--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Sun Aug 06 18:51:32 2017 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy Sun Aug 06 18:56:47 2017 +0200
@@ -1,30 +1,13 @@
(* Title: HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
Author: Johannes Hölzl, TU München
Author: Robert Himmelmann, TU München
+ Huge cleanup by LCP
*)
theory Equivalence_Lebesgue_Henstock_Integration
imports Lebesgue_Measure Henstock_Kurzweil_Integration Complete_Measure Set_Integral
begin
-lemma finite_product_dependent: (*FIXME DELETE*)
- assumes "finite s"
- and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
- shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
- using assms
-proof induct
- case (insert x s)
- have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
- (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
- show ?case
- unfolding *
- apply (rule finite_UnI)
- using insert
- apply auto
- done
-qed auto
-
-
lemma le_left_mono: "x \<le> y \<Longrightarrow> y \<le> a \<longrightarrow> x \<le> (a::'a::preorder)"
by (auto intro: order_trans)
@@ -90,7 +73,7 @@
obtain d
where "gauge d"
and integral_f: "\<forall>p. p tagged_division_of cbox x y \<and> d fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - I) < e"
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R f x) - I) < e"
using \<open>0<e\<close> f unfolding has_integral by auto
define C where "C X m = X \<inter> {x. ball x (1/Suc m) \<subseteq> d x}" for X m
@@ -194,14 +177,14 @@
then show "\<Union>{k. \<exists>x. (x, k) \<in> ?p} = cbox x y"
using p(1) by auto
qed
- ultimately have I: "norm ((\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) - I) < e"
+ ultimately have I: "norm ((\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) - I) < e"
using integral_f by auto
- have "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
- (\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)"
+ have "(\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) =
+ (\<Sum>(x,k) \<in> ?T ` (p \<inter> s). content k *\<^sub>R f x) + (\<Sum>(x,k) \<in> p - s. content k *\<^sub>R f x)"
using p(1)[THEN tagged_division_ofD(1)]
by (safe intro!: sum.union_inter_neutral) (auto simp: s_def T_def)
- also have "(\<Sum>(x, k)\<in>?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k))"
+ also have "(\<Sum>(x,k) \<in> ?T ` (p \<inter> s). content k *\<^sub>R f x) = (\<Sum>(x,k) \<in> p \<inter> s. content k *\<^sub>R f (T X k))"
proof (subst sum.reindex_nontrivial, safe)
fix x1 x2 k assume 1: "(x1, k) \<in> p" "(x1, k) \<in> s" and 2: "(x2, k) \<in> p" "(x2, k) \<in> s"
and eq: "content k *\<^sub>R f (T X k) \<noteq> 0"
@@ -209,8 +192,8 @@
show "x1 = x2"
by (auto simp: content_eq_0_interior)
qed (use p in \<open>auto intro!: sum.cong\<close>)
- finally have eq: "(\<Sum>(x, k)\<in>?p. content k *\<^sub>R f x) =
- (\<Sum>(x, k)\<in>p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x, k)\<in>p - s. content k *\<^sub>R f x)" .
+ finally have eq: "(\<Sum>(x,k) \<in> ?p. content k *\<^sub>R f x) =
+ (\<Sum>(x,k) \<in> p \<inter> s. content k *\<^sub>R f (T X k)) + (\<Sum>(x,k) \<in> p - s. content k *\<^sub>R f x)" .
have in_T: "(x, k) \<in> s \<Longrightarrow> T X k \<in> X" for x k
using in_s[of x k] by (auto simp: C_def)
@@ -224,7 +207,7 @@
have [simp]: "finite p"
using tagged_division_ofD(1)[OF p(1)] .
- have "(M - 3*e) * (b - a) \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k) * (b - a)"
+ have "(M - 3*e) * (b - a) \<le> (\<Sum>(x,k) \<in> p \<inter> s. content k) * (b - a)"
proof (intro mult_right_mono)
have fin: "?\<mu> (E \<inter> \<Union>{k\<in>snd`p. k \<inter> C X m = {}}) < \<infinity>" for X
using \<open>?\<mu> E < \<infinity>\<close> by (rule le_less_trans[rotated]) (auto intro!: emeasure_mono \<open>E \<in> sets ?L\<close>)
@@ -287,15 +270,15 @@
finally show "M - 3 * e \<le> (\<Sum>(x, y)\<in>p \<inter> s. content y)"
using \<open>0 < e\<close> by (simp add: split_beta)
qed (use \<open>a < b\<close> in auto)
- also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * (b - a))"
+ also have "\<dots> = (\<Sum>(x,k) \<in> p \<inter> s. content k * (b - a))"
by (simp add: sum_distrib_right split_beta')
- also have "\<dots> \<le> (\<Sum>(x, k)\<in>p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
+ also have "\<dots> \<le> (\<Sum>(x,k) \<in> p \<inter> s. content k * (f (T ?F k) - f (T ?E k)))"
using parts(3) by (auto intro!: sum_mono mult_left_mono diff_mono)
- also have "\<dots> = (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x, k)\<in>p \<inter> s. content k * f (T ?E k))"
+ also have "\<dots> = (\<Sum>(x,k) \<in> p \<inter> s. content k * f (T ?F k)) - (\<Sum>(x,k) \<in> p \<inter> s. content k * f (T ?E k))"
by (auto intro!: sum.cong simp: field_simps sum_subtractf[symmetric])
- also have "\<dots> = (\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x)"
+ also have "\<dots> = (\<Sum>(x,k) \<in> ?B. content k *\<^sub>R f x) - (\<Sum>(x,k) \<in> ?A. content k *\<^sub>R f x)"
by (subst (1 2) parts) auto
- also have "\<dots> \<le> norm ((\<Sum>(x, k)\<in>?B. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>?A. content k *\<^sub>R f x))"
+ also have "\<dots> \<le> norm ((\<Sum>(x,k) \<in> ?B. content k *\<^sub>R f x) - (\<Sum>(x,k) \<in> ?A. content k *\<^sub>R f x))"
by auto
also have "\<dots> \<le> e + e"
using parts(1)[of ?E] parts(1)[of ?F] by (intro norm_diff_triangle_le[of _ I]) auto
@@ -1202,13 +1185,13 @@
assume "0 < e"
have "S \<in> lmeasurable"
using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets fmeasurableI_null_sets)
- have e22: "0 < e / 2 / (2 * B * real DIM('M)) ^ DIM('N)"
+ have e22: "0 < e/2 / (2 * B * real DIM('M)) ^ DIM('N)"
using \<open>0 < e\<close> \<open>0 < B\<close> by (simp add: divide_simps)
obtain T
where "open T" "S \<subseteq> T" "T \<in> lmeasurable"
- and "measure lebesgue T \<le> measure lebesgue S + e / 2 / (2 * B * DIM('M)) ^ DIM('N)"
+ and "measure lebesgue T \<le> measure lebesgue S + e/2 / (2 * B * DIM('M)) ^ DIM('N)"
by (rule lmeasurable_outer_open [OF \<open>S \<in> lmeasurable\<close> e22])
- then have T: "measure lebesgue T \<le> e / 2 / (2 * B * DIM('M)) ^ DIM('N)"
+ then have T: "measure lebesgue T \<le> e/2 / (2 * B * DIM('M)) ^ DIM('N)"
using \<open>negligible S\<close> by (simp add: negligible_iff_null_sets measure_eq_0_null_sets)
have "\<exists>r. 0 < r \<and> r \<le> 1/2 \<and>
(x \<in> S \<longrightarrow> (\<forall>y. norm(y - x) < r
@@ -1289,7 +1272,7 @@
qed
have countbl: "countable (fbx ` \<D>)"
using \<open>countable \<D>\<close> by blast
- have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> e / 2" if "\<D>' \<subseteq> \<D>" "finite \<D>'" for \<D>'
+ have "(\<Sum>k\<in>fbx`\<D>'. measure lebesgue k) \<le> e/2" if "\<D>' \<subseteq> \<D>" "finite \<D>'" for \<D>'
proof -
have BM_ge0: "0 \<le> B * (DIM('M) * prj1 (vf X - uf X))" if "X \<in> \<D>'" for X
using \<open>0 < B\<close> \<open>\<D>' \<subseteq> \<D>\<close> that vu_pos by fastforce
@@ -1333,7 +1316,7 @@
qed
also have "\<dots> = (2 * B * DIM('M)) ^ DIM('N) * sum (measure lebesgue) \<D>'"
by (simp add: sum_distrib_left)
- also have "\<dots> \<le> e / 2"
+ also have "\<dots> \<le> e/2"
proof -
have div: "\<D>' division_of \<Union>\<D>'"
apply (auto simp: \<open>finite \<D>'\<close> \<open>{} \<notin> \<D>'\<close> division_of_def)
@@ -1366,13 +1349,13 @@
using \<open>0 < B\<close>
apply (simp add: algebra_simps)
done
- also have "\<dots> \<le> e / 2"
+ also have "\<dots> \<le> e/2"
using T \<open>0 < B\<close> by (simp add: field_simps)
finally show ?thesis .
qed
finally show ?thesis .
qed
- then have e2: "sum (measure lebesgue) \<G> \<le> e / 2" if "\<G> \<subseteq> fbx ` \<D>" "finite \<G>" for \<G>
+ then have e2: "sum (measure lebesgue) \<G> \<le> e/2" if "\<G> \<subseteq> fbx ` \<D>" "finite \<G>" for \<G>
by (metis finite_subset_image that)
show "\<exists>W\<in>lmeasurable. f ` S \<subseteq> W \<and> measure lebesgue W < e"
proof (intro bexI conjI)
@@ -1616,28 +1599,27 @@
apply (rule norm_triangle_ineq3)
done
-text\<open>FIXME: needs refactoring and use of Sigma\<close>
-lemma bounded_variation_absolutely_integrable_interval:
+proposition bounded_variation_absolutely_integrable_interval:
fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
assumes f: "f integrable_on cbox a b"
- and *: "\<forall>d. d division_of (cbox a b) \<longrightarrow> sum (\<lambda>k. norm(integral k f)) d \<le> B"
+ and *: "\<And>d. d division_of (cbox a b) \<Longrightarrow> sum (\<lambda>K. norm(integral K f)) d \<le> B"
shows "f absolutely_integrable_on cbox a b"
proof -
- let ?f = "\<lambda>d. \<Sum>k\<in>d. norm (integral k f)" and ?D = "{d. d division_of (cbox a b)}"
+ let ?f = "\<lambda>d. \<Sum>K\<in>d. norm (integral K f)" and ?D = "{d. d division_of (cbox a b)}"
have D_1: "?D \<noteq> {}"
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"
- have *: "\<exists>d. gauge d \<and>
+ have *: "\<exists>\<gamma>. gauge \<gamma> \<and>
(\<forall>p. p tagged_division_of cbox a b \<and>
- d fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - ?S) < e)"
+ \<gamma> fine p \<longrightarrow>
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e)"
if e: "e > 0" for e
proof -
- have "?S - e / 2 < ?S" using \<open>e > 0\<close> by simp
- then obtain d where d: "d division_of (cbox a b)" "?S - e / 2 < (\<Sum>k\<in>d. norm (integral k f))"
+ have "?S - e/2 < ?S" using \<open>e > 0\<close> by simp
+ then obtain d where d: "d division_of (cbox a b)" "?S - e/2 < (\<Sum>k\<in>d. norm (integral k f))"
unfolding less_cSUP_iff[OF D] by auto
note d' = division_ofD[OF this(1)]
@@ -1661,72 +1643,65 @@
by metis
have "e/2 > 0"
using e by auto
- from henstock_lemma[OF assms(1) this]
- obtain g where g: "gauge g"
- "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; g fine p\<rbrakk>
- \<Longrightarrow> (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
+ with henstock_lemma[OF f]
+ obtain \<gamma> where g: "gauge \<gamma>"
+ "\<And>p. \<lbrakk>p tagged_partial_division_of cbox a b; \<gamma> fine p\<rbrakk>
+ \<Longrightarrow> (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x - integral k f)) < e/2"
by (metis (no_types, lifting))
- let ?g = "\<lambda>x. g x \<inter> ball x (k x)"
+ let ?g = "\<lambda>x. \<gamma> x \<inter> ball x (k x)"
show ?thesis
- apply (rule_tac x="?g" in exI)
- apply safe
- proof -
+ proof (intro exI conjI allI impI)
show "gauge ?g"
- using g(1) k(1)
- unfolding gauge_def
- by auto
+ using g(1) k(1) by (auto simp: gauge_def)
+ next
fix p
- assume "p tagged_division_of (cbox a b)" and "?g fine p"
- note p = this(1) conjunctD2[OF this(2)[unfolded fine_Int]]
+ assume "p tagged_division_of (cbox a b) \<and> ?g fine p"
+ then have p: "p tagged_division_of cbox a b" "\<gamma> fine p" "(\<lambda>x. ball x (k x)) fine p"
+ by (auto simp: fine_Int)
note p' = tagged_division_ofD[OF p(1)]
define p' where "p' = {(x,k) | x k. \<exists>i l. x \<in> i \<and> i \<in> d \<and> (x,l) \<in> p \<and> k = i \<inter> l}"
- have gp': "g fine p'"
- using p(2)
- unfolding p'_def fine_def
- by auto
+ have gp': "\<gamma> fine p'"
+ using p(2) by (auto simp: p'_def fine_def)
have p'': "p' tagged_division_of (cbox a b)"
proof (rule tagged_division_ofI)
show "finite p'"
- apply (rule finite_subset[of _ "(\<lambda>(k,(x,l)). (x,k \<inter> l)) `
- {(k,xl) | k xl. k \<in> d \<and> xl \<in> 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) \<in> p'"
- then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l"
+ proof (rule finite_subset)
+ show "p' \<subseteq> (\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p)"
+ by (force simp: p'_def image_iff)
+ show "finite ((\<lambda>(k, x, l). (x, k \<inter> l)) ` (d \<times> p))"
+ by (simp add: d'(1) p'(1))
+ qed
+ next
+ fix x K
+ assume "(x, K) \<in> p'"
+ then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> K = i \<inter> l"
unfolding p'_def by auto
- then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "k = i \<inter> l" by blast
- show "x \<in> k" and "k \<subseteq> cbox a b"
+ then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "K = i \<inter> l" by blast
+ show "x \<in> K" and "K \<subseteq> cbox a b"
using p'(2-3)[OF il(3)] il by auto
- show "\<exists>a b. k = cbox a b"
+ show "\<exists>a b. K = cbox a b"
unfolding il using p'(4)[OF il(3)] d'(4)[OF il(2)]
by (meson Int_interval)
next
- fix x1 k1
- assume "(x1, k1) \<in> p'"
- then have "\<exists>i l. x1 \<in> i \<and> i \<in> d \<and> (x1, l) \<in> p \<and> k1 = i \<inter> l"
+ fix x1 K1
+ assume "(x1, K1) \<in> p'"
+ then have "\<exists>i l. x1 \<in> i \<and> i \<in> d \<and> (x1, l) \<in> p \<and> K1 = i \<inter> l"
unfolding p'_def by auto
- then obtain i1 l1 where il1: "x1 \<in> i1" "i1 \<in> d" "(x1, l1) \<in> p" "k1 = i1 \<inter> l1" by blast
- fix x2 k2
- assume "(x2,k2)\<in>p'"
- then have "\<exists>i l. x2 \<in> i \<and> i \<in> d \<and> (x2, l) \<in> p \<and> k2 = i \<inter> l"
+ then obtain i1 l1 where il1: "x1 \<in> i1" "i1 \<in> d" "(x1, l1) \<in> p" "K1 = i1 \<inter> l1" by blast
+ fix x2 K2
+ assume "(x2,K2) \<in> p'"
+ then have "\<exists>i l. x2 \<in> i \<and> i \<in> d \<and> (x2, l) \<in> p \<and> K2 = i \<inter> l"
unfolding p'_def by auto
- then obtain i2 l2 where il2: "x2 \<in> i2" "i2 \<in> d" "(x2, l2) \<in> p" "k2 = i2 \<inter> l2" by blast
- assume "(x1, k1) \<noteq> (x2, k2)"
+ then obtain i2 l2 where il2: "x2 \<in> i2" "i2 \<in> d" "(x2, l2) \<in> p" "K2 = i2 \<inter> l2" by blast
+ assume "(x1, K1) \<noteq> (x2, K2)"
then have "interior i1 \<inter> interior i2 = {} \<or> interior l1 \<inter> interior l2 = {}"
using d'(5)[OF il1(2) il2(2)] p'(5)[OF il1(3) il2(3)] by (auto simp: il1 il2)
- then show "interior k1 \<inter> interior k2 = {}"
+ then show "interior K1 \<inter> interior K2 = {}"
unfolding il1 il2 by auto
next
have *: "\<forall>(x, X) \<in> p'. X \<subseteq> cbox a b"
- unfolding p'_def using d' by auto
- have "y \<in> \<Union>{k. \<exists>x. (x, k) \<in> p'}" if y: "y \<in> cbox a b" for y
+ unfolding p'_def using d' by blast
+ have "y \<in> \<Union>{K. \<exists>x. (x, K) \<in> p'}" if y: "y \<in> cbox a b" for y
proof -
obtain x l where xl: "(x, l) \<in> p" "y \<in> l"
using y unfolding p'(6)[symmetric] by auto
@@ -1735,10 +1710,10 @@
have "x \<in> i"
using fineD[OF p(3) xl(1)] using k(2) i xl by auto
then show ?thesis
- apply (rule_tac X="i \<inter> l" in UnionI)
- using i xl by (auto simp: p'_def)
+ unfolding p'_def
+ by (rule_tac X="i \<inter> l" in UnionI) (use i xl in auto)
qed
- show "\<Union>{k. \<exists>x. (x, k) \<in> p'} = cbox a b"
+ show "\<Union>{K. \<exists>x. (x, K) \<in> p'} = cbox a b"
proof
show "\<Union>{k. \<exists>x. (x, k) \<in> p'} \<subseteq> cbox a b"
using * by auto
@@ -1747,26 +1722,23 @@
proof
fix y
assume y: "y \<in> cbox a b"
- obtain x l where xl: "(x, l) \<in> p" "y \<in> l"
+ obtain x L where xl: "(x, L) \<in> p" "y \<in> L"
using y unfolding p'(6)[symmetric] by auto
- obtain i where i: "i \<in> d" "y \<in> i"
+ obtain I where i: "I \<in> d" "y \<in> I"
using y unfolding d'(6)[symmetric] by auto
- have "x \<in> i"
+ have "x \<in> I"
using fineD[OF p(3) xl(1)] using k(2) i xl by auto
then show "y \<in> \<Union>{k. \<exists>x. (x, k) \<in> p'}"
- apply (rule_tac X="i \<inter> l" in UnionI)
+ apply (rule_tac X="I \<inter> L" in UnionI)
using i xl by (auto simp: p'_def)
qed
qed
qed
- then have sum_less_e2: "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x - integral k f)) < e / 2"
+ then have sum_less_e2: "(\<Sum>(x,K) \<in> p'. norm (content K *\<^sub>R f x - integral K f)) < e/2"
using g(2) gp' tagged_division_of_def by blast
- then have XXX: "\<bar>(\<Sum>(x,k)\<in>p'. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e / 2"
- unfolding split_def
- using p'' by (force intro!: absdiff_norm_less)
- have p'alt: "p' = {(x,(i \<inter> l)) | x i l. (x,l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}}"
+ have p'alt: "p' = {(x, I \<inter> L) | x I L. (x,L) \<in> p \<and> I \<in> d \<and> I \<inter> L \<noteq> {}}"
proof (safe, goal_cases)
case prems: (2 _ _ x i l)
have "x \<in> i"
@@ -1783,79 +1755,75 @@
then show ?case
using prems(3) by auto
next
- fix x k
- assume "(x, k) \<in> p'"
- then have "\<exists>i l. x \<in> i \<and> i \<in> d \<and> (x, l) \<in> p \<and> k = i \<inter> l"
+ fix x K
+ assume "(x, K) \<in> p'"
+ then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "K = i \<inter> l"
unfolding p'_def by auto
- then obtain i l where il: "x \<in> i" "i \<in> d" "(x, l) \<in> p" "k = i \<inter> l" by blast
- then show "\<exists>y i l. (x, k) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
+ then show "\<exists>y i l. (x, K) = (y, i \<inter> l) \<and> (y, l) \<in> p \<and> i \<in> d \<and> i \<inter> l \<noteq> {}"
using p'(2) by fastforce
qed
- have sum_p': "(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
+ have sum_p': "(\<Sum>(x,K) \<in> p'. norm (integral K f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
apply (subst sum.over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
apply (auto intro: integral_null simp: content_eq_0_interior)
done
- note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]]
-
+ have snd_p_div: "snd ` p division_of cbox a b"
+ by (rule division_of_tagged_division[OF p(1)])
+ note snd_p = division_ofD[OF snd_p_div]
+ have fin_d_sndp: "finite (d \<times> snd ` p)"
+ by (simp add: d'(1) snd_p(1))
- have *: "\<And>sni sni' sf sf'. \<lbrakk>\<bar>sf' - sni'\<bar> < e / 2; ?S - e / 2 < sni; sni' \<le> ?S;
- sni \<le> sni'; sf' = sf\<rbrakk> \<Longrightarrow> \<bar>sf - ?S\<bar> < e"
+ have *: "\<And>sni sni' sf sf'. \<lbrakk>\<bar>sf' - sni'\<bar> < e/2; ?S - e/2 < sni; sni' \<le> ?S;
+ sni \<le> sni'; sf' = sf\<rbrakk> \<Longrightarrow> \<bar>sf - ?S\<bar> < e"
by arith
- show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - ?S) < e"
+ show "norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - ?S) < e"
unfolding real_norm_def
proof (rule *)
- show "\<bar>(\<Sum>(x,k)\<in>p'. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e / 2"
- by (rule XXX)
- show "(\<Sum>(x, k)\<in>p'. norm (integral k f)) \<le>?S"
+ show "\<bar>(\<Sum>(x,K)\<in>p'. norm (content K *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p'. norm (integral k f))\<bar> < e/2"
+ using p'' sum_less_e2 unfolding split_def by (force intro!: absdiff_norm_less)
+ show "(\<Sum>(x,k) \<in> p'. norm (integral k f)) \<le>?S"
by (auto simp: sum_p' division_of_tagged_division[OF p''] D intro!: cSUP_upper)
- show "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>(x, k)\<in>p'. norm (integral k f))"
+ show "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>(x,k) \<in> p'. norm (integral k f))"
proof -
- have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} =
- (\<lambda>(k,l). k \<inter> l) ` {(k,l)|k l. k \<in> d \<and> l \<in> snd ` p}"
+ have *: "{k \<inter> l | k l. k \<in> d \<and> l \<in> snd ` p} = (\<lambda>(k,l). k \<inter> l) ` (d \<times> snd ` p)"
by auto
- have "(\<Sum>k\<in>d. norm (integral k f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
- proof (rule sum_mono, goal_cases)
- case k: (1 k)
- from d'(4)[OF this] obtain u v where uv: "k = cbox u v" by metis
+ have "(\<Sum>K\<in>d. norm (integral K f)) \<le> (\<Sum>i\<in>d. \<Sum>l\<in>snd ` p. norm (integral (i \<inter> l) f))"
+ proof (rule sum_mono)
+ fix K assume k: "K \<in> d"
+ from d'(4)[OF this] obtain u v where uv: "K = cbox u v" by metis
define d' where "d' = {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}"
- note uvab = d'(2)[OF k[unfolded uv]]
+ have uvab: "cbox u v \<subseteq> cbox a b"
+ using d(1) k uv by blast
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) \<le> sum (\<lambda>k. norm (integral k f)) d'"
- unfolding uv
+ unfolding d'_def by (rule division_inter_1 [OF snd_p_div uvab])
+ moreover then have "norm (\<Sum>i\<in>d'. integral i f) \<le> (\<Sum>k\<in>d'. norm (integral k f))"
+ by (simp add: sum_norm_le)
+ ultimately have "norm (integral K f) \<le> sum (\<lambda>k. norm (integral k f)) d'"
apply (subst integral_combine_division_topdown[of _ _ d'])
- apply (rule integrable_on_subcbox[OF assms(1) uvab])
- apply assumption
- apply (rule sum_norm_le)
- apply auto
+ apply (auto simp: uv intro: integrable_on_subcbox[OF assms(1) uvab])
done
- also have "\<dots> = (\<Sum>k\<in>{k \<inter> l |l. l \<in> snd ` p}. norm (integral k f))"
+ also have "\<dots> = (\<Sum>I\<in>{K \<inter> L |L. L \<in> snd ` p}. norm (integral I f))"
proof -
- have *: "norm (integral i f) = 0"
- if "i \<in> {cbox u v \<inter> l |l. l \<in> snd ` p}"
- "i \<notin> {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}" for i
+ have *: "norm (integral I f) = 0"
+ if "I \<in> {cbox u v \<inter> l |l. l \<in> snd ` p}"
+ "I \<notin> {cbox u v \<inter> l |l. l \<in> snd ` p \<and> cbox u v \<inter> l \<noteq> {}}" for I
using that by auto
show ?thesis
apply (rule sum.mono_neutral_left)
apply (simp add: snd_p(1))
unfolding d'_def uv using * by auto
qed
- also have "\<dots> = (\<Sum>l\<in>snd ` p. norm (integral (k \<inter> l) f))"
+ also have "\<dots> = (\<Sum>l\<in>snd ` p. norm (integral (K \<inter> l) f))"
proof -
- have *: "norm (integral (k \<inter> l) f) = 0"
- if "l \<in> snd ` p" "y \<in> snd ` p" "l \<noteq> y" "k \<inter> l = k \<inter> y" for l y
+ have *: "norm (integral (K \<inter> l) f) = 0"
+ if "l \<in> snd ` p" "y \<in> snd ` p" "l \<noteq> y" "K \<inter> l = K \<inter> y" for l y
proof -
- have "interior (k \<inter> l) \<subseteq> interior (l \<inter> y)"
+ have "interior (K \<inter> l) \<subseteq> interior (l \<inter> y)"
by (metis Int_lower2 interior_mono le_inf_iff that(4))
- then have "interior (k \<inter> l) = {}"
+ then have "interior (K \<inter> l) = {}"
by (simp add: snd_p(5) that)
moreover from d'(4)[OF k] snd_p(4)[OF that(1)]
obtain u1 v1 u2 v2
- where uv: "k = cbox u1 u2" "l = cbox v1 v2" by metis
+ where uv: "K = cbox u1 u2" "l = cbox v1 v2" by metis
ultimately show ?thesis
using that integral_null
unfolding uv Int_interval content_eq_0_interior
@@ -1868,12 +1836,12 @@
apply (rule p')
using * by auto
qed
- finally show ?case .
+ finally show "norm (integral K f) \<le> (\<Sum>l\<in>snd ` p. norm (integral (K \<inter> l) f))" .
qed
- also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> (snd ` p). norm (integral (i\<inter>l) f))"
+ also have "\<dots> = (\<Sum>(i,l) \<in> d \<times> snd ` p. norm (integral (i\<inter>l) f))"
by (simp add: sum.cartesian_product)
- also have "\<dots> = (\<Sum>x\<in>{(i, l) |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral (case_prod op \<inter> x) f))"
- by (force simp: split_def Sigma_def intro!: sum.cong)
+ also have "\<dots> = (\<Sum>x \<in> d \<times> snd ` p. norm (integral (case_prod op \<inter> x) f))"
+ by (force simp: split_def intro!: sum.cong)
also have "\<dots> = (\<Sum>k\<in>{i \<inter> l |i l. i \<in> d \<and> l \<in> snd ` p}. norm (integral k f))"
proof -
have eq0: " (integral (l1 \<inter> k1) f) = 0"
@@ -1896,142 +1864,123 @@
qed
show ?thesis
unfolding *
- apply (rule sum.reindex_nontrivial [symmetric, unfolded o_def])
- apply (rule finite_product_dependent [OF \<open>finite d\<close>])
- apply (rule finite_imageI [OF \<open>finite p\<close>])
+ apply (rule sum.reindex_nontrivial [OF fin_d_sndp, symmetric, unfolded o_def])
apply clarsimp
by (metis eq0 fst_conv snd_conv)
qed
- also have "\<dots> = (\<Sum>(x, k)\<in>p'. norm (integral k f))"
+ also have "\<dots> = (\<Sum>(x,k) \<in> p'. norm (integral k f))"
proof -
have 0: "integral (ia \<inter> snd (a, b)) f = 0"
- if "ia \<inter> snd (a, b) \<notin> snd ` p'" "ia \<in> d" "(a, b) \<in> p" for ia a b
+ if "ia \<inter> snd (a, b) \<notin> snd ` p'" "ia \<in> d" "(a, b) \<in> p" for ia a b
proof -
have "ia \<inter> b = {}"
- using that
- unfolding p'alt image_iff Bex_def not_ex
+ using that unfolding p'alt image_iff Bex_def not_ex
apply (erule_tac x="(a, ia \<inter> b)" in allE)
apply auto
done
- then show ?thesis
- by auto
+ then show ?thesis by auto
qed
have 1: "\<exists>i l. snd (a, b) = i \<inter> l \<and> i \<in> d \<and> l \<in> snd ` p" if "(a, b) \<in> p'" for a b
- proof -
- have "\<exists>n N Na. (a, b) = (n, N \<inter> Na) \<and> (n, Na) \<in> p \<and> N \<in> d \<and> N \<inter> Na \<noteq> {}"
- using that p'alt by blast
- then show "\<exists>N Na. snd (a, b) = N \<inter> Na \<and> N \<in> d \<and> Na \<in> snd ` p"
- by (metis (no_types) imageI prod.sel(2))
- qed
+ using that
+ apply (clarsimp simp: p'_def image_iff)
+ by (metis (no_types, hide_lams) snd_conv)
show ?thesis
unfolding sum_p'
apply (rule sum.mono_neutral_right)
- apply (subst *)
- apply (rule finite_imageI[OF finite_product_dependent])
- apply fact
- apply (rule finite_imageI[OF p'(1)])
+ apply (metis * finite_imageI[OF fin_d_sndp])
using 0 1 by auto
qed
finally show ?thesis .
qed
- show "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))"
+ show "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
proof -
let ?S = "{(x, i \<inter> l) |x i l. (x, l) \<in> p \<and> i \<in> d}"
have *: "?S = (\<lambda>(xl,i). (fst xl, snd xl \<inter> i)) ` (p \<times> d)"
by force
- have pdfin: "finite (p \<times> d)"
+ have fin_pd: "finite (p \<times> d)"
using finite_cartesian_product[OF p'(1) d'(1)] by metis
- have "(\<Sum>(x, k)\<in>p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x, k)\<in>?S. \<bar>content k\<bar> * norm (f x))"
+ have "(\<Sum>(x,k) \<in> p'. norm (content k *\<^sub>R f x)) = (\<Sum>(x,k) \<in> ?S. \<bar>content k\<bar> * norm (f x))"
unfolding norm_scaleR
apply (rule sum.mono_neutral_left)
apply (subst *)
- apply (rule finite_imageI)
- apply fact
- unfolding p'alt apply blast
+ apply (rule finite_imageI [OF fin_pd])
+ unfolding p'alt apply auto
by fastforce
also have "\<dots> = (\<Sum>((x,l),i)\<in>p \<times> d. \<bar>content (l \<inter> i)\<bar> * norm (f x))"
- unfolding *
- apply (subst sum.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) \<in> p" "(x2, l2) \<in> p" "k1 \<in> d" "k2 \<in> d"
- "x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "\<not> ((x1 = x2 \<and> l1 = l2) \<and> 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 \<noteq> l2 \<or> k1 \<noteq> k2"
- by auto
- then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> 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
+ have "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
+ if "(x1, l1) \<in> p" "(x2, l2) \<in> p" "k1 \<in> d" "k2 \<in> d"
+ "x1 = x2" "l1 \<inter> k1 = l2 \<inter> k2" "x1 \<noteq> x2 \<or> l1 \<noteq> l2 \<or> k1 \<noteq> k2"
+ for x1 l1 k1 x2 l2 k2
+ proof -
+ obtain u1 v1 u2 v2 where uv: "k1 = cbox u1 u2" "l1 = cbox v1 v2"
+ by (meson \<open>(x1, l1) \<in> p\<close> \<open>k1 \<in> d\<close> d(1) division_ofD(4) p'(4))
+ have "l1 \<noteq> l2 \<or> k1 \<noteq> k2"
+ using that by auto
+ then have "interior k1 \<inter> interior k2 = {} \<or> interior l1 \<inter> interior l2 = {}"
+ apply (rule disjE)
+ using that p'(5) d'(5) by auto
+ moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
+ unfolding that ..
+ ultimately have "interior (l1 \<inter> k1) = {}"
+ by auto
+ then show "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
+ unfolding uv Int_interval content_eq_0_interior[symmetric] by auto
+ qed
+ then show ?thesis
+ unfolding *
+ apply (subst sum.reindex_nontrivial [OF fin_pd])
+ unfolding split_paired_all o_def split_def prod.inject
+ apply force+
done
- moreover have "interior (l1 \<inter> k1) = interior (l2 \<inter> k2)"
- unfolding as ..
- ultimately have "interior (l1 \<inter> k1) = {}"
- by auto
- then show "\<bar>content (l1 \<inter> k1)\<bar> * norm (f x1) = 0"
- unfolding uv Int_interval
- unfolding content_eq_0_interior[symmetric]
- by auto
- qed safe
- also have "\<dots> = (\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x))"
- apply (subst sum_Sigma_product[symmetric])
- apply (rule p')
- apply (rule d')
- apply (rule sum.cong)
- apply (rule refl)
- unfolding split_paired_all split_conv
+ qed
+ also have "\<dots> = (\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x))"
proof -
- fix x l
- assume as: "(x, l) \<in> p"
- note xl = p'(2-4)[OF this]
- from this(3) guess u v by (elim exE) note uv=this
- have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
- by (simp add: Int_commute uv)
- also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
+ have sumeq: "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
+ if "(x, l) \<in> p" for x l
proof -
- have eq0: "content (k \<inter> cbox u v) = 0"
- if "k \<in> d" "y \<in> d" "k \<noteq> y" and eq: "k \<inter> cbox u v = y \<inter> cbox u v" for k y
+ note xl = p'(2-4)[OF that]
+ then obtain u v where uv: "l = cbox u v" by blast
+ have "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar>) = (\<Sum>k\<in>d. content (k \<inter> cbox u v))"
+ by (simp add: Int_commute uv)
+ also have "\<dots> = sum content {k \<inter> cbox u v| k. k \<in> d}"
proof -
- from d'(4)[OF that(1)] d'(4)[OF that(2)]
- obtain \<alpha> \<beta> where \<alpha>: "k \<inter> cbox u v = cbox \<alpha> \<beta>"
- by (meson Int_interval)
- have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
- by (simp add: d'(5) that)
- also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
- by auto
- also have "\<dots> = interior (k \<inter> cbox u v)"
- unfolding eq by auto
- finally show ?thesis
- unfolding \<alpha> content_eq_0_interior ..
+ have eq0: "content (k \<inter> cbox u v) = 0"
+ if "k \<in> d" "y \<in> d" "k \<noteq> y" and eq: "k \<inter> cbox u v = y \<inter> cbox u v" for k y
+ proof -
+ from d'(4)[OF that(1)] d'(4)[OF that(2)]
+ obtain \<alpha> \<beta> where \<alpha>: "k \<inter> cbox u v = cbox \<alpha> \<beta>"
+ by (meson Int_interval)
+ have "{} = interior ((k \<inter> y) \<inter> cbox u v)"
+ by (simp add: d'(5) that)
+ also have "\<dots> = interior (y \<inter> (k \<inter> cbox u v))"
+ by auto
+ also have "\<dots> = interior (k \<inter> cbox u v)"
+ unfolding eq by auto
+ finally show ?thesis
+ unfolding \<alpha> content_eq_0_interior ..
+ qed
+ then show ?thesis
+ unfolding Setcompr_eq_image
+ apply (rule sum.reindex_nontrivial [OF \<open>finite d\<close>, unfolded o_def, symmetric])
+ by auto
+ qed
+ also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
+ apply (rule sum.mono_neutral_right)
+ unfolding Setcompr_eq_image
+ apply (rule finite_imageI [OF \<open>finite d\<close>])
+ apply (fastforce simp: inf.commute)+
+ done
+ finally show "(\<Sum>i\<in>d. content (l \<inter> i) * norm (f x)) = content l * norm (f x)"
+ unfolding sum_distrib_right[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
- then show ?thesis
- unfolding Setcompr_eq_image
- apply (rule sum.reindex_nontrivial [OF \<open>finite d\<close>, unfolded o_def, symmetric])
- by auto
+ show ?thesis
+ by (subst sum_Sigma_product[symmetric]) (auto intro!: sumeq sum.cong p' d')
qed
- also have "\<dots> = sum content {cbox u v \<inter> k |k. k \<in> d \<and> cbox u v \<inter> k \<noteq> {}}"
- apply (rule sum.mono_neutral_right)
- unfolding Setcompr_eq_image
- apply (rule finite_imageI [OF \<open>finite d\<close>])
- apply (fastforce simp: inf.commute)+
- done
- finally show "(\<Sum>i\<in>d. \<bar>content (l \<inter> i)\<bar> * norm (f x)) = content l *\<^sub>R norm (f x)"
- unfolding sum_distrib_right[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 ?thesis .
+ finally show ?thesis .
qed
qed (rule d)
qed
@@ -2147,23 +2096,23 @@
using \<open>e > 0\<close> by auto
from * [OF this] obtain d1 where
d1: "gauge d1" "\<forall>p. p tagged_division_of (cbox a b) \<and> d1 fine p \<longrightarrow>
- norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm (f x))) < e / 2"
+ norm ((\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm (f x))) < e/2"
by auto
from henstock_lemma [OF f(1) \<open>e/2>0\<close>] obtain d2 where
d2: "gauge d2" "\<forall>p. p tagged_partial_division_of (cbox a b) \<and> d2 fine p \<longrightarrow>
- (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x - integral k f)) < e / 2" .
+ (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>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_Int [OF d1(1) d2(1)], of a b])
(auto simp add: fine_Int)
- have *: "\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> ?S \<longrightarrow> \<bar>sf - si\<bar> < e / 2 \<longrightarrow>
- \<bar>sf' - di\<bar> < e / 2 \<longrightarrow> di < ?S + e"
+ have *: "\<And>sf sf' si di. sf' = sf \<longrightarrow> si \<le> ?S \<longrightarrow> \<bar>sf - si\<bar> < e/2 \<longrightarrow>
+ \<bar>sf' - di\<bar> < e/2 \<longrightarrow> di < ?S + e"
by arith
show "integral (cbox a b) (\<lambda>x. if x \<in> UNIV then norm (f x) else 0) < ?S + e"
apply (subst if_P)
apply rule
proof (rule *[rule_format])
- show "\<bar>(\<Sum>(x,k)\<in>p. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p. norm (integral k f))\<bar> < e / 2"
+ show "\<bar>(\<Sum>(x,k)\<in>p. norm (content k *\<^sub>R f x)) - (\<Sum>(x,k)\<in>p. norm (integral k f))\<bar> < e/2"
unfolding split_def
apply (rule absdiff_norm_less)
using d2(2)[rule_format,of p]
@@ -2171,19 +2120,16 @@
unfolding tagged_division_of_def split_def
apply auto
done
- show "\<bar>(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm(f x))\<bar> < e / 2"
+ show "\<bar>(\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) - integral (cbox a b) (\<lambda>x. norm(f x))\<bar> < e/2"
using d1(2)[rule_format,OF conjI[OF p(1,2)]]
by (simp only: real_norm_def)
- show "(\<Sum>(x, k)\<in>p. content k *\<^sub>R norm (f x)) = (\<Sum>(x, k)\<in>p. norm (content k *\<^sub>R f x))"
+ show "(\<Sum>(x,k) \<in> p. content k *\<^sub>R norm (f x)) = (\<Sum>(x,k) \<in> p. norm (content k *\<^sub>R f x))"
apply (rule sum.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 "(\<Sum>(x, k)\<in>p. norm (integral k f)) \<le> ?S"
+ by simp
+ show "(\<Sum>(x,k) \<in> p. norm (integral k f)) \<le> ?S"
using partial_division_of_tagged_division[of p "cbox a b"] p(1)
apply (subst sum.over_tagged_division_lemma[OF p(1)])
apply (simp add: content_eq_0_interior)
@@ -2656,11 +2602,12 @@
using f nonneg by (intro DERIV_nonneg_imp_nondecreasing[of x y F]) (auto intro: order_trans)
then have F_le_T: "a \<le> x \<Longrightarrow> F x \<le> T" for x
by (intro tendsto_lowerbound[OF lim])
- (auto simp: trivial_limit_at_top_linorder eventually_at_top_linorder)
+ (auto simp: eventually_at_top_linorder)
have "(SUP i::nat. ?f i x) = ?fR x" for x
proof (rule LIMSEQ_unique[OF LIMSEQ_SUP])
- from reals_Archimedean2[of "x - a"] guess n ..
+ obtain n where "x - a < real n"
+ using reals_Archimedean2[of "x - a"] ..
then have "eventually (\<lambda>n. ?f n x = ?fR x) sequentially"
by (auto intro!: eventually_sequentiallyI[where c=n] split: split_indicator)
then show "(\<lambda>n. ?f n x) \<longlonglongrightarrow> ?fR x"
--- a/src/HOL/ex/Normalization_by_Evaluation.thy Sun Aug 06 18:51:32 2017 +0200
+++ b/src/HOL/ex/Normalization_by_Evaluation.thy Sun Aug 06 18:56:47 2017 +0200
@@ -68,20 +68,20 @@
lemma "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs"
by normalization rule
lemma "rev [a, b, c] = [c, b, a]" by normalization
-value "rev (a#b#cs) = rev cs @ [b, a]"
-value "map (%F. F [a,b,c::'x]) (map map [f,g,h])"
-value "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"
-value "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"
+value [nbe] "rev (a#b#cs) = rev cs @ [b, a]"
+value [nbe] "map (%F. F [a,b,c::'x]) (map map [f,g,h])"
+value [nbe] "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"
+value [nbe] "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"
lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]"
by normalization
-value "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False"
-value "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P"
+value [nbe] "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False"
+value [nbe] "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P"
lemma "let x = y in [x, x] = [y, y]" by normalization
lemma "Let y (%x. [x,x]) = [y, y]" by normalization
-value "case n of Z \<Rightarrow> True | S x \<Rightarrow> False"
+value [nbe] "case n of Z \<Rightarrow> True | S x \<Rightarrow> False"
lemma "(%(x,y). add x y) (S z,S z) = S (add z (S z))" by normalization
-value "filter (%x. x) ([True,False,x]@xs)"
-value "filter Not ([True,False,x]@xs)"
+value [nbe] "filter (%x. x) ([True,False,x]@xs)"
+value [nbe] "filter Not ([True,False,x]@xs)"
lemma "[x,y,z] @ [a,b,c] = [x, y, z, a, b, c]" by normalization
lemma "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" by normalization
@@ -106,7 +106,7 @@
lemma "[Suc 0, 0] = [Suc 0, 0]" by normalization
lemma "max (Suc 0) 0 = Suc 0" by normalization
lemma "(42::rat) / 1704 = 1 / 284 + 3 / 142" by normalization
-value "Suc 0 \<in> set ms"
+value [nbe] "Suc 0 \<in> set ms"
(* non-left-linear patterns, equality by extensionality *)
@@ -115,13 +115,13 @@
lemma "(f o g) x = f (g x)" by normalization
lemma "(f o id) x = f x" by normalization
lemma "(id :: bool \<Rightarrow> bool) = id" by normalization
-value "(\<lambda>x. x)"
+value [nbe] "(\<lambda>x. x)"
(* Church numerals: *)
-value "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
-value "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
-value "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
+value [nbe] "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
+value [nbe] "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
+value [nbe] "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
(* handling of type classes in connection with equality *)