# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1501081665 -3600
# Node ID 33a47f2d9edc8984d6feb80e3af1c1f825468e66
# Parent  1ec601d9c829dc597d6924b489b1499a5b5a146f
New theory of Equiintegrability / Continuity of the indefinite integral / improper integration

diff -r 1ec601d9c829 -r 33a47f2d9edc src/HOL/Analysis/Analysis.thy
--- a/src/HOL/Analysis/Analysis.thy	Wed Jul 26 13:36:36 2017 +0100
+++ b/src/HOL/Analysis/Analysis.thy	Wed Jul 26 16:07:45 2017 +0100
@@ -1,6 +1,7 @@
 theory Analysis
 imports
   Lebesgue_Integral_Substitution
+  Improper_Integral
   Embed_Measure
   Complete_Measure
   Radon_Nikodym
diff -r 1ec601d9c829 -r 33a47f2d9edc src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed Jul 26 13:36:36 2017 +0100
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed Jul 26 16:07:45 2017 +0100
@@ -46,6 +46,19 @@
 lemma content_cbox': "cbox a b \<noteq> {} \<Longrightarrow> content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   by (simp add: box_ne_empty inner_diff)
 
+lemma content_cbox_if: "content (cbox a b) = (if cbox a b = {} then 0 else \<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
+  by (simp add: content_cbox')
+
+lemma content_division_of:
+  assumes "K \<in> \<D>" "\<D> division_of S"
+  shows "content K = (\<Prod>i \<in> Basis. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)"
+proof -
+  obtain a b where "K = cbox a b"
+    using cbox_division_memE assms by metis
+  then show ?thesis
+    using assms by (force simp: division_of_def content_cbox')
+qed
+
 lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
   by simp
 
diff -r 1ec601d9c829 -r 33a47f2d9edc src/HOL/Analysis/Improper_Integral.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Improper_Integral.thy	Wed Jul 26 16:07:45 2017 +0100
@@ -0,0 +1,1663 @@
+section\<open>Continuity of the indefinite integral; improper integral theorem\<close>
+
+theory "Improper_Integral"
+  imports Equivalence_Lebesgue_Henstock_Integration
+begin
+
+subsection\<open>Equiintegrability\<close>
+
+text\<open>The definition here only really makes sense for an elementary set. 
+     We just use compact intervals in applications below.\<close>
+
+definition equiintegrable_on (infixr "equiintegrable'_on" 46)
+  where "F equiintegrable_on I \<equiv>
+         (\<forall>f \<in> F. f integrable_on I) \<and>
+         (\<forall>e > 0. \<exists>\<gamma>. gauge \<gamma> \<and>
+                    (\<forall>f \<D>. f \<in> F \<and> \<D> tagged_division_of I \<and> \<gamma> fine \<D>
+                          \<longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < e))"
+
+lemma equiintegrable_on_integrable:
+     "\<lbrakk>F equiintegrable_on I; f \<in> F\<rbrakk> \<Longrightarrow> f integrable_on I"
+  using equiintegrable_on_def by metis
+
+lemma equiintegrable_on_sing [simp]:
+     "{f} equiintegrable_on cbox a b \<longleftrightarrow> f integrable_on cbox a b"
+  by (simp add: equiintegrable_on_def has_integral_integral has_integral integrable_on_def)
+    
+lemma equiintegrable_on_subset: "\<lbrakk>F equiintegrable_on I; G \<subseteq> F\<rbrakk> \<Longrightarrow> G equiintegrable_on I"
+  unfolding equiintegrable_on_def Ball_def
+  by (erule conj_forward imp_forward all_forward ex_forward | blast)+
+
+lemma equiintegrable_on_Un:
+  assumes "F equiintegrable_on I" "G equiintegrable_on I"
+  shows "(F \<union> G) equiintegrable_on I"
+  unfolding equiintegrable_on_def
+proof (intro conjI impI allI)
+  show "\<forall>f\<in>F \<union> G. f integrable_on I"
+    using assms unfolding equiintegrable_on_def by blast
+  show "\<exists>\<gamma>. gauge \<gamma> \<and>
+            (\<forall>f \<D>. f \<in> F \<union> G \<and>
+                   \<D> tagged_division_of I \<and> \<gamma> fine \<D> \<longrightarrow>
+                   norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>)"
+         if "\<epsilon> > 0" for \<epsilon>
+  proof -
+    obtain \<gamma>1 where "gauge \<gamma>1"
+      and \<gamma>1: "\<And>f \<D>. f \<in> F \<and> \<D> tagged_division_of I \<and> \<gamma>1 fine \<D>
+                    \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>"
+      using assms \<open>\<epsilon> > 0\<close> unfolding equiintegrable_on_def by auto
+    obtain \<gamma>2 where  "gauge \<gamma>2"
+      and \<gamma>2: "\<And>f \<D>. f \<in> G \<and> \<D> tagged_division_of I \<and> \<gamma>2 fine \<D>
+                    \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>"
+      using assms \<open>\<epsilon> > 0\<close> unfolding equiintegrable_on_def by auto
+    have "gauge (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x)"
+      using \<open>gauge \<gamma>1\<close> \<open>gauge \<gamma>2\<close> by blast
+    moreover have "\<forall>f \<D>. f \<in> F \<union> G \<and> \<D> tagged_division_of I \<and> (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x) fine \<D> \<longrightarrow>
+          norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>"
+      using \<gamma>1 \<gamma>2 by (auto simp: fine_Int)
+    ultimately show ?thesis
+      by (intro exI conjI) assumption+
+  qed
+qed
+
+
+lemma equiintegrable_on_insert:
+  assumes "f integrable_on cbox a b" "F equiintegrable_on cbox a b"
+  shows "(insert f F) equiintegrable_on cbox a b"
+  by (metis assms equiintegrable_on_Un equiintegrable_on_sing insert_is_Un)
+
+
+text\<open> Basic combining theorems for the interval of integration.\<close>
+
+lemma equiintegrable_on_null [simp]:
+   "content(cbox a b) = 0 \<Longrightarrow> F equiintegrable_on cbox a b"
+  apply (auto simp: equiintegrable_on_def)
+  by (metis gauge_trivial norm_eq_zero sum_content_null)
+
+
+text\<open> Main limit theorem for an equiintegrable sequence.\<close>
+
+theorem equiintegrable_limit:
+  fixes g :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
+  assumes feq: "range f equiintegrable_on cbox a b"
+      and to_g: "\<And>x. x \<in> cbox a b \<Longrightarrow> (\<lambda>n. f n x) \<longlonglongrightarrow> g x"
+    shows "g integrable_on cbox a b \<and> (\<lambda>n. integral (cbox a b) (f n)) \<longlonglongrightarrow> integral (cbox a b) g"
+proof -
+  have "Cauchy (\<lambda>n. integral(cbox a b) (f n))"
+  proof (clarsimp simp add: Cauchy_def)
+    fix e::real
+    assume "0 < e"
+    then have e3: "0 < e/3"
+      by simp
+    then obtain \<gamma> where "gauge \<gamma>"
+         and \<gamma>: "\<And>n \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk>
+                       \<Longrightarrow> norm((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f n x) - integral (cbox a b) (f n)) < e/3"
+      using feq unfolding equiintegrable_on_def
+      by (meson image_eqI iso_tuple_UNIV_I)
+    obtain \<D> where \<D>: "\<D> tagged_division_of (cbox a b)" and "\<gamma> fine \<D>"  "finite \<D>"
+      by (meson \<open>gauge \<gamma>\<close> fine_division_exists tagged_division_of_finite)
+    with \<gamma> have \<delta>T: "\<And>n. dist ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x)) (integral (cbox a b) (f n)) < e/3"
+      by (force simp: dist_norm)
+    have "(\<lambda>n. \<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x) \<longlonglongrightarrow> (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x)"
+      using \<D> to_g by (auto intro!: tendsto_sum tendsto_scaleR)
+    then have "Cauchy (\<lambda>n. \<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x)"
+      by (meson convergent_eq_Cauchy)
+    with e3 obtain M where
+      M: "\<And>m n. \<lbrakk>m\<ge>M; n\<ge>M\<rbrakk> \<Longrightarrow> dist (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f m x) (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x)
+                      < e/3"
+      unfolding Cauchy_def by blast
+    have "\<And>m n. \<lbrakk>m\<ge>M; n\<ge>M;
+                 dist (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f m x) (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x) < e/3\<rbrakk>
+                \<Longrightarrow> dist (integral (cbox a b) (f m)) (integral (cbox a b) (f n)) < e"
+       by (metis \<delta>T dist_commute dist_triangle_third [OF _ _ \<delta>T])
+    then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (integral (cbox a b) (f m)) (integral (cbox a b) (f n)) < e"
+      using M by auto
+  qed
+  then obtain L where L: "(\<lambda>n. integral (cbox a b) (f n)) \<longlonglongrightarrow> L"
+    by (meson convergent_eq_Cauchy)
+  have "(g has_integral L) (cbox a b)"
+  proof (clarsimp simp: has_integral)
+    fix e::real assume "0 < e"
+    then have e2: "0 < e/2"
+      by simp
+    then obtain \<gamma> where "gauge \<gamma>"
+      and \<gamma>: "\<And>n \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk>
+                    \<Longrightarrow> norm((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x) - integral (cbox a b) (f n)) < e/2"
+      using feq unfolding equiintegrable_on_def
+      by (meson image_eqI iso_tuple_UNIV_I)
+    moreover
+    have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - L) < e"
+              if "\<D> tagged_division_of cbox a b" "\<gamma> fine \<D>" for \<D>
+    proof -
+      have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - L) \<le> e/2"
+      proof (rule Lim_norm_ubound)
+        show "(\<lambda>n. (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f n x) - integral (cbox a b) (f n)) \<longlonglongrightarrow> (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - L"
+          using to_g that L
+          by (intro tendsto_diff tendsto_sum) (auto simp: tag_in_interval tendsto_scaleR)
+        show "\<forall>\<^sub>F n in sequentially.
+                norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f n x) - integral (cbox a b) (f n)) \<le> e/2"
+          by (intro eventuallyI less_imp_le \<gamma> that)
+      qed auto
+      with \<open>0 < e\<close> show ?thesis
+        by linarith
+    qed
+    ultimately
+    show "\<exists>\<gamma>. gauge \<gamma> \<and>
+             (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
+                  norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - L) < e)"
+      by meson
+  qed
+  with L show ?thesis
+    by (simp add: \<open>(\<lambda>n. integral (cbox a b) (f n)) \<longlonglongrightarrow> L\<close> has_integral_integrable_integral)
+qed
+
+
+lemma equiintegrable_reflect:
+  assumes "F equiintegrable_on cbox a b"
+  shows "(\<lambda>f. f \<circ> uminus) ` F equiintegrable_on cbox (-b) (-a)"
+proof -
+  have "\<exists>\<gamma>. gauge \<gamma> \<and>
+            (\<forall>f \<D>. f \<in> (\<lambda>f. f \<circ> uminus) ` F \<and> \<D> tagged_division_of cbox (- b) (- a) \<and> \<gamma> fine \<D> \<longrightarrow>
+                   norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral (cbox (- b) (- a)) f) < e)"
+       if "gauge \<gamma>" and
+           \<gamma>: "\<And>f \<D>. \<lbrakk>f \<in> F; \<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk> \<Longrightarrow>
+                     norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral (cbox a b) f) < e" for e \<gamma>
+  proof (intro exI, safe)
+    show "gauge (\<lambda>x. uminus ` \<gamma> (-x))"
+      by (metis \<open>gauge \<gamma>\<close> gauge_reflect)
+    show "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R (f \<circ> uminus) x) - integral (cbox (- b) (- a)) (f \<circ> uminus)) < e"
+      if "f \<in> F" and tag: "\<D> tagged_division_of cbox (- b) (- a)"
+         and fine: "(\<lambda>x. uminus ` \<gamma> (- x)) fine \<D>" for f \<D>
+    proof -
+      have 1: "(\<lambda>(x,K). (- x, uminus ` K)) ` \<D> tagged_partial_division_of cbox a b"
+        if "\<D> tagged_partial_division_of cbox (- b) (- a)"
+      proof -
+        have "- y \<in> cbox a b"
+          if "\<And>x K. (x,K) \<in> \<D> \<Longrightarrow> x \<in> K \<and> K \<subseteq> cbox (- b) (- a) \<and> (\<exists>a b. K = cbox a b)"
+             "(x, Y) \<in> \<D>" "y \<in> Y" for x Y y
+        proof -
+          have "y \<in> uminus ` cbox a b"
+            using that by auto
+          then show "- y \<in> cbox a b"
+            by force
+        qed
+        with that show ?thesis
+          by (fastforce simp: tagged_partial_division_of_def interior_negations image_iff)
+      qed
+      have 2: "\<exists>K. (\<exists>x. (x,K) \<in> (\<lambda>(x,K). (- x, uminus ` K)) ` \<D>) \<and> x \<in> K"
+              if "\<Union>{K. \<exists>x. (x,K) \<in> \<D>} = cbox (- b) (- a)" "x \<in> cbox a b" for x
+      proof -
+        have xm: "x \<in> uminus ` \<Union>{A. \<exists>a. (a, A) \<in> \<D>}"
+          by (simp add: that)
+        then obtain a X where "-x \<in> X" "(a, X) \<in> \<D>"
+          by auto
+        then show ?thesis
+          by (metis (no_types, lifting) add.inverse_inverse image_iff pair_imageI)
+      qed
+      have 3: "\<And>x X y. \<lbrakk>\<D> tagged_partial_division_of cbox (- b) (- a); (x, X) \<in> \<D>; y \<in> X\<rbrakk> \<Longrightarrow> - y \<in> cbox a b"
+        by (metis (no_types, lifting) equation_minus_iff imageE subsetD tagged_partial_division_ofD(3) uminus_interval_vector)
+      have tag': "(\<lambda>(x,K). (- x, uminus ` K)) ` \<D> tagged_division_of cbox a b"
+        using tag  by (auto simp: tagged_division_of_def dest: 1 2 3)
+      have fine': "\<gamma> fine (\<lambda>(x,K). (- x, uminus ` K)) ` \<D>"
+        using fine by (fastforce simp: fine_def)
+      have inj: "inj_on (\<lambda>(x,K). (- x, uminus ` K)) \<D>"
+        unfolding inj_on_def by force
+      have eq: "content (uminus ` I) = content I"
+               if I: "(x, I) \<in> \<D>" and fnz: "f (- x) \<noteq> 0" for x I
+      proof -
+        obtain a b where "I = cbox a b"
+          using tag I that by (force simp: tagged_division_of_def tagged_partial_division_of_def)
+        then show ?thesis
+          using content_image_affinity_cbox [of "-1" 0] by auto
+      qed
+      have "(\<Sum>(x,K) \<in> (\<lambda>(x,K). (- x, uminus ` K)) ` \<D>.  content K *\<^sub>R f x) =
+            (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f (- x))"
+        apply (simp add: sum.reindex [OF inj])
+        apply (auto simp: eq intro!: sum.cong)
+        done
+      then show ?thesis
+        using \<gamma> [OF \<open>f \<in> F\<close> tag' fine'] integral_reflect
+        by (metis (mono_tags, lifting) Henstock_Kurzweil_Integration.integral_cong comp_apply split_def sum.cong)
+    qed
+  qed
+  then show ?thesis
+    using assms by (auto simp: equiintegrable_on_def integrable_eq)
+qed
+
+subsection\<open>Subinterval restrictions for equiintegrable families\<close>
+
+text\<open>First, some technical lemmas about minimizing a "flat" part of a sum over a division.\<close>
+
+lemma lemma0:
+  assumes "i \<in> Basis"
+    shows "content (cbox u v) / (interval_upperbound (cbox u v) \<bullet> i - interval_lowerbound (cbox u v) \<bullet> i) =
+           (if content (cbox u v) = 0 then 0
+            else \<Prod>j \<in> Basis - {i}. interval_upperbound (cbox u v) \<bullet> j - interval_lowerbound (cbox u v) \<bullet> j)"
+proof (cases "content (cbox u v) = 0")
+  case True
+  then show ?thesis by simp
+next
+  case False
+  then show ?thesis
+    using prod.subset_diff [of "{i}" Basis] assms
+      by (force simp: content_cbox_if divide_simps  split: if_split_asm)
+qed
+
+
+lemma content_division_lemma1:
+  assumes div: "\<D> division_of S" and S: "S \<subseteq> cbox a b" and i: "i \<in> Basis"
+      and mt: "\<And>K. K \<in> \<D> \<Longrightarrow> content K \<noteq> 0"
+      and disj: "(\<forall>K \<in> \<D>. K \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}) \<or> (\<forall>K \<in> \<D>. K \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {})"
+   shows "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
+          \<le> content(cbox a b)"   (is "?lhs \<le> ?rhs")
+proof -
+  have "finite \<D>"
+    using div by blast
+  define extend where
+    "extend \<equiv> \<lambda>K. cbox (\<Sum>j \<in> Basis. if j = i then (a \<bullet> i) *\<^sub>R i else (interval_lowerbound K \<bullet> j) *\<^sub>R j)
+                       (\<Sum>j \<in> Basis. if j = i then (b \<bullet> i) *\<^sub>R i else (interval_upperbound K \<bullet> j) *\<^sub>R j)"
+  have div_subset_cbox: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<subseteq> cbox a b"
+    using S div by auto
+  have "\<And>K. K \<in> \<D> \<Longrightarrow> K \<noteq> {}"
+    using div by blast
+  have extend: "extend K \<noteq> {}" "extend K \<subseteq> cbox a b" if K: "K \<in> \<D>" for K
+  proof -
+    obtain u v where K: "K = cbox u v" "K \<noteq> {}" "K \<subseteq> cbox a b"
+      using K cbox_division_memE [OF _ div] by (meson div_subset_cbox)
+    with i show "extend K \<noteq> {}" "extend K \<subseteq> cbox a b"
+      apply (auto simp: extend_def subset_box box_ne_empty sum_if_inner)
+      by fastforce
+  qed
+  have int_extend_disjoint:
+       "interior(extend K1) \<inter> interior(extend K2) = {}" if K: "K1 \<in> \<D>" "K2 \<in> \<D>" "K1 \<noteq> K2" for K1 K2
+  proof -
+    obtain u v where K1: "K1 = cbox u v" "K1 \<noteq> {}" "K1 \<subseteq> cbox a b"
+      using K cbox_division_memE [OF _ div] by (meson div_subset_cbox)
+    obtain w z where K2: "K2 = cbox w z" "K2 \<noteq> {}" "K2 \<subseteq> cbox a b"
+      using K cbox_division_memE [OF _ div] by (meson div_subset_cbox)
+    have cboxes: "cbox u v \<in> \<D>" "cbox w z \<in> \<D>" "cbox u v \<noteq> cbox w z"
+      using K1 K2 that by auto
+    with div have "interior (cbox u v) \<inter> interior (cbox w z) = {}"
+      by blast
+    moreover
+    have "\<exists>x. x \<in> box u v \<and> x \<in> box w z"
+         if "x \<in> interior (extend K1)" "x \<in> interior (extend K2)" for x
+    proof -
+      have "a \<bullet> i < x \<bullet> i" "x \<bullet> i < b \<bullet> i"
+       and ux: "\<And>k. k \<in> Basis - {i} \<Longrightarrow> u \<bullet> k < x \<bullet> k"
+       and xv: "\<And>k. k \<in> Basis - {i} \<Longrightarrow> x \<bullet> k < v \<bullet> k"
+       and wx: "\<And>k. k \<in> Basis - {i} \<Longrightarrow> w \<bullet> k < x \<bullet> k"
+       and xz: "\<And>k. k \<in> Basis - {i} \<Longrightarrow> x \<bullet> k < z \<bullet> k"
+        using that K1 K2 i by (auto simp: extend_def box_ne_empty sum_if_inner mem_box)
+      have "box u v \<noteq> {}" "box w z \<noteq> {}"
+        using cboxes interior_cbox by (auto simp: content_eq_0_interior dest: mt)
+      then obtain q s
+        where q: "\<And>k. k \<in> Basis \<Longrightarrow> w \<bullet> k < q \<bullet> k \<and> q \<bullet> k < z \<bullet> k"
+          and s: "\<And>k. k \<in> Basis \<Longrightarrow> u \<bullet> k < s \<bullet> k \<and> s \<bullet> k < v \<bullet> k"
+        by (meson all_not_in_conv mem_box(1))
+      show ?thesis  using disj
+      proof
+        assume "\<forall>K\<in>\<D>. K \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}"
+        then have uva: "(cbox u v) \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}"
+             and  wza: "(cbox w z) \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}"
+          using cboxes by (auto simp: content_eq_0_interior)
+        then obtain r t where "r \<bullet> i = a \<bullet> i" and r: "\<And>k. k \<in> Basis \<Longrightarrow> w \<bullet> k \<le> r \<bullet> k \<and> r \<bullet> k \<le> z \<bullet> k"
+                        and "t \<bullet> i = a \<bullet> i" and t: "\<And>k. k \<in> Basis \<Longrightarrow> u \<bullet> k \<le> t \<bullet> k \<and> t \<bullet> k \<le> v \<bullet> k"
+          by (fastforce simp: mem_box)
+        have u: "u \<bullet> i < q \<bullet> i"
+          using i K2(1) K2(3) \<open>t \<bullet> i = a \<bullet> i\<close> q s t [OF i] by (force simp: subset_box)
+        have w: "w \<bullet> i < s \<bullet> i"
+          using i K1(1) K1(3) \<open>r \<bullet> i = a \<bullet> i\<close> s r [OF i] by (force simp: subset_box)
+        let ?x = "(\<Sum>j \<in> Basis. if j = i then min (q \<bullet> i) (s \<bullet> i) *\<^sub>R i else (x \<bullet> j) *\<^sub>R j)"
+        show ?thesis
+        proof (intro exI conjI)
+          show "?x \<in> box u v"
+            using \<open>i \<in> Basis\<close> s apply (clarsimp simp: mem_box)
+            apply (subst sum_if_inner; simp)+
+            apply (fastforce simp: u ux xv)
+            done
+          show "?x \<in> box w z"
+            using \<open>i \<in> Basis\<close> q apply (clarsimp simp: mem_box)
+            apply (subst sum_if_inner; simp)+
+            apply (fastforce simp: w wx xz)
+            done
+        qed
+      next
+        assume "\<forall>K\<in>\<D>. K \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {}"
+        then have uva: "(cbox u v) \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {}"
+             and  wza: "(cbox w z) \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {}"
+          using cboxes by (auto simp: content_eq_0_interior)
+        then obtain r t where "r \<bullet> i = b \<bullet> i" and r: "\<And>k. k \<in> Basis \<Longrightarrow> w \<bullet> k \<le> r \<bullet> k \<and> r \<bullet> k \<le> z \<bullet> k"
+                        and "t \<bullet> i = b \<bullet> i" and t: "\<And>k. k \<in> Basis \<Longrightarrow> u \<bullet> k \<le> t \<bullet> k \<and> t \<bullet> k \<le> v \<bullet> k"
+          by (fastforce simp: mem_box)
+        have z: "s \<bullet> i < z \<bullet> i"
+          using K1(1) K1(3) \<open>r \<bullet> i = b \<bullet> i\<close> r [OF i] i s  by (force simp: subset_box)
+        have v: "q \<bullet> i < v \<bullet> i"
+          using K2(1) K2(3) \<open>t \<bullet> i = b \<bullet> i\<close> t [OF i] i q  by (force simp: subset_box)
+        let ?x = "(\<Sum>j \<in> Basis. if j = i then max (q \<bullet> i) (s \<bullet> i) *\<^sub>R i else (x \<bullet> j) *\<^sub>R j)"
+        show ?thesis
+        proof (intro exI conjI)
+          show "?x \<in> box u v"
+            using \<open>i \<in> Basis\<close> s apply (clarsimp simp: mem_box)
+            apply (subst sum_if_inner; simp)+
+            apply (fastforce simp: v ux xv)
+            done
+          show "?x \<in> box w z"
+            using \<open>i \<in> Basis\<close> q apply (clarsimp simp: mem_box)
+            apply (subst sum_if_inner; simp)+
+            apply (fastforce simp: z wx xz)
+            done
+        qed
+      qed
+    qed
+    ultimately show ?thesis by auto
+  qed
+  have "?lhs = (\<Sum>K\<in>\<D>. (b \<bullet> i - a \<bullet> i) * content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
+    by (simp add: sum_distrib_left)
+  also have "\<dots> = sum (content \<circ> extend) \<D>"
+  proof (rule sum.cong [OF refl])
+    fix K assume "K \<in> \<D>"
+    then obtain u v where K: "K = cbox u v" "cbox u v \<noteq> {}" "K \<subseteq> cbox a b"
+      using cbox_division_memE [OF _ div] div_subset_cbox by metis
+    then have uv: "u \<bullet> i < v \<bullet> i"
+      using mt [OF \<open>K \<in> \<D>\<close>] \<open>i \<in> Basis\<close> content_eq_0 by fastforce
+    have "insert i (Basis \<inter> -{i}) = Basis"
+      using \<open>i \<in> Basis\<close> by auto
+    then have "(b \<bullet> i - a \<bullet> i) * content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)
+             = (b \<bullet> i - a \<bullet> i) * (\<Prod>i \<in> insert i (Basis \<inter> -{i}). v \<bullet> i - u \<bullet> i) / (interval_upperbound (cbox u v) \<bullet> i - interval_lowerbound (cbox u v) \<bullet> i)"
+      using K box_ne_empty(1) content_cbox by fastforce
+    also have "... = (\<Prod>x\<in>Basis. if x = i then b \<bullet> x - a \<bullet> x
+                      else (interval_upperbound (cbox u v) - interval_lowerbound (cbox u v)) \<bullet> x)"
+      using \<open>i \<in> Basis\<close> K uv by (simp add: prod.If_cases) (simp add: algebra_simps)
+    also have "... = (\<Prod>k\<in>Basis.
+                        (\<Sum>j\<in>Basis. if j = i then (b \<bullet> i - a \<bullet> i) *\<^sub>R i else ((interval_upperbound (cbox u v) - interval_lowerbound (cbox u v)) \<bullet> j) *\<^sub>R j) \<bullet> k)"
+      using \<open>i \<in> Basis\<close> by (subst prod.cong [OF refl sum_if_inner]; simp)
+    also have "... = (\<Prod>k\<in>Basis.
+                        (\<Sum>j\<in>Basis. if j = i then (b \<bullet> i) *\<^sub>R i else (interval_upperbound (cbox u v) \<bullet> j) *\<^sub>R j) \<bullet> k -
+                        (\<Sum>j\<in>Basis. if j = i then (a \<bullet> i) *\<^sub>R i else (interval_lowerbound (cbox u v) \<bullet> j) *\<^sub>R j) \<bullet> k)"
+      apply (rule prod.cong [OF refl])
+      using \<open>i \<in> Basis\<close>
+      apply (subst sum_if_inner; simp add: algebra_simps)+
+      done
+    also have "... = (content \<circ> extend) K"
+      using \<open>i \<in> Basis\<close> K box_ne_empty
+      apply (simp add: extend_def)
+      apply (subst content_cbox, auto)
+      done
+    finally show "(b \<bullet> i - a \<bullet> i) * content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)
+         = (content \<circ> extend) K" .
+  qed
+  also have "... = sum content (extend ` \<D>)"
+  proof -
+    have "\<lbrakk>K1 \<in> \<D>; K2 \<in> \<D>; K1 \<noteq> K2; extend K1 = extend K2\<rbrakk> \<Longrightarrow> content (extend K1) = 0" for K1 K2
+      using int_extend_disjoint [of K1 K2] extend_def by (simp add: content_eq_0_interior)
+    then show ?thesis
+      by (simp add: comm_monoid_add_class.sum.reindex_nontrivial [OF \<open>finite \<D>\<close>])
+  qed
+  also have "... \<le> ?rhs"
+  proof (rule subadditive_content_division)
+    show "extend ` \<D> division_of UNION \<D> extend"
+      using int_extend_disjoint apply (auto simp: division_of_def \<open>finite \<D>\<close> extend)
+      using extend_def apply blast
+      done
+    show "UNION \<D> extend \<subseteq> cbox a b"
+      using extend by fastforce
+  qed
+  finally show ?thesis .
+qed
+
+
+proposition sum_content_area_over_thin_division:
+  assumes div: "\<D> division_of S" and S: "S \<subseteq> cbox a b" and i: "i \<in> Basis"
+    and "a \<bullet> i \<le> c" "c \<le> b \<bullet> i"
+    and nonmt: "\<And>K. K \<in> \<D> \<Longrightarrow> K \<inter> {x. x \<bullet> i = c} \<noteq> {}"
+  shows "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
+          \<le> 2 * content(cbox a b)"
+proof (cases "content(cbox a b) = 0")
+  case True
+  have "(\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) = 0"
+    using S div by (force intro!: sum.neutral content_0_subset [OF True])
+  then show ?thesis
+    by (auto simp: True)
+next
+  case False
+  then have "content(cbox a b) > 0"
+    using zero_less_measure_iff by blast
+  then have "a \<bullet> i < b \<bullet> i" if "i \<in> Basis" for i
+    using content_pos_lt_eq that by blast
+  have "finite \<D>"
+    using div by blast
+  define Dlec where "Dlec \<equiv> {L \<in> (\<lambda>L. L \<inter> {x. x \<bullet> i \<le> c}) ` \<D>. content L \<noteq> 0}"
+  define Dgec where "Dgec \<equiv> {L \<in> (\<lambda>L. L \<inter> {x. x \<bullet> i \<ge> c}) ` \<D>. content L \<noteq> 0}"
+  define a' where "a' \<equiv> (\<Sum>j\<in>Basis. (if j = i then c else a \<bullet> j) *\<^sub>R j)"
+  define b' where "b' \<equiv> (\<Sum>j\<in>Basis. (if j = i then c else b \<bullet> j) *\<^sub>R j)"
+  have Dlec_cbox: "\<And>K. K \<in> Dlec \<Longrightarrow> \<exists>a b. K = cbox a b"
+    using interval_split [OF i] div by (fastforce simp: Dlec_def division_of_def)
+  then have lec_is_cbox: "\<lbrakk>content (L \<inter> {x. x \<bullet> i \<le> c}) \<noteq> 0; L \<in> \<D>\<rbrakk> \<Longrightarrow> \<exists>a b. L \<inter> {x. x \<bullet> i \<le> c} = cbox a b" for L
+    using Dlec_def by blast
+  have Dgec_cbox: "\<And>K. K \<in> Dgec \<Longrightarrow> \<exists>a b. K = cbox a b"
+    using interval_split [OF i] div by (fastforce simp: Dgec_def division_of_def)
+  then have gec_is_cbox: "\<lbrakk>content (L \<inter> {x. x \<bullet> i \<ge> c}) \<noteq> 0; L \<in> \<D>\<rbrakk> \<Longrightarrow> \<exists>a b. L \<inter> {x. x \<bullet> i \<ge> c} = cbox a b" for L
+    using Dgec_def by blast
+  have "(b' \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>Dlec. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<le> content(cbox a b')"
+  proof (rule content_division_lemma1)
+    show "Dlec division_of \<Union>Dlec"
+      unfolding division_of_def
+    proof (intro conjI ballI Dlec_cbox)
+      show "\<And>K1 K2. \<lbrakk>K1 \<in> Dlec; K2 \<in> Dlec\<rbrakk> \<Longrightarrow> K1 \<noteq> K2 \<longrightarrow> interior K1 \<inter> interior K2 = {}"
+        by (clarsimp simp: Dlec_def) (use div in auto)
+    qed (use \<open>finite \<D>\<close> Dlec_def in auto)
+    show "\<Union>Dlec \<subseteq> cbox a b'"
+      using Dlec_def div S by (auto simp: b'_def division_of_def mem_box)
+    show "(\<forall>K\<in>Dlec. K \<inter> {x. x \<bullet> i = a \<bullet> i} \<noteq> {}) \<or> (\<forall>K\<in>Dlec. K \<inter> {x. x \<bullet> i = b' \<bullet> i} \<noteq> {})"
+      using nonmt by (fastforce simp: Dlec_def b'_def sum_if_inner i)
+  qed (use i Dlec_def in auto)
+  moreover
+  have "(\<Sum>K\<in>Dlec. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) =
+        (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K)"
+    apply (subst sum.reindex_nontrivial [OF \<open>finite \<D>\<close>, symmetric], simp)
+     apply (metis division_split_left_inj [OF div] lec_is_cbox content_eq_0_interior)
+    unfolding Dlec_def using \<open>finite \<D>\<close> apply (auto simp: sum.mono_neutral_left)
+    done
+  moreover have "(b' \<bullet> i - a \<bullet> i) = (c - a \<bullet> i)"
+    by (simp add: b'_def sum_if_inner i)
+  ultimately
+  have lec: "(c - a \<bullet> i) * (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K)
+             \<le> content(cbox a b')"
+    by simp
+
+  have "(b \<bullet> i - a' \<bullet> i) * (\<Sum>K\<in>Dgec. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<le> content(cbox a' b)"
+  proof (rule content_division_lemma1)
+    show "Dgec division_of \<Union>Dgec"
+      unfolding division_of_def
+    proof (intro conjI ballI Dgec_cbox)
+      show "\<And>K1 K2. \<lbrakk>K1 \<in> Dgec; K2 \<in> Dgec\<rbrakk> \<Longrightarrow> K1 \<noteq> K2 \<longrightarrow> interior K1 \<inter> interior K2 = {}"
+        by (clarsimp simp: Dgec_def) (use div in auto)
+    qed (use \<open>finite \<D>\<close> Dgec_def in auto)
+    show "\<Union>Dgec \<subseteq> cbox a' b"
+      using Dgec_def div S by (auto simp: a'_def division_of_def mem_box)
+    show "(\<forall>K\<in>Dgec. K \<inter> {x. x \<bullet> i = a' \<bullet> i} \<noteq> {}) \<or> (\<forall>K\<in>Dgec. K \<inter> {x. x \<bullet> i = b \<bullet> i} \<noteq> {})"
+      using nonmt by (fastforce simp: Dgec_def a'_def sum_if_inner i)
+  qed (use i Dgec_def in auto)
+  moreover
+  have "(\<Sum>K\<in>Dgec. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) =
+        (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)"
+    apply (subst sum.reindex_nontrivial [OF \<open>finite \<D>\<close>, symmetric], simp)
+     apply (metis division_split_right_inj [OF div] gec_is_cbox content_eq_0_interior)
+    unfolding Dgec_def using \<open>finite \<D>\<close> apply (auto simp: sum.mono_neutral_left)
+    done
+  moreover have "(b \<bullet> i - a' \<bullet> i) = (b \<bullet> i - c)"
+    by (simp add: a'_def sum_if_inner i)
+  ultimately
+  have gec: "(b \<bullet> i - c) * (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)
+             \<le> content(cbox a' b)"
+    by simp
+  show ?thesis
+  proof (cases "c = a \<bullet> i \<or> c = b \<bullet> i")
+    case True
+    then show ?thesis
+    proof
+      assume c: "c = a \<bullet> i"
+      then have "a' = a"
+        apply (simp add: sum_if_inner i a'_def cong: if_cong)
+        using euclidean_representation [of a] sum.cong [OF refl, of Basis "\<lambda>i. (a \<bullet> i) *\<^sub>R i"] by presburger
+      then have "content (cbox a' b) \<le> 2 * content (cbox a b)"  by simp
+      moreover
+      have eq: "(\<Sum>K\<in>\<D>. content (K \<inter> {x. a \<bullet> i \<le> x \<bullet> i}) /
+                  (interval_upperbound (K \<inter> {x. a \<bullet> i \<le> x \<bullet> i}) \<bullet> i - interval_lowerbound (K \<inter> {x. a \<bullet> i \<le> x \<bullet> i}) \<bullet> i))
+              = (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
+               (is "sum ?f _ = sum ?g _")
+      proof (rule sum.cong [OF refl])
+        fix K assume "K \<in> \<D>"
+        then have "a \<bullet> i \<le> x \<bullet> i" if "x \<in> K" for x
+          by (metis S UnionI div division_ofD(6) i mem_box(2) subsetCE that)
+        then have "K \<inter> {x. a \<bullet> i \<le> x \<bullet> i} = K"
+          by blast
+        then show "?f K = ?g K"
+          by simp
+      qed
+      ultimately show ?thesis
+        using gec c eq by auto
+    next
+      assume c: "c = b \<bullet> i"
+      then have "b' = b"
+        apply (simp add: sum_if_inner i b'_def cong: if_cong)
+        using euclidean_representation [of b] sum.cong [OF refl, of Basis "\<lambda>i. (b \<bullet> i) *\<^sub>R i"] by presburger
+      then have "content (cbox a b') \<le> 2 * content (cbox a b)"  by simp
+      moreover
+      have eq: "(\<Sum>K\<in>\<D>. content (K \<inter> {x. x \<bullet> i \<le> b \<bullet> i}) /
+                  (interval_upperbound (K \<inter> {x. x \<bullet> i \<le> b \<bullet> i}) \<bullet> i - interval_lowerbound (K \<inter> {x. x \<bullet> i \<le> b \<bullet> i}) \<bullet> i))
+              = (\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
+               (is "sum ?f _ = sum ?g _")
+      proof (rule sum.cong [OF refl])
+        fix K assume "K \<in> \<D>"
+        then have "x \<bullet> i \<le> b \<bullet> i" if "x \<in> K" for x
+          by (metis S UnionI div division_ofD(6) i mem_box(2) subsetCE that)
+        then have "K \<inter> {x. x \<bullet> i \<le> b \<bullet> i} = K"
+          by blast
+        then show "?f K = ?g K"
+          by simp
+      qed
+      ultimately show ?thesis
+        using lec c eq by auto
+    qed
+  next
+    case False
+    have prod_if: "(\<Prod>k\<in>Basis \<inter> - {i}. f k) = (\<Prod>k\<in>Basis. f k) / f i" if "f i \<noteq> (0::real)" for f
+      using that mk_disjoint_insert [OF i]
+      apply (clarsimp simp add: divide_simps)
+      by (metis Int_insert_left_if0 finite_Basis finite_insert le_iff_inf mult.commute order_refl prod.insert subset_Compl_singleton)
+    have abc: "a \<bullet> i < c" "c < b \<bullet> i"
+      using False assms by auto
+    then have "(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K)
+                  \<le> content(cbox a b') / (c - a \<bullet> i)"
+              "(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)
+                 \<le> content(cbox a' b) / (b \<bullet> i - c)"
+      using lec gec by (simp_all add: divide_simps mult.commute)
+    moreover
+    have "(\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
+          \<le> (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K) +
+            (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)"
+           (is "?lhs \<le> ?rhs")
+    proof -
+      have "?lhs \<le>
+            (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K +
+                    ((\<lambda>K. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)"
+            (is "sum ?f _ \<le> sum ?g _")
+      proof (rule sum_mono)
+        fix K assume "K \<in> \<D>"
+        then obtain u v where uv: "K = cbox u v"
+          using div by blast
+        obtain u' v' where uv': "cbox u v \<inter> {x. x \<bullet> i \<le> c} = cbox u v'"
+                                "cbox u v \<inter> {x. c \<le> x \<bullet> i} = cbox u' v"
+                                "\<And>k. k \<in> Basis \<Longrightarrow> u' \<bullet> k = (if k = i then max (u \<bullet> i) c else u \<bullet> k)"
+                                "\<And>k. k \<in> Basis \<Longrightarrow> v' \<bullet> k = (if k = i then min (v \<bullet> i) c else v \<bullet> k)"
+          using i by (auto simp: interval_split)
+        have *: "\<lbrakk>content (cbox u v') = 0; content (cbox u' v) = 0\<rbrakk> \<Longrightarrow> content (cbox u v) = 0"
+                "content (cbox u' v) \<noteq> 0 \<Longrightarrow> content (cbox u v) \<noteq> 0"
+                "content (cbox u v') \<noteq> 0 \<Longrightarrow> content (cbox u v) \<noteq> 0"
+          using i uv uv' by (auto simp: content_eq_0 le_max_iff_disj min_le_iff_disj split: if_split_asm intro: order_trans)
+        show "?f K \<le> ?g K"
+          using i uv uv' apply (clarsimp simp add: lemma0 * intro!: prod_nonneg)
+          by (metis content_eq_0 le_less_linear order.strict_implies_order)
+      qed
+      also have "... = ?rhs"
+        by (simp add: sum.distrib)
+      finally show ?thesis .
+    qed
+    moreover have "content (cbox a b') / (c - a \<bullet> i) = content (cbox a b) / (b \<bullet> i - a \<bullet> i)"
+      using i abc
+      apply (simp add: field_simps a'_def b'_def measure_lborel_cbox_eq inner_diff)
+      apply (auto simp: if_distrib if_distrib [of "\<lambda>f. f x" for x] prod.If_cases [of Basis "\<lambda>x. x = i", simplified] prod_if field_simps)
+      done
+    moreover have "content (cbox a' b) / (b \<bullet> i - c) = content (cbox a b) / (b \<bullet> i - a \<bullet> i)"
+      using i abc
+      apply (simp add: field_simps a'_def b'_def measure_lborel_cbox_eq inner_diff)
+      apply (auto simp: if_distrib prod.If_cases [of Basis "\<lambda>x. x = i", simplified] prod_if field_simps)
+      done
+    ultimately
+    have "(\<Sum>K\<in>\<D>. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
+          \<le> 2 * content (cbox a b) / (b \<bullet> i - a \<bullet> i)"
+      by linarith
+    then show ?thesis
+      using abc by (simp add: divide_simps mult.commute)
+  qed
+qed
+
+
+
+
+proposition bounded_equiintegral_over_thin_tagged_partial_division:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes F: "F equiintegrable_on cbox a b" and f: "f \<in> F" and "0 < \<epsilon>"
+      and norm_f: "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm(h x) \<le> norm(f x)"
+  obtains \<gamma> where "gauge \<gamma>"
+             "\<And>c i S h. \<lbrakk>c \<in> cbox a b; i \<in> Basis; S tagged_partial_division_of cbox a b;
+                         \<gamma> fine S; h \<in> F; \<And>x K. (x,K) \<in> S \<Longrightarrow> (K \<inter> {x. x \<bullet> i = c \<bullet> i} \<noteq> {})\<rbrakk>
+                        \<Longrightarrow> (\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>"
+proof (cases "content(cbox a b) = 0")
+  case True
+  show ?thesis
+  proof
+    show "gauge (\<lambda>x. ball x 1)"
+      by (simp add: gauge_trivial)
+    show "(\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>"
+         if "S tagged_partial_division_of cbox a b" "(\<lambda>x. ball x 1) fine S" for S and h:: "'a \<Rightarrow> 'b"
+    proof -
+      have "(\<Sum>(x,K) \<in> S. norm (integral K h)) = 0"
+          using that True content_0_subset
+          by (fastforce simp: tagged_partial_division_of_def intro: sum.neutral)
+      with \<open>0 < \<epsilon>\<close> show ?thesis
+        by simp
+    qed
+  qed
+next
+  case False
+  then have contab_gt0:  "content(cbox a b) > 0"
+    by (simp add: zero_less_measure_iff)
+  then have a_less_b: "\<And>i. i \<in> Basis \<Longrightarrow> a\<bullet>i < b\<bullet>i"
+    by (auto simp: content_pos_lt_eq)
+  obtain \<gamma>0 where "gauge \<gamma>0"
+            and \<gamma>0: "\<And>S h. \<lbrakk>S tagged_partial_division_of cbox a b; \<gamma>0 fine S; h \<in> F\<rbrakk>
+                           \<Longrightarrow> (\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) < \<epsilon>/2"
+  proof -
+    obtain \<gamma> where "gauge \<gamma>"
+               and \<gamma>: "\<And>f \<D>. \<lbrakk>f \<in> F; \<D> tagged_division_of cbox a b; \<gamma> fine \<D>\<rbrakk>
+                              \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral (cbox a b) f)
+                                  < \<epsilon>/(5 * (Suc DIM('b)))"
+    proof -
+      have e5: "\<epsilon>/(5 * (Suc DIM('b))) > 0"
+        using \<open>\<epsilon> > 0\<close> by auto
+      then show ?thesis
+        using F that by (auto simp: equiintegrable_on_def)
+    qed
+    show ?thesis
+    proof
+      show "gauge \<gamma>"
+        by (rule \<open>gauge \<gamma>\<close>)
+      show "(\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) < \<epsilon>/2"
+           if "S tagged_partial_division_of cbox a b" "\<gamma> fine S" "h \<in> F" for S h
+      proof -
+        have "(\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) \<le> 2 * real DIM('b) * (\<epsilon>/(5 * Suc DIM('b)))"
+        proof (rule henstock_lemma_part2 [of h a b])
+          show "h integrable_on cbox a b"
+            using that F equiintegrable_on_def by metis
+          show "gauge \<gamma>"
+            by (rule \<open>gauge \<gamma>\<close>)
+        qed (use that \<open>\<epsilon> > 0\<close> \<gamma> in auto)
+        also have "... < \<epsilon>/2"
+          using \<open>\<epsilon> > 0\<close> by (simp add: divide_simps)
+        finally show ?thesis .
+      qed
+    qed
+  qed
+  define \<gamma> where "\<gamma> \<equiv> \<lambda>x. \<gamma>0 x \<inter>
+                          ball x ((\<epsilon>/8 / (norm(f x) + 1)) * (INF m:Basis. b \<bullet> m - a \<bullet> m) / content(cbox a b))"
+  have "gauge (\<lambda>x. ball x
+                    (\<epsilon> * (INF m:Basis. b \<bullet> m - a \<bullet> m) / ((8 * norm (f x) + 8) * content (cbox a b))))"
+    using \<open>0 < content (cbox a b)\<close> \<open>0 < \<epsilon>\<close> a_less_b
+    apply (auto simp: gauge_def divide_simps mult_less_0_iff zero_less_mult_iff add_nonneg_eq_0_iff finite_less_Inf_iff)
+    apply (meson add_nonneg_nonneg mult_nonneg_nonneg norm_ge_zero not_less zero_le_numeral)
+    done
+  then have "gauge \<gamma>"
+    unfolding \<gamma>_def using \<open>gauge \<gamma>0\<close> gauge_Int by auto
+  moreover
+  have "(\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>"
+       if "c \<in> cbox a b" "i \<in> Basis" and S: "S tagged_partial_division_of cbox a b"
+          and "\<gamma> fine S" "h \<in> F" and ne: "\<And>x K. (x,K) \<in> S \<Longrightarrow> K \<inter> {x. x \<bullet> i = c \<bullet> i} \<noteq> {}" for c i S h
+  proof -
+    have "cbox c b \<subseteq> cbox a b"
+      by (meson mem_box(2) order_refl subset_box(1) that(1))
+    have "finite S"
+      using S by blast
+    have "\<gamma>0 fine S" and fineS:
+         "(\<lambda>x. ball x (\<epsilon> * (INF m:Basis. b \<bullet> m - a \<bullet> m) / ((8 * norm (f x) + 8) * content (cbox a b)))) fine S"
+      using \<open>\<gamma> fine S\<close> by (auto simp: \<gamma>_def fine_Int)
+    then have "(\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) < \<epsilon>/2"
+      by (intro \<gamma>0 that fineS)
+    moreover have "(\<Sum>(x,K) \<in> S. norm (integral K h) - norm (content K *\<^sub>R h x - integral K h)) \<le> \<epsilon>/2"
+    proof -
+      have "(\<Sum>(x,K) \<in> S. norm (integral K h) - norm (content K *\<^sub>R h x - integral K h))
+            \<le> (\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x))"
+      proof (clarify intro!: sum_mono)
+        fix x K
+        assume xK: "(x,K) \<in> S"
+        have "norm (integral K h) - norm (content K *\<^sub>R h x - integral K h) \<le> norm (integral K h - (integral K h - content K *\<^sub>R h x))"
+          by (metis norm_minus_commute norm_triangle_ineq2)
+        also have "... \<le> norm (content K *\<^sub>R h x)"
+          by simp
+        finally show "norm (integral K h) - norm (content K *\<^sub>R h x - integral K h) \<le> norm (content K *\<^sub>R h x)" .
+      qed
+      also have "... \<le> (\<Sum>(x,K) \<in> S. \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) *
+                                    content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
+      proof (clarify intro!: sum_mono)
+        fix x K
+        assume xK: "(x,K) \<in> S"
+        then have x: "x \<in> cbox a b"
+          using S unfolding tagged_partial_division_of_def by (meson subset_iff)
+        let ?\<Delta> = "interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i"
+        show "norm (content K *\<^sub>R h x) \<le> \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) * content K / ?\<Delta>"
+        proof (cases "content K = 0")
+          case True
+          then show ?thesis by simp
+        next
+          case False
+          then have Kgt0: "content K > 0"
+            using zero_less_measure_iff by blast
+          moreover
+          obtain u v where uv: "K = cbox u v"
+            using S \<open>(x,K) \<in> S\<close> by blast
+          then have u_less_v: "\<And>i. i \<in> Basis \<Longrightarrow> u \<bullet> i < v \<bullet> i"
+            using content_pos_lt_eq uv Kgt0 by blast
+          then have dist_uv: "dist u v > 0"
+            using that by auto
+          ultimately have "norm (h x) \<le> (\<epsilon> * (b \<bullet> i - a \<bullet> i)) / (4 * content (cbox a b) * ?\<Delta>)"
+          proof -
+            have "dist x u < \<epsilon> * (INF m:Basis. b \<bullet> m - a \<bullet> m) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
+                 "dist x v < \<epsilon> * (INF m:Basis. b \<bullet> m - a \<bullet> m) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
+              using fineS u_less_v uv xK
+              by (force simp: fine_def mem_box field_simps dest!: bspec)+
+            moreover have "\<epsilon> * (INF m:Basis. b \<bullet> m - a \<bullet> m) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2
+                  \<le> \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
+              apply (intro mult_left_mono divide_right_mono)
+              using \<open>i \<in> Basis\<close> \<open>0 < \<epsilon>\<close> apply (auto simp: intro!: cInf_le_finite)
+              done
+            ultimately
+            have "dist x u < \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
+                 "dist x v < \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
+              by linarith+
+            then have duv: "dist u v < \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * (norm (f x) + 1) * content (cbox a b))"
+              using dist_triangle_half_r by blast
+            have uvi: "\<bar>v \<bullet> i - u \<bullet> i\<bar> \<le> norm (v - u)"
+              by (metis inner_commute inner_diff_right \<open>i \<in> Basis\<close> Basis_le_norm)
+            have "norm (h x) \<le> norm (f x)"
+              using x that by (auto simp: norm_f)
+            also have "... < (norm (f x) + 1)"
+              by simp
+            also have "... < \<epsilon> * (b \<bullet> i - a \<bullet> i) / dist u v / (4 * content (cbox a b))"
+              using duv dist_uv contab_gt0
+              apply (simp add: divide_simps algebra_simps mult_less_0_iff zero_less_mult_iff split: if_split_asm)
+              by (meson add_nonneg_nonneg linorder_not_le measure_nonneg mult_nonneg_nonneg norm_ge_zero zero_le_numeral)
+            also have "... = \<epsilon> * (b \<bullet> i - a \<bullet> i) / norm (v - u) / (4 * content (cbox a b))"
+              by (simp add: dist_norm norm_minus_commute)
+            also have "... \<le> \<epsilon> * (b \<bullet> i - a \<bullet> i) / \<bar>v \<bullet> i - u \<bullet> i\<bar> / (4 * content (cbox a b))"
+              apply (intro mult_right_mono divide_left_mono divide_right_mono uvi)
+              using \<open>0 < \<epsilon>\<close> a_less_b [OF \<open>i \<in> Basis\<close>] u_less_v [OF \<open>i \<in> Basis\<close>] contab_gt0
+              by (auto simp: less_eq_real_def zero_less_mult_iff that)
+            also have "... = \<epsilon> * (b \<bullet> i - a \<bullet> i)
+                       / (4 * content (cbox a b) * ?\<Delta>)"
+              using uv False that(2) u_less_v by fastforce
+            finally show ?thesis by simp
+          qed
+          with Kgt0 have "norm (content K *\<^sub>R h x) \<le> content K * ((\<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b)) / ?\<Delta>)"
+            using mult_left_mono by fastforce
+          also have "... = \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) *
+                           content K / ?\<Delta>"
+            by (simp add: divide_simps)
+          finally show ?thesis .
+        qed
+      qed
+      also have "... = (\<Sum>K\<in>snd ` S. \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) * content K
+                                     / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))"
+        apply (rule sum.over_tagged_division_lemma [OF tagged_partial_division_of_union_self [OF S]])
+        apply (simp add: box_eq_empty(1) content_eq_0)
+        done
+      also have "... = \<epsilon>/2 * ((b \<bullet> i - a \<bullet> i) / (2 * content (cbox a b)) * (\<Sum>K\<in>snd ` S. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)))"
+        by (simp add: sum_distrib_left mult.assoc)
+      also have "... \<le> (\<epsilon>/2) * 1"
+      proof (rule mult_left_mono)
+        have "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>snd ` S. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i))
+              \<le> 2 * content (cbox a b)"
+        proof (rule sum_content_area_over_thin_division)
+          show "snd ` S division_of \<Union>(snd ` S)"
+            by (auto intro: S tagged_partial_division_of_union_self division_of_tagged_division)
+          show "UNION S snd \<subseteq> cbox a b"
+            using S by force
+          show "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> b \<bullet> i"
+            using mem_box(2) that by blast+
+        qed (use that in auto)
+        then show "(b \<bullet> i - a \<bullet> i) / (2 * content (cbox a b)) * (\<Sum>K\<in>snd ` S. content K / (interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i)) \<le> 1"
+          by (simp add: contab_gt0)
+      qed (use \<open>0 < \<epsilon>\<close> in auto)
+      finally show ?thesis by simp
+    qed
+    then have "(\<Sum>(x,K) \<in> S. norm (integral K h)) - (\<Sum>(x,K) \<in> S. norm (content K *\<^sub>R h x - integral K h)) \<le> \<epsilon>/2"
+      by (simp add: Groups_Big.sum_subtractf [symmetric])
+    ultimately show "(\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>"
+      by linarith
+  qed
+  ultimately show ?thesis using that by auto
+qed
+
+
+
+proposition equiintegrable_halfspace_restrictions_le:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes F: "F equiintegrable_on cbox a b" and f: "f \<in> F"
+    and norm_f: "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm(h x) \<le> norm(f x)"
+  shows "(\<Union>i \<in> Basis. \<Union>c. \<Union>h \<in> F. {(\<lambda>x. if x \<bullet> i \<le> c then h x else 0)})
+         equiintegrable_on cbox a b"
+proof (cases "content(cbox a b) = 0")
+  case True
+  then show ?thesis by simp
+next
+  case False
+  then have "content(cbox a b) > 0"
+    using zero_less_measure_iff by blast
+  then have "a \<bullet> i < b \<bullet> i" if "i \<in> Basis" for i
+    using content_pos_lt_eq that by blast
+  have int_F: "f integrable_on cbox a b" if "f \<in> F" for f
+    using F that by (simp add: equiintegrable_on_def)
+  let ?CI = "\<lambda>K h x. content K *\<^sub>R h x - integral K h"
+  show ?thesis
+    unfolding equiintegrable_on_def
+  proof (intro conjI; clarify)
+    show int_lec: "\<lbrakk>i \<in> Basis; h \<in> F\<rbrakk> \<Longrightarrow> (\<lambda>x. if x \<bullet> i \<le> c then h x else 0) integrable_on cbox a b" for i c h
+      using integrable_restrict_Int [of "{x. x \<bullet> i \<le> c}" h]
+      apply (auto simp: interval_split Int_commute mem_box intro!: integrable_on_subcbox int_F)
+      by (metis (full_types, hide_lams) min.bounded_iff)
+    show "\<exists>\<gamma>. gauge \<gamma> \<and>
+              (\<forall>f T. f \<in> (\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if x \<bullet> i \<le> c then h x else 0}) \<and>
+                     T tagged_division_of cbox a b \<and> \<gamma> fine T \<longrightarrow>
+                     norm ((\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) - integral (cbox a b) f) < \<epsilon>)"
+      if "\<epsilon> > 0" for \<epsilon>
+    proof -
+      obtain \<gamma>0 where "gauge \<gamma>0" and \<gamma>0:
+        "\<And>c i S h. \<lbrakk>c \<in> cbox a b; i \<in> Basis; S tagged_partial_division_of cbox a b;
+                        \<gamma>0 fine S; h \<in> F; \<And>x K. (x,K) \<in> S \<Longrightarrow> (K \<inter> {x. x \<bullet> i = c \<bullet> i} \<noteq> {})\<rbrakk>
+                       \<Longrightarrow> (\<Sum>(x,K) \<in> S. norm (integral K h)) < \<epsilon>/12"
+        apply (rule bounded_equiintegral_over_thin_tagged_partial_division [OF F f, of \<open>\<epsilon>/12\<close>])
+        using \<open>\<epsilon> > 0\<close> by (auto simp: norm_f)
+      obtain \<gamma>1 where "gauge \<gamma>1"
+        and \<gamma>1: "\<And>h T. \<lbrakk>h \<in> F; T tagged_division_of cbox a b; \<gamma>1 fine T\<rbrakk>
+                              \<Longrightarrow> norm ((\<Sum>(x,K) \<in> T. content K *\<^sub>R h x) - integral (cbox a b) h)
+                                  < \<epsilon>/(7 * (Suc DIM('b)))"
+      proof -
+        have e5: "\<epsilon>/(7 * (Suc DIM('b))) > 0"
+          using \<open>\<epsilon> > 0\<close> by auto
+        then show ?thesis
+          using F that by (auto simp: equiintegrable_on_def)
+      qed
+      have h_less3: "(\<Sum>(x,K) \<in> T. norm (?CI K h x)) < \<epsilon>/3"
+        if "T tagged_partial_division_of cbox a b" "\<gamma>1 fine T" "h \<in> F" for T h
+      proof -
+        have "(\<Sum>(x,K) \<in> T. norm (?CI K h x)) \<le> 2 * real DIM('b) * (\<epsilon>/(7 * Suc DIM('b)))"
+        proof (rule henstock_lemma_part2 [of h a b])
+          show "h integrable_on cbox a b"
+            using that F equiintegrable_on_def by metis
+          show "gauge \<gamma>1"
+            by (rule \<open>gauge \<gamma>1\<close>)
+        qed (use that \<open>\<epsilon> > 0\<close> \<gamma>1 in auto)
+        also have "... < \<epsilon>/3"
+          using \<open>\<epsilon> > 0\<close> by (simp add: divide_simps)
+        finally show ?thesis .
+      qed
+      have *: "norm ((\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) - integral (cbox a b) f) < \<epsilon>"
+                if f: "f = (\<lambda>x. if x \<bullet> i \<le> c then h x else 0)"
+                and T: "T tagged_division_of cbox a b"
+                and fine: "(\<lambda>x. \<gamma>0 x \<inter> \<gamma>1 x) fine T" and "i \<in> Basis" "h \<in> F" for f T i c h
+      proof (cases "a \<bullet> i \<le> c \<and> c \<le> b \<bullet> i")
+        case True
+        have "finite T"
+          using T by blast
+        define T' where "T' \<equiv> {(x,K) \<in> T. K \<inter> {x. x \<bullet> i \<le> c} \<noteq> {}}"
+        then have "T' \<subseteq> T"
+          by auto
+        then have "finite T'"
+          using \<open>finite T\<close> infinite_super by blast
+        have T'_tagged: "T' tagged_partial_division_of cbox a b"
+          by (meson T \<open>T' \<subseteq> T\<close> tagged_division_of_def tagged_partial_division_subset)
+        have fine': "\<gamma>0 fine T'" "\<gamma>1 fine T'"
+          using \<open>T' \<subseteq> T\<close> fine_Int fine_subset fine by blast+
+        have int_KK': "(\<Sum>(x,K) \<in> T. integral K f) = (\<Sum>(x,K) \<in> T'. integral K f)"
+          apply (rule sum.mono_neutral_right [OF \<open>finite T\<close> \<open>T' \<subseteq> T\<close>])
+          using f \<open>finite T\<close> \<open>T' \<subseteq> T\<close>
+          using integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h]
+          apply (auto simp: T'_def Int_commute)
+          done
+        have "(\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) = (\<Sum>(x,K) \<in> T'. content K *\<^sub>R f x)"
+          apply (rule sum.mono_neutral_right [OF \<open>finite T\<close> \<open>T' \<subseteq> T\<close>])
+          using T f \<open>finite T\<close> \<open>T' \<subseteq> T\<close> apply (force simp: T'_def)
+          done
+        moreover have "norm ((\<Sum>(x,K) \<in> T'. content K *\<^sub>R f x) - integral (cbox a b) f) < \<epsilon>"
+        proof -
+          have *: "norm y < \<epsilon>" if "norm x < \<epsilon>/3" "norm(x - y) \<le> 2 * \<epsilon>/3" for x y::'b
+          proof -
+            have "norm y \<le> norm x + norm(x - y)"
+              by (metis norm_minus_commute norm_triangle_sub)
+            also have "\<dots> < \<epsilon>/3 + 2*\<epsilon>/3"
+              using that by linarith
+            also have "... = \<epsilon>"
+              by simp
+            finally show ?thesis .
+          qed
+          have "norm (\<Sum>(x,K) \<in> T'. ?CI K h x)
+                \<le> (\<Sum>(x,K) \<in> T'. norm (?CI K h x))"
+            by (simp add: norm_sum split_def)
+          also have "... < \<epsilon>/3"
+            by (intro h_less3 T'_tagged fine' that)
+          finally have "norm (\<Sum>(x,K) \<in> T'. ?CI K h x) < \<epsilon>/3" .
+          moreover have "integral (cbox a b) f = (\<Sum>(x,K) \<in> T. integral K f)"
+            using int_lec that by (auto simp: integral_combine_tagged_division_topdown)
+          moreover have "norm (\<Sum>(x,K) \<in> T'. ?CI K h x - ?CI K f x)
+                \<le> 2*\<epsilon>/3"
+          proof -
+            define T'' where "T'' \<equiv> {(x,K) \<in> T'. ~ (K \<subseteq> {x. x \<bullet> i \<le> c})}"
+            then have "T'' \<subseteq> T'"
+              by auto
+            then have "finite T''"
+              using \<open>finite T'\<close> infinite_super by blast
+            have T''_tagged: "T'' tagged_partial_division_of cbox a b"
+              using T'_tagged \<open>T'' \<subseteq> T'\<close> tagged_partial_division_subset by blast
+            have fine'': "\<gamma>0 fine T''" "\<gamma>1 fine T''"
+              using \<open>T'' \<subseteq> T'\<close> fine' by (blast intro: fine_subset)+
+            have "(\<Sum>(x,K) \<in> T'. ?CI K h x - ?CI K f x)
+                = (\<Sum>(x,K) \<in> T''. ?CI K h x - ?CI K f x)"
+            proof (clarify intro!: sum.mono_neutral_right [OF \<open>finite T'\<close> \<open>T'' \<subseteq> T'\<close>])
+              fix x K
+              assume "(x,K) \<in> T'" "(x,K) \<notin> T''"
+              then have "x \<in> K" "x \<bullet> i \<le> c" "{x. x \<bullet> i \<le> c} \<inter> K = K"
+                using T''_def T'_tagged by blast+
+              then show "?CI K h x - ?CI K f x = 0"
+                using integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h] by (auto simp: f)
+            qed
+            moreover have "norm (\<Sum>(x,K) \<in> T''. ?CI K h x - ?CI K f x) \<le> 2*\<epsilon>/3"
+            proof -
+              define A where "A \<equiv> {(x,K) \<in> T''. x \<bullet> i \<le> c}"
+              define B where "B \<equiv> {(x,K) \<in> T''. x \<bullet> i > c}"
+              then have "A \<subseteq> T''" "B \<subseteq> T''" and disj: "A \<inter> B = {}" and T''_eq: "T'' = A \<union> B"
+                by (auto simp: A_def B_def)
+              then have "finite A" "finite B"
+                using \<open>finite T''\<close>  by (auto intro: finite_subset)
+              have A_tagged: "A tagged_partial_division_of cbox a b"
+                using T''_tagged \<open>A \<subseteq> T''\<close> tagged_partial_division_subset by blast
+              have fineA: "\<gamma>0 fine A" "\<gamma>1 fine A"
+                using \<open>A \<subseteq> T''\<close> fine'' by (blast intro: fine_subset)+
+              have B_tagged: "B tagged_partial_division_of cbox a b"
+                using T''_tagged \<open>B \<subseteq> T''\<close> tagged_partial_division_subset by blast
+              have fineB: "\<gamma>0 fine B" "\<gamma>1 fine B"
+                using \<open>B \<subseteq> T''\<close> fine'' by (blast intro: fine_subset)+
+              have "norm (\<Sum>(x,K) \<in> T''. ?CI K h x - ?CI K f x)
+                          \<le> (\<Sum>(x,K) \<in> T''. norm (?CI K h x - ?CI K f x))"
+                by (simp add: norm_sum split_def)
+              also have "... = (\<Sum>(x,K) \<in> A. norm (?CI K h x - ?CI K f x)) +
+                               (\<Sum>(x,K) \<in> B. norm (?CI K h x - ?CI K f x))"
+                by (simp add: sum.union_disjoint T''_eq disj \<open>finite A\<close> \<open>finite B\<close>)
+              also have "... = (\<Sum>(x,K) \<in> A. norm (integral K h - integral K f)) +
+                               (\<Sum>(x,K) \<in> B. norm (?CI K h x + integral K f))"
+                by (auto simp: A_def B_def f norm_minus_commute intro!: sum.cong arg_cong2 [where f= "op+"])
+              also have "... \<le> (\<Sum>(x,K)\<in>A. norm (integral K h)) +
+                                 (\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A. norm (integral K h))
+                             + ((\<Sum>(x,K)\<in>B. norm (?CI K h x)) +
+                                (\<Sum>(x,K)\<in>B. norm (integral K h)) +
+                                  (\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B. norm (integral K h)))"
+              proof (rule add_mono)
+                show "(\<Sum>(x,K)\<in>A. norm (integral K h - integral K f))
+                        \<le> (\<Sum>(x,K)\<in>A. norm (integral K h)) +
+                           (\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A.
+                              norm (integral K h))"
+                proof (subst sum.reindex_nontrivial [OF \<open>finite A\<close>], clarsimp)
+                  fix x K L
+                  assume "(x,K) \<in> A" "(x,L) \<in> A"
+                    and int_ne0: "integral (L \<inter> {x. x \<bullet> i \<le> c}) h \<noteq> 0"
+                    and eq: "K \<inter> {x. x \<bullet> i \<le> c} = L \<inter> {x. x \<bullet> i \<le> c}"
+                  have False if "K \<noteq> L"
+                  proof -
+                    obtain u v where uv: "L = cbox u v"
+                      using T'_tagged \<open>(x, L) \<in> A\<close> \<open>A \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by blast
+                    have "A tagged_division_of UNION A snd"
+                      using A_tagged tagged_partial_division_of_union_self by auto
+                    then have "interior (K \<inter> {x. x \<bullet> i \<le> c}) = {}"
+                      apply (rule tagged_division_split_left_inj [OF _ \<open>(x,K) \<in> A\<close> \<open>(x,L) \<in> A\<close>])
+                      using that eq \<open>i \<in> Basis\<close> by auto
+                    then show False
+                      using interval_split [OF \<open>i \<in> Basis\<close>] int_ne0 content_eq_0_interior eq uv by fastforce
+                  qed
+                  then show "K = L" by blast
+                next
+                  show "(\<Sum>(x,K) \<in> A. norm (integral K h - integral K f))
+                          \<le> (\<Sum>(x,K) \<in> A. norm (integral K h)) +
+                             sum ((\<lambda>(x,K). norm (integral K h)) \<circ> (\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c}))) A"
+                    using integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h] f
+                    by (auto simp: Int_commute A_def [symmetric] sum.distrib [symmetric] intro!: sum_mono norm_triangle_ineq4)
+                qed
+              next
+                show "(\<Sum>(x,K)\<in>B. norm (?CI K h x + integral K f))
+                      \<le> (\<Sum>(x,K)\<in>B. norm (?CI K h x)) + (\<Sum>(x,K)\<in>B. norm (integral K h)) +
+                         (\<Sum>(x,K)\<in>(\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B. norm (integral K h))"
+                proof (subst sum.reindex_nontrivial [OF \<open>finite B\<close>], clarsimp)
+                  fix x K L
+                  assume "(x,K) \<in> B" "(x,L) \<in> B"
+                    and int_ne0: "integral (L \<inter> {x. c \<le> x \<bullet> i}) h \<noteq> 0"
+                    and eq: "K \<inter> {x. c \<le> x \<bullet> i} = L \<inter> {x. c \<le> x \<bullet> i}"
+                  have False if "K \<noteq> L"
+                  proof -
+                    obtain u v where uv: "L = cbox u v"
+                      using T'_tagged \<open>(x, L) \<in> B\<close> \<open>B \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by blast
+                    have "B tagged_division_of UNION B snd"
+                      using B_tagged tagged_partial_division_of_union_self by auto
+                    then have "interior (K \<inter> {x. c \<le> x \<bullet> i}) = {}"
+                      apply (rule tagged_division_split_right_inj [OF _ \<open>(x,K) \<in> B\<close> \<open>(x,L) \<in> B\<close>])
+                      using that eq \<open>i \<in> Basis\<close> by auto
+                    then show False
+                      using interval_split [OF \<open>i \<in> Basis\<close>] int_ne0
+                        content_eq_0_interior eq uv by fastforce
+                  qed
+                  then show "K = L" by blast
+                next
+                  show "(\<Sum>(x,K) \<in> B. norm (?CI K h x + integral K f))
+                        \<le> (\<Sum>(x,K) \<in> B. norm (?CI K h x)) +
+                           (\<Sum>(x,K) \<in> B. norm (integral K h)) + sum ((\<lambda>(x,K). norm (integral K h)) \<circ> (\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i}))) B"
+                  proof (clarsimp simp: B_def [symmetric] sum.distrib [symmetric] intro!: sum_mono)
+                    fix x K
+                    assume "(x,K) \<in> B"
+                    have *: "i = i1 + i2 \<Longrightarrow> norm(c + i1) \<le> norm c + norm i + norm(i2)"
+                      for i::'b and c i1 i2
+                      by (metis add.commute add.left_commute add_diff_cancel_right' dual_order.refl norm_add_rule_thm norm_triangle_ineq4)
+                    obtain u v where uv: "K = cbox u v"
+                      using T'_tagged \<open>(x,K) \<in> B\<close> \<open>B \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by blast
+                    have "h integrable_on cbox a b"
+                      by (simp add: int_F \<open>h \<in> F\<close>)
+                    then have huv: "h integrable_on cbox u v"
+                      apply (rule integrable_on_subcbox)
+                      using B_tagged \<open>(x,K) \<in> B\<close> uv by blast
+                    have "integral K h = integral K f + integral (K \<inter> {x. c \<le> x \<bullet> i}) h"
+                      using integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h] f uv \<open>i \<in> Basis\<close>
+                      by (simp add: Int_commute integral_split [OF huv \<open>i \<in> Basis\<close>])
+                  then show "norm (?CI K h x + integral K f)
+                             \<le> norm (?CI K h x) + norm (integral K h) + norm (integral (K \<inter> {x. c \<le> x \<bullet> i}) h)"
+                    by (rule *)
+                qed
+              qed
+            qed
+            also have "... \<le> 2*\<epsilon>/3"
+            proof -
+              have overlap: "K \<inter> {x. x \<bullet> i = c} \<noteq> {}" if "(x,K) \<in> T''" for x K
+              proof -
+                obtain y y' where y: "y' \<in> K" "c < y' \<bullet> i" "y \<in> K" "y \<bullet> i \<le> c"
+                  using that  T''_def T'_def \<open>(x,K) \<in> T''\<close> by fastforce
+                obtain u v where uv: "K = cbox u v"
+                  using T''_tagged \<open>(x,K) \<in> T''\<close> by blast
+                then have "connected K"
+                  by (simp add: is_interval_cbox is_interval_connected)
+                then have "(\<exists>z \<in> K. z \<bullet> i = c)"
+                  using y connected_ivt_component by fastforce
+                then show ?thesis
+                  by fastforce
+              qed
+              have **: "\<lbrakk>x < \<epsilon>/12; y < \<epsilon>/12; z \<le> \<epsilon>/2\<rbrakk> \<Longrightarrow> x + y + z \<le> 2 * \<epsilon>/3" for x y z
+                by auto
+              show ?thesis
+              proof (rule **)
+                have cb_ab: "(\<Sum>j \<in> Basis. if j = i then c *\<^sub>R i else (a \<bullet> j) *\<^sub>R j) \<in> cbox a b"
+                  using \<open>i \<in> Basis\<close> True \<open>\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i < b \<bullet> i\<close>
+                  apply (clarsimp simp add: mem_box)
+                  apply (subst sum_if_inner | force)+
+                  done
+                show "(\<Sum>(x,K) \<in> A. norm (integral K h)) < \<epsilon>/12"
+                  apply (rule \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> A_tagged fineA(1) \<open>h \<in> F\<close>])
+                  using \<open>i \<in> Basis\<close> \<open>A \<subseteq> T''\<close> overlap
+                  apply (subst sum_if_inner | force)+
+                  done
+                have 1: "(\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A tagged_partial_division_of cbox a b"
+                  using \<open>finite A\<close> \<open>i \<in> Basis\<close>
+                  apply (auto simp: tagged_partial_division_of_def)
+                  using A_tagged apply (auto simp: A_def)
+                  using interval_split(1) by blast
+                have 2: "\<gamma>0 fine (\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A"
+                  using fineA(1) fine_def by fastforce
+                show "(\<Sum>(x,K) \<in> (\<lambda>(x,K). (x,K \<inter> {x. x \<bullet> i \<le> c})) ` A. norm (integral K h)) < \<epsilon>/12"
+                  apply (rule \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> 1 2 \<open>h \<in> F\<close>])
+                  using \<open>i \<in> Basis\<close> apply (subst sum_if_inner | force)+
+                  using overlap apply (auto simp: A_def)
+                  done
+                have *: "\<lbrakk>x < \<epsilon>/3; y < \<epsilon>/12; z < \<epsilon>/12\<rbrakk> \<Longrightarrow> x + y + z \<le> \<epsilon>/2" for x y z
+                  by auto
+                show "(\<Sum>(x,K) \<in> B. norm (?CI K h x)) +
+                      (\<Sum>(x,K) \<in> B. norm (integral K h)) +
+                      (\<Sum>(x,K) \<in> (\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B. norm (integral K h))
+                      \<le> \<epsilon>/2"
+                proof (rule *)
+                  show "(\<Sum>(x,K) \<in> B. norm (?CI K h x)) < \<epsilon>/3"
+                    by (intro h_less3 B_tagged fineB that)
+                  show "(\<Sum>(x,K) \<in> B. norm (integral K h)) < \<epsilon>/12"
+                    apply (rule \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> B_tagged fineB(1) \<open>h \<in> F\<close>])
+                    using \<open>i \<in> Basis\<close> \<open>B \<subseteq> T''\<close> overlap by (subst sum_if_inner | force)+
+                  have 1: "(\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B tagged_partial_division_of cbox a b"
+                    using \<open>finite B\<close> \<open>i \<in> Basis\<close>
+                    apply (auto simp: tagged_partial_division_of_def)
+                    using B_tagged apply (auto simp: B_def)
+                    using interval_split(2) by blast
+                  have 2: "\<gamma>0 fine (\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B"
+                    using fineB(1) fine_def by fastforce
+                  show "(\<Sum>(x,K) \<in> (\<lambda>(x,K). (x,K \<inter> {x. c \<le> x \<bullet> i})) ` B. norm (integral K h)) < \<epsilon>/12"
+                    apply (rule \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> 1 2 \<open>h \<in> F\<close>])
+                    using \<open>i \<in> Basis\<close> apply (subst sum_if_inner | force)+
+                    using overlap apply (auto simp: B_def)
+                    done
+                qed
+              qed
+            qed
+            finally show ?thesis .
+          qed
+          ultimately show ?thesis by metis
+        qed
+        ultimately show ?thesis
+          by (simp add: sum_subtractf [symmetric] int_KK' *)
+      qed
+        ultimately show ?thesis by metis
+      next
+        case False
+        then consider "c < a \<bullet> i" | "b \<bullet> i < c"
+          by auto
+        then show ?thesis
+        proof cases
+          case 1
+          then have f0: "f x = 0" if "x \<in> cbox a b" for x
+            using that f \<open>i \<in> Basis\<close> mem_box(2) by force
+          then have int_f0: "integral (cbox a b) f = 0"
+            by (simp add: integral_cong)
+          have f0_tag: "f x = 0" if "(x,K) \<in> T" for x K
+            using T f0 that by (force simp: tagged_division_of_def)
+          then have "(\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) = 0"
+            by (metis (mono_tags, lifting) real_vector.scale_eq_0_iff split_conv sum.neutral surj_pair)
+          then show ?thesis
+            using \<open>0 < \<epsilon>\<close> by (simp add: int_f0)
+      next
+          case 2
+          then have fh: "f x = h x" if "x \<in> cbox a b" for x
+            using that f \<open>i \<in> Basis\<close> mem_box(2) by force
+          then have int_f: "integral (cbox a b) f = integral (cbox a b) h"
+            using integral_cong by blast
+          have fh_tag: "f x = h x" if "(x,K) \<in> T" for x K
+            using T fh that by (force simp: tagged_division_of_def)
+          then have "(\<Sum>(x,K) \<in> T. content K *\<^sub>R f x) = (\<Sum>(x,K) \<in> T. content K *\<^sub>R h x)"
+            by (metis (mono_tags, lifting) split_cong sum.cong)
+          with \<open>0 < \<epsilon>\<close> show ?thesis
+            apply (simp add: int_f)
+            apply (rule less_trans [OF \<gamma>1])
+            using that fine_Int apply (force simp: divide_simps)+
+            done
+        qed
+      qed
+      have  "gauge (\<lambda>x. \<gamma>0 x \<inter> \<gamma>1 x)"
+        by (simp add: \<open>gauge \<gamma>0\<close> \<open>gauge \<gamma>1\<close> gauge_Int)
+      then show ?thesis
+        by (auto intro: *)
+    qed
+  qed
+qed
+
+
+
+corollary equiintegrable_halfspace_restrictions_ge:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes F: "F equiintegrable_on cbox a b" and f: "f \<in> F"
+    and norm_f: "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm(h x) \<le> norm(f x)"
+  shows "(\<Union>i \<in> Basis. \<Union>c. \<Union>h \<in> F. {(\<lambda>x. if x \<bullet> i \<ge> c then h x else 0)})
+         equiintegrable_on cbox a b"
+proof -
+  have *: "(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>(\<lambda>f. f \<circ> uminus) ` F. {\<lambda>x. if x \<bullet> i \<le> c then h x else 0})
+           equiintegrable_on  cbox (- b) (- a)"
+  proof (rule equiintegrable_halfspace_restrictions_le)
+    show "(\<lambda>f. f \<circ> uminus) ` F equiintegrable_on cbox (- b) (- a)"
+      using F equiintegrable_reflect by blast
+    show "f \<circ> uminus \<in> (\<lambda>f. f \<circ> uminus) ` F"
+      using f by auto
+    show "\<And>h x. \<lbrakk>h \<in> (\<lambda>f. f \<circ> uminus) ` F; x \<in> cbox (- b) (- a)\<rbrakk> \<Longrightarrow> norm (h x) \<le> norm ((f \<circ> uminus) x)"
+      using f apply (clarsimp simp:)
+      by (metis add.inverse_inverse image_eqI norm_f uminus_interval_vector)
+  qed
+  have eq: "(\<lambda>f. f \<circ> uminus) `
+            (\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if x \<bullet> i \<le> c then (h \<circ> uminus) x else 0}) =
+            (\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if c \<le> x \<bullet> i then h x else 0})"
+    apply (auto simp: o_def cong: if_cong)
+    using minus_le_iff apply fastforce
+    apply (rule_tac x="\<lambda>x. if c \<le> (-x) \<bullet> i then h(-x) else 0" in image_eqI)
+    using le_minus_iff apply fastforce+
+    done
+  show ?thesis
+    using equiintegrable_reflect [OF *] by (auto simp: eq)
+qed
+
+
+proposition equiintegrable_closed_interval_restrictions:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  assumes f: "f integrable_on cbox a b"
+  shows "(\<Union>c d. {(\<lambda>x. if x \<in> cbox c d then f x else 0)}) equiintegrable_on cbox a b"
+proof -
+  let ?g = "\<lambda>B c d x. if \<forall>i\<in>B. c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i then f x else 0"
+  have *: "insert f (\<Union>c d. {?g B c d}) equiintegrable_on cbox a b" if "B \<subseteq> Basis" for B
+  proof -
+    have "finite B"
+      using finite_Basis finite_subset \<open>B \<subseteq> Basis\<close> by blast
+    then show ?thesis using \<open>B \<subseteq> Basis\<close>
+    proof (induction B)
+      case empty
+      with f show ?case by auto
+    next
+      case (insert i B)
+      then have "i \<in> Basis"
+        by auto
+      have *: "norm (h x) \<le> norm (f x)"
+        if "h \<in> insert f (\<Union>c d. {?g B c d})" "x \<in> cbox a b" for h x
+        using that by auto
+      have "(\<Union>i\<in>Basis. 
+                \<Union>\<xi>. \<Union>h\<in>insert f (\<Union>i\<in>Basis. \<Union>\<psi>. \<Union>h\<in>insert f (\<Union>c d. {?g B c d}). {\<lambda>x. if x \<bullet> i \<le> \<psi> then h x else 0}). 
+                {\<lambda>x. if \<xi> \<le> x \<bullet> i then h x else 0}) 
+             equiintegrable_on cbox a b"
+      proof (rule equiintegrable_halfspace_restrictions_ge [where f=f])
+        show "insert f (\<Union>i\<in>Basis. \<Union>\<xi>. \<Union>h\<in>insert f (\<Union>c d. {?g B c d}).
+              {\<lambda>x. if x \<bullet> i \<le> \<xi> then h x else 0}) equiintegrable_on cbox a b"
+          apply (intro * f equiintegrable_on_insert equiintegrable_halfspace_restrictions_le [OF insert.IH insertI1])
+          using insert.prems apply auto
+          done
+        show"norm(h x) \<le> norm(f x)"
+          if "h \<in> insert f (\<Union>i\<in>Basis. \<Union>\<xi>. \<Union>h\<in>insert f (\<Union>c d. {?g B c d}). {\<lambda>x. if x \<bullet> i \<le> \<xi> then h x else 0})" 
+             "x \<in> cbox a b" for h x
+          using that by auto
+      qed auto
+      then have "insert f (\<Union>i\<in>Basis. 
+                \<Union>\<xi>. \<Union>h\<in>insert f (\<Union>i\<in>Basis. \<Union>\<psi>. \<Union>h\<in>insert f (\<Union>c d. {?g B c d}). {\<lambda>x. if x \<bullet> i \<le> \<psi> then h x else 0}). 
+                {\<lambda>x. if \<xi> \<le> x \<bullet> i then h x else 0}) 
+             equiintegrable_on cbox a b"
+        by (blast intro: f equiintegrable_on_insert)
+      then show ?case
+        apply (rule equiintegrable_on_subset, clarify)
+        using \<open>i \<in> Basis\<close> apply simp
+        apply (drule_tac x=i in bspec, assumption)
+        apply (drule_tac x="c \<bullet> i" in spec, clarify)
+        apply (drule_tac x=i in bspec, assumption)
+        apply (drule_tac x="d \<bullet> i" in spec)
+        apply (clarsimp simp add: fun_eq_iff)
+        apply (drule_tac x=c in spec)
+        apply (drule_tac x=d in spec)
+        apply (simp add: split: if_split_asm)
+        done
+    qed
+  qed
+  show ?thesis
+    by (rule equiintegrable_on_subset [OF * [OF subset_refl]]) (auto simp: mem_box)
+qed
+  
+
+
+subsection\<open>Continuity of the indefinite integral\<close>
+
+proposition indefinite_integral_continuous:
+  fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
+  assumes int_f: "f integrable_on cbox a b"
+      and c: "c \<in> cbox a b" and d: "d \<in> cbox a b" "0 < \<epsilon>"
+  obtains \<delta> where "0 < \<delta>"
+              "\<And>c' d'. \<lbrakk>c' \<in> cbox a b; d' \<in> cbox a b; norm(c' - c) \<le> \<delta>; norm(d' - d) \<le> \<delta>\<rbrakk>
+                        \<Longrightarrow> norm(integral(cbox c' d') f - integral(cbox c d) f) < \<epsilon>"
+proof -
+  { assume "\<exists>c' d'. c' \<in> cbox a b \<and> d' \<in> cbox a b \<and> norm(c' - c) \<le> \<delta> \<and> norm(d' - d) \<le> \<delta> \<and>
+                    norm(integral(cbox c' d') f - integral(cbox c d) f) \<ge> \<epsilon>"
+                    (is "\<exists>c' d'. ?\<Phi> c' d' \<delta>") if "0 < \<delta>" for \<delta>
+    then have "\<exists>c' d'. ?\<Phi> c' d' (1 / Suc n)" for n
+      by simp
+    then obtain u v where "\<And>n. ?\<Phi> (u n) (v n) (1 / Suc n)"
+      by metis
+    then have u: "u n \<in> cbox a b" and norm_u: "norm(u n - c) \<le> 1 / Suc n"
+         and  v: "v n \<in> cbox a b" and norm_v: "norm(v n - d) \<le> 1 / Suc n"
+         and \<epsilon>: "\<epsilon> \<le> norm (integral (cbox (u n) (v n)) f - integral (cbox c d) f)" for n
+      by blast+
+    then have False
+    proof -
+      have uvn: "cbox (u n) (v n) \<subseteq> cbox a b" for n
+        by (meson u v mem_box(2) subset_box(1))
+      define S where "S \<equiv> \<Union>i \<in> Basis. {x. x \<bullet> i = c \<bullet> i} \<union> {x. x \<bullet> i = d \<bullet> i}"
+      have "negligible S"
+        unfolding S_def by force
+      then have int_f': "(\<lambda>x. if x \<in> S then 0 else f x) integrable_on cbox a b"
+        by (rule integrable_spike) (auto intro: assms)
+      have get_n: "\<exists>n. \<forall>m\<ge>n. x \<in> cbox (u m) (v m) \<longleftrightarrow> x \<in> cbox c d" if x: "x \<notin> S" for x
+      proof -
+        define \<epsilon> where "\<epsilon> \<equiv> Min ((\<lambda>i. min \<bar>x \<bullet> i - c \<bullet> i\<bar> \<bar>x \<bullet> i - d \<bullet> i\<bar>) ` Basis)"
+        have "\<epsilon> > 0"
+          using \<open>x \<notin> S\<close> by (auto simp: S_def \<epsilon>_def)
+        then obtain n where "n \<noteq> 0" and n: "1 / (real n) < \<epsilon>"
+          by (metis inverse_eq_divide real_arch_inverse)
+        have emin: "\<epsilon> \<le> min \<bar>x \<bullet> i - c \<bullet> i\<bar> \<bar>x \<bullet> i - d \<bullet> i\<bar>" if "i \<in> Basis" for i
+          unfolding \<epsilon>_def
+          apply (rule Min.coboundedI)
+          using that by force+
+        have "1 / real (Suc n) < \<epsilon>"
+          using n \<open>n \<noteq> 0\<close> \<open>\<epsilon> > 0\<close> by (simp add: field_simps)
+        have "x \<in> cbox (u m) (v m) \<longleftrightarrow> x \<in> cbox c d" if "m \<ge> n" for m
+        proof -
+          have *: "\<lbrakk>\<bar>u - c\<bar> \<le> n; \<bar>v - d\<bar> \<le> n; N < \<bar>x - c\<bar>; N < \<bar>x - d\<bar>; n \<le> N\<rbrakk>
+                   \<Longrightarrow> u \<le> x \<and> x \<le> v \<longleftrightarrow> c \<le> x \<and> x \<le> d" for N n u v c d and x::real
+            by linarith
+          have "(u m \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> v m \<bullet> i) = (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i)"
+            if "i \<in> Basis" for i
+          proof (rule *)
+            show "\<bar>u m \<bullet> i - c \<bullet> i\<bar> \<le> 1 / Suc m"
+              using norm_u [of m]
+              by (metis (full_types) order_trans Basis_le_norm inner_commute inner_diff_right that)
+            show "\<bar>v m \<bullet> i - d \<bullet> i\<bar> \<le> 1 / real (Suc m)"
+              using norm_v [of m]
+              by (metis (full_types) order_trans Basis_le_norm inner_commute inner_diff_right that)
+            show "1/n < \<bar>x \<bullet> i - c \<bullet> i\<bar>" "1/n < \<bar>x \<bullet> i - d \<bullet> i\<bar>"
+              using n \<open>n \<noteq> 0\<close> emin [OF \<open>i \<in> Basis\<close>]
+              by (simp_all add: inverse_eq_divide)
+            show "1 / real (Suc m) \<le> 1 / real n"
+              using \<open>n \<noteq> 0\<close> \<open>m \<ge> n\<close> by (simp add: divide_simps)
+          qed
+          then show ?thesis by (simp add: mem_box)
+        qed
+        then show ?thesis by blast
+      qed
+      have 1: "range (\<lambda>n x. if x \<in> cbox (u n) (v n) then if x \<in> S then 0 else f x else 0) equiintegrable_on cbox a b"
+        by (blast intro: equiintegrable_on_subset [OF equiintegrable_closed_interval_restrictions [OF int_f']])
+      have 2: "(\<lambda>n. if x \<in> cbox (u n) (v n) then if x \<in> S then 0 else f x else 0)
+               \<longlonglongrightarrow> (if x \<in> cbox c d then if x \<in> S then 0 else f x else 0)" for x
+        by (fastforce simp: dest: get_n intro: Lim_eventually eventually_sequentiallyI)
+      have [simp]: "cbox c d \<inter> cbox a b = cbox c d"
+        using c d by (force simp: mem_box)
+      have [simp]: "cbox (u n) (v n) \<inter> cbox a b = cbox (u n) (v n)" for n
+        using u v by (fastforce simp: mem_box intro: order.trans)
+      have "\<And>y A. y \<in> A - S \<Longrightarrow> f y = (\<lambda>x. if x \<in> S then 0 else f x) y"
+        by simp
+      then have "\<And>A. integral A (\<lambda>x. if x \<in> S then 0 else f (x)) = integral A (\<lambda>x. f (x))"
+        by (blast intro: integral_spike [OF \<open>negligible S\<close>])
+      moreover
+      obtain N where "dist (integral (cbox (u N) (v N)) (\<lambda>x. if x \<in> S then 0 else f x))
+                           (integral (cbox c d) (\<lambda>x. if x \<in> S then 0 else f x)) < \<epsilon>"
+        using equiintegrable_limit [OF 1 2] \<open>0 < \<epsilon>\<close> by (force simp: integral_restrict_Int lim_sequentially)
+      ultimately have "dist (integral (cbox (u N) (v N)) f) (integral (cbox c d) f) < \<epsilon>"
+        by simp
+      then show False
+        by (metis dist_norm not_le \<epsilon>)
+    qed
+  }
+  then show ?thesis
+    by (meson not_le that)
+qed
+
+corollary indefinite_integral_uniformly_continuous:
+  fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
+  assumes "f integrable_on cbox a b"
+  shows "uniformly_continuous_on (cbox (Pair a a) (Pair b b)) (\<lambda>y. integral (cbox (fst y) (snd y)) f)"
+proof -
+  show ?thesis
+  proof (rule compact_uniformly_continuous, clarsimp simp add: continuous_on_iff)
+    fix c d and \<epsilon>::real
+    assume c: "c \<in> cbox a b" and d: "d \<in> cbox a b" and "0 < \<epsilon>"
+    obtain \<delta> where "0 < \<delta>" and \<delta>:
+              "\<And>c' d'. \<lbrakk>c' \<in> cbox a b; d' \<in> cbox a b; norm(c' - c) \<le> \<delta>; norm(d' - d) \<le> \<delta>\<rbrakk>
+                                  \<Longrightarrow> norm(integral(cbox c' d') f -
+                                           integral(cbox c d) f) < \<epsilon>"
+      using indefinite_integral_continuous \<open>0 < \<epsilon>\<close> assms c d by blast
+    show "\<exists>\<delta> > 0. \<forall>x' \<in> cbox (a, a) (b, b).
+                   dist x' (c, d) < \<delta> \<longrightarrow>
+                   dist (integral (cbox (fst x') (snd x')) f)
+                        (integral (cbox c d) f)
+                   < \<epsilon>"
+      using \<open>0 < \<delta>\<close>
+      by (force simp: dist_norm intro: \<delta> order_trans [OF norm_fst_le] order_trans [OF norm_snd_le] less_imp_le)
+  qed auto
+qed
+
+
+corollary bounded_integrals_over_subintervals:
+  fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
+  assumes "f integrable_on cbox a b"
+  shows "bounded {integral (cbox c d) f |c d. cbox c d \<subseteq> cbox a b}"
+proof -
+  have "bounded ((\<lambda>y. integral (cbox (fst y) (snd y)) f) ` cbox (a, a) (b, b))"
+       (is "bounded ?I")
+    by (blast intro: bounded_cbox bounded_uniformly_continuous_image indefinite_integral_uniformly_continuous [OF assms])
+  then obtain B where "B > 0" and B: "\<And>x. x \<in> ?I \<Longrightarrow> norm x \<le> B"
+    by (auto simp: bounded_pos)
+  have "norm x \<le> B" if "x = integral (cbox c d) f" "cbox c d \<subseteq> cbox a b" for x c d
+  proof (cases "cbox c d = {}")
+    case True
+    with \<open>0 < B\<close> that show ?thesis by auto
+  next
+    case False
+    show ?thesis
+      apply (rule B)
+      using that \<open>B > 0\<close> False apply (clarsimp simp: image_def)
+      by (metis cbox_Pair_iff interval_subset_is_interval is_interval_cbox prod.sel)
+  qed
+  then show ?thesis
+    by (blast intro: boundedI)
+qed
+
+
+text\<open>An existence theorem for "improper" integrals.
+Hake's theorem implies that if the integrals over subintervals have a limit, the integral exists.
+We only need to assume that the integrals are bounded, and we get absolute integrability,
+but we also need a (rather weak) bound assumption on the function.\<close>
+
+theorem absolutely_integrable_improper:
+  fixes f :: "'M::euclidean_space \<Rightarrow> 'N::euclidean_space"
+  assumes int_f: "\<And>c d. cbox c d \<subseteq> box a b \<Longrightarrow> f integrable_on cbox c d"
+      and bo: "bounded {integral (cbox c d) f |c d. cbox c d \<subseteq> box a b}"
+      and absi: "\<And>i. i \<in> Basis
+          \<Longrightarrow> \<exists>g. g absolutely_integrable_on cbox a b \<and>
+                  ((\<forall>x \<in> cbox a b. f x \<bullet> i \<le> g x) \<or> (\<forall>x \<in> cbox a b. f x \<bullet> i \<ge> g x))"
+      shows "f absolutely_integrable_on cbox a b"
+proof (cases "content(cbox a b) = 0")
+  case True
+  then show ?thesis
+    by auto
+next
+  case False
+  then have pos: "content(cbox a b) > 0"
+    using zero_less_measure_iff by blast
+  show ?thesis
+    unfolding absolutely_integrable_componentwise_iff [where f = f]
+  proof
+    fix j::'N
+    assume "j \<in> Basis"
+    then obtain g where absint_g: "g absolutely_integrable_on cbox a b"
+                    and g: "(\<forall>x \<in> cbox a b. f x \<bullet> j \<le> g x) \<or> (\<forall>x \<in> cbox a b. f x \<bullet> j \<ge> g x)"
+      using absi by blast
+    have int_gab: "g integrable_on cbox a b"
+      using absint_g set_lebesgue_integral_eq_integral(1) by blast
+    have 1: "cbox (a + (b - a) /\<^sub>R real (Suc n)) (b - (b - a) /\<^sub>R real (Suc n)) \<subseteq> box a b" for n
+      apply (rule subset_box_imp)
+      using pos apply (auto simp: content_pos_lt_eq algebra_simps)
+      done
+    have 2: "cbox (a + (b - a) /\<^sub>R real (Suc n)) (b - (b - a) /\<^sub>R real (Suc n)) \<subseteq>
+             cbox (a + (b - a) /\<^sub>R real (Suc n + 1)) (b - (b - a) /\<^sub>R real (Suc n + 1))" for n
+      apply (rule subset_box_imp)
+      using pos apply (simp add: content_pos_lt_eq algebra_simps)
+        apply (simp add: divide_simps)
+      apply (auto simp: field_simps)
+      done
+    have getN: "\<exists>N::nat. \<forall>k. k \<ge> N \<longrightarrow> x \<in> cbox (a + (b - a) /\<^sub>R real k) (b - (b - a) /\<^sub>R real k)"
+      if x: "x \<in> box a b" for x
+    proof -
+      let ?\<Delta> = "(\<Union>i \<in> Basis. {((x - a) \<bullet> i) / ((b - a) \<bullet> i), (b - x) \<bullet> i / ((b - a) \<bullet> i)})"
+      obtain N where N: "real N > 1 / Inf ?\<Delta>"
+        using reals_Archimedean2 by blast
+      moreover have \<Delta>: "Inf ?\<Delta> > 0"
+        using that by (auto simp: finite_less_Inf_iff mem_box algebra_simps divide_simps)
+      ultimately have "N > 0"
+        using of_nat_0_less_iff by fastforce
+      show ?thesis
+      proof (intro exI impI allI)
+        fix k assume "N \<le> k"
+        with \<open>0 < N\<close> have "k > 0"
+          by linarith
+        have xa_gt: "(x - a) \<bullet> i > ((b - a) \<bullet> i) / (real k)" if "i \<in> Basis" for i
+        proof -
+          have *: "Inf ?\<Delta> \<le> ((x - a) \<bullet> i) / ((b - a) \<bullet> i)"
+            using that by (force intro: cInf_le_finite)
+          have "1 / Inf ?\<Delta> \<ge> ((b - a) \<bullet> i) / ((x - a) \<bullet> i)"
+            using le_imp_inverse_le [OF * \<Delta>]
+            by (simp add: field_simps)
+          with N have "k > ((b - a) \<bullet> i) / ((x - a) \<bullet> i)"
+            using \<open>N \<le> k\<close> by linarith
+          with x that show ?thesis
+            by (auto simp: mem_box algebra_simps divide_simps)
+        qed
+        have bx_gt: "(b - x) \<bullet> i > ((b - a) \<bullet> i) / k" if "i \<in> Basis" for i
+        proof -
+          have *: "Inf ?\<Delta> \<le> ((b - x) \<bullet> i) / ((b - a) \<bullet> i)"
+            using that by (force intro: cInf_le_finite)
+          have "1 / Inf ?\<Delta> \<ge> ((b - a) \<bullet> i) / ((b - x) \<bullet> i)"
+            using le_imp_inverse_le [OF * \<Delta>]
+            by (simp add: field_simps)
+          with N have "k > ((b - a) \<bullet> i) / ((b - x) \<bullet> i)"
+            using \<open>N \<le> k\<close> by linarith
+          with x that show ?thesis
+            by (auto simp: mem_box algebra_simps divide_simps)
+        qed
+        show "x \<in> cbox (a + (b - a) /\<^sub>R k) (b - (b - a) /\<^sub>R k)"
+          using that \<Delta> \<open>k > 0\<close>
+          by (auto simp: mem_box algebra_simps divide_inverse dest: xa_gt bx_gt)
+      qed
+    qed
+    obtain Bf where "Bf > 0" and Bf: "\<And>c d. cbox c d \<subseteq> box a b \<Longrightarrow> norm (integral (cbox c d) f) \<le> Bf"
+      using bo unfolding bounded_pos by blast
+    obtain Bg where "Bg > 0" and Bg:"\<And>c d. cbox c d \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox c d) g\<bar> \<le> Bg"
+      using bounded_integrals_over_subintervals [OF int_gab] unfolding bounded_pos real_norm_def by blast
+    show "(\<lambda>x. f x \<bullet> j) absolutely_integrable_on cbox a b"
+      using g
+    proof     --\<open>A lot of duplication in the two proofs\<close>
+      assume fg [rule_format]: "\<forall>x\<in>cbox a b. f x \<bullet> j \<le> g x"
+      have "(\<lambda>x. (f x \<bullet> j)) = (\<lambda>x. g x - (g x - (f x \<bullet> j)))"
+        by simp
+      moreover have "(\<lambda>x. g x - (g x - (f x \<bullet> j))) integrable_on cbox a b"
+      proof (rule Henstock_Kurzweil_Integration.integrable_diff [OF int_gab])
+        let ?\<phi> = "\<lambda>k x. if x \<in> cbox (a + (b - a) /\<^sub>R (Suc k)) (b - (b - a) /\<^sub>R (Suc k))
+                        then g x - f x \<bullet> j else 0"
+        have "(\<lambda>x. g x - f x \<bullet> j) integrable_on box a b"
+        proof (rule monotone_convergence_increasing [of ?\<phi>, THEN conjunct1], safe)
+          have *: "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k)) \<inter> box a b
+                 = cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))" for k
+            using box_subset_cbox "1" by fastforce
+          show "?\<phi> k integrable_on box a b" for k
+            apply (simp add: integrable_restrict_Int integral_restrict_Int *)
+            apply (rule integrable_diff [OF integrable_on_subcbox [OF int_gab]])
+            using "*" box_subset_cbox apply blast
+            by (metis "1" int_f integrable_component of_nat_Suc)
+          have cb12: "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))
+                    \<subseteq> cbox (a + (b - a) /\<^sub>R (2 + real k)) (b - (b - a) /\<^sub>R (2 + real k))" for k
+            using False content_eq_0
+            apply (simp add: subset_box algebra_simps)
+            apply (simp add: divide_simps)
+            apply (fastforce simp: field_simps)
+            done
+          show "?\<phi> k x \<le> ?\<phi> (Suc k) x" if "x \<in> box a b" for k x
+            using cb12 box_subset_cbox that by (force simp: intro!: fg)
+          show "(\<lambda>k. ?\<phi> k x) \<longlonglongrightarrow> g x - f x \<bullet> j" if x: "x \<in> box a b" for x
+          proof (rule Lim_eventually)
+            obtain N::nat where N: "\<And>k. k \<ge> N \<Longrightarrow> x \<in> cbox (a + (b - a) /\<^sub>R real k) (b - (b - a) /\<^sub>R real k)"
+              using getN [OF x] by blast
+            show "\<forall>\<^sub>F k in sequentially. ?\<phi> k x = g x - f x \<bullet> j"
+            proof
+              fix k::nat assume "N \<le> k"
+              have "x \<in> cbox (a + (b - a) /\<^sub>R (Suc k)) (b - (b - a) /\<^sub>R (Suc k))"
+                by (metis \<open>N \<le> k\<close> le_Suc_eq N)
+              then show "?\<phi> k x = g x - f x \<bullet> j"
+                by simp
+            qed
+          qed
+          have "\<bar>integral (box a b)
+                      (\<lambda>x. if x \<in> cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))
+                           then g x - f x \<bullet> j else 0)\<bar> \<le> Bg + Bf" for k
+          proof -
+            let ?I = "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))"
+            have I_int [simp]: "?I \<inter> box a b = ?I"
+              using 1 by (simp add: Int_absorb2)
+            have int_fI: "f integrable_on ?I"
+              apply (rule integrable_subinterval [OF int_f order_refl])
+              using "*" box_subset_cbox by blast
+            then have "(\<lambda>x. f x \<bullet> j) integrable_on ?I"
+              by (simp add: integrable_component)
+            moreover have "g integrable_on ?I"
+              apply (rule integrable_subinterval [OF int_gab])
+              using "*" box_subset_cbox by blast
+            moreover
+            have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ?I f)"
+              by (simp add: Basis_le_norm int_fI \<open>j \<in> Basis\<close>)
+            with 1 I_int have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> Bf"
+              by (blast intro: order_trans [OF _ Bf])
+            ultimately show ?thesis
+              apply (simp add: integral_restrict_Int integral_diff)
+              using "*" box_subset_cbox by (blast intro: Bg add_mono order_trans [OF abs_triangle_ineq4])
+          qed
+          then show "bounded {integral (box a b) (?\<phi> k)| k. True}"
+            apply (simp add: bounded_pos)
+            apply (rule_tac x="Bg+Bf" in exI)
+            using \<open>0 < Bf\<close> \<open>0 < Bg\<close>  apply auto
+            done
+        qed
+        then show "(\<lambda>x. g x - f x \<bullet> j) integrable_on cbox a b"
+          by (simp add: integrable_on_open_interval)
+      qed
+      ultimately have "(\<lambda>x. f x \<bullet> j) integrable_on cbox a b"
+        by auto
+      then show ?thesis
+        apply (rule absolutely_integrable_component_ubound [OF _ absint_g])
+        by (simp add: fg)
+    next
+      assume gf [rule_format]: "\<forall>x\<in>cbox a b. g x \<le> f x \<bullet> j"
+      have "(\<lambda>x. (f x \<bullet> j)) = (\<lambda>x. ((f x \<bullet> j) - g x) + g x)"
+        by simp
+      moreover have "(\<lambda>x. (f x \<bullet> j - g x) + g x) integrable_on cbox a b"
+      proof (rule Henstock_Kurzweil_Integration.integrable_add [OF _ int_gab])
+        let ?\<phi> = "\<lambda>k x. if x \<in> cbox (a + (b - a) /\<^sub>R (Suc k)) (b - (b - a) /\<^sub>R (Suc k))
+                        then f x \<bullet> j - g x else 0"
+        have "(\<lambda>x. f x \<bullet> j - g x) integrable_on box a b"
+        proof (rule monotone_convergence_increasing [of ?\<phi>, THEN conjunct1], safe)
+          have *: "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k)) \<inter> box a b
+                 = cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))" for k
+            using box_subset_cbox "1" by fastforce
+          show "?\<phi> k integrable_on box a b" for k
+            apply (simp add: integrable_restrict_Int integral_restrict_Int *)
+            apply (rule integrable_diff)
+              apply (metis "1" int_f integrable_component of_nat_Suc)
+             apply (rule integrable_on_subcbox [OF int_gab])
+            using "*" box_subset_cbox apply blast
+              done
+          have cb12: "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))
+                    \<subseteq> cbox (a + (b - a) /\<^sub>R (2 + real k)) (b - (b - a) /\<^sub>R (2 + real k))" for k
+            using False content_eq_0
+            apply (simp add: subset_box algebra_simps)
+            apply (simp add: divide_simps)
+            apply (fastforce simp: field_simps)
+            done
+          show "?\<phi> k x \<le> ?\<phi> (Suc k) x" if "x \<in> box a b" for k x
+            using cb12 box_subset_cbox that by (force simp: intro!: gf)
+          show "(\<lambda>k. ?\<phi> k x) \<longlonglongrightarrow> f x \<bullet> j - g x" if x: "x \<in> box a b" for x
+          proof (rule Lim_eventually)
+            obtain N::nat where N: "\<And>k. k \<ge> N \<Longrightarrow> x \<in> cbox (a + (b - a) /\<^sub>R real k) (b - (b - a) /\<^sub>R real k)"
+              using getN [OF x] by blast
+            show "\<forall>\<^sub>F k in sequentially. ?\<phi> k x = f x \<bullet> j - g x"
+            proof
+              fix k::nat assume "N \<le> k"
+              have "x \<in> cbox (a + (b - a) /\<^sub>R (Suc k)) (b - (b - a) /\<^sub>R (Suc k))"
+                by (metis \<open>N \<le> k\<close> le_Suc_eq N)
+              then show "?\<phi> k x = f x \<bullet> j - g x"
+                by simp
+            qed
+          qed
+          have "\<bar>integral (box a b)
+                      (\<lambda>x. if x \<in> cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))
+                           then f x \<bullet> j - g x else 0)\<bar> \<le> Bf + Bg" for k
+          proof -
+            let ?I = "cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))"
+            have I_int [simp]: "?I \<inter> box a b = ?I"
+              using 1 by (simp add: Int_absorb2)
+            have int_fI: "f integrable_on ?I"
+              apply (rule integrable_subinterval [OF int_f order_refl])
+              using "*" box_subset_cbox by blast
+            then have "(\<lambda>x. f x \<bullet> j) integrable_on ?I"
+              by (simp add: integrable_component)
+            moreover have "g integrable_on ?I"
+              apply (rule integrable_subinterval [OF int_gab])
+              using "*" box_subset_cbox by blast
+            moreover
+            have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ?I f)"
+              by (simp add: Basis_le_norm int_fI \<open>j \<in> Basis\<close>)
+            with 1 I_int have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> Bf"
+              by (blast intro: order_trans [OF _ Bf])
+            ultimately show ?thesis
+              apply (simp add: integral_restrict_Int integral_diff)
+              using "*" box_subset_cbox by (blast intro: Bg add_mono order_trans [OF abs_triangle_ineq4])
+          qed
+          then show "bounded {integral (box a b) (?\<phi> k)| k. True}"
+            apply (simp add: bounded_pos)
+            apply (rule_tac x="Bf+Bg" in exI)
+            using \<open>0 < Bf\<close> \<open>0 < Bg\<close>  by auto
+        qed
+        then show "(\<lambda>x. f x \<bullet> j - g x) integrable_on cbox a b"
+          by (simp add: integrable_on_open_interval)
+      qed
+      ultimately have "(\<lambda>x. f x \<bullet> j) integrable_on cbox a b"
+        by auto
+      then show ?thesis
+        apply (rule absolutely_integrable_component_lbound [OF absint_g])
+        by (simp add: gf)
+    qed
+  qed
+qed
+
+end
+  
\ No newline at end of file
diff -r 1ec601d9c829 -r 33a47f2d9edc src/HOL/Analysis/Tagged_Division.thy
--- a/src/HOL/Analysis/Tagged_Division.thy	Wed Jul 26 13:36:36 2017 +0100
+++ b/src/HOL/Analysis/Tagged_Division.thy	Wed Jul 26 16:07:45 2017 +0100
@@ -308,7 +308,7 @@
   using assms unfolding division_of_def by auto
 
 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
-  unfolding division_of_def by auto
+  by auto
 
 lemma division_of_self[intro]: "cbox a b \<noteq> {} \<Longrightarrow> {cbox a b} division_of (cbox a b)"
   unfolding division_of_def by auto
@@ -357,6 +357,11 @@
   "d division_of i \<Longrightarrow> (\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. cbox a b \<in> d \<longrightarrow> P (cbox a b))"
   unfolding division_of_def by fastforce
 
+lemma cbox_division_memE:
+  assumes \<D>: "K \<in> \<D>" "\<D> division_of S"
+  obtains a b where "K = cbox a b" "K \<noteq> {}" "K \<subseteq> S"
+  using assms unfolding division_of_def by metis
+
 lemma division_of_subset:
   assumes "p division_of (\<Union>p)"
     and "q \<subseteq> p"