable_bound [of "e / content ?CBOX / 2"]])
using cbp e' nf'
apply (auto simp: integrable_diff f_int_uwvz integrable_const)
done
have int_integrable: "(λx. integral (cbox w z) (λy. f (x, y))) integrable_on cbox u v"
using assms integral_integrable_2dim continuous_on_subset uwvz_sub(2) by blast
have normint_wz:
"⋀x. x ∈ cbox u v ⟹
norm (integral (cbox w z) (λy. f (x, y)) - integral (cbox w z) (λy. f (t1, t2)))
≤ e * content (cbox w z) / content (cbox (a, c) (b, d)) / 2"
apply (simp only: integral_diff [symmetric] f_int_uv integrable_const)
apply (rule order_trans [OF integrable_bound [of "e / content ?CBOX / 2"]])
using cbp e' nf'
apply (auto simp: integrable_diff f_int_uv integrable_const)
done
have "norm (integral (cbox u v)
(λx. integral (cbox w z) (λy. f (x,y)) - integral (cbox w z) (λy. f (t1,t2))))
≤ e * content (cbox w z) / content ?CBOX / 2 * content (cbox u v)"
apply (rule integrable_bound [OF _ _ normint_wz])
using cbp e'
apply (auto simp: divide_simps content_pos_le integrable_diff int_integrable integrable_const)
done
also have "... ≤ e * content (cbox (u,w) (v,z)) / content ?CBOX / 2"
by (simp add: content_Pair divide_simps)
finally have 2: "norm (integral (cbox u v) (λx. integral (cbox w z) (λy. f (x,y))) -
integral (cbox u v) (λx. integral (cbox w z) (λy. f (t1,t2))))
≤ e * content (cbox (u,w) (v,z)) / content ?CBOX / 2"
by (simp only: integral_diff [symmetric] int_integrable integrable_const)
have norm_xx: "⟦x' = y'; norm(x - x') ≤ e/2; norm(y - y') ≤ e/2⟧ ⟹ norm(x - y) ≤ e" for x::'c and y x' y' e
using norm_triangle_mono [of "x-y'" "e/2" "y'-y" "e/2"] real_sum_of_halves
by (simp add: norm_minus_commute)
have "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (λx. integral (cbox w z) (λy. f (x,y))))
≤ e * content (cbox (u,w) (v,z)) / content ?CBOX"
by (rule norm_xx [OF integral_Pair_const 1 2])
} note * = this
show "norm (integral ?CBOX f - integral (cbox a b) (λx. integral (cbox c d) (λy. f (x,y)))) ≤ e"
using compact_uniformly_continuous [OF assms compact_cbox]
apply (simp add: uniformly_continuous_on_def dist_norm)
apply (drule_tac x="e / 2 / content?CBOX" in spec)
using cbp ‹0 < e›
apply (auto simp: zero_less_mult_iff)
apply (rename_tac k)
apply (rule_tac e1=k in fine_division_exists [OF gauge_ball, where a = "(a,c)" and b = "(b,d)"])
apply assumption
apply (rule op_acbd)
apply (erule division_of_tagged_division)
using *
apply auto
done
qed
then show ?thesis
by simp
qed
lemma swap_continuous:
assumes "continuous_on (cbox (a,c) (b,d)) (λ(x,y). f x y)"
shows "continuous_on (cbox (c,a) (d,b)) (λ(x, y). f y x)"
proof -
have "(λ(x, y). f y x) = (λ(x, y). f x y) ∘ prod.swap"
by auto
then show ?thesis
apply (rule ssubst)
apply (rule continuous_on_compose)
apply (simp add: split_def)
apply (rule continuous_intros | simp add: assms)+
done
qed
lemma integral_swap_2dim:
fixes f :: "['a::euclidean_space, 'b::euclidean_space] ⇒ 'c::banach"
assumes "continuous_on (cbox (a,c) (b,d)) (λ(x,y). f x y)"
shows "integral (cbox (a, c) (b, d)) (λ(x, y). f x y) = integral (cbox (c, a) (d, b)) (λ(x, y). f y x)"
proof -
have "((λ(x, y). f x y) has_integral integral (cbox (c, a) (d, b)) (λ(x, y). f y x)) (prod.swap ` (cbox (c, a) (d, b)))"
apply (rule has_integral_twiddle [of 1 prod.swap prod.swap "λ(x,y). f y x" "integral (cbox (c, a) (d, b)) (λ(x, y). f y x)", simplified])
apply (auto simp: isCont_swap content_Pair has_integral_integral [symmetric] integrable_continuous swap_continuous assms)
done
then show ?thesis
by force
qed
theorem integral_swap_continuous:
fixes f :: "['a::euclidean_space, 'b::euclidean_space] ⇒ 'c::banach"
assumes "continuous_on (cbox (a,c) (b,d)) (λ(x,y). f x y)"
shows "integral (cbox a b) (λx. integral (cbox c d) (f x)) =
integral (cbox c d) (λy. integral (cbox a b) (λx. f x y))"
proof -
have "integral (cbox a b) (λx. integral (cbox c d) (f x)) = integral (cbox (a, c) (b, d)) (λ(x, y). f x y)"
using integral_prod_continuous [OF assms] by auto
also have "... = integral (cbox (c, a) (d, b)) (λ(x, y). f y x)"
by (rule integral_swap_2dim [OF assms])
also have "... = integral (cbox c d) (λy. integral (cbox a b) (λx. f x y))"
using integral_prod_continuous [OF swap_continuous] assms
by auto
finally show ?thesis .
qed
end