diff --git a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
@@ -1,7591 +1,7592 @@
 (*  Author:     John Harrison
     Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light)
                 Huge cleanup by LCP
 *)
 
 section \<open>Henstock-Kurzweil Gauge Integration in Many Dimensions\<close>
 
 theory Henstock_Kurzweil_Integration
 imports
   Lebesgue_Measure Tagged_Division
 begin
 
 lemma norm_diff2: "\<lbrakk>y = y1 + y2; x = x1 + x2; e = e1 + e2; norm(y1 - x1) \<le> e1; norm(y2 - x2) \<le> e2\<rbrakk>
   \<Longrightarrow> norm(y-x) \<le> e"
   using norm_triangle_mono [of "y1 - x1" "e1" "y2 - x2" "e2"]
   by (simp add: add_diff_add)
 
 lemma setcomp_dot1: "{z. P (z \<bullet> (i,0))} = {(x,y). P(x \<bullet> i)}"
   by auto
 
 lemma setcomp_dot2: "{z. P (z \<bullet> (0,i))} = {(x,y). P(y \<bullet> i)}"
   by auto
 
 lemma Sigma_Int_Paircomp1: "(Sigma A B) \<inter> {(x, y). P x} = Sigma (A \<inter> {x. P x}) B"
   by blast
 
 lemma Sigma_Int_Paircomp2: "(Sigma A B) \<inter> {(x, y). P y} = Sigma A (\<lambda>z. B z \<inter> {y. P y})"
   by blast
 (* END MOVE *)
 
 subsection \<open>Content (length, area, volume...) of an interval\<close>
 
 abbreviation content :: "'a::euclidean_space set \<Rightarrow> real"
   where "content s \<equiv> measure lborel s"
 
 lemma content_cbox_cases:
   "content (cbox a b) = (if \<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i then prod (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis else 0)"
   by (simp add: measure_lborel_cbox_eq inner_diff)
 
 lemma content_cbox: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   unfolding content_cbox_cases by simp
 
 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_cbox_cart:
    "cbox a b \<noteq> {} \<Longrightarrow> content(cbox a b) = prod (\<lambda>i. b$i - a$i) UNIV"
   by (simp add: content_cbox_if Basis_vec_def cart_eq_inner_axis axis_eq_axis prod.UNION_disjoint)
 
 lemma content_cbox_if_cart:
    "content(cbox a b) = (if cbox a b = {} then 0 else prod (\<lambda>i. b$i - a$i) UNIV)"
   by (simp add: content_cbox_cart)
 
 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
 
 lemma abs_eq_content: "\<bar>y - x\<bar> = (if x\<le>y then content {x..y} else content {y..x})"
   by (auto simp: content_real)
 
 lemma content_singleton: "content {a} = 0"
   by simp
 
 lemma content_unit[iff]: "content (cbox 0 (One::'a::euclidean_space)) = 1"
   by simp
 
 lemma content_pos_le [iff]: "0 \<le> content X"
   by simp
 
 corollary\<^marker>\<open>tag unimportant\<close> content_nonneg [simp]: "\<not> content (cbox a b) < 0"
   using not_le by blast
 
 lemma content_pos_lt: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i \<Longrightarrow> 0 < content (cbox a b)"
   by (auto simp: less_imp_le inner_diff box_eq_empty intro!: prod_pos)
 
 lemma content_eq_0: "content (cbox a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
   by (auto simp: content_cbox_cases not_le intro: less_imp_le antisym eq_refl)
 
 lemma content_eq_0_interior: "content (cbox a b) = 0 \<longleftrightarrow> interior(cbox a b) = {}"
   unfolding content_eq_0 interior_cbox box_eq_empty by auto
 
 lemma content_pos_lt_eq: "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
   by (auto simp add: content_cbox_cases less_le prod_nonneg)
 
 lemma content_empty [simp]: "content {} = 0"
   by simp
 
 lemma content_real_if [simp]: "content {a..b} = (if a \<le> b then b - a else 0)"
   by (simp add: content_real)
 
 lemma content_subset: "cbox a b \<subseteq> cbox c d \<Longrightarrow> content (cbox a b) \<le> content (cbox c d)"
   unfolding measure_def
   by (intro enn2real_mono emeasure_mono) (auto simp: emeasure_lborel_cbox_eq)
 
 lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
 
 lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
   unfolding measure_lborel_cbox_eq Basis_prod_def
   apply (subst prod.union_disjoint)
   apply (auto simp: bex_Un ball_Un)
   apply (subst (1 2) prod.reindex_nontrivial)
   apply auto
   done
 
 lemma content_cbox_pair_eq0_D:
    "content (cbox (a,c) (b,d)) = 0 \<Longrightarrow> content (cbox a b) = 0 \<or> content (cbox c d) = 0"
   by (simp add: content_Pair)
 
 lemma content_cbox_plus:
   fixes x :: "'a::euclidean_space"
   shows "content(cbox x (x + h *\<^sub>R One)) = (if h \<ge> 0 then h ^ DIM('a) else 0)"
   by (simp add: algebra_simps content_cbox_if box_eq_empty)
 
 lemma content_0_subset: "content(cbox a b) = 0 \<Longrightarrow> s \<subseteq> cbox a b \<Longrightarrow> content s = 0"
   using emeasure_mono[of s "cbox a b" lborel]
   by (auto simp: measure_def enn2real_eq_0_iff emeasure_lborel_cbox_eq)
 
 lemma content_ball_pos:
   assumes "r > 0"
   shows   "content (ball c r) > 0"
 proof -
   from rational_boxes[OF assms, of c] obtain a b where ab: "c \<in> box a b" "box a b \<subseteq> ball c r"
     by auto
   from ab have "0 < content (box a b)"
     by (subst measure_lborel_box_eq) (auto intro!: prod_pos simp: algebra_simps box_def)
   have "emeasure lborel (box a b) \<le> emeasure lborel (ball c r)"
     using ab by (intro emeasure_mono) auto
   also have "emeasure lborel (box a b) = ennreal (content (box a b))"
     using emeasure_lborel_box_finite[of a b] by (intro emeasure_eq_ennreal_measure) auto
   also have "emeasure lborel (ball c r) = ennreal (content (ball c r))"
     using emeasure_lborel_ball_finite[of c r] by (intro emeasure_eq_ennreal_measure) auto
   finally show ?thesis
     using \<open>content (box a b) > 0\<close> by simp
 qed
 
 lemma content_cball_pos:
   assumes "r > 0"
   shows   "content (cball c r) > 0"
 proof -
   from rational_boxes[OF assms, of c] obtain a b where ab: "c \<in> box a b" "box a b \<subseteq> ball c r"
     by auto
   from ab have "0 < content (box a b)"
     by (subst measure_lborel_box_eq) (auto intro!: prod_pos simp: algebra_simps box_def)
   have "emeasure lborel (box a b) \<le> emeasure lborel (ball c r)"
     using ab by (intro emeasure_mono) auto
   also have "\<dots> \<le> emeasure lborel (cball c r)"
     by (intro emeasure_mono) auto
   also have "emeasure lborel (box a b) = ennreal (content (box a b))"
     using emeasure_lborel_box_finite[of a b] by (intro emeasure_eq_ennreal_measure) auto
   also have "emeasure lborel (cball c r) = ennreal (content (cball c r))"
     using emeasure_lborel_cball_finite[of c r] by (intro emeasure_eq_ennreal_measure) auto
   finally show ?thesis
     using \<open>content (box a b) > 0\<close> by simp
 qed
 
 lemma content_split:
   fixes a :: "'a::euclidean_space"
   assumes "k \<in> Basis"
   shows "content (cbox a b) = content(cbox a b \<inter> {x. x\<bullet>k \<le> c}) + content(cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
   \<comment> \<open>Prove using measure theory\<close>
 proof (cases "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i")
   case True
   have 1: "\<And>X Y Z. (\<Prod>i\<in>Basis. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>Basis-{k}. Z i (Y i))"
     by (simp add: if_distrib prod.delta_remove assms)
   note simps = interval_split[OF assms] content_cbox_cases
   have 2: "(\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i) = (\<Prod>i\<in>Basis-{k}. b\<bullet>i - a\<bullet>i) * (b\<bullet>k - a\<bullet>k)"
     by (metis (no_types, lifting) assms finite_Basis mult.commute prod.remove)
   have "\<And>x. min (b \<bullet> k) c = max (a \<bullet> k) c \<Longrightarrow>
     x * (b\<bullet>k - a\<bullet>k) = x * (max (a \<bullet> k) c - a \<bullet> k) + x * (b \<bullet> k - max (a \<bullet> k) c)"
     by  (auto simp add: field_simps)
   moreover
   have **: "(\<Prod>i\<in>Basis. ((\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) *\<^sub>R i) \<bullet> i - a \<bullet> i)) =
       (\<Prod>i\<in>Basis. (if i = k then min (b \<bullet> k) c else b \<bullet> i) - a \<bullet> i)"
     "(\<Prod>i\<in>Basis. b \<bullet> i - ((\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i)) =
       (\<Prod>i\<in>Basis. b \<bullet> i - (if i = k then max (a \<bullet> k) c else a \<bullet> i))"
     by (auto intro!: prod.cong)
   have "\<not> a \<bullet> k \<le> c \<Longrightarrow> \<not> c \<le> b \<bullet> k \<Longrightarrow> False"
     unfolding not_le using True assms by auto
   ultimately show ?thesis
     using assms unfolding simps ** 1[of "\<lambda>i x. b\<bullet>i - x"] 1[of "\<lambda>i x. x - a\<bullet>i"] 2
     by auto
 next
   case False
   then have "cbox a b = {}"
     unfolding box_eq_empty by (auto simp: not_le)
   then show ?thesis
     by (auto simp: not_le)
 qed
 
 lemma division_of_content_0:
   assumes "content (cbox a b) = 0" "d division_of (cbox a b)" "K \<in> d"
   shows "content K = 0"
   unfolding forall_in_division[OF assms(2)]
   by (meson assms content_0_subset division_of_def)
 
 lemma sum_content_null:
   assumes "content (cbox a b) = 0"
     and "p tagged_division_of (cbox a b)"
   shows "(\<Sum>(x,K)\<in>p. content K *\<^sub>R f x) = (0::'a::real_normed_vector)"
 proof (rule sum.neutral, rule)
   fix y
   assume y: "y \<in> p"
   obtain x K where xk: "y = (x, K)"
     using surj_pair[of y] by blast
   then obtain c d where k: "K = cbox c d" "K \<subseteq> cbox a b"
     by (metis assms(2) tagged_division_ofD(3) tagged_division_ofD(4) y)
   have "(\<lambda>(x',K'). content K' *\<^sub>R f x') y = content K *\<^sub>R f x"
     unfolding xk by auto
   also have "\<dots> = 0"
     using assms(1) content_0_subset k(2) by auto
   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
 qed
 
 global_interpretation sum_content: operative plus 0 content
   rewrites "comm_monoid_set.F plus 0 = sum"
 proof -
   interpret operative plus 0 content
     by standard (auto simp add: content_split [symmetric] content_eq_0_interior)
   show "operative plus 0 content"
     by standard
   show "comm_monoid_set.F plus 0 = sum"
     by (simp add: sum_def)
 qed
 
 lemma additive_content_division: "d division_of (cbox a b) \<Longrightarrow> sum content d = content (cbox a b)"
   by (fact sum_content.division)
 
 lemma additive_content_tagged_division:
   "d tagged_division_of (cbox a b) \<Longrightarrow> sum (\<lambda>(x,l). content l) d = content (cbox a b)"
   by (fact sum_content.tagged_division)
 
 lemma subadditive_content_division:
   assumes "\<D> division_of S" "S \<subseteq> cbox a b"
   shows "sum content \<D> \<le> content(cbox a b)"
 proof -
   have "\<D> division_of \<Union>\<D>" "\<Union>\<D> \<subseteq> cbox a b"
     using assms by auto
   then obtain \<D>' where "\<D> \<subseteq> \<D>'" "\<D>' division_of cbox a b"
     using partial_division_extend_interval by metis
   then have "sum content \<D> \<le> sum content \<D>'"
     using sum_mono2 by blast
   also have "... \<le> content(cbox a b)"
     by (simp add: \<open>\<D>' division_of cbox a b\<close> additive_content_division less_eq_real_def)
   finally show ?thesis .
 qed
 
 lemma content_real_eq_0: "content {a..b::real} = 0 \<longleftrightarrow> a \<ge> b"
   by (metis atLeastatMost_empty_iff2 content_empty content_real diff_self eq_iff le_cases le_iff_diff_le_0)
 
 lemma property_empty_interval: "\<forall>a b. content (cbox a b) = 0 \<longrightarrow> P (cbox a b) \<Longrightarrow> P {}"
   using content_empty unfolding empty_as_interval by auto
 
 lemma interval_bounds_nz_content [simp]:
   assumes "content (cbox a b) \<noteq> 0"
   shows "interval_upperbound (cbox a b) = b"
     and "interval_lowerbound (cbox a b) = a"
   by (metis assms content_empty interval_bounds')+
 
 subsection \<open>Gauge integral\<close>
 
 text \<open>Case distinction to define it first on compact intervals first, then use a limit. This is only
 much later unified. In Fremlin: Measure Theory, Volume 4I this is generalized using residual sets.\<close>
 
 definition has_integral :: "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
   (infixr "has'_integral" 46)
   where "(f has_integral I) s \<longleftrightarrow>
     (if \<exists>a b. s = cbox a b
       then ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter s)
       else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
         (\<exists>z. ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R (if x \<in> s then f x else 0)) \<longlongrightarrow> z) (division_filter (cbox a b)) \<and>
           norm (z - I) < e)))"
 
 lemma has_integral_cbox:
   "(f has_integral I) (cbox a b) \<longleftrightarrow> ((\<lambda>p. \<Sum>(x,k)\<in>p. content k *\<^sub>R f x) \<longlongrightarrow> I) (division_filter (cbox a b))"
   by (auto simp add: has_integral_def)
 
 lemma has_integral:
   "(f has_integral y) (cbox a b) \<longleftrightarrow>
     (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
       (\<forall>\<D>. \<D> tagged_division_of (cbox a b) \<and> \<gamma> fine \<D> \<longrightarrow>
         norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) \<D> - y) < e))"
   by (auto simp: dist_norm eventually_division_filter has_integral_def tendsto_iff)
 
 lemma has_integral_real:
   "(f has_integral y) {a..b::real} \<longleftrightarrow>
     (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
       (\<forall>\<D>. \<D> tagged_division_of {a..b} \<and> \<gamma> fine \<D> \<longrightarrow>
         norm (sum (\<lambda>(x,k). content(k) *\<^sub>R f x) \<D> - y) < e))"
   unfolding box_real[symmetric] by (rule has_integral)
 
 lemma has_integralD[dest]:
   assumes "(f has_integral y) (cbox a b)"
     and "e > 0"
   obtains \<gamma>
     where "gauge \<gamma>"
       and "\<And>\<D>. \<D> tagged_division_of (cbox a b) \<Longrightarrow> \<gamma> fine \<D> \<Longrightarrow>
         norm ((\<Sum>(x,k)\<in>\<D>. content k *\<^sub>R f x) - y) < e"
   using assms unfolding has_integral by auto
 
 lemma has_integral_alt:
   "(f has_integral y) i \<longleftrightarrow>
     (if \<exists>a b. i = cbox a b
      then (f has_integral y) i
      else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)))"
   by (subst has_integral_def) (auto simp add: has_integral_cbox)
 
 lemma has_integral_altD:
   assumes "(f has_integral y) i"
     and "\<not> (\<exists>a b. i = cbox a b)"
     and "e>0"
   obtains B where "B > 0"
     and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - y) < e)"
   using assms has_integral_alt[of f y i] by auto
 
 definition integrable_on (infixr "integrable'_on" 46)
   where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
 
 definition "integral i f = (SOME y. (f has_integral y) i \<or> \<not> f integrable_on i \<and> y=0)"
 
 lemma integrable_integral[intro]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
   unfolding integrable_on_def integral_def by (metis (mono_tags, lifting) someI_ex)
 
 lemma not_integrable_integral: "\<not> f integrable_on i \<Longrightarrow> integral i f = 0"
   unfolding integrable_on_def integral_def by blast
 
 lemma has_integral_integrable[dest]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
   unfolding integrable_on_def by auto
 
 lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   by auto
 
 subsection \<open>Basic theorems about integrals\<close>
 
 lemma has_integral_eq_rhs: "(f has_integral j) S \<Longrightarrow> i = j \<Longrightarrow> (f has_integral i) S"
   by (rule forw_subst)
 
 lemma has_integral_unique_cbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   shows "(f has_integral k1) (cbox a b) \<Longrightarrow> (f has_integral k2) (cbox a b) \<Longrightarrow> k1 = k2"
     by (auto simp: has_integral_cbox intro: tendsto_unique[OF division_filter_not_empty])    
 
 lemma has_integral_unique:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "(f has_integral k1) i" "(f has_integral k2) i"
   shows "k1 = k2"
 proof (rule ccontr)
   let ?e = "norm (k1 - k2)/2"
   let ?F = "(\<lambda>x. if x \<in> i then f x else 0)"
   assume "k1 \<noteq> k2"
   then have e: "?e > 0"
     by auto
   have nonbox: "\<not> (\<exists>a b. i = cbox a b)"
     using \<open>k1 \<noteq> k2\<close> assms has_integral_unique_cbox by blast
   obtain B1 where B1:
       "0 < B1"
       "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?F has_integral z) (cbox a b) \<and> norm (z - k1) < norm (k1 - k2)/2"
     by (rule has_integral_altD[OF assms(1) nonbox,OF e]) blast
   obtain B2 where B2:
       "0 < B2"
       "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?F has_integral z) (cbox a b) \<and> norm (z - k2) < norm (k1 - k2)/2"
     by (rule has_integral_altD[OF assms(2) nonbox,OF e]) blast
   obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
     by (metis Un_subset_iff bounded_Un bounded_ball bounded_subset_cbox_symmetric)
   obtain w where w: "(?F has_integral w) (cbox a b)" "norm (w - k1) < norm (k1 - k2)/2"
     using B1(2)[OF ab(1)] by blast
   obtain z where z: "(?F has_integral z) (cbox a b)" "norm (z - k2) < norm (k1 - k2)/2"
     using B2(2)[OF ab(2)] by blast
   have "z = w"
     using has_integral_unique_cbox[OF w(1) z(1)] by auto
   then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
     using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
     by (auto simp add: norm_minus_commute)
   also have "\<dots> < norm (k1 - k2)/2 + norm (k1 - k2)/2"
     by (metis add_strict_mono z(2) w(2))
   finally show False by auto
 qed
 
 lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
   unfolding integral_def
   by (rule some_equality) (auto intro: has_integral_unique)
 
 lemma has_integral_iff: "(f has_integral i) S \<longleftrightarrow> (f integrable_on S \<and> integral S f = i)"
   by blast
 
 lemma eq_integralD: "integral k f = y \<Longrightarrow> (f has_integral y) k \<or> \<not> f integrable_on k \<and> y=0"
   unfolding integral_def integrable_on_def
   apply (erule subst)
   apply (rule someI_ex)
   by blast
 
 lemma has_integral_const [intro]:
   fixes a b :: "'a::euclidean_space"
   shows "((\<lambda>x. c) has_integral (content (cbox a b) *\<^sub>R c)) (cbox a b)"
   using eventually_division_filter_tagged_division[of "cbox a b"]
      additive_content_tagged_division[of _ a b]
   by (auto simp: has_integral_cbox split_beta' scaleR_sum_left[symmetric]
            elim!: eventually_mono intro!: tendsto_cong[THEN iffD1, OF _ tendsto_const])
 
 lemma has_integral_const_real [intro]:
   fixes a b :: real
   shows "((\<lambda>x. c) has_integral (content {a..b} *\<^sub>R c)) {a..b}"
   by (metis box_real(2) has_integral_const)
 
 lemma has_integral_integrable_integral: "(f has_integral i) s \<longleftrightarrow> f integrable_on s \<and> integral s f = i"
   by blast
 
 lemma integral_const [simp]:
   fixes a b :: "'a::euclidean_space"
   shows "integral (cbox a b) (\<lambda>x. c) = content (cbox a b) *\<^sub>R c"
   by (rule integral_unique) (rule has_integral_const)
 
 lemma integral_const_real [simp]:
   fixes a b :: real
   shows "integral {a..b} (\<lambda>x. c) = content {a..b} *\<^sub>R c"
   by (metis box_real(2) integral_const)
 
 lemma has_integral_is_0_cbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "\<And>x. x \<in> cbox a b \<Longrightarrow> f x = 0"
   shows "(f has_integral 0) (cbox a b)"
     unfolding has_integral_cbox
     using eventually_division_filter_tagged_division[of "cbox a b"] assms
     by (subst tendsto_cong[where g="\<lambda>_. 0"])
        (auto elim!: eventually_mono intro!: sum.neutral simp: tag_in_interval)
 
 lemma has_integral_is_0:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "\<And>x. x \<in> S \<Longrightarrow> f x = 0"
   shows "(f has_integral 0) S"
 proof (cases "(\<exists>a b. S = cbox a b)")
   case True with assms has_integral_is_0_cbox show ?thesis
     by blast
 next
   case False
   have *: "(\<lambda>x. if x \<in> S then f x else 0) = (\<lambda>x. 0)"
     by (auto simp add: assms)
   show ?thesis
     using has_integral_is_0_cbox False
     by (subst has_integral_alt) (force simp add: *)
 qed
 
 lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) S"
   by (rule has_integral_is_0) auto
 
 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) S \<longleftrightarrow> i = 0"
   using has_integral_unique[OF has_integral_0] by auto
 
 lemma has_integral_linear_cbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes f: "(f has_integral y) (cbox a b)"
     and h: "bounded_linear h"
   shows "((h \<circ> f) has_integral (h y)) (cbox a b)"
 proof -
   interpret bounded_linear h using h .
   show ?thesis
     unfolding has_integral_cbox using tendsto [OF f [unfolded has_integral_cbox]]
     by (simp add: sum scaleR split_beta')
 qed
 
 lemma has_integral_linear:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes f: "(f has_integral y) S"
     and h: "bounded_linear h"
   shows "((h \<circ> f) has_integral (h y)) S"
 proof (cases "(\<exists>a b. S = cbox a b)")
   case True with f h has_integral_linear_cbox show ?thesis 
     by blast
 next
   case False
   interpret bounded_linear h using h .
   from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
     by blast
   let ?S = "\<lambda>f x. if x \<in> S then f x else 0"
   show ?thesis
   proof (subst has_integral_alt, clarsimp simp: False)
     fix e :: real
     assume e: "e > 0"
     have *: "0 < e/B" using e B(1) by simp
     obtain M where M:
       "M > 0"
       "\<And>a b. ball 0 M \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?S f has_integral z) (cbox a b) \<and> norm (z - y) < e/B"
       using has_integral_altD[OF f False *] by blast
     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. (?S(h \<circ> f) has_integral z) (cbox a b) \<and> norm (z - h y) < e)"
     proof (rule exI, intro allI conjI impI)
       show "M > 0" using M by metis
     next
       fix a b::'n
       assume sb: "ball 0 M \<subseteq> cbox a b"
       obtain z where z: "(?S f has_integral z) (cbox a b)" "norm (z - y) < e/B"
         using M(2)[OF sb] by blast
       have *: "?S(h \<circ> f) = h \<circ> ?S f"
         using zero by auto
       show "\<exists>z. (?S(h \<circ> f) has_integral z) (cbox a b) \<and> norm (z - h y) < e"
       proof (intro exI conjI)
         show "(?S(h \<circ> f) has_integral h z) (cbox a b)"
           by (simp add: * has_integral_linear_cbox[OF z(1) h])
         show "norm (h z - h y) < e"
           by (metis B diff le_less_trans pos_less_divide_eq z(2))
       qed
     qed
   qed
 qed
 
 lemma has_integral_scaleR_left:
   "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) S"
   using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
 
 lemma integrable_on_scaleR_left:
   assumes "f integrable_on A"
   shows "(\<lambda>x. f x *\<^sub>R y) integrable_on A"
   using assms has_integral_scaleR_left unfolding integrable_on_def by blast
 
 lemma has_integral_mult_left:
   fixes c :: "_ :: real_normed_algebra"
   shows "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) S"
   using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
 
 lemma has_integral_divide:
   fixes c :: "_ :: real_normed_div_algebra"
   shows "(f has_integral y) S \<Longrightarrow> ((\<lambda>x. f x / c) has_integral (y / c)) S"
   unfolding divide_inverse by (simp add: has_integral_mult_left)
 
 text\<open>The case analysis eliminates the condition \<^term>\<open>f integrable_on S\<close> at the cost
      of the type class constraint \<open>division_ring\<close>\<close>
 corollary integral_mult_left [simp]:
   fixes c:: "'a::{real_normed_algebra,division_ring}"
   shows "integral S (\<lambda>x. f x * c) = integral S f * c"
 proof (cases "f integrable_on S \<or> c = 0")
   case True then show ?thesis
     by (force intro: has_integral_mult_left)
 next
   case False then have "\<not> (\<lambda>x. f x * c) integrable_on S"
     using has_integral_mult_left [of "(\<lambda>x. f x * c)" _ S "inverse c"]
     by (auto simp add: mult.assoc)
   with False show ?thesis by (simp add: not_integrable_integral)
 qed
 
 corollary integral_mult_right [simp]:
   fixes c:: "'a::{real_normed_field}"
   shows "integral S (\<lambda>x. c * f x) = c * integral S f"
 by (simp add: mult.commute [of c])
 
 corollary integral_divide [simp]:
   fixes z :: "'a::real_normed_field"
   shows "integral S (\<lambda>x. f x / z) = integral S (\<lambda>x. f x) / z"
 using integral_mult_left [of S f "inverse z"]
   by (simp add: divide_inverse_commute)
 
 lemma has_integral_mult_right:
   fixes c :: "'a :: real_normed_algebra"
   shows "(f has_integral y) i \<Longrightarrow> ((\<lambda>x. c * f x) has_integral (c * y)) i"
   using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def)
 
 lemma has_integral_cmul: "(f has_integral k) S \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) S"
   unfolding o_def[symmetric]
   by (metis has_integral_linear bounded_linear_scaleR_right)
 
 lemma has_integral_cmult_real:
   fixes c :: real
   assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"
   shows "((\<lambda>x. c * f x) has_integral c * x) A"
 proof (cases "c = 0")
   case True
   then show ?thesis by simp
 next
   case False
   from has_integral_cmul[OF assms[OF this], of c] show ?thesis
     unfolding real_scaleR_def .
 qed
 
 lemma has_integral_neg: "(f has_integral k) S \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral -k) S"
   by (drule_tac c="-1" in has_integral_cmul) auto
 
 lemma has_integral_neg_iff: "((\<lambda>x. - f x) has_integral k) S \<longleftrightarrow> (f has_integral - k) S"
   using has_integral_neg[of f "- k"] has_integral_neg[of "\<lambda>x. - f x" k] by auto
 
 lemma has_integral_add_cbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "(f has_integral k) (cbox a b)" "(g has_integral l) (cbox a b)"
   shows "((\<lambda>x. f x + g x) has_integral (k + l)) (cbox a b)"
   using assms
     unfolding has_integral_cbox
     by (simp add: split_beta' scaleR_add_right sum.distrib[abs_def] tendsto_add)
 
 lemma has_integral_add:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes f: "(f has_integral k) S" and g: "(g has_integral l) S"
   shows "((\<lambda>x. f x + g x) has_integral (k + l)) S"
 proof (cases "\<exists>a b. S = cbox a b")
   case True with has_integral_add_cbox assms show ?thesis
     by blast 
 next
   let ?S = "\<lambda>f x. if x \<in> S then f x else 0"
   case False
   then show ?thesis
   proof (subst has_integral_alt, clarsimp, goal_cases)
     case (1 e)
     then have e2: "e/2 > 0"
       by auto
     obtain Bf where "0 < Bf"
       and Bf: "\<And>a b. ball 0 Bf \<subseteq> cbox a b \<Longrightarrow>
                      \<exists>z. (?S f has_integral z) (cbox a b) \<and> norm (z - k) < e/2"
       using has_integral_altD[OF f False e2] by blast
     obtain Bg where "0 < Bg"
       and Bg: "\<And>a b. ball 0 Bg \<subseteq> (cbox a b) \<Longrightarrow>
                      \<exists>z. (?S g has_integral z) (cbox a b) \<and> norm (z - l) < e/2"
       using has_integral_altD[OF g False e2] by blast
     show ?case
     proof (rule_tac x="max Bf Bg" in exI, clarsimp simp add: max.strict_coboundedI1 \<open>0 < Bf\<close>)
       fix a b
       assume "ball 0 (max Bf Bg) \<subseteq> cbox a (b::'n)"
       then have fs: "ball 0 Bf \<subseteq> cbox a (b::'n)" and gs: "ball 0 Bg \<subseteq> cbox a (b::'n)"
         by auto
       obtain w where w: "(?S f has_integral w) (cbox a b)" "norm (w - k) < e/2"
         using Bf[OF fs] by blast
       obtain z where z: "(?S g has_integral z) (cbox a b)" "norm (z - l) < e/2"
         using Bg[OF gs] by blast
       have *: "\<And>x. (if x \<in> S then f x + g x else 0) = (?S f x) + (?S g x)"
         by auto
       show "\<exists>z. (?S(\<lambda>x. f x + g x) has_integral z) (cbox a b) \<and> norm (z - (k + l)) < e"
       proof (intro exI conjI)
         show "(?S(\<lambda>x. f x + g x) has_integral (w + z)) (cbox a b)"
           by (simp add: has_integral_add_cbox[OF w(1) z(1), unfolded *[symmetric]])
         show "norm (w + z - (k + l)) < e"
           by (metis dist_norm dist_triangle_add_half w(2) z(2))
       qed
     qed
   qed
 qed
 
 lemma has_integral_diff:
   "(f has_integral k) S \<Longrightarrow> (g has_integral l) S \<Longrightarrow>
     ((\<lambda>x. f x - g x) has_integral (k - l)) S"
   using has_integral_add[OF _ has_integral_neg, of f k S g l]
   by (auto simp: algebra_simps)
 
 lemma integral_0 [simp]:
   "integral S (\<lambda>x::'n::euclidean_space. 0::'m::real_normed_vector) = 0"
   by (rule integral_unique has_integral_0)+
 
 lemma integral_add: "f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow>
     integral S (\<lambda>x. f x + g x) = integral S f + integral S g"
   by (rule integral_unique) (metis integrable_integral has_integral_add)
 
 lemma integral_cmul [simp]: "integral S (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral S f"
 proof (cases "f integrable_on S \<or> c = 0")
   case True with has_integral_cmul integrable_integral show ?thesis
     by fastforce
 next
   case False then have "\<not> (\<lambda>x. c *\<^sub>R f x) integrable_on S"
     using has_integral_cmul [of "(\<lambda>x. c *\<^sub>R f x)" _ S "inverse c"] by auto
   with False show ?thesis by (simp add: not_integrable_integral)
 qed
 
 lemma integral_mult:
   fixes K::real
   shows "f integrable_on X \<Longrightarrow> K * integral X f = integral X (\<lambda>x. K * f x)"
   unfolding real_scaleR_def[symmetric] integral_cmul ..
 
 lemma integral_neg [simp]: "integral S (\<lambda>x. - f x) = - integral S f"
 proof (cases "f integrable_on S")
   case True then show ?thesis
     by (simp add: has_integral_neg integrable_integral integral_unique)
 next
   case False then have "\<not> (\<lambda>x. - f x) integrable_on S"
     using has_integral_neg [of "(\<lambda>x. - f x)" _ S ] by auto
   with False show ?thesis by (simp add: not_integrable_integral)
 qed
 
 lemma integral_diff: "f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow>
     integral S (\<lambda>x. f x - g x) = integral S f - integral S g"
   by (rule integral_unique) (metis integrable_integral has_integral_diff)
 
 lemma integrable_0: "(\<lambda>x. 0) integrable_on S"
   unfolding integrable_on_def using has_integral_0 by auto
 
 lemma integrable_add: "f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_add)
 
 lemma integrable_cmul: "f integrable_on S \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_cmul)
 
 lemma integrable_on_scaleR_iff [simp]:
   fixes c :: real
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. c *\<^sub>R f x) integrable_on S \<longleftrightarrow> f integrable_on S"
   using integrable_cmul[of "\<lambda>x. c *\<^sub>R f x" S "1 / c"] integrable_cmul[of f S c] \<open>c \<noteq> 0\<close>
   by auto
 
 lemma integrable_on_cmult_iff [simp]:
   fixes c :: real
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. c * f x) integrable_on S \<longleftrightarrow> f integrable_on S"
   using integrable_on_scaleR_iff [of c f] assms by simp
 
 lemma integrable_on_cmult_left:
   assumes "f integrable_on S"
   shows "(\<lambda>x. of_real c * f x) integrable_on S"
     using integrable_cmul[of f S "of_real c"] assms
     by (simp add: scaleR_conv_of_real)
 
 lemma integrable_neg: "f integrable_on S \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_neg)
 
 lemma integrable_neg_iff: "(\<lambda>x. -f(x)) integrable_on S \<longleftrightarrow> f integrable_on S"
   using integrable_neg by fastforce
 
 lemma integrable_diff:
   "f integrable_on S \<Longrightarrow> g integrable_on S \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_diff)
 
 lemma integrable_linear:
   "f integrable_on S \<Longrightarrow> bounded_linear h \<Longrightarrow> (h \<circ> f) integrable_on S"
   unfolding integrable_on_def by(auto intro: has_integral_linear)
 
 lemma integral_linear:
   "f integrable_on S \<Longrightarrow> bounded_linear h \<Longrightarrow> integral S (h \<circ> f) = h (integral S f)"
   by (meson has_integral_iff has_integral_linear)
 
 lemma integrable_on_cnj_iff:
   "(\<lambda>x. cnj (f x)) integrable_on A \<longleftrightarrow> f integrable_on A"
   using integrable_linear[OF _ bounded_linear_cnj, of f A]
         integrable_linear[OF _ bounded_linear_cnj, of "cnj \<circ> f" A]
   by (auto simp: o_def)
 
 lemma integral_cnj: "cnj (integral A f) = integral A (\<lambda>x. cnj (f x))"
   by (cases "f integrable_on A")
      (simp_all add: integral_linear[OF _ bounded_linear_cnj, symmetric]
                     o_def integrable_on_cnj_iff not_integrable_integral)
 
 lemma integral_component_eq[simp]:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "f integrable_on S"
   shows "integral S (\<lambda>x. f x \<bullet> k) = integral S f \<bullet> k"
   unfolding integral_linear[OF assms(1) bounded_linear_inner_left,unfolded o_def] ..
 
 lemma has_integral_sum:
   assumes "finite T"
     and "\<And>a. a \<in> T \<Longrightarrow> ((f a) has_integral (i a)) S"
   shows "((\<lambda>x. sum (\<lambda>a. f a x) T) has_integral (sum i T)) S"
   using assms(1) subset_refl[of T]
 proof (induct rule: finite_subset_induct)
   case empty
   then show ?case by auto
 next
   case (insert x F)
   with assms show ?case
     by (simp add: has_integral_add)
 qed
 
 lemma integral_sum:
   "\<lbrakk>finite I;  \<And>a. a \<in> I \<Longrightarrow> f a integrable_on S\<rbrakk> \<Longrightarrow>
    integral S (\<lambda>x. \<Sum>a\<in>I. f a x) = (\<Sum>a\<in>I. integral S (f a))"
   by (simp add: has_integral_sum integrable_integral integral_unique)
 
 lemma integrable_sum:
   "\<lbrakk>finite I;  \<And>a. a \<in> I \<Longrightarrow> f a integrable_on S\<rbrakk> \<Longrightarrow> (\<lambda>x. \<Sum>a\<in>I. f a x) integrable_on S"
   unfolding integrable_on_def using has_integral_sum[of I] by metis
 
 lemma has_integral_eq:
   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
     and "(f has_integral k) s"
   shows "(g has_integral k) s"
   using has_integral_diff[OF assms(2), of "\<lambda>x. f x - g x" 0]
   using has_integral_is_0[of s "\<lambda>x. f x - g x"]
   using assms(1)
   by auto
 
 lemma integrable_eq: "\<lbrakk>f integrable_on s; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> g integrable_on s"
   unfolding integrable_on_def
   using has_integral_eq[of s f g] has_integral_eq by blast
 
 lemma has_integral_cong:
   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
   shows "(f has_integral i) s = (g has_integral i) s"
   using has_integral_eq[of s f g] has_integral_eq[of s g f] assms
   by auto
 
 lemma integral_cong:
   assumes "\<And>x. x \<in> s \<Longrightarrow> f x = g x"
   shows "integral s f = integral s g"
   unfolding integral_def
 by (metis (full_types, hide_lams) assms has_integral_cong integrable_eq)
 
 lemma integrable_on_cmult_left_iff [simp]:
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. of_real c * f x) integrable_on s \<longleftrightarrow> f integrable_on s"
         (is "?lhs = ?rhs")
 proof
   assume ?lhs
   then have "(\<lambda>x. of_real (1 / c) * (of_real c * f x)) integrable_on s"
     using integrable_cmul[of "\<lambda>x. of_real c * f x" s "1 / of_real c"]
     by (simp add: scaleR_conv_of_real)
   then have "(\<lambda>x. (of_real (1 / c) * of_real c * f x)) integrable_on s"
     by (simp add: algebra_simps)
   with \<open>c \<noteq> 0\<close> show ?rhs
     by (metis (no_types, lifting) integrable_eq mult.left_neutral nonzero_divide_eq_eq of_real_1 of_real_mult)
 qed (blast intro: integrable_on_cmult_left)
 
 lemma integrable_on_cmult_right:
   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
   assumes "f integrable_on s"
   shows "(\<lambda>x. f x * of_real c) integrable_on s"
 using integrable_on_cmult_left [OF assms] by (simp add: mult.commute)
 
 lemma integrable_on_cmult_right_iff [simp]:
   fixes f :: "_ \<Rightarrow> 'b :: {comm_ring,real_algebra_1,real_normed_vector}"
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. f x * of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
 using integrable_on_cmult_left_iff [OF assms] by (simp add: mult.commute)
 
 lemma integrable_on_cdivide:
   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
   assumes "f integrable_on s"
   shows "(\<lambda>x. f x / of_real c) integrable_on s"
 by (simp add: integrable_on_cmult_right divide_inverse assms flip: of_real_inverse)
 
 lemma integrable_on_cdivide_iff [simp]:
   fixes f :: "_ \<Rightarrow> 'b :: real_normed_field"
   assumes "c \<noteq> 0"
   shows "(\<lambda>x. f x / of_real c) integrable_on s \<longleftrightarrow> f integrable_on s"
 by (simp add: divide_inverse assms flip: of_real_inverse)
 
 lemma has_integral_null [intro]: "content(cbox a b) = 0 \<Longrightarrow> (f has_integral 0) (cbox a b)"
   unfolding has_integral_cbox
   using eventually_division_filter_tagged_division[of "cbox a b"]
   by (subst tendsto_cong[where g="\<lambda>_. 0"]) (auto elim: eventually_mono intro: sum_content_null)
 
 lemma has_integral_null_real [intro]: "content {a..b::real} = 0 \<Longrightarrow> (f has_integral 0) {a..b}"
   by (metis box_real(2) has_integral_null)
 
 lemma has_integral_null_eq[simp]: "content (cbox a b) = 0 \<Longrightarrow> (f has_integral i) (cbox a b) \<longleftrightarrow> i = 0"
   by (auto simp add: has_integral_null dest!: integral_unique)
 
 lemma integral_null [simp]: "content (cbox a b) = 0 \<Longrightarrow> integral (cbox a b) f = 0"
   by (metis has_integral_null integral_unique)
 
 lemma integrable_on_null [intro]: "content (cbox a b) = 0 \<Longrightarrow> f integrable_on (cbox a b)"
   by (simp add: has_integral_integrable)
 
 lemma has_integral_empty[intro]: "(f has_integral 0) {}"
   by (meson ex_in_conv has_integral_is_0)
 
 lemma has_integral_empty_eq[simp]: "(f has_integral i) {} \<longleftrightarrow> i = 0"
   by (auto simp add: has_integral_empty has_integral_unique)
 
 lemma integrable_on_empty[intro]: "f integrable_on {}"
   unfolding integrable_on_def by auto
 
 lemma integral_empty[simp]: "integral {} f = 0"
   by (rule integral_unique) (rule has_integral_empty)
 
 lemma has_integral_refl[intro]:
   fixes a :: "'a::euclidean_space"
   shows "(f has_integral 0) (cbox a a)"
     and "(f has_integral 0) {a}"
 proof -
   show "(f has_integral 0) (cbox a a)"
      by (rule has_integral_null) simp
   then show "(f has_integral 0) {a}"
     by simp
 qed
 
 lemma integrable_on_refl[intro]: "f integrable_on cbox a a"
   unfolding integrable_on_def by auto
 
 lemma integral_refl [simp]: "integral (cbox a a) f = 0"
   by (rule integral_unique) auto
 
 lemma integral_singleton [simp]: "integral {a} f = 0"
   by auto
 
 lemma integral_blinfun_apply:
   assumes "f integrable_on s"
   shows "integral s (\<lambda>x. blinfun_apply h (f x)) = blinfun_apply h (integral s f)"
   by (subst integral_linear[symmetric, OF assms blinfun.bounded_linear_right]) (simp add: o_def)
 
 lemma blinfun_apply_integral:
   assumes "f integrable_on s"
   shows "blinfun_apply (integral s f) x = integral s (\<lambda>y. blinfun_apply (f y) x)"
   by (metis (no_types, lifting) assms blinfun.prod_left.rep_eq integral_blinfun_apply integral_cong)
 
 lemma has_integral_componentwise_iff:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "(f has_integral y) A \<longleftrightarrow> (\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
 proof safe
   fix b :: 'b assume "(f has_integral y) A"
   from has_integral_linear[OF this(1) bounded_linear_inner_left, of b]
     show "((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A" by (simp add: o_def)
 next
   assume "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
   hence "\<forall>b\<in>Basis. (((\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. f x \<bullet> b)) has_integral ((y \<bullet> b) *\<^sub>R b)) A"
     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
   hence "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. (y \<bullet> b) *\<^sub>R b)) A"
     by (intro has_integral_sum) (simp_all add: o_def)
   thus "(f has_integral y) A" by (simp add: euclidean_representation)
 qed
 
 lemma has_integral_componentwise:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "(\<And>b. b \<in> Basis \<Longrightarrow> ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A) \<Longrightarrow> (f has_integral y) A"
   by (subst has_integral_componentwise_iff) blast
 
 lemma integrable_componentwise_iff:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "f integrable_on A \<longleftrightarrow> (\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
 proof
   assume "f integrable_on A"
   then obtain y where "(f has_integral y) A" by (auto simp: integrable_on_def)
   hence "(\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral (y \<bullet> b)) A)"
     by (subst (asm) has_integral_componentwise_iff)
   thus "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)" by (auto simp: integrable_on_def)
 next
   assume "(\<forall>b\<in>Basis. (\<lambda>x. f x \<bullet> b) integrable_on A)"
   then obtain y where "\<forall>b\<in>Basis. ((\<lambda>x. f x \<bullet> b) has_integral y b) A"
     unfolding integrable_on_def by (subst (asm) bchoice_iff) blast
   hence "\<forall>b\<in>Basis. (((\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. f x \<bullet> b)) has_integral (y b *\<^sub>R b)) A"
     by (intro ballI has_integral_linear) (simp_all add: bounded_linear_scaleR_left)
   hence "((\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b) has_integral (\<Sum>b\<in>Basis. y b *\<^sub>R b)) A"
     by (intro has_integral_sum) (simp_all add: o_def)
   thus "f integrable_on A" by (auto simp: integrable_on_def o_def euclidean_representation)
 qed
 
 lemma integrable_componentwise:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   shows "(\<And>b. b \<in> Basis \<Longrightarrow> (\<lambda>x. f x \<bullet> b) integrable_on A) \<Longrightarrow> f integrable_on A"
   by (subst integrable_componentwise_iff) blast
 
 lemma integral_componentwise:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: euclidean_space"
   assumes "f integrable_on A"
   shows "integral A f = (\<Sum>b\<in>Basis. integral A (\<lambda>x. (f x \<bullet> b) *\<^sub>R b))"
 proof -
   from assms have integrable: "\<forall>b\<in>Basis. (\<lambda>x. x *\<^sub>R b) \<circ> (\<lambda>x. (f x \<bullet> b)) integrable_on A"
     by (subst (asm) integrable_componentwise_iff, intro integrable_linear ballI)
        (simp_all add: bounded_linear_scaleR_left)
   have "integral A f = integral A (\<lambda>x. \<Sum>b\<in>Basis. (f x \<bullet> b) *\<^sub>R b)"
     by (simp add: euclidean_representation)
   also from integrable have "\<dots> = (\<Sum>a\<in>Basis. integral A (\<lambda>x. (f x \<bullet> a) *\<^sub>R a))"
     by (subst integral_sum) (simp_all add: o_def)
   finally show ?thesis .
 qed
 
 lemma integrable_component:
   "f integrable_on A \<Longrightarrow> (\<lambda>x. f x \<bullet> (y :: 'b :: euclidean_space)) integrable_on A"
   by (drule integrable_linear[OF _ bounded_linear_inner_left[of y]]) (simp add: o_def)
 
 
 
 subsection \<open>Cauchy-type criterion for integrability\<close>
 
 proposition integrable_Cauchy:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::{real_normed_vector,complete_space}"
   shows "f integrable_on cbox a b \<longleftrightarrow>
         (\<forall>e>0. \<exists>\<gamma>. gauge \<gamma> \<and>
           (\<forall>\<D>1 \<D>2. \<D>1 tagged_division_of (cbox a b) \<and> \<gamma> fine \<D>1 \<and>
             \<D>2 tagged_division_of (cbox a b) \<and> \<gamma> fine \<D>2 \<longrightarrow>
             norm ((\<Sum>(x,K)\<in>\<D>1. content K *\<^sub>R f x) - (\<Sum>(x,K)\<in>\<D>2. content K *\<^sub>R f x)) < e))"
   (is "?l = (\<forall>e>0. \<exists>\<gamma>. ?P e \<gamma>)")
 proof (intro iffI allI impI)
   assume ?l
   then obtain y
     where y: "\<And>e. e > 0 \<Longrightarrow>
         \<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 f x) - y) < e)"
     by (auto simp: integrable_on_def has_integral)
   show "\<exists>\<gamma>. ?P e \<gamma>" if "e > 0" for e
   proof -
     have "e/2 > 0" using that by auto
     with y obtain \<gamma> where "gauge \<gamma>"
       and \<gamma>: "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<Longrightarrow>
                   norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f x) - y) < e/2"
       by meson
     show ?thesis
     apply (rule_tac x=\<gamma> in exI, clarsimp simp: \<open>gauge \<gamma>\<close>)
         by (blast intro!: \<gamma> dist_triangle_half_l[where y=y,unfolded dist_norm])
     qed
 next
   assume "\<forall>e>0. \<exists>\<gamma>. ?P e \<gamma>"
   then have "\<forall>n::nat. \<exists>\<gamma>. ?P (1 / (n + 1)) \<gamma>"
     by auto
   then obtain \<gamma> :: "nat \<Rightarrow> 'n \<Rightarrow> 'n set" where \<gamma>:
     "\<And>m. gauge (\<gamma> m)"
     "\<And>m \<D>1 \<D>2. \<lbrakk>\<D>1 tagged_division_of cbox a b;
               \<gamma> m fine \<D>1; \<D>2 tagged_division_of cbox a b; \<gamma> m fine \<D>2\<rbrakk>
               \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> \<D>2. content K *\<^sub>R f x))
                   < 1 / (m + 1)"
     by metis
   have "gauge (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..n}})" for n
     using \<gamma> by (intro gauge_Inter) auto
   then have "\<forall>n. \<exists>p. p tagged_division_of (cbox a b) \<and> (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..n}}) fine p"
     by (meson fine_division_exists)
   then obtain p where p: "\<And>z. p z tagged_division_of cbox a b"
                          "\<And>z. (\<lambda>x. \<Inter>{\<gamma> i x |i. i \<in> {0..z}}) fine p z"
     by meson
   have dp: "\<And>i n. i\<le>n \<Longrightarrow> \<gamma> i fine p n"
     using p unfolding fine_Inter
     using atLeastAtMost_iff by blast
   have "Cauchy (\<lambda>n. sum (\<lambda>(x,K). content K *\<^sub>R (f x)) (p n))"
   proof (rule CauchyI)
     fix e::real
     assume "0 < e"
     then obtain N where "N \<noteq> 0" and N: "inverse (real N) < e"
       using real_arch_inverse[of e] by blast
     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < e"
     proof (intro exI allI impI)
       fix m n
       assume mn: "N \<le> m" "N \<le> n"
       have "norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < 1 / (real N + 1)"
         by (simp add: p(1) dp mn \<gamma>)
       also have "... < e"
         using  N \<open>N \<noteq> 0\<close> \<open>0 < e\<close> by (auto simp: field_simps)
       finally show "norm ((\<Sum>(x,K) \<in> p m. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x)) < e" .
     qed
   qed
   then obtain y where y: "\<exists>no. \<forall>n\<ge>no. norm ((\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x) - y) < r" if "r > 0" for r
     by (auto simp: convergent_eq_Cauchy[symmetric] dest: LIMSEQ_D)
   show ?l
     unfolding integrable_on_def has_integral
   proof (rule_tac x=y in exI, clarify)
     fix e :: real
     assume "e>0"
     then have e2: "e/2 > 0" by auto
     then obtain N1::nat where N1: "N1 \<noteq> 0" "inverse (real N1) < e/2"
       using real_arch_inverse by blast
     obtain N2::nat where N2: "\<And>n. n \<ge> N2 \<Longrightarrow> norm ((\<Sum>(x,K) \<in> p n. content K *\<^sub>R f x) - y) < e/2"
       using y[OF e2] by metis
     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 f x) - y) < e)"
     proof (intro exI conjI allI impI)
       show "gauge (\<gamma> (N1+N2))"
         using \<gamma> by auto
       show "norm ((\<Sum>(x,K) \<in> q. content K *\<^sub>R f x) - y) < e"
            if "q tagged_division_of cbox a b \<and> \<gamma> (N1+N2) fine q" for q
       proof (rule norm_triangle_half_r)
         have "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> q. content K *\<^sub>R f x))
                < 1 / (real (N1+N2) + 1)"
           by (rule \<gamma>; simp add: dp p that)
         also have "... < e/2"
           using N1 \<open>0 < e\<close> by (auto simp: field_simps intro: less_le_trans)
         finally show "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> q. content K *\<^sub>R f x)) < e/2" .
         show "norm ((\<Sum>(x,K) \<in> p (N1+N2). content K *\<^sub>R f x) - y) < e/2"
           using N2 le_add_same_cancel2 by blast
       qed
     qed
   qed
 qed
 
 
 subsection \<open>Additivity of integral on abutting intervals\<close>
 
 lemma tagged_division_split_left_inj_content:
   assumes \<D>: "\<D> tagged_division_of S"
     and "(x1, K1) \<in> \<D>" "(x2, K2) \<in> \<D>" "K1 \<noteq> K2" "K1 \<inter> {x. x\<bullet>k \<le> c} = K2 \<inter> {x. x\<bullet>k \<le> c}" "k \<in> Basis"
   shows "content (K1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
 proof -
   from tagged_division_ofD(4)[OF \<D> \<open>(x1, K1) \<in> \<D>\<close>] obtain a b where K1: "K1 = cbox a b"
     by auto
   then have "interior (K1 \<inter> {x. x \<bullet> k \<le> c}) = {}"
     by (metis tagged_division_split_left_inj assms)
   then show ?thesis
     unfolding K1 interval_split[OF \<open>k \<in> Basis\<close>] by (auto simp: content_eq_0_interior)
 qed
 
 lemma tagged_division_split_right_inj_content:
   assumes \<D>: "\<D> tagged_division_of S"
     and "(x1, K1) \<in> \<D>" "(x2, K2) \<in> \<D>" "K1 \<noteq> K2" "K1 \<inter> {x. x\<bullet>k \<ge> c} = K2 \<inter> {x. x\<bullet>k \<ge> c}" "k \<in> Basis"
   shows "content (K1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
 proof -
   from tagged_division_ofD(4)[OF \<D> \<open>(x1, K1) \<in> \<D>\<close>] obtain a b where K1: "K1 = cbox a b"
     by auto
   then have "interior (K1 \<inter> {x. c \<le> x \<bullet> k}) = {}"
     by (metis tagged_division_split_right_inj assms)
   then show ?thesis
     unfolding K1 interval_split[OF \<open>k \<in> Basis\<close>]
     by (auto simp: content_eq_0_interior)
 qed
 
 
 proposition has_integral_split:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes fi: "(f has_integral i) (cbox a b \<inter> {x. x\<bullet>k \<le> c})"
       and fj: "(f has_integral j) (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
       and k: "k \<in> Basis"
 shows "(f has_integral (i + j)) (cbox a b)"
   unfolding has_integral
 proof clarify
   fix e::real
   assume "0 < e"
   then have e: "e/2 > 0"
     by auto
     obtain \<gamma>1 where \<gamma>1: "gauge \<gamma>1"
       and \<gamma>1norm:
         "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; \<gamma>1 fine \<D>\<rbrakk>
              \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - i) < e/2"
        apply (rule has_integralD[OF fi[unfolded interval_split[OF k]] e])
        apply (simp add: interval_split[symmetric] k)
       done
     obtain \<gamma>2 where \<gamma>2: "gauge \<gamma>2"
       and \<gamma>2norm:
         "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; \<gamma>2 fine \<D>\<rbrakk>
              \<Longrightarrow> norm ((\<Sum>(x, k) \<in> \<D>. content k *\<^sub>R f x) - j) < e/2"
        apply (rule has_integralD[OF fj[unfolded interval_split[OF k]] e])
        apply (simp add: interval_split[symmetric] k)
        done
   let ?\<gamma> = "\<lambda>x. if x\<bullet>k = c then (\<gamma>1 x \<inter> \<gamma>2 x) else ball x \<bar>x\<bullet>k - c\<bar> \<inter> \<gamma>1 x \<inter> \<gamma>2 x"
   have "gauge ?\<gamma>"
     using \<gamma>1 \<gamma>2 unfolding gauge_def by auto
   then 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 f x) - (i + j)) < e)"
   proof (rule_tac x="?\<gamma>" in exI, safe)
     fix p
     assume p: "p tagged_division_of (cbox a b)" and "?\<gamma> fine p"
     have ab_eqp: "cbox a b = \<Union>{K. \<exists>x. (x, K) \<in> p}"
       using p by blast
     have xk_le_c: "x\<bullet>k \<le> c" if as: "(x,K) \<in> p" and K: "K \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}" for x K
     proof (rule ccontr)
       assume **: "\<not> x \<bullet> k \<le> c"
       then have "K \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
         using \<open>?\<gamma> fine p\<close> as by (fastforce simp: not_le algebra_simps)
       with K obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<le> c"
         by blast
       then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
         using Basis_le_norm[OF k, of "x - y"]
         by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
       with y show False
         using ** by (auto simp add: field_simps)
     qed
     have xk_ge_c: "x\<bullet>k \<ge> c" if as: "(x,K) \<in> p" and K: "K \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}" for x K
     proof (rule ccontr)
       assume **: "\<not> x \<bullet> k \<ge> c"
       then have "K \<subseteq> ball x \<bar>x \<bullet> k - c\<bar>"
         using \<open>?\<gamma> fine p\<close> as by (fastforce simp: not_le algebra_simps)
       with K obtain y where y: "y \<in> ball x \<bar>x \<bullet> k - c\<bar>" "y\<bullet>k \<ge> c"
         by blast
       then have "\<bar>x \<bullet> k - y \<bullet> k\<bar> < \<bar>x \<bullet> k - c\<bar>"
         using Basis_le_norm[OF k, of "x - y"]
         by (auto simp add: dist_norm inner_diff_left intro: le_less_trans)
       with y show False
         using ** by (auto simp add: field_simps)
     qed
     have fin_finite: "finite {(x,f K) | x K. (x,K) \<in> s \<and> P x K}"
       if "finite s" for s and f :: "'a set \<Rightarrow> 'a set" and P :: "'a \<Rightarrow> 'a set \<Rightarrow> bool"
     proof -
       from that have "finite ((\<lambda>(x,K). (x, f K)) ` s)"
         by auto
       then show ?thesis
         by (rule rev_finite_subset) auto
     qed
     { fix \<G> :: "'a set \<Rightarrow> 'a set"
       fix i :: "'a \<times> 'a set"
       assume "i \<in> (\<lambda>(x, k). (x, \<G> k)) ` p - {(x, \<G> k) |x k. (x, k) \<in> p \<and> \<G> k \<noteq> {}}"
       then obtain x K where xk: "i = (x, \<G> K)"  "(x,K) \<in> p"
                                  "(x, \<G> K) \<notin> {(x, \<G> K) |x K. (x,K) \<in> p \<and> \<G> K \<noteq> {}}"
         by auto
       have "content (\<G> K) = 0"
         using xk using content_empty by auto
       then have "(\<lambda>(x,K). content K *\<^sub>R f x) i = 0"
         unfolding xk split_conv by auto
     } note [simp] = this
     have "finite p"
       using p by blast
     let ?M1 = "{(x, K \<inter> {x. x\<bullet>k \<le> c}) |x K. (x,K) \<in> p \<and> K \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
     have \<gamma>1_fine: "\<gamma>1 fine ?M1"
       using \<open>?\<gamma> fine p\<close> by (fastforce simp: fine_def split: if_split_asm)
     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2"
     proof (rule \<gamma>1norm [OF tagged_division_ofI \<gamma>1_fine])
       show "finite ?M1"
         by (rule fin_finite) (use p in blast)
       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = cbox a b \<inter> {x. x\<bullet>k \<le> c}"
         by (auto simp: ab_eqp)
 
       fix x L
       assume xL: "(x, L) \<in> ?M1"
       then obtain x' L' where xL': "x = x'" "L = L' \<inter> {x. x \<bullet> k \<le> c}"
                                    "(x', L') \<in> p" "L' \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
         by blast
       then obtain a' b' where ab': "L' = cbox a' b'"
         using p by blast
       show "x \<in> L" "L \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<le> c}"
         using p xk_le_c xL' by auto
       show "\<exists>a b. L = cbox a b"
         using p xL' ab' by (auto simp add: interval_split[OF k,where c=c])
 
       fix y R
       assume yR: "(y, R) \<in> ?M1"
       then obtain y' R' where yR': "y = y'" "R = R' \<inter> {x. x \<bullet> k \<le> c}"
                                    "(y', R') \<in> p" "R' \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
         by blast
       assume as: "(x, L) \<noteq> (y, R)"
       show "interior L \<inter> interior R = {}"
       proof (cases "L' = R' \<longrightarrow> x' = y'")
         case False
         have "interior R' = {}"
           by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3))
         then show ?thesis
           using yR' by simp
       next
         case True
         then have "L' \<noteq> R'"
           using as unfolding xL' yR' by auto
         have "interior L' \<inter> interior R' = {}"
           by (metis (no_types) Pair_inject \<open>L' \<noteq> R'\<close> p tagged_division_ofD(5) xL'(3) yR'(3))
         then show ?thesis
           using xL'(2) yR'(2) by auto
       qed
     qed
     moreover
     let ?M2 = "{(x,K \<inter> {x. x\<bullet>k \<ge> c}) |x K. (x,K) \<in> p \<and> K \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
     have \<gamma>2_fine: "\<gamma>2 fine ?M2"
       using \<open>?\<gamma> fine p\<close> by (fastforce simp: fine_def split: if_split_asm)
     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2"
     proof (rule \<gamma>2norm [OF tagged_division_ofI \<gamma>2_fine])
       show "finite ?M2"
         by (rule fin_finite) (use p in blast)
       show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = cbox a b \<inter> {x. x\<bullet>k \<ge> c}"
         by (auto simp: ab_eqp)
 
       fix x L
       assume xL: "(x, L) \<in> ?M2"
       then obtain x' L' where xL': "x = x'" "L = L' \<inter> {x. x \<bullet> k \<ge> c}"
                                    "(x', L') \<in> p" "L' \<inter> {x. x \<bullet> k \<ge> c} \<noteq> {}"
         by blast
       then obtain a' b' where ab': "L' = cbox a' b'"
         using p by blast
       show "x \<in> L" "L \<subseteq> cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
         using p xk_ge_c xL' by auto
       show "\<exists>a b. L = cbox a b"
         using p xL' ab' by (auto simp add: interval_split[OF k,where c=c])
 
       fix y R
       assume yR: "(y, R) \<in> ?M2"
       then obtain y' R' where yR': "y = y'" "R = R' \<inter> {x. x \<bullet> k \<ge> c}"
                                    "(y', R') \<in> p" "R' \<inter> {x. x \<bullet> k \<ge> c} \<noteq> {}"
         by blast
       assume as: "(x, L) \<noteq> (y, R)"
       show "interior L \<inter> interior R = {}"
       proof (cases "L' = R' \<longrightarrow> x' = y'")
         case False
         have "interior R' = {}"
           by (metis (no_types) False Pair_inject inf.idem tagged_division_ofD(5) [OF p] xL'(3) yR'(3))
         then show ?thesis
           using yR' by simp
       next
         case True
         then have "L' \<noteq> R'"
           using as unfolding xL' yR' by auto
         have "interior L' \<inter> interior R' = {}"
           by (metis (no_types) Pair_inject \<open>L' \<noteq> R'\<close> p tagged_division_ofD(5) xL'(3) yR'(3))
         then show ?thesis
           using xL'(2) yR'(2) by auto
       qed
     qed
     ultimately
     have "norm (((\<Sum>(x,K) \<in> ?M1. content K *\<^sub>R f x) - i) + ((\<Sum>(x,K) \<in> ?M2. content K *\<^sub>R f x) - j)) < e/2 + e/2"
       using norm_add_less by blast
     moreover have "((\<Sum>(x,K) \<in> ?M1. content K *\<^sub>R f x) - i) +
                    ((\<Sum>(x,K) \<in> ?M2. content K *\<^sub>R f x) - j) =
                    (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)"
     proof -
       have eq0: "\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'b) = 0"
          by auto
       have cont_eq: "\<And>g. (\<lambda>(x,l). content l *\<^sub>R f x) \<circ> (\<lambda>(x,l). (x,g l)) = (\<lambda>(x,l). content (g l) *\<^sub>R f x)"
         by auto
       have *: "\<And>\<G> :: 'a set \<Rightarrow> 'a set.
                   (\<Sum>(x,K)\<in>{(x, \<G> K) |x K. (x,K) \<in> p \<and> \<G> K \<noteq> {}}. content K *\<^sub>R f x) =
                   (\<Sum>(x,K)\<in>(\<lambda>(x,K). (x, \<G> K)) ` p. content K *\<^sub>R f x)"
         by (rule sum.mono_neutral_left) (auto simp: \<open>finite p\<close>)
       have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) =
         (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)"
         by auto
       moreover have "\<dots> = (\<Sum>(x,K) \<in> p. content (K \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
         (\<Sum>(x,K) \<in> p. content (K \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
         unfolding *
         apply (subst (1 2) sum.reindex_nontrivial)
            apply (auto intro!: k p eq0 tagged_division_split_left_inj_content tagged_division_split_right_inj_content
                        simp: cont_eq \<open>finite p\<close>)
         done
       moreover have "\<And>x. x \<in> p \<Longrightarrow> (\<lambda>(a,B). content (B \<inter> {a. a \<bullet> k \<le> c}) *\<^sub>R f a) x +
                                 (\<lambda>(a,B). content (B \<inter> {a. c \<le> a \<bullet> k}) *\<^sub>R f a) x =
                                 (\<lambda>(a,B). content B *\<^sub>R f a) x"
       proof clarify
         fix a B
         assume "(a, B) \<in> p"
         with p obtain u v where uv: "B = cbox u v" by blast
         then show "content (B \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f a + content (B \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f a = content B *\<^sub>R f a"
           by (auto simp: scaleR_left_distrib uv content_split[OF k,of u v c])
       qed
       ultimately show ?thesis
         by (auto simp: sum.distrib[symmetric])
       qed
     ultimately show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e"
       by auto
   qed
 qed
 
 
 subsection \<open>A sort of converse, integrability on subintervals\<close>
 
 lemma has_integral_separate_sides:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes f: "(f has_integral i) (cbox a b)"
     and "e > 0"
     and k: "k \<in> Basis"
   obtains d where "gauge d"
     "\<forall>p1 p2. p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<and> d fine p1 \<and>
         p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) \<and> d fine p2 \<longrightarrow>
         norm ((sum (\<lambda>(x,k). content k *\<^sub>R f x) p1 + sum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e"
 proof -
   obtain \<gamma> where d: "gauge \<gamma>"
       "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma> fine p\<rbrakk>
             \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < e"
     using has_integralD[OF f \<open>e > 0\<close>] by metis
   { fix p1 p2
     assume tdiv1: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" and "\<gamma> fine p1"
     note p1=tagged_division_ofD[OF this(1)] 
     assume tdiv2: "p2 tagged_division_of (cbox a b) \<inter> {x. c \<le> x \<bullet> k}" and "\<gamma> fine p2"
     note p2=tagged_division_ofD[OF this(1)] 
     note tagged_division_Un_interval[OF tdiv1 tdiv2] 
     note p12 = tagged_division_ofD[OF this] this
     { fix a b
       assume ab: "(a, b) \<in> p1 \<inter> p2"
       have "(a, b) \<in> p1"
         using ab by auto
       obtain u v where uv: "b = cbox u v"
         using \<open>(a, b) \<in> p1\<close> p1(4) by moura
       have "b \<subseteq> {x. x\<bullet>k = c}"
         using ab p1(3)[of a b] p2(3)[of a b] by fastforce
       moreover
       have "interior {x::'a. x \<bullet> k = c} = {}"
       proof (rule ccontr)
         assume "\<not> ?thesis"
         then obtain x where x: "x \<in> interior {x::'a. x\<bullet>k = c}"
           by auto
         then obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball x \<epsilon> \<subseteq> {x. x \<bullet> k = c}"
           using mem_interior by metis
         have x: "x\<bullet>k = c"
           using x interior_subset by fastforce
         have *: "\<And>i. i \<in> Basis \<Longrightarrow> \<bar>(x - (x + (\<epsilon>/2) *\<^sub>R k)) \<bullet> i\<bar> = (if i = k then \<epsilon>/2 else 0)"
           using \<open>0 < \<epsilon>\<close> k by (auto simp: inner_simps inner_not_same_Basis)
         have "(\<Sum>i\<in>Basis. \<bar>(x - (x + (\<epsilon>/2 ) *\<^sub>R k)) \<bullet> i\<bar>) =
               (\<Sum>i\<in>Basis. (if i = k then \<epsilon>/2 else 0))"
           using "*" by (blast intro: sum.cong)
         also have "\<dots> < \<epsilon>"
           by (subst sum.delta) (use \<open>0 < \<epsilon>\<close> in auto)
         finally have "x + (\<epsilon>/2) *\<^sub>R k \<in> ball x \<epsilon>"
           unfolding mem_ball dist_norm by(rule le_less_trans[OF norm_le_l1])
         then have "x + (\<epsilon>/2) *\<^sub>R k \<in> {x. x\<bullet>k = c}"
           using \<epsilon> by auto
         then show False
           using \<open>0 < \<epsilon>\<close> x k by (auto simp: inner_simps)
       qed
       ultimately have "content b = 0"
         unfolding uv content_eq_0_interior
         using interior_mono by blast
       then have "content b *\<^sub>R f a = 0"
         by auto
     }
     then have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) =
                norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"
       by (subst sum.union_inter_neutral) (auto simp: p1 p2)
     also have "\<dots> < e"
       using d(2) p12 by (simp add: fine_Un k \<open>\<gamma> fine p1\<close> \<open>\<gamma> fine p2\<close>)
     finally have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" .
    }
   then show ?thesis
     using d(1) that by auto
 qed
 
 lemma integrable_split [intro]:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
   assumes f: "f integrable_on cbox a b"
       and k: "k \<in> Basis"
     shows "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<le> c})"   (is ?thesis1)
     and   "f integrable_on (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"   (is ?thesis2)
 proof -
   obtain y where y: "(f has_integral y) (cbox a b)"
     using f by blast
   define a' where "a' = (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i)*\<^sub>R i)"
   define b' where "b' = (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i)*\<^sub>R i)"
   have "\<exists>d. gauge d \<and>
             (\<forall>p1 p2. p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p1 \<and>
                      p2 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> d fine p2 \<longrightarrow>
                      norm ((\<Sum>(x,K) \<in> p1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p2. content K *\<^sub>R f x)) < e)"
     if "e > 0" for e
   proof -
     have "e/2 > 0" using that by auto
   with has_integral_separate_sides[OF y this k, of c]
   obtain d
     where "gauge d"
          and d: "\<And>p1 p2. \<lbrakk>p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; d fine p1;
                           p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; d fine p2\<rbrakk>
                   \<Longrightarrow> norm ((\<Sum>(x,K)\<in>p1. content K *\<^sub>R f x) + (\<Sum>(x,K)\<in>p2. content K *\<^sub>R f x) - y) < e/2"
     by metis
   show ?thesis
     proof (rule_tac x=d in exI, clarsimp simp add: \<open>gauge d\<close>)
       fix p1 p2
       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p1"
                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<le> c}" "d fine p2"
       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
       proof (rule fine_division_exists[OF \<open>gauge d\<close>, of a' b])
         fix p
         assume "p tagged_division_of cbox a' b" "d fine p"
         then show ?thesis
           using as norm_triangle_half_l[OF d[of p1 p] d[of p2 p]]
           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
           by (auto simp add: algebra_simps)
       qed
     qed
   qed
   with f show ?thesis1
     by (simp add: interval_split[OF k] integrable_Cauchy)
   have "\<exists>d. gauge d \<and>
             (\<forall>p1 p2. p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p1 \<and>
                      p2 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<ge> c} \<and> d fine p2 \<longrightarrow>
                      norm ((\<Sum>(x,K) \<in> p1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p2. content K *\<^sub>R f x)) < e)"
     if "e > 0" for e
   proof -
     have "e/2 > 0" using that by auto
   with has_integral_separate_sides[OF y this k, of c]
   obtain d
     where "gauge d"
          and d: "\<And>p1 p2. \<lbrakk>p1 tagged_division_of cbox a b \<inter> {x. x \<bullet> k \<le> c}; d fine p1;
                           p2 tagged_division_of cbox a b \<inter> {x. c \<le> x \<bullet> k}; d fine p2\<rbrakk>
                   \<Longrightarrow> norm ((\<Sum>(x,K)\<in>p1. content K *\<^sub>R f x) + (\<Sum>(x,K)\<in>p2. content K *\<^sub>R f x) - y) < e/2"
     by metis
   show ?thesis
     proof (rule_tac x=d in exI, clarsimp simp add: \<open>gauge d\<close>)
       fix p1 p2
       assume as: "p1 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p1"
                  "p2 tagged_division_of (cbox a b) \<inter> {x. x \<bullet> k \<ge> c}" "d fine p2"
       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"
       proof (rule fine_division_exists[OF \<open>gauge d\<close>, of a b'])
         fix p
         assume "p tagged_division_of cbox a b'" "d fine p"
         then show ?thesis
           using as norm_triangle_half_l[OF d[of p p1] d[of p p2]]
           unfolding interval_split[OF k] b'_def[symmetric] a'_def[symmetric]
           by (auto simp add: algebra_simps)
       qed
     qed
   qed
   with f show ?thesis2
     by (simp add: interval_split[OF k] integrable_Cauchy)
 qed
 
 lemma operative_integralI:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   shows "operative (lift_option (+)) (Some 0)
     (\<lambda>i. if f integrable_on i then Some (integral i f) else None)"
 proof -
   interpret comm_monoid "lift_option plus" "Some (0::'b)"
     by (rule comm_monoid_lift_option)
       (rule add.comm_monoid_axioms)
   show ?thesis
   proof
     fix a b c
     fix k :: 'a
     assume k: "k \<in> Basis"
     show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) =
           lift_option (+) (if f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c} then Some (integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f) else None)
           (if f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k} then Some (integral (cbox a b \<inter> {x. c \<le> x \<bullet> k}) f) else None)"
     proof (cases "f integrable_on cbox a b")
       case True
       with k show ?thesis
         by (auto simp: integrable_split intro: integral_unique [OF has_integral_split[OF _ _ k]])
     next
     case False
       have "\<not> (f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}) \<or> \<not> ( f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k})"
       proof (rule ccontr)
         assume "\<not> ?thesis"
         then have "f integrable_on cbox a b"
           unfolding integrable_on_def
           apply (rule_tac x="integral (cbox a b \<inter> {x. x \<bullet> k \<le> c}) f + integral (cbox a b \<inter> {x. x \<bullet> k \<ge> c}) f" in exI)
           apply (auto intro: has_integral_split[OF _ _ k])
           done
         then show False
           using False by auto
       qed
       then show ?thesis
         using False by auto
     qed
   next
     fix a b :: 'a
     assume "box a b = {}"
     then show "(if f integrable_on cbox a b then Some (integral (cbox a b) f) else None) = Some 0"
       using has_integral_null_eq
       by (auto simp: integrable_on_null content_eq_0_interior)
   qed
 qed
 
 subsection \<open>Bounds on the norm of Riemann sums and the integral itself\<close>
 
 lemma dsum_bound:
   assumes p: "p division_of (cbox a b)"
     and "norm c \<le> e"
   shows "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content(cbox a b)"
 proof -
   have sumeq: "(\<Sum>i\<in>p. \<bar>content i\<bar>) = sum content p"
     by simp
   have e: "0 \<le> e"
     using assms(2) norm_ge_zero order_trans by blast
   have "norm (sum (\<lambda>l. content l *\<^sub>R c) p) \<le> (\<Sum>i\<in>p. norm (content i *\<^sub>R c))"
     using norm_sum by blast
   also have "...  \<le> e * (\<Sum>i\<in>p. \<bar>content i\<bar>)"
     by (simp add: sum_distrib_left[symmetric] mult.commute assms(2) mult_right_mono sum_nonneg)
   also have "... \<le> e * content (cbox a b)"
     by (metis additive_content_division p eq_iff sumeq)
   finally show ?thesis .
 qed
 
 lemma rsum_bound:
   assumes p: "p tagged_division_of (cbox a b)"
       and "\<forall>x\<in>cbox a b. norm (f x) \<le> e"
     shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content (cbox a b)"
 proof (cases "cbox a b = {}")
   case True show ?thesis
     using p unfolding True tagged_division_of_trivial by auto
 next
   case False
   then have e: "e \<ge> 0"
     by (meson ex_in_conv assms(2) norm_ge_zero order_trans)
   have sum_le: "sum (content \<circ> snd) p \<le> content (cbox a b)"
     unfolding additive_content_tagged_division[OF p, symmetric] split_def
     by (auto intro: eq_refl)
   have con: "\<And>xk. xk \<in> p \<Longrightarrow> 0 \<le> content (snd xk)"
     using tagged_division_ofD(4) [OF p] content_pos_le
     by force
   have "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> (\<Sum>i\<in>p. norm (case i of (x, k) \<Rightarrow> content k *\<^sub>R f x))"
     by (rule norm_sum)
   also have "...  \<le> e * content (cbox a b)"
   proof -
     have "\<And>xk. xk \<in> p \<Longrightarrow> norm (f (fst xk)) \<le> e"
       using assms(2) p tag_in_interval by force
     moreover have "(\<Sum>i\<in>p. \<bar>content (snd i)\<bar> * e) \<le> e * content (cbox a b)"
       unfolding sum_distrib_right[symmetric]
       using con sum_le by (auto simp: mult.commute intro: mult_left_mono [OF _ e])
     ultimately show ?thesis
       unfolding split_def norm_scaleR
       by (metis (no_types, lifting) mult_left_mono[OF _ abs_ge_zero]   order_trans[OF sum_mono])
   qed
   finally show ?thesis .
 qed
 
 lemma rsum_diff_bound:
   assumes "p tagged_division_of (cbox a b)"
     and "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e"
   shows "norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - sum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le>
          e * content (cbox a b)"
   using order_trans[OF _ rsum_bound[OF assms]]
   by (simp add: split_def scaleR_diff_right sum_subtractf eq_refl)
 
 lemma has_integral_bound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B"
       and f: "(f has_integral i) (cbox a b)"
       and "\<And>x. x\<in>cbox a b \<Longrightarrow> norm (f x) \<le> B"
     shows "norm i \<le> B * content (cbox a b)"
 proof (rule ccontr)
   assume "\<not> ?thesis"
   then have "norm i - B * content (cbox a b) > 0"
     by auto
   with f[unfolded has_integral]
   obtain \<gamma> where "gauge \<gamma>" and \<gamma>:
     "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma> fine p\<rbrakk>
           \<Longrightarrow> norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) - i) < norm i - B * content (cbox a b)"
     by metis
   then obtain p where p: "p tagged_division_of cbox a b" and "\<gamma> fine p"
     using fine_division_exists by blast
   have "\<And>s B. norm s \<le> B \<Longrightarrow> \<not> norm (s - i) < norm i - B"
     unfolding not_less
     by (metis diff_left_mono dist_commute dist_norm norm_triangle_ineq2 order_trans)
   then show False
     using \<gamma> [OF p \<open>\<gamma> fine p\<close>] rsum_bound[OF p] assms by metis
 qed
 
 corollary integrable_bound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B"
       and "f integrable_on (cbox a b)"
       and "\<And>x. x\<in>cbox a b \<Longrightarrow> norm (f x) \<le> B"
     shows "norm (integral (cbox a b) f) \<le> B * content (cbox a b)"
 by (metis integrable_integral has_integral_bound assms)
 
 
 subsection \<open>Similar theorems about relationship among components\<close>
 
 lemma rsum_component_le:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes p: "p tagged_division_of (cbox a b)"
       and "\<And>x. x \<in> cbox a b \<Longrightarrow> (f x)\<bullet>i \<le> (g x)\<bullet>i"
     shows "(\<Sum>(x, K)\<in>p. content K *\<^sub>R f x) \<bullet> i \<le> (\<Sum>(x, K)\<in>p. content K *\<^sub>R g x) \<bullet> i"
 unfolding inner_sum_left
 proof (rule sum_mono, clarify)
   fix x K
   assume ab: "(x, K) \<in> p"
   with p obtain u v where K: "K = cbox u v"
     by blast
   then show "(content K *\<^sub>R f x) \<bullet> i \<le> (content K *\<^sub>R g x) \<bullet> i"
     by (metis ab assms inner_scaleR_left measure_nonneg mult_left_mono tag_in_interval)
 qed
 
 lemma has_integral_component_le:
   fixes f g :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes k: "k \<in> Basis"
   assumes "(f has_integral i) S" "(g has_integral j) S"
     and f_le_g: "\<And>x. x \<in> S \<Longrightarrow> (f x)\<bullet>k \<le> (g x)\<bullet>k"
   shows "i\<bullet>k \<le> j\<bullet>k"
 proof -
   have ik_le_jk: "i\<bullet>k \<le> j\<bullet>k"
     if f_i: "(f has_integral i) (cbox a b)"
     and g_j: "(g has_integral j) (cbox a b)"
     and le: "\<forall>x\<in>cbox a b. (f x)\<bullet>k \<le> (g x)\<bullet>k"
     for a b i and j :: 'b and f g :: "'a \<Rightarrow> 'b"
   proof (rule ccontr)
     assume "\<not> ?thesis"
     then have *: "0 < (i\<bullet>k - j\<bullet>k) / 3"
       by auto
     obtain \<gamma>1 where "gauge \<gamma>1" 
       and \<gamma>1: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma>1 fine p\<rbrakk>
                 \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < (i \<bullet> k - j \<bullet> k) / 3"
       using f_i[unfolded has_integral,rule_format,OF *] by fastforce 
     obtain \<gamma>2 where "gauge \<gamma>2" 
       and \<gamma>2: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; \<gamma>2 fine p\<rbrakk>
                 \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - j) < (i \<bullet> k - j \<bullet> k) / 3"
       using g_j[unfolded has_integral,rule_format,OF *] by fastforce 
     obtain p where p: "p tagged_division_of cbox a b" and "\<gamma>1 fine p" "\<gamma>2 fine p"
        using fine_division_exists[OF gauge_Int[OF \<open>gauge \<gamma>1\<close> \<open>gauge \<gamma>2\<close>], of a b] unfolding fine_Int
        by metis
     then have "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
          "\<bar>((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - j) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
       using le_less_trans[OF Basis_le_norm[OF k]] k \<gamma>1 \<gamma>2 by metis+ 
     then show False
       unfolding inner_simps
       using rsum_component_le[OF p] le
       by (fastforce simp add: abs_real_def split: if_split_asm)
   qed
   show ?thesis
   proof (cases "\<exists>a b. S = cbox a b")
     case True
     with ik_le_jk assms show ?thesis
       by auto
   next
     case False
     show ?thesis
     proof (rule ccontr)
       assume "\<not> i\<bullet>k \<le> j\<bullet>k"
       then have ij: "(i\<bullet>k - j\<bullet>k) / 3 > 0"
         by auto
       obtain B1 where "0 < B1" 
            and B1: "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
                     \<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b) \<and>
                         norm (z - i) < (i \<bullet> k - j \<bullet> k) / 3"
         using has_integral_altD[OF _ False ij] assms by blast
       obtain B2 where "0 < B2" 
            and B2: "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
                     \<exists>z. ((\<lambda>x. if x \<in> S then g x else 0) has_integral z) (cbox a b) \<and>
                         norm (z - j) < (i \<bullet> k - j \<bullet> k) / 3"
         using has_integral_altD[OF _ False ij] assms by blast
       have "bounded (ball 0 B1 \<union> ball (0::'a) B2)"
         unfolding bounded_Un by(rule conjI bounded_ball)+
       from bounded_subset_cbox_symmetric[OF this] 
       obtain a b::'a where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
         by (meson Un_subset_iff)
       then obtain w1 w2 where int_w1: "((\<lambda>x. if x \<in> S then f x else 0) has_integral w1) (cbox a b)"
                         and norm_w1:  "norm (w1 - i) < (i \<bullet> k - j \<bullet> k) / 3"
                         and int_w2: "((\<lambda>x. if x \<in> S then g x else 0) has_integral w2) (cbox a b)"
                         and norm_w2: "norm (w2 - j) < (i \<bullet> k - j \<bullet> k) / 3"
         using B1 B2 by blast
       have *: "\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False"
         by (simp add: abs_real_def split: if_split_asm)
       have "\<bar>(w1 - i) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
            "\<bar>(w2 - j) \<bullet> k\<bar> < (i \<bullet> k - j \<bullet> k) / 3"
         using Basis_le_norm k le_less_trans norm_w1 norm_w2 by blast+
       moreover
       have "w1\<bullet>k \<le> w2\<bullet>k"
         using ik_le_jk int_w1 int_w2 f_le_g by auto
       ultimately show False
         unfolding inner_simps by(rule *)
     qed
   qed
 qed
 
 lemma integral_component_le:
   fixes g f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "k \<in> Basis"
     and "f integrable_on S" "g integrable_on S"
     and "\<And>x. x \<in> S \<Longrightarrow> (f x)\<bullet>k \<le> (g x)\<bullet>k"
   shows "(integral S f)\<bullet>k \<le> (integral S g)\<bullet>k"
   using has_integral_component_le assms by blast
 
 lemma has_integral_component_nonneg:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "k \<in> Basis"
     and "(f has_integral i) S"
     and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> (f x)\<bullet>k"
   shows "0 \<le> i\<bullet>k"
   using has_integral_component_le[OF assms(1) has_integral_0 assms(2)]
   using assms(3-)
   by auto
 
 lemma integral_component_nonneg:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "k \<in> Basis"
     and  "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> (f x)\<bullet>k"
   shows "0 \<le> (integral S f)\<bullet>k"
 proof (cases "f integrable_on S")
   case True show ?thesis
     using True assms has_integral_component_nonneg by blast
 next
   case False then show ?thesis by (simp add: not_integrable_integral)
 qed
 
 lemma has_integral_component_neg:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "k \<in> Basis"
     and "(f has_integral i) S"
     and "\<And>x. x \<in> S \<Longrightarrow> (f x)\<bullet>k \<le> 0"
   shows "i\<bullet>k \<le> 0"
   using has_integral_component_le[OF assms(1,2) has_integral_0] assms(2-)
   by auto
 
 lemma has_integral_component_lbound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "(f has_integral i) (cbox a b)"
     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
     and "k \<in> Basis"
   shows "B * content (cbox a b) \<le> i\<bullet>k"
   using has_integral_component_le[OF assms(3) has_integral_const assms(1),of "(\<Sum>i\<in>Basis. B *\<^sub>R i)::'b"] assms(2-)
   by (auto simp add: field_simps)
 
 lemma has_integral_component_ubound:
   fixes f::"'a::euclidean_space => 'b::euclidean_space"
   assumes "(f has_integral i) (cbox a b)"
     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
     and "k \<in> Basis"
   shows "i\<bullet>k \<le> B * content (cbox a b)"
   using has_integral_component_le[OF assms(3,1) has_integral_const, of "\<Sum>i\<in>Basis. B *\<^sub>R i"] assms(2-)
   by (auto simp add: field_simps)
 
 lemma integral_component_lbound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "f integrable_on cbox a b"
     and "\<forall>x\<in>cbox a b. B \<le> f(x)\<bullet>k"
     and "k \<in> Basis"
   shows "B * content (cbox a b) \<le> (integral(cbox a b) f)\<bullet>k"
   using assms has_integral_component_lbound by blast
 
 lemma integral_component_lbound_real:
   assumes "f integrable_on {a ::real..b}"
     and "\<forall>x\<in>{a..b}. B \<le> f(x)\<bullet>k"
     and "k \<in> Basis"
   shows "B * content {a..b} \<le> (integral {a..b} f)\<bullet>k"
   using assms
   by (metis box_real(2) integral_component_lbound)
 
 lemma integral_component_ubound:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
   assumes "f integrable_on cbox a b"
     and "\<forall>x\<in>cbox a b. f x\<bullet>k \<le> B"
     and "k \<in> Basis"
   shows "(integral (cbox a b) f)\<bullet>k \<le> B * content (cbox a b)"
   using assms has_integral_component_ubound by blast
 
 lemma integral_component_ubound_real:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes "f integrable_on {a..b}"
     and "\<forall>x\<in>{a..b}. f x\<bullet>k \<le> B"
     and "k \<in> Basis"
   shows "(integral {a..b} f)\<bullet>k \<le> B * content {a..b}"
   using assms
   by (metis box_real(2) integral_component_ubound)
 
 subsection \<open>Uniform limit of integrable functions is integrable\<close>
 
 lemma real_arch_invD:
   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
   by (subst(asm) real_arch_inverse)
 
 
 lemma integrable_uniform_limit:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
   shows "f integrable_on cbox a b"
 proof (cases "content (cbox a b) > 0")
   case False then show ?thesis
     using has_integral_null by (simp add: content_lt_nz integrable_on_def)
 next
   case True
   have "1 / (real n + 1) > 0" for n
     by auto
   then have "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> 1 / (real n + 1)) \<and> g integrable_on cbox a b" for n
     using assms by blast
   then obtain g where g_near_f: "\<And>n x. x \<in> cbox a b \<Longrightarrow> norm (f x - g n x) \<le> 1 / (real n + 1)"
                   and int_g: "\<And>n. g n integrable_on cbox a b"
     by metis
   then obtain h where h: "\<And>n. (g n has_integral h n) (cbox a b)"
     unfolding integrable_on_def by metis
   have "Cauchy h"
     unfolding Cauchy_def
   proof clarify
     fix e :: real
     assume "e>0"
     then have "e/4 / content (cbox a b) > 0"
       using True by (auto simp: field_simps)
     then obtain M where "M \<noteq> 0" and M: "1 / (real M) < e/4 / content (cbox a b)"
       by (metis inverse_eq_divide real_arch_inverse)
     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (h m) (h n) < e"
     proof (rule exI [where x=M], clarify)
       fix m n
       assume m: "M \<le> m" and n: "M \<le> n"
       have "e/4>0" using \<open>e>0\<close> by auto
       then obtain gm gn where "gauge gm" "gauge gn"
               and gm: "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> gm fine \<D> 
                             \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g m x) - h m) < e/4"
               and gn: "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> gn fine \<D> \<Longrightarrow>
                       norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g n x) - h n) < e/4"
         using h[unfolded has_integral] by meson
       then obtain \<D> where \<D>: "\<D> tagged_division_of cbox a b" "(\<lambda>x. gm x \<inter> gn x) fine \<D>"
         by (metis (full_types) fine_division_exists gauge_Int)
       have triangle3: "norm (i1 - i2) < e"
         if no: "norm(s2 - s1) \<le> e/2" "norm (s1 - i1) < e/4" "norm (s2 - i2) < e/4" for s1 s2 i1 and i2::'b
       proof -
         have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"
           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]
           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"]
           by (auto simp: algebra_simps)
         also have "\<dots> < e"
           using no by (auto simp: algebra_simps norm_minus_commute)
         finally show ?thesis .
       qed
       have finep: "gm fine \<D>" "gn fine \<D>"
         using fine_Int \<D>  by auto
       have norm_le: "norm (g n x - g m x) \<le> 2 / real M" if x: "x \<in> cbox a b" for x
       proof -
         have "norm (f x - g n x) + norm (f x - g m x) \<le> 1 / (real n + 1) + 1 / (real m + 1)"          
           using g_near_f[OF x, of n] g_near_f[OF x, of m] by simp
         also have "\<dots> \<le> 1 / (real M) + 1 / (real M)"
           using \<open>M \<noteq> 0\<close> m n by (intro add_mono; force simp: field_split_simps)
         also have "\<dots> = 2 / real M"
           by auto
         finally show "norm (g n x - g m x) \<le> 2 / real M"
           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]
           by (auto simp: algebra_simps simp add: norm_minus_commute)
       qed
       have "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g n x) - (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g m x)) \<le> 2 / real M * content (cbox a b)"
         by (blast intro: norm_le rsum_diff_bound[OF \<D>(1), where e="2 / real M"])
       also have "... \<le> e/2"
         using M True
         by (auto simp: field_simps)
       finally have le_e2: "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g n x) - (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g m x)) \<le> e/2" .
       then show "dist (h m) (h n) < e"
         unfolding dist_norm using gm gn \<D> finep by (auto intro!: triangle3)
     qed
   qed
   then obtain s where s: "h \<longlonglongrightarrow> s"
     using convergent_eq_Cauchy[symmetric] by blast
   show ?thesis
     unfolding integrable_on_def has_integral
   proof (rule_tac x=s in exI, clarify)
     fix e::real
     assume e: "0 < e"
     then have "e/3 > 0" by auto
     then obtain N1 where N1: "\<forall>n\<ge>N1. norm (h n - s) < e/3"
       using LIMSEQ_D [OF s] by metis
     from e True have "e/3 / content (cbox a b) > 0"
       by (auto simp: field_simps)
     then obtain N2 :: nat
          where "N2 \<noteq> 0" and N2: "1 / (real N2) < e/3 / content (cbox a b)"
       by (metis inverse_eq_divide real_arch_inverse)
     obtain g' where "gauge g'"
             and g': "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> g' fine \<D> \<Longrightarrow>
                     norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g (N1 + N2) x) - h (N1 + N2)) < e/3"
       by (metis h has_integral \<open>e/3 > 0\<close>)
     have *: "norm (sf - s) < e" 
         if no: "norm (sf - sg) \<le> e/3" "norm(h - s) < e/3" "norm (sg - h) < e/3" for sf sg h
     proof -
       have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - h) + norm (h - s)"
         using norm_triangle_ineq[of "sf - sg" "sg - s"]
         using norm_triangle_ineq[of "sg -  h" " h - s"]
         by (auto simp: algebra_simps)
       also have "\<dots> < e"
         using no by (auto simp: algebra_simps norm_minus_commute)
       finally show ?thesis .
     qed
     { fix \<D>
       assume ptag: "\<D> tagged_division_of (cbox a b)" and "g' fine \<D>"
       then have norm_less: "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g (N1 + N2) x) - h (N1 + N2)) < e/3"
         using g' by blast
       have "content (cbox a b) < e/3 * (of_nat N2)"
         using \<open>N2 \<noteq> 0\<close> N2 using True by (auto simp: field_split_simps)
       moreover have "e/3 * of_nat N2 \<le> e/3 * (of_nat (N1 + N2) + 1)"
         using \<open>e>0\<close> by auto
       ultimately have "content (cbox a b) < e/3 * (of_nat (N1 + N2) + 1)"
         by linarith
       then have le_e3: "1 / (real (N1 + N2) + 1) * content (cbox a b) \<le> e/3"
         unfolding inverse_eq_divide
         by (auto simp: field_simps)
       have ne3: "norm (h (N1 + N2) - s) < e/3"
         using N1 by auto
       have "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g (N1 + N2) x))
             \<le> 1 / (real (N1 + N2) + 1) * content (cbox a b)"
         by (blast intro: g_near_f rsum_diff_bound[OF ptag])
       then have "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - s) < e"
         by (rule *[OF order_trans [OF _ le_e3] ne3 norm_less])
     }
     then show "\<exists>d. gauge d \<and>
              (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> d fine \<D> \<longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - s) < e)"
       by (blast intro: g' \<open>gauge g'\<close>)
   qed
 qed
 
 lemmas integrable_uniform_limit_real = integrable_uniform_limit [where 'a=real, simplified]
 
 
 subsection \<open>Negligible sets\<close>
 
 definition "negligible (s:: 'a::euclidean_space set) \<longleftrightarrow>
   (\<forall>a b. ((indicator s :: 'a\<Rightarrow>real) has_integral 0) (cbox a b))"
 
 
 subsubsection \<open>Negligibility of hyperplane\<close>
 
 lemma content_doublesplit:
   fixes a :: "'a::euclidean_space"
   assumes "0 < e"
     and k: "k \<in> Basis"
   obtains d where "0 < d" and "content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) < e"
 proof cases
   assume *: "a \<bullet> k \<le> c \<and> c \<le> b \<bullet> k \<and> (\<forall>j\<in>Basis. a \<bullet> j \<le> b \<bullet> j)"
   define a' where "a' d = (\<Sum>j\<in>Basis. (if j = k then max (a\<bullet>j) (c - d) else a\<bullet>j) *\<^sub>R j)" for d
   define b' where "b' d = (\<Sum>j\<in>Basis. (if j = k then min (b\<bullet>j) (c + d) else b\<bullet>j) *\<^sub>R j)" for d
 
   have "((\<lambda>d. \<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) \<longlongrightarrow> (\<Prod>j\<in>Basis. (b' 0 - a' 0) \<bullet> j)) (at_right 0)"
     by (auto simp: b'_def a'_def intro!: tendsto_min tendsto_max tendsto_eq_intros)
   also have "(\<Prod>j\<in>Basis. (b' 0 - a' 0) \<bullet> j) = 0"
     using k *
     by (intro prod_zero bexI[OF _ k])
        (auto simp: b'_def a'_def inner_diff inner_sum_left inner_not_same_Basis intro!: sum.cong)
   also have "((\<lambda>d. \<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) \<longlongrightarrow> 0) (at_right 0) =
     ((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)"
   proof (intro tendsto_cong eventually_at_rightI)
     fix d :: real assume d: "d \<in> {0<..<1}"
     have "cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d} = cbox (a' d) (b' d)" for d
       using * d k by (auto simp add: cbox_def set_eq_iff Int_def ball_conj_distrib abs_diff_le_iff a'_def b'_def)
     moreover have "j \<in> Basis \<Longrightarrow> a' d \<bullet> j \<le> b' d \<bullet> j" for j
       using * d k by (auto simp: a'_def b'_def)
     ultimately show "(\<Prod>j\<in>Basis. (b' d - a' d) \<bullet> j) = content (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d})"
       by simp
   qed simp
   finally have "((\<lambda>d. content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) \<longlongrightarrow> 0) (at_right 0)" .
   from order_tendstoD(2)[OF this \<open>0<e\<close>]
   obtain d' where "0 < d'" and d': "\<And>y. y > 0 \<Longrightarrow> y < d' \<Longrightarrow> content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> y}) < e"
     by (subst (asm) eventually_at_right[of _ 1]) auto
   show ?thesis
     by (rule that[of "d'/2"], insert \<open>0<d'\<close> d'[of "d'/2"], auto)
 next
   assume *: "\<not> (a \<bullet> k \<le> c \<and> c \<le> b \<bullet> k \<and> (\<forall>j\<in>Basis. a \<bullet> j \<le> b \<bullet> j))"
   then have "(\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j) \<or> (c < a \<bullet> k \<or> b \<bullet> k < c)"
     by (auto simp: not_le)
   show thesis
   proof cases
     assume "\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j"
     then have [simp]: "cbox a b = {}"
       using box_ne_empty(1)[of a b] by auto
     show ?thesis
       by (rule that[of 1]) (simp_all add: \<open>0<e\<close>)
   next
     assume "\<not> (\<exists>j\<in>Basis. b \<bullet> j < a \<bullet> j)"
     with * have "c < a \<bullet> k \<or> b \<bullet> k < c"
       by auto
     then show thesis
     proof
       assume c: "c < a \<bullet> k"
       moreover have "x \<in> cbox a b \<Longrightarrow> c \<le> x \<bullet> k" for x
         using k c by (auto simp: cbox_def)
       ultimately have "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> (a \<bullet> k - c)/2} = {}"
         using k by (auto simp: cbox_def)
       with \<open>0<e\<close> c that[of "(a \<bullet> k - c)/2"] show ?thesis
         by auto
     next
       assume c: "b \<bullet> k < c"
       moreover have "x \<in> cbox a b \<Longrightarrow> x \<bullet> k \<le> c" for x
         using k c by (auto simp: cbox_def)
       ultimately have "cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> (c - b \<bullet> k)/2} = {}"
         using k by (auto simp: cbox_def)
       with \<open>0<e\<close> c that[of "(c - b \<bullet> k)/2"] show ?thesis
         by auto
     qed
   qed
 qed
 
 
 proposition negligible_standard_hyperplane[intro]:
   fixes k :: "'a::euclidean_space"
   assumes k: "k \<in> Basis"
   shows "negligible {x. x\<bullet>k = c}"
   unfolding negligible_def has_integral
 proof clarsimp
   fix a b and e::real assume "e > 0"
   with k obtain d where "0 < d" and d: "content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) < e"
     by (metis content_doublesplit)
   let ?i = "indicator {x::'a. x\<bullet>k = c} :: 'a\<Rightarrow>real"
   show "\<exists>\<gamma>. gauge \<gamma> \<and>
            (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
                  \<bar>\<Sum>(x,K) \<in> \<D>. content K * ?i x\<bar> < e)"
   proof (intro exI, safe)
     show "gauge (\<lambda>x. ball x d)"
       using \<open>0 < d\<close> by blast
   next
     fix \<D>
     assume p: "\<D> tagged_division_of (cbox a b)" "(\<lambda>x. ball x d) fine \<D>"
     have "content L = content (L \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})" 
       if "(x, L) \<in> \<D>" "?i x \<noteq> 0" for x L
     proof -
       have xk: "x\<bullet>k = c"
         using that by (simp add: indicator_def split: if_split_asm)
       have "L \<subseteq> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
       proof 
         fix y
         assume y: "y \<in> L"
         have "L \<subseteq> ball x d"
           using p(2) that(1) by auto
         then have "norm (x - y) < d"
           by (simp add: dist_norm subset_iff y)
         then have "\<bar>(x - y) \<bullet> k\<bar> < d"
           using k norm_bound_Basis_lt by blast
         then show "y \<in> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
           unfolding inner_simps xk by auto
       qed 
       then show "content L = content (L \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
         by (metis inf.orderE)
     qed
     then have *: "(\<Sum>(x,K)\<in>\<D>. content K * ?i x) = (\<Sum>(x,K)\<in>\<D>. content (K \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> d}) *\<^sub>R ?i x)"
       by (force simp add: split_paired_all intro!: sum.cong [OF refl])
     note p'= tagged_division_ofD[OF p(1)] and p''=division_of_tagged_division[OF p(1)]
     have "(\<Sum>(x,K)\<in>\<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * indicator {x. x \<bullet> k = c} x) < e"
     proof -
       have "(\<Sum>(x,K)\<in>\<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x,K)\<in>\<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}))"
         by (force simp add: indicator_def intro!: sum_mono)
       also have "\<dots> < e"
       proof (subst sum.over_tagged_division_lemma[OF p(1)])
         fix u v::'a
         assume "box u v = {}"
         then have *: "content (cbox u v) = 0"
           unfolding content_eq_0_interior by simp
         have "cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<subseteq> cbox u v"
           by auto
         then have "content (cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<le> content (cbox u v)"
           unfolding interval_doublesplit[OF k] by (rule content_subset)
         then show "content (cbox u v \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0"
           unfolding * interval_doublesplit[OF k]
           by (blast intro: antisym)
       next
         have "(\<Sum>l\<in>snd ` \<D>. content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) =
           sum content ((\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` \<D>. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}})"
         proof (subst (2) sum.reindex_nontrivial)
           fix x y assume "x \<in> {l \<in> snd ` \<D>. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}" "y \<in> {l \<in> snd ` \<D>. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}"
             "x \<noteq> y" and eq: "x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} = y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}"
           then obtain x' y' where "(x', x) \<in> \<D>" "x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}" "(y', y) \<in> \<D>" "y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}"
             by (auto)
           from p'(5)[OF \<open>(x', x) \<in> \<D>\<close> \<open>(y', y) \<in> \<D>\<close>] \<open>x \<noteq> y\<close> have "interior (x \<inter> y) = {}"
             by auto
           moreover have "interior ((x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) \<inter> (y \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) \<subseteq> interior (x \<inter> y)"
             by (auto intro: interior_mono)
           ultimately have "interior (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = {}"
             by (auto simp: eq)
           then show "content (x \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) = 0"
             using p'(4)[OF \<open>(x', x) \<in> \<D>\<close>] by (auto simp: interval_doublesplit[OF k] content_eq_0_interior simp del: interior_Int)
         qed (insert p'(1), auto intro!: sum.mono_neutral_right)
         also have "\<dots> \<le> norm (\<Sum>l\<in>(\<lambda>l. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})`{l\<in>snd ` \<D>. l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d} \<noteq> {}}. content l *\<^sub>R 1::real)"
           by simp
         also have "\<dots> \<le> 1 * content (cbox a b \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})"
           using division_doublesplit[OF p'' k, unfolded interval_doublesplit[OF k]]
           unfolding interval_doublesplit[OF k] by (intro dsum_bound) auto
         also have "\<dots> < e"
           using d by simp
         finally show "(\<Sum>K\<in>snd ` \<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) < e" .
       qed
       finally show "(\<Sum>(x, K)\<in>\<D>. content (K \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d}) * ?i x) < e" .
     qed
     then show "\<bar>\<Sum>(x, K)\<in>\<D>. content K * ?i x\<bar> < e"
       unfolding *  by (simp add: sum_nonneg split: prod.split)
   qed
 qed
 
 corollary negligible_standard_hyperplane_cart:
   fixes k :: "'a::finite"
   shows "negligible {x. x$k = (0::real)}"
   by (simp add: cart_eq_inner_axis negligible_standard_hyperplane)
 
 
 subsubsection \<open>Hence the main theorem about negligible sets\<close>
 
 
 lemma has_integral_negligible_cbox:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes negs: "negligible S"
     and 0: "\<And>x. x \<notin> S \<Longrightarrow> f x = 0"
   shows "(f has_integral 0) (cbox a b)"
   unfolding has_integral
 proof clarify
   fix e::real
   assume "e > 0"
   then have nn_gt0: "e/2 / ((real n+1) * (2 ^ n)) > 0" for n
     by simp
   then have "\<exists>\<gamma>. gauge \<gamma> \<and>
                  (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> \<longrightarrow>
                       \<bar>\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R indicator S x\<bar>
                       < e/2 / ((real n + 1) * 2 ^ n))" for n
     using negs [unfolded negligible_def has_integral] by auto
   then obtain \<gamma> where 
     gd: "\<And>n. gauge (\<gamma> n)"
     and \<gamma>: "\<And>n \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; \<gamma> n fine \<D>\<rbrakk>
                   \<Longrightarrow> \<bar>\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R indicator S x\<bar> < e/2 / ((real n + 1) * 2 ^ n)"
     by metis
   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 f x) - 0) < e)"
   proof (intro exI, safe)
     show "gauge (\<lambda>x. \<gamma> (nat \<lfloor>norm (f x)\<rfloor>) x)"
       using gd by (auto simp: gauge_def)
 
     show "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - 0) < e"
       if "\<D> tagged_division_of (cbox a b)" "(\<lambda>x. \<gamma> (nat \<lfloor>norm (f x)\<rfloor>) x) fine \<D>" for \<D>
     proof (cases "\<D> = {}")
       case True with \<open>0 < e\<close> show ?thesis by simp
     next
       case False
       obtain N where "Max ((\<lambda>(x, K). norm (f x)) ` \<D>) \<le> real N"
         using real_arch_simple by blast
       then have N: "\<And>x. x \<in> (\<lambda>(x, K). norm (f x)) ` \<D> \<Longrightarrow> x \<le> real N"
         by (meson Max_ge that(1) dual_order.trans finite_imageI tagged_division_of_finite)
       have "\<forall>i. \<exists>q. q tagged_division_of (cbox a b) \<and> (\<gamma> i) fine q \<and> (\<forall>(x,K) \<in> \<D>. K \<subseteq> (\<gamma> i) x \<longrightarrow> (x, K) \<in> q)"
         by (auto intro: tagged_division_finer[OF that(1) gd])
       from choice[OF this] 
       obtain q where q: "\<And>n. q n tagged_division_of cbox a b"
                         "\<And>n. \<gamma> n fine q n"
                         "\<And>n x K. \<lbrakk>(x, K) \<in> \<D>; K \<subseteq> \<gamma> n x\<rbrakk> \<Longrightarrow> (x, K) \<in> q n"
         by fastforce
       have "finite \<D>"
         using that(1) by blast
       then have sum_le_inc: "\<lbrakk>finite T; \<And>x y. (x,y) \<in> T \<Longrightarrow> (0::real) \<le> g(x,y);
                       \<And>y. y\<in>\<D> \<Longrightarrow> \<exists>x. (x,y) \<in> T \<and> f(y) \<le> g(x,y)\<rbrakk> \<Longrightarrow> sum f \<D> \<le> sum g T" for f g T
         by (rule sum_le_included[of \<D> T g snd f]; force)
       have "norm (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) \<le> (\<Sum>(x,K) \<in> \<D>. norm (content K *\<^sub>R f x))"
         unfolding split_def by (rule norm_sum)
       also have "... \<le> (\<Sum>(i, j) \<in> Sigma {..N + 1} q.
                           (real i + 1) * (case j of (x, K) \<Rightarrow> content K *\<^sub>R indicator S x))"
       proof (rule sum_le_inc, safe)
         show "finite (Sigma {..N+1} q)"
           by (meson finite_SigmaI finite_atMost tagged_division_of_finite q(1)) 
       next
         fix x K
         assume xk: "(x, K) \<in> \<D>"
         define n where "n = nat \<lfloor>norm (f x)\<rfloor>"
         have *: "norm (f x) \<in> (\<lambda>(x, K). norm (f x)) ` \<D>"
           using xk by auto
         have nfx: "real n \<le> norm (f x)" "norm (f x) \<le> real n + 1"
           unfolding n_def by auto
         then have "n \<in> {0..N + 1}"
           using N[OF *] by auto
         moreover have "K \<subseteq> \<gamma> (nat \<lfloor>norm (f x)\<rfloor>) x"
           using that(2) xk by auto
         moreover then have "(x, K) \<in> q (nat \<lfloor>norm (f x)\<rfloor>)"
           by (simp add: q(3) xk)
         moreover then have "(x, K) \<in> q n"
           using n_def by blast
         moreover
         have "norm (content K *\<^sub>R f x) \<le> (real n + 1) * (content K * indicator S x)"
         proof (cases "x \<in> S")
           case False
           then show ?thesis by (simp add: 0)
         next
           case True
           have *: "content K \<ge> 0"
             using tagged_division_ofD(4)[OF that(1) xk] by auto
           moreover have "content K * norm (f x) \<le> content K * (real n + 1)"
             by (simp add: mult_left_mono nfx(2))
           ultimately show ?thesis
             using nfx True by (auto simp: field_simps)
         qed
         ultimately show "\<exists>y. (y, x, K) \<in> (Sigma {..N + 1} q) \<and> norm (content K *\<^sub>R f x) \<le>
           (real y + 1) * (content K *\<^sub>R indicator S x)"
           by force
       qed auto
       also have "... = (\<Sum>i\<le>N + 1. \<Sum>j\<in>q i. (real i + 1) * (case j of (x, K) \<Rightarrow> content K *\<^sub>R indicator S x))"
         using q(1) by (intro sum_Sigma_product [symmetric]) auto
       also have "... \<le> (\<Sum>i\<le>N + 1. (real i + 1) * \<bar>\<Sum>(x,K) \<in> q i. content K *\<^sub>R indicator S x\<bar>)"
         by (rule sum_mono) (simp add: sum_distrib_left [symmetric])
       also have "... \<le> (\<Sum>i\<le>N + 1. e/2/2 ^ i)"
       proof (rule sum_mono)
         show "(real i + 1) * \<bar>\<Sum>(x,K) \<in> q i. content K *\<^sub>R indicator S x\<bar> \<le> e/2/2 ^ i"
           if "i \<in> {..N + 1}" for i
           using \<gamma>[of "q i" i] q by (simp add: divide_simps mult.left_commute)
       qed
       also have "... = e/2 * (\<Sum>i\<le>N + 1. (1/2) ^ i)"
         unfolding sum_distrib_left by (metis divide_inverse inverse_eq_divide power_one_over)
       also have "\<dots> < e/2 * 2"
       proof (rule mult_strict_left_mono)
         have "sum (power (1/2)) {..N + 1} = sum (power (1/2::real)) {..<N + 2}"
           using lessThan_Suc_atMost by auto
         also have "... < 2"
           by (auto simp: geometric_sum)
         finally show "sum (power (1/2::real)) {..N + 1} < 2" .
       qed (use \<open>0 < e\<close> in auto)
       finally  show ?thesis by auto
     qed
   qed
 qed
 
 
 proposition has_integral_negligible:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes negs: "negligible S"
     and "\<And>x. x \<in> (T - S) \<Longrightarrow> f x = 0"
   shows "(f has_integral 0) T"
 proof (cases "\<exists>a b. T = cbox a b")
   case True
   then have "((\<lambda>x. if x \<in> T then f x else 0) has_integral 0) T"
     using assms by (auto intro!: has_integral_negligible_cbox)
   then show ?thesis
     by (rule has_integral_eq [rotated]) auto
 next
   case False
   let ?f = "(\<lambda>x. if x \<in> T then f x else 0)"
   have "((\<lambda>x. if x \<in> T then f x else 0) has_integral 0) T"
     apply (auto simp: False has_integral_alt [of ?f])
     apply (rule_tac x=1 in exI, auto)
     apply (rule_tac x=0 in exI, simp add: has_integral_negligible_cbox [OF negs] assms)
     done
   then show ?thesis
     by (rule_tac f="?f" in has_integral_eq) auto
 qed
 
 lemma
   assumes "negligible S"
   shows integrable_negligible: "f integrable_on S" and integral_negligible: "integral S f = 0"
   using has_integral_negligible [OF assms]
   by (auto simp: has_integral_iff)
 
 lemma has_integral_spike:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "negligible S"
     and gf: "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
     and fint: "(f has_integral y) T"
   shows "(g has_integral y) T"
 proof -
   have *: "(g has_integral y) (cbox a b)"
        if "(f has_integral y) (cbox a b)" "\<forall>x \<in> cbox a b - S. g x = f x" for a b f and g:: "'b \<Rightarrow> 'a" and y
   proof -
     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) (cbox a b)"
       using that by (intro has_integral_add has_integral_negligible) (auto intro!: \<open>negligible S\<close>)
     then show ?thesis
       by auto
   qed
   have \<section>: "\<And>a b z. \<lbrakk>\<And>x. x \<in> T \<and> x \<notin> S \<Longrightarrow> g x = f x;
                      ((\<lambda>x. if x \<in> T then f x else 0) has_integral z) (cbox a b)\<rbrakk>
                     \<Longrightarrow> ((\<lambda>x. if x \<in> T then g x else 0) has_integral z) (cbox a b)"
       by (auto dest!: *[where f="\<lambda>x. if x\<in>T then f x else 0" and g="\<lambda>x. if x \<in> T then g x else 0"])
   show ?thesis
     using fint gf
     apply (subst has_integral_alt)
     apply (subst (asm) has_integral_alt)
     apply (auto split: if_split_asm)
      apply (blast dest: *)
     using \<section> by meson
 qed
 
 lemma has_integral_spike_eq:
   assumes "negligible S"
     and gf: "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
   shows "(f has_integral y) T \<longleftrightarrow> (g has_integral y) T"
     using has_integral_spike [OF \<open>negligible S\<close>] gf
     by metis
 
 lemma integrable_spike:
   assumes "f integrable_on T" "negligible S" "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
     shows "g integrable_on T"
   using assms unfolding integrable_on_def by (blast intro: has_integral_spike)
 
 lemma integral_spike:
   assumes "negligible S"
     and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
   shows "integral T f = integral T g"
   using has_integral_spike_eq[OF assms]
     by (auto simp: integral_def integrable_on_def)
 
 
 subsection \<open>Some other trivialities about negligible sets\<close>
 
 lemma negligible_subset:
   assumes "negligible s" "t \<subseteq> s"
   shows "negligible t"
   unfolding negligible_def
     by (metis (no_types) Diff_iff assms contra_subsetD has_integral_negligible indicator_simps(2))
 
 lemma negligible_diff[intro?]:
   assumes "negligible s"
   shows "negligible (s - t)"
   using assms by (meson Diff_subset negligible_subset)
 
 lemma negligible_Int:
   assumes "negligible s \<or> negligible t"
   shows "negligible (s \<inter> t)"
   using assms negligible_subset by force
 
 lemma negligible_Un:
   assumes "negligible S" and T: "negligible T"
   shows "negligible (S \<union> T)"
 proof -
   have "(indicat_real (S \<union> T) has_integral 0) (cbox a b)"
     if S0: "(indicat_real S has_integral 0) (cbox a b)" 
       and  "(indicat_real T has_integral 0) (cbox a b)" for a b
   proof (subst has_integral_spike_eq[OF T])
     show "indicat_real S x = indicat_real (S \<union> T) x" if "x \<in> cbox a b - T" for x
       by (metis Diff_iff Un_iff indicator_def that)
     show "(indicat_real S has_integral 0) (cbox a b)"
       by (simp add: S0)
   qed
   with assms show ?thesis
     unfolding negligible_def by blast
 qed
 
 lemma negligible_Un_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> negligible s \<and> negligible t"
   using negligible_Un negligible_subset by blast
 
 lemma negligible_sing[intro]: "negligible {a::'a::euclidean_space}"
   using negligible_standard_hyperplane[OF SOME_Basis, of "a \<bullet> (SOME i. i \<in> Basis)"] negligible_subset by blast
 
 lemma negligible_insert[simp]: "negligible (insert a s) \<longleftrightarrow> negligible s"
   by (metis insert_is_Un negligible_Un_eq negligible_sing)
 
 lemma negligible_empty[iff]: "negligible {}"
   using negligible_insert by blast
 
 text\<open>Useful in this form for backchaining\<close>
 lemma empty_imp_negligible: "S = {} \<Longrightarrow> negligible S"
   by simp
 
 lemma negligible_finite[intro]:
   assumes "finite s"
   shows "negligible s"
   using assms by (induct s) auto
 
 lemma negligible_Union[intro]:
   assumes "finite \<T>"
     and "\<And>t. t \<in> \<T> \<Longrightarrow> negligible t"
   shows "negligible(\<Union>\<T>)"
   using assms by induct auto
 
 lemma negligible: "negligible S \<longleftrightarrow> (\<forall>T. (indicat_real S has_integral 0) T)"
 proof (intro iffI allI)
   fix T
   assume "negligible S"
   then show "(indicator S has_integral 0) T"
     by (meson Diff_iff has_integral_negligible indicator_simps(2))
 qed (simp add: negligible_def)
 
 subsection \<open>Finite case of the spike theorem is quite commonly needed\<close>
 
 lemma has_integral_spike_finite:
   assumes "finite S"
     and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
     and "(f has_integral y) T"
   shows "(g has_integral y) T"
   using assms has_integral_spike negligible_finite by blast
 
 lemma has_integral_spike_finite_eq:
   assumes "finite S"
     and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
   shows "((f has_integral y) T \<longleftrightarrow> (g has_integral y) T)"
   by (metis assms has_integral_spike_finite)
 
 lemma integrable_spike_finite:
   assumes "finite S"
     and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"
     and "f integrable_on T"
   shows "g integrable_on T"
   using assms has_integral_spike_finite by blast
 
 lemma has_integral_bound_spike_finite:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B" "finite S"
       and f: "(f has_integral i) (cbox a b)"
       and leB: "\<And>x. x \<in> cbox a b - S \<Longrightarrow> norm (f x) \<le> B"
     shows "norm i \<le> B * content (cbox a b)"
 proof -
   define g where "g \<equiv> (\<lambda>x. if x \<in> S then 0 else f x)"
   then have "\<And>x. x \<in> cbox a b - S \<Longrightarrow> norm (g x) \<le> B"
     using leB by simp
   moreover have "(g has_integral i) (cbox a b)"
     using has_integral_spike_finite [OF \<open>finite S\<close> _ f]
     by (simp add: g_def)
   ultimately show ?thesis
     by (simp add: \<open>0 \<le> B\<close> g_def has_integral_bound)
 qed
 
 corollary has_integral_bound_real:
   fixes f :: "real \<Rightarrow> 'b::real_normed_vector"
   assumes "0 \<le> B" "finite S"
       and "(f has_integral i) {a..b}"
       and "\<And>x. x \<in> {a..b} - S \<Longrightarrow> norm (f x) \<le> B"
     shows "norm i \<le> B * content {a..b}"
   by (metis assms box_real(2) has_integral_bound_spike_finite)
 
 
 subsection \<open>In particular, the boundary of an interval is negligible\<close>
 
 lemma negligible_frontier_interval: "negligible(cbox (a::'a::euclidean_space) b - box a b)"
 proof -
   let ?A = "\<Union>((\<lambda>k. {x. x\<bullet>k = a\<bullet>k} \<union> {x::'a. x\<bullet>k = b\<bullet>k}) ` Basis)"
   have "negligible ?A"
     by (force simp add: negligible_Union[OF finite_imageI])
   moreover have "cbox a b - box a b \<subseteq> ?A"
     by (force simp add: mem_box)
   ultimately show ?thesis
     by (rule negligible_subset)
 qed
 
 lemma has_integral_spike_interior:
   assumes f: "(f has_integral y) (cbox a b)" and gf: "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x"
   shows "(g has_integral y) (cbox a b)"
   by (meson Diff_iff gf has_integral_spike[OF negligible_frontier_interval _ f])
   
 lemma has_integral_spike_interior_eq:
   assumes "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x"
   shows "(f has_integral y) (cbox a b) \<longleftrightarrow> (g has_integral y) (cbox a b)"
   by (metis assms has_integral_spike_interior)
 
 lemma integrable_spike_interior:
   assumes "\<And>x. x \<in> box a b \<Longrightarrow> g x = f x"
     and "f integrable_on cbox a b"
   shows "g integrable_on cbox a b"
   using assms has_integral_spike_interior_eq by blast
 
 
 subsection \<open>Integrability of continuous functions\<close>
 
 lemma operative_approximableI:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "0 \<le> e"
   shows "operative conj True (\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i)"
 proof -
   interpret comm_monoid conj True
     by standard auto
   show ?thesis
   proof (standard, safe)
     fix a b :: 'b
     show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
       if "box a b = {}" for a b
       using assms that
       by (metis content_eq_0_interior integrable_on_null interior_cbox norm_zero right_minus_eq)
     {
       fix c g and k :: 'b
       assume fg: "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" and g: "g integrable_on cbox a b"
       assume k: "k \<in> Basis"
       show "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
            "\<exists>g. (\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}"
         using fg g k by auto
     }
     show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
       if fg1: "\<forall>x\<in>cbox a b \<inter> {x. x \<bullet> k \<le> c}. norm (f x - g1 x) \<le> e" 
         and g1: "g1 integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}"
         and fg2: "\<forall>x\<in>cbox a b \<inter> {x. c \<le> x \<bullet> k}. norm (f x - g2 x) \<le> e" 
         and g2: "g2 integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k}" 
         and k: "k \<in> Basis"
       for c k g1 g2
     proof -
       let ?g = "\<lambda>x. if x\<bullet>k = c then f x else if x\<bullet>k \<le> c then g1 x else g2 x"
       show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
       proof (intro exI conjI ballI)
         show "norm (f x - ?g x) \<le> e" if "x \<in> cbox a b" for x
           by (auto simp: that assms fg1 fg2)
         show "?g integrable_on cbox a b"
         proof -
           have "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c}" "?g integrable_on cbox a b \<inter> {x. x \<bullet> k \<ge> c}"
             by(rule integrable_spike[OF _ negligible_standard_hyperplane[of k c]], use k g1 g2 in auto)+
           with has_integral_split[OF _ _ k] show ?thesis
             unfolding integrable_on_def by blast
         qed
       qed
     qed
   qed
 qed
 
 lemma comm_monoid_set_F_and: "comm_monoid_set.F (\<and>) True f s \<longleftrightarrow> (finite s \<longrightarrow> (\<forall>x\<in>s. f x))"
 proof -
   interpret bool: comm_monoid_set \<open>(\<and>)\<close> True ..
   show ?thesis
     by (induction s rule: infinite_finite_induct) auto
 qed
 
 lemma approximable_on_division:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "0 \<le> e"
     and d: "d division_of (cbox a b)"
     and f: "\<forall>i\<in>d. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
   obtains g where "\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e" "g integrable_on cbox a b"
 proof -
   interpret operative conj True "\<lambda>i. \<exists>g. (\<forall>x\<in>i. norm (f x - g (x::'b)) \<le> e) \<and> g integrable_on i"
     using \<open>0 \<le> e\<close> by (rule operative_approximableI)
   from f local.division [OF d] that show thesis
     by auto
 qed
 
 lemma integrable_continuous:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "continuous_on (cbox a b) f"
   shows "f integrable_on cbox a b"
 proof (rule integrable_uniform_limit)
   fix e :: real
   assume e: "e > 0"
   then obtain d where "0 < d" and d: "\<And>x x'. \<lbrakk>x \<in> cbox a b; x' \<in> cbox a b; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
     using compact_uniformly_continuous[OF assms compact_cbox] unfolding uniformly_continuous_on_def by metis
   obtain p where ptag: "p tagged_division_of cbox a b" and finep: "(\<lambda>x. ball x d) fine p"
     using fine_division_exists[OF gauge_ball[OF \<open>0 < d\<close>], of a b] .
   have *: "\<forall>i\<in>snd ` p. \<exists>g. (\<forall>x\<in>i. norm (f x - g x) \<le> e) \<and> g integrable_on i"
   proof (safe, unfold snd_conv)
     fix x l
     assume as: "(x, l) \<in> p"
     obtain a b where l: "l = cbox a b"
       using as ptag by blast
     then have x: "x \<in> cbox a b"
       using as ptag by auto
     show "\<exists>g. (\<forall>x\<in>l. norm (f x - g x) \<le> e) \<and> g integrable_on l"
     proof (intro exI conjI strip)
       show "(\<lambda>y. f x) integrable_on l"
         unfolding integrable_on_def l by blast
     next
       fix y
       assume y: "y \<in> l"
       then have "y \<in> ball x d"
         using as finep by fastforce
       then show "norm (f y - f x) \<le> e"
         using d x y as l
         by (metis dist_commute dist_norm less_imp_le mem_ball ptag subsetCE tagged_division_ofD(3))
     qed
   qed
   from e have "e \<ge> 0"
     by auto
   from approximable_on_division[OF this division_of_tagged_division[OF ptag] *]
   show "\<exists>g. (\<forall>x\<in>cbox a b. norm (f x - g x) \<le> e) \<and> g integrable_on cbox a b"
     by metis
 qed
 
 lemma integrable_continuous_interval:
   fixes f :: "'b::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "continuous_on {a..b} f"
   shows "f integrable_on {a..b}"
   by (metis assms integrable_continuous interval_cbox)
 
 lemmas integrable_continuous_real = integrable_continuous_interval[where 'b=real]
 
 lemma integrable_continuous_closed_segment:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "continuous_on (closed_segment a b) f"
   shows "f integrable_on (closed_segment a b)"
   using assms
   by (auto intro!: integrable_continuous_interval simp: closed_segment_eq_real_ivl)
 
 
 subsection \<open>Specialization of additivity to one dimension\<close>
 
 
 subsection \<open>A useful lemma allowing us to factor out the content size\<close>
 
 lemma has_integral_factor_content:
   "(f has_integral i) (cbox a b) \<longleftrightarrow>
     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
       norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content (cbox a b)))"
 proof (cases "content (cbox a b) = 0")
   case True
   have "\<And>e p. p tagged_division_of cbox a b \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)) \<le> e * content (cbox a b)"
     unfolding sum_content_null[OF True] True by force
   moreover have "i = 0" 
     if "\<And>e. e > 0 \<Longrightarrow> \<exists>d. gauge d \<and>
               (\<forall>p. p tagged_division_of cbox a b \<and>
                    d fine p \<longrightarrow>
                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) \<le> e * content (cbox a b))"
     using that [of 1]
     by (force simp add: True sum_content_null[OF True] intro: fine_division_exists[of _ a b])
   ultimately show ?thesis
     unfolding has_integral_null_eq[OF True]
     by (force simp add: )
 next
   case False
   then have F: "0 < content (cbox a b)"
     using zero_less_measure_iff by blast
   let ?P = "\<lambda>e opp. \<exists>d. gauge d \<and>
     (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow> opp (norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i)) e)"
   show ?thesis
   proof (subst has_integral, safe)
     fix e :: real
     assume e: "e > 0"
     show "?P (e * content (cbox a b)) (\<le>)" if \<section>[rule_format]: "\<forall>\<epsilon>>0. ?P \<epsilon> (<)"
       using \<section> [of "e * content (cbox a b)"]
       by (meson F e less_imp_le mult_pos_pos)
     show "?P e (<)" if \<section>[rule_format]:  "\<forall>\<epsilon>>0. ?P (\<epsilon> * content (cbox a b)) (\<le>)"
       using \<section> [of "e/2 / content (cbox a b)"]
         using F e by (force simp add: algebra_simps)
   qed
 qed
 
 lemma has_integral_factor_content_real:
   "(f has_integral i) {a..b::real} \<longleftrightarrow>
     (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b}  \<and> d fine p \<longrightarrow>
       norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) p - i) \<le> e * content {a..b} ))"
   unfolding box_real[symmetric]
   by (rule has_integral_factor_content)
 
 
 subsection \<open>Fundamental theorem of calculus\<close>
 
 lemma interval_bounds_real:
   fixes q b :: real
   assumes "a \<le> b"
   shows "Sup {a..b} = b"
     and "Inf {a..b} = a"
   using assms by auto
 
 theorem fundamental_theorem_of_calculus:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "a \<le> b" 
       and vecd: "\<And>x. x \<in> {a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x within {a..b})"
   shows "(f' has_integral (f b - f a)) {a..b}"
   unfolding has_integral_factor_content box_real[symmetric]
 proof safe
   fix e :: real
   assume "e > 0"
   then have "\<forall>x. \<exists>d>0. x \<in> {a..b} \<longrightarrow>
          (\<forall>y\<in>{a..b}. norm (y-x) < d \<longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e * norm (y-x))"
     using vecd[unfolded has_vector_derivative_def has_derivative_within_alt] by blast
   then obtain d where d: "\<And>x. 0 < d x"
                  "\<And>x y. \<lbrakk>x \<in> {a..b}; y \<in> {a..b}; norm (y-x) < d x\<rbrakk>
                         \<Longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e * norm (y-x)"
     by metis  
   show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
     norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b))"
   proof (rule exI, safe)
     show "gauge (\<lambda>x. ball x (d x))"
       using d(1) gauge_ball_dependent by blast
   next
     fix p
     assume ptag: "p tagged_division_of cbox a b" and finep: "(\<lambda>x. ball x (d x)) fine p"
     have ba: "b - a = (\<Sum>(x,K)\<in>p. Sup K - Inf K)" "f b - f a = (\<Sum>(x,K)\<in>p. f(Sup K) - f(Inf K))"
       using additive_tagged_division_1[where f= "\<lambda>x. x"] additive_tagged_division_1[where f= f]
             \<open>a \<le> b\<close> ptag by auto
     have "norm (\<Sum>(x, K) \<in> p. (content K *\<^sub>R f' x) - (f (Sup K) - f (Inf K)))
           \<le> (\<Sum>n\<in>p. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))"
     proof (rule sum_norm_le,safe)
       fix x K
       assume "(x, K) \<in> p"
       then have "x \<in> K" and kab: "K \<subseteq> cbox a b"
         using ptag by blast+
       then obtain u v where k: "K = cbox u v" and "x \<in> K" and kab: "K \<subseteq> cbox a b"
         using ptag \<open>(x, K) \<in> p\<close> by auto 
       have "u \<le> v"
         using \<open>x \<in> K\<close> unfolding k by auto
       have ball: "\<forall>y\<in>K. y \<in> ball x (d x)"
         using finep \<open>(x, K) \<in> p\<close> by blast
       have "u \<in> K" "v \<in> K"
         by (simp_all add: \<open>u \<le> v\<close> k)
       have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) = norm (f u - f x - (u - x) *\<^sub>R f' x - (f v - f x - (v - x) *\<^sub>R f' x))"
         by (auto simp add: algebra_simps)
       also have "... \<le> norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)"
         by (rule norm_triangle_ineq4)
       finally have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) \<le>
         norm (f u - f x - (u - x) *\<^sub>R f' x) + norm (f v - f x - (v - x) *\<^sub>R f' x)" .
       also have "\<dots> \<le> e * norm (u - x) + e * norm (v - x)"
       proof (rule add_mono)
         show "norm (f u - f x - (u - x) *\<^sub>R f' x) \<le> e * norm (u - x)"
         proof (rule d)
           show "norm (u - x) < d x"
             using \<open>u \<in> K\<close> ball by (auto simp add: dist_real_def)
         qed (use \<open>x \<in> K\<close> \<open>u \<in> K\<close> kab in auto)
         show "norm (f v - f x - (v - x) *\<^sub>R f' x) \<le> e * norm (v - x)"
         proof (rule d)
           show "norm (v - x) < d x"
             using \<open>v \<in> K\<close> ball by (auto simp add: dist_real_def)
         qed (use \<open>x \<in> K\<close> \<open>v \<in> K\<close> kab in auto)
       qed
       also have "\<dots> \<le> e * (Sup K - Inf K)"
         using \<open>x \<in> K\<close> by (auto simp: k interval_bounds_real[OF \<open>u \<le> v\<close>] field_simps)
       finally show "norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \<le> e * (Sup K - Inf K)"
         using \<open>u \<le> v\<close> by (simp add: k)
     qed
     with \<open>a \<le> b\<close> show "norm ((\<Sum>(x, K)\<in>p. content K *\<^sub>R f' x) - (f b - f a)) \<le> e * content (cbox a b)"
       by (auto simp: ba split_def sum_subtractf [symmetric] sum_distrib_left)
   qed
 qed
 
 lemma ident_has_integral:
   fixes a::real
   assumes "a \<le> b"
   shows "((\<lambda>x. x) has_integral (b\<^sup>2 - a\<^sup>2)/2) {a..b}"
 proof -
   have "((\<lambda>x. x) has_integral inverse 2 * b\<^sup>2 - inverse 2 * a\<^sup>2) {a..b}"
     unfolding power2_eq_square
     by (rule fundamental_theorem_of_calculus [OF assms] derivative_eq_intros | simp)+
   then show ?thesis
     by (simp add: field_simps)
 qed
 
 lemma integral_ident [simp]:
   fixes a::real
   assumes "a \<le> b"
   shows "integral {a..b} (\<lambda>x. x) = (if a \<le> b then (b\<^sup>2 - a\<^sup>2)/2 else 0)"
   by (metis assms ident_has_integral integral_unique)
 
 lemma ident_integrable_on:
   fixes a::real
   shows "(\<lambda>x. x) integrable_on {a..b}"
 by (metis atLeastatMost_empty_iff integrable_on_def has_integral_empty ident_has_integral)
 
 lemma integral_sin [simp]:
   fixes a::real
   assumes "a \<le> b" shows "integral {a..b} sin = cos a - cos b"
 proof -
   have "(sin has_integral (- cos b - - cos a)) {a..b}"
   proof (rule fundamental_theorem_of_calculus)
     show "((\<lambda>a. - cos a) has_vector_derivative sin x) (at x within {a..b})" for x
       unfolding has_field_derivative_iff_has_vector_derivative [symmetric]
       by (rule derivative_eq_intros | force)+
   qed (use assms in auto)
   then show ?thesis
     by (simp add: integral_unique)
 qed
 
 lemma integral_cos [simp]:
   fixes a::real
   assumes "a \<le> b" shows "integral {a..b} cos = sin b - sin a"
 proof -
   have "(cos has_integral (sin b - sin a)) {a..b}"
   proof (rule fundamental_theorem_of_calculus)
     show "(sin has_vector_derivative cos x) (at x within {a..b})" for x
       unfolding has_field_derivative_iff_has_vector_derivative [symmetric]
       by (rule derivative_eq_intros | force)+
   qed (use assms in auto)
   then show ?thesis
     by (simp add: integral_unique)
 qed
 
 lemma has_integral_sin_nx: "((\<lambda>x. sin(real_of_int n * x)) has_integral 0) {-pi..pi}"
 proof (cases "n = 0")
   case False
   have "((\<lambda>x. sin (n * x)) has_integral (- cos (n * pi)/n - - cos (n * - pi)/n)) {-pi..pi}"
   proof (rule fundamental_theorem_of_calculus)
     show "((\<lambda>x. - cos (n * x) / n) has_vector_derivative sin (n * a)) (at a within {-pi..pi})"
       if "a \<in> {-pi..pi}" for a :: real
       using that False
       unfolding has_vector_derivative_def
       by (intro derivative_eq_intros | force)+
   qed auto
   then show ?thesis
     by simp
 qed auto
 
 lemma integral_sin_nx:
    "integral {-pi..pi} (\<lambda>x. sin(x * real_of_int n)) = 0"
   using has_integral_sin_nx [of n] by (force simp: mult.commute)
 
 lemma has_integral_cos_nx:
   "((\<lambda>x. cos(real_of_int n * x)) has_integral (if n = 0 then 2 * pi else 0)) {-pi..pi}"
 proof (cases "n = 0")
   case True
   then show ?thesis
     using has_integral_const_real [of "1::real" "-pi" pi] by auto
 next
   case False
   have "((\<lambda>x. cos (n * x)) has_integral (sin (n * pi)/n - sin (n * - pi)/n)) {-pi..pi}"
   proof (rule fundamental_theorem_of_calculus)
     show "((\<lambda>x. sin (n * x) / n) has_vector_derivative cos (n * x)) (at x within {-pi..pi})"
       if "x \<in> {-pi..pi}"
       for x :: real
       using that False
       unfolding has_vector_derivative_def
       by (intro derivative_eq_intros | force)+
   qed auto
   with False show ?thesis
     by (simp add: mult.commute)
 qed
 
 lemma integral_cos_nx:
    "integral {-pi..pi} (\<lambda>x. cos(x * real_of_int n)) = (if n = 0 then 2 * pi else 0)"
   using has_integral_cos_nx [of n] by (force simp: mult.commute)
 
 
 subsection \<open>Taylor series expansion\<close>
 
 lemma mvt_integral:
   fixes f::"'a::real_normed_vector\<Rightarrow>'b::banach"
   assumes f'[derivative_intros]:
     "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"
   assumes line_in: "\<And>t. t \<in> {0..1} \<Longrightarrow> x + t *\<^sub>R y \<in> S"
   shows "f (x + y) - f x = integral {0..1} (\<lambda>t. f' (x + t *\<^sub>R y) y)" (is ?th1)
 proof -
   from assms have subset: "(\<lambda>xa. x + xa *\<^sub>R y) ` {0..1} \<subseteq> S" by auto
   note [derivative_intros] =
     has_derivative_subset[OF _ subset]
     has_derivative_in_compose[where f="(\<lambda>xa. x + xa *\<^sub>R y)" and g = f]
   note [continuous_intros] =
     continuous_on_compose2[where f="(\<lambda>xa. x + xa *\<^sub>R y)"]
     continuous_on_subset[OF _ subset]
   have "\<And>t. t \<in> {0..1} \<Longrightarrow>
     ((\<lambda>t. f (x + t *\<^sub>R y)) has_vector_derivative f' (x + t *\<^sub>R y) y)
     (at t within {0..1})"
     using assms
     by (auto simp: has_vector_derivative_def
         linear_cmul[OF has_derivative_linear[OF f'], symmetric]
       intro!: derivative_eq_intros)
   from fundamental_theorem_of_calculus[rule_format, OF _ this]
   show ?th1
     by (auto intro!: integral_unique[symmetric])
 qed
 
 lemma (in bounded_bilinear) sum_prod_derivatives_has_vector_derivative:
   assumes "p>0"
   and f0: "Df 0 = f"
   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})"
   and g0: "Dg 0 = g"
   and Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})"
   and ivl: "a \<le> t" "t \<le> b"
   shows "((\<lambda>t. \<Sum>i<p. (-1)^i *\<^sub>R prod (Df i t) (Dg (p - Suc i) t))
     has_vector_derivative
       prod (f t) (Dg p t) - (-1)^p *\<^sub>R prod (Df p t) (g t))
     (at t within {a..b})"
   using assms
 proof cases
   assume p: "p \<noteq> 1"
   define p' where "p' = p - 2"
   from assms p have p': "{..<p} = {..Suc p'}" "p = Suc (Suc p')"
     by (auto simp: p'_def)
   have *: "\<And>i. i \<le> p' \<Longrightarrow> Suc (Suc p' - i) = (Suc (Suc p') - i)"
     by auto
   let ?f = "\<lambda>i. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg ((p - i)) t))"
   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
     prod (Df (Suc i) t) (Dg (p - Suc i) t))) =
     (\<Sum>i\<le>(Suc p'). ?f i - ?f (Suc i))"
     by (auto simp: algebra_simps p'(2) numeral_2_eq_2 * lessThan_Suc_atMost)
   also note sum_telescope
   finally
   have "(\<Sum>i<p. (-1) ^ i *\<^sub>R (prod (Df i t) (Dg (Suc (p - Suc i)) t) +
     prod (Df (Suc i) t) (Dg (p - Suc i) t)))
     = prod (f t) (Dg p t) - (- 1) ^ p *\<^sub>R prod (Df p t) (g t)"
     unfolding p'[symmetric]
     by (simp add: assms)
   thus ?thesis
     using assms
     by (auto intro!: derivative_eq_intros has_vector_derivative)
 qed (auto intro!: derivative_eq_intros has_vector_derivative)
 
 lemma
   fixes f::"real\<Rightarrow>'a::banach"
   assumes "p>0"
   and f0: "Df 0 = f"
   and Df: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Df m has_vector_derivative Df (Suc m) t) (at t within {a..b})"
   and ivl: "a \<le> b"
   defines "i \<equiv> \<lambda>x. ((b - x) ^ (p - 1) / fact (p - 1)) *\<^sub>R Df p x"
   shows Taylor_has_integral:
     "(i has_integral f b - (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a)) {a..b}"
   and Taylor_integral:
     "f b = (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a) + integral {a..b} i"
   and Taylor_integrable:
     "i integrable_on {a..b}"
 proof goal_cases
   case 1
   interpret bounded_bilinear "scaleR::real\<Rightarrow>'a\<Rightarrow>'a"
     by (rule bounded_bilinear_scaleR)
   define g where "g s = (b - s)^(p - 1)/fact (p - 1)" for s
   define Dg where [abs_def]:
     "Dg n s = (if n < p then (-1)^n * (b - s)^(p - 1 - n) / fact (p - 1 - n) else 0)" for n s
   have g0: "Dg 0 = g"
     using \<open>p > 0\<close>
     by (auto simp add: Dg_def field_split_simps g_def split: if_split_asm)
   {
     fix m
     assume "p > Suc m"
     hence "p - Suc m = Suc (p - Suc (Suc m))"
       by auto
     hence "real (p - Suc m) * fact (p - Suc (Suc m)) = fact (p - Suc m)"
       by auto
   } note fact_eq = this
   have Dg: "\<And>m t. m < p \<Longrightarrow> a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (Dg m has_vector_derivative Dg (Suc m) t) (at t within {a..b})"
     unfolding Dg_def
     by (auto intro!: derivative_eq_intros simp: has_vector_derivative_def fact_eq field_split_simps)
   let ?sum = "\<lambda>t. \<Sum>i<p. (- 1) ^ i *\<^sub>R Dg i t *\<^sub>R Df (p - Suc i) t"
   from sum_prod_derivatives_has_vector_derivative[of _ Dg _ _ _ Df,
       OF \<open>p > 0\<close> g0 Dg f0 Df]
   have deriv: "\<And>t. a \<le> t \<Longrightarrow> t \<le> b \<Longrightarrow>
     (?sum has_vector_derivative
       g t *\<^sub>R Df p t - (- 1) ^ p *\<^sub>R Dg p t *\<^sub>R f t) (at t within {a..b})"
     by auto
   from fundamental_theorem_of_calculus[rule_format, OF \<open>a \<le> b\<close> deriv]
   have "(i has_integral ?sum b - ?sum a) {a..b}"
     using atLeastatMost_empty'[simp del]
     by (simp add: i_def g_def Dg_def)
   also
   have one: "(- 1) ^ p' * (- 1) ^ p' = (1::real)"
     and "{..<p} \<inter> {i. p = Suc i} = {p - 1}"
     for p'
     using \<open>p > 0\<close>
     by (auto simp: power_mult_distrib[symmetric])
   then have "?sum b = f b"
     using Suc_pred'[OF \<open>p > 0\<close>]
     by (simp add: diff_eq_eq Dg_def power_0_left le_Suc_eq if_distrib
         if_distribR sum.If_cases f0)
   also
   have "{..<p} = (\<lambda>x. p - x - 1) ` {..<p}"
   proof safe
     fix x
     assume "x < p"
     thus "x \<in> (\<lambda>x. p - x - 1) ` {..<p}"
       by (auto intro!: image_eqI[where x = "p - x - 1"])
   qed simp
   from _ this
   have "?sum a = (\<Sum>i<p. ((b-a) ^ i / fact i) *\<^sub>R Df i a)"
     by (rule sum.reindex_cong) (auto simp add: inj_on_def Dg_def one)
   finally show c: ?case .
   case 2 show ?case using c integral_unique
     by (metis (lifting) add.commute diff_eq_eq integral_unique)
   case 3 show ?case using c by force
 qed
 
 
 subsection \<open>Only need trivial subintervals if the interval itself is trivial\<close>
 
 proposition division_of_nontrivial:
   fixes \<D> :: "'a::euclidean_space set set"
   assumes sdiv: "\<D> division_of (cbox a b)"
      and cont0: "content (cbox a b) \<noteq> 0"
   shows "{k. k \<in> \<D> \<and> content k \<noteq> 0} division_of (cbox a b)"
   using sdiv
 proof (induction "card \<D>" arbitrary: \<D> rule: less_induct)
   case less
   note \<D> = division_ofD[OF less.prems]
   {
     presume *: "{k \<in> \<D>. content k \<noteq> 0} \<noteq> \<D> \<Longrightarrow> ?case"
     then show ?case
       using less.prems by fastforce
   }
   assume noteq: "{k \<in> \<D>. content k \<noteq> 0} \<noteq> \<D>"
   then obtain K c d where "K \<in> \<D>" and contk: "content K = 0" and keq: "K = cbox c d"
     using \<D>(4) by blast 
   then have "card \<D> > 0"
     unfolding card_gt_0_iff using less by auto
   then have card: "card (\<D> - {K}) < card \<D>"
     using less \<open>K \<in> \<D>\<close> by (simp add: \<D>(1))
   have closed: "closed (\<Union>(\<D> - {K}))"
     using less.prems by auto
   have "x islimpt \<Union>(\<D> - {K})" if "x \<in> K" for x 
     unfolding islimpt_approachable
   proof (intro allI impI)
     fix e::real
     assume "e > 0"
     obtain i where i: "c\<bullet>i = d\<bullet>i" "i\<in>Basis"
       using contk \<D>(3) [OF \<open>K \<in> \<D>\<close>] unfolding box_ne_empty keq
       by (meson content_eq_0 dual_order.antisym)
     then have xi: "x\<bullet>i = d\<bullet>i"
       using \<open>x \<in> K\<close> unfolding keq mem_box by (metis antisym)
     define y where "y = (\<Sum>j\<in>Basis. (if j = i then if c\<bullet>i \<le> (a\<bullet>i + b\<bullet>i)/2 then c\<bullet>i +
       min e (b\<bullet>i - c\<bullet>i)/2 else c\<bullet>i - min e (c\<bullet>i - a\<bullet>i)/2 else x\<bullet>j) *\<^sub>R j)"
     show "\<exists>x'\<in>\<Union>(\<D> - {K}). x' \<noteq> x \<and> dist x' x < e"
     proof (intro bexI conjI)
       have "d \<in> cbox c d"
         using \<D>(3)[OF \<open>K \<in> \<D>\<close>] by (simp add: box_ne_empty(1) keq mem_box(2))
       then have "d \<in> cbox a b"
         using \<D>(2)[OF \<open>K \<in> \<D>\<close>] by (auto simp: keq)
       then have di: "a \<bullet> i \<le> d \<bullet> i \<and> d \<bullet> i \<le> b \<bullet> i"
         using \<open>i \<in> Basis\<close> mem_box(2) by blast
       then have xyi: "y\<bullet>i \<noteq> x\<bullet>i"
         unfolding y_def i xi using \<open>e > 0\<close> cont0 \<open>i \<in> Basis\<close>
         by (auto simp: content_eq_0 elim!: ballE[of _ _ i])
       then show "y \<noteq> x"
         unfolding euclidean_eq_iff[where 'a='a] using i by auto
       have "norm (y-x) \<le> (\<Sum>b\<in>Basis. \<bar>(y - x) \<bullet> b\<bar>)"
         by (rule norm_le_l1)
       also have "... = \<bar>(y - x) \<bullet> i\<bar> + (\<Sum>b \<in> Basis - {i}. \<bar>(y - x) \<bullet> b\<bar>)"
         by (meson finite_Basis i(2) sum.remove)
       also have "... <  e + sum (\<lambda>i. 0) Basis"
       proof (rule add_less_le_mono)
         show "\<bar>(y-x) \<bullet> i\<bar> < e"
           using di \<open>e > 0\<close> y_def i xi by (auto simp: inner_simps)
         show "(\<Sum>i\<in>Basis - {i}. \<bar>(y-x) \<bullet> i\<bar>) \<le> (\<Sum>i\<in>Basis. 0)"
           unfolding y_def by (auto simp: inner_simps)
       qed 
       finally have "norm (y-x) < e + sum (\<lambda>i. 0) Basis" .
       then show "dist y x < e"
         unfolding dist_norm by auto
       have "y \<notin> K"
         unfolding keq mem_box using i(1) i(2) xi xyi by fastforce
       moreover have "y \<in> \<Union>\<D>"
         using subsetD[OF \<D>(2)[OF \<open>K \<in> \<D>\<close>] \<open>x \<in> K\<close>] \<open>e > 0\<close> di i
         by (auto simp: \<D> mem_box y_def field_simps elim!: ballE[of _ _ i])
       ultimately show "y \<in> \<Union>(\<D> - {K})" by auto
     qed
   qed
   then have "K \<subseteq> \<Union>(\<D> - {K})"
     using closed closed_limpt by blast
   then have "\<Union>(\<D> - {K}) = cbox a b"
     unfolding \<D>(6)[symmetric] by auto
   then have "\<D> - {K} division_of cbox a b"
     by (metis Diff_subset less.prems division_of_subset \<D>(6))
   then have "{ka \<in> \<D> - {K}. content ka \<noteq> 0} division_of (cbox a b)"
     using card less.hyps by blast
   moreover have "{ka \<in> \<D> - {K}. content ka \<noteq> 0} = {K \<in> \<D>. content K \<noteq> 0}"
     using contk by auto
   ultimately show ?case by auto
 qed
 
 
 subsection \<open>Integrability on subintervals\<close>
 
 lemma operative_integrableI:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "0 \<le> e"
   shows "operative conj True (\<lambda>i. f integrable_on i)"
 proof -
   interpret comm_monoid conj True
   proof qed
   show ?thesis
   proof
     show "\<And>a b. box a b = {} \<Longrightarrow> (f integrable_on cbox a b) = True"
       by (simp add: content_eq_0_interior integrable_on_null)
     show "\<And>a b c k.
              k \<in> Basis \<Longrightarrow>
              (f integrable_on cbox a b) \<longleftrightarrow>
              (f integrable_on cbox a b \<inter> {x. x \<bullet> k \<le> c} \<and> f integrable_on cbox a b \<inter> {x. c \<le> x \<bullet> k})"
       unfolding integrable_on_def by (auto intro!: has_integral_split)
   qed
 qed
 
 lemma integrable_subinterval:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on cbox a b"
     and cd: "cbox c d \<subseteq> cbox a b"
   shows "f integrable_on cbox c d"
 proof -
   interpret operative conj True "\<lambda>i. f integrable_on i"
     using order_refl by (rule operative_integrableI)
   show ?thesis
   proof (cases "cbox c d = {}")
     case True
     then show ?thesis
       using division [symmetric] f by (auto simp: comm_monoid_set_F_and)
   next
     case False
     then show ?thesis
       by (metis cd comm_monoid_set_F_and division division_of_finite f partial_division_extend_1)
   qed
 qed
 
 lemma integrable_subinterval_real:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}"
     and "{c..d} \<subseteq> {a..b}"
   shows "f integrable_on {c..d}"
   by (metis assms box_real(2) integrable_subinterval)
 
 subsection \<open>Combining adjacent intervals in 1 dimension\<close>
 
 lemma has_integral_combine:
   fixes a b c :: real and j :: "'a::banach"
   assumes "a \<le> c"
       and "c \<le> b"
       and ac: "(f has_integral i) {a..c}"
       and cb: "(f has_integral j) {c..b}"
   shows "(f has_integral (i + j)) {a..b}"
 proof -
   interpret operative_real "lift_option plus" "Some 0"
     "\<lambda>i. if f integrable_on i then Some (integral i f) else None"
     using operative_integralI by (rule operative_realI)
   from \<open>a \<le> c\<close> \<open>c \<le> b\<close> ac cb coalesce_less_eq
   have *: "lift_option (+)
              (if f integrable_on {a..c} then Some (integral {a..c} f) else None)
              (if f integrable_on {c..b} then Some (integral {c..b} f) else None) =
             (if f integrable_on {a..b} then Some (integral {a..b} f) else None)"
     by (auto simp: split: if_split_asm)
   then have "f integrable_on cbox a b"
     using ac cb by (auto split: if_split_asm)
   with *
   show ?thesis
     using ac cb by (auto simp add: integrable_on_def integral_unique split: if_split_asm)
 qed
 
 lemma integral_combine:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "a \<le> c"
     and "c \<le> b"
     and ab: "f integrable_on {a..b}"
   shows "integral {a..c} f + integral {c..b} f = integral {a..b} f"
 proof -
   have "(f has_integral integral {a..c} f) {a..c}"
     using ab \<open>c \<le> b\<close> integrable_subinterval_real by fastforce
   moreover
   have "(f has_integral integral {c..b} f) {c..b}"
     using ab \<open>a \<le> c\<close> integrable_subinterval_real by fastforce
   ultimately have "(f has_integral integral {a..c} f + integral {c..b} f) {a..b}"
     using \<open>a \<le> c\<close> \<open>c \<le> b\<close> has_integral_combine by blast
   then show ?thesis
     by (simp add: has_integral_integrable_integral)
 qed
 
 lemma integrable_combine:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "a \<le> c"
     and "c \<le> b"
     and "f integrable_on {a..c}"
     and "f integrable_on {c..b}"
   shows "f integrable_on {a..b}"
   using assms
   unfolding integrable_on_def
   by (auto intro!: has_integral_combine)
 
 lemma integral_minus_sets:
   fixes f::"real \<Rightarrow> 'a::banach"
   shows "c \<le> a \<Longrightarrow> c \<le> b \<Longrightarrow> f integrable_on {c .. max a b} \<Longrightarrow>
     integral {c .. a} f - integral {c .. b} f =
     (if a \<le> b then - integral {a .. b} f else integral {b .. a} f)"
   using integral_combine[of c a b f]  integral_combine[of c b a f]
   by (auto simp: algebra_simps max_def)
 
 lemma integral_minus_sets':
   fixes f::"real \<Rightarrow> 'a::banach"
   shows "c \<ge> a \<Longrightarrow> c \<ge> b \<Longrightarrow> f integrable_on {min a b .. c} \<Longrightarrow>
     integral {a .. c} f - integral {b .. c} f =
     (if a \<le> b then integral {a .. b} f else - integral {b .. a} f)"
   using integral_combine[of b a c f] integral_combine[of a b c f]
   by (auto simp: algebra_simps min_def)
 
 subsection \<open>Reduce integrability to "local" integrability\<close>
 
 lemma integrable_on_little_subintervals:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::banach"
   assumes "\<forall>x\<in>cbox a b. \<exists>d>0. \<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow>
     f integrable_on cbox u v"
   shows "f integrable_on cbox a b"
 proof -
   interpret operative conj True "\<lambda>i. f integrable_on i"
     using order_refl by (rule operative_integrableI)
   have "\<forall>x. \<exists>d>0. x\<in>cbox a b \<longrightarrow> (\<forall>u v. x \<in> cbox u v \<and> cbox u v \<subseteq> ball x d \<and> cbox u v \<subseteq> cbox a b \<longrightarrow>
     f integrable_on cbox u v)"
     using assms by (metis zero_less_one)
   then obtain d where d: "\<And>x. 0 < d x"
      "\<And>x u v. \<lbrakk>x \<in> cbox a b; x \<in> cbox u v; cbox u v \<subseteq> ball x (d x); cbox u v \<subseteq> cbox a b\<rbrakk> 
                \<Longrightarrow> f integrable_on cbox u v"
     by metis
   obtain p where p: "p tagged_division_of cbox a b" "(\<lambda>x. ball x (d x)) fine p"
     using fine_division_exists[OF gauge_ball_dependent,of d a b] d(1) by blast 
   then have sndp: "snd ` p division_of cbox a b"
     by (metis division_of_tagged_division)
   have "f integrable_on k" if "(x, k) \<in> p" for x k
     using tagged_division_ofD(2-4)[OF p(1) that] fineD[OF p(2) that] d[of x] by auto
   then show ?thesis
     unfolding division [symmetric, OF sndp] comm_monoid_set_F_and
     by auto
 qed
 
 
 subsection \<open>Second FTC or existence of antiderivative\<close>
 
 lemma integrable_const[intro]: "(\<lambda>x. c) integrable_on cbox a b"
   unfolding integrable_on_def by blast
 
 lemma integral_has_vector_derivative_continuous_at:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on {a..b}"
      and x: "x \<in> {a..b} - S"
      and "finite S"
      and fx: "continuous (at x within ({a..b} - S)) f"
  shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f x) (at x within ({a..b} - S))"
 proof -
   let ?I = "\<lambda>a b. integral {a..b} f"
   { fix e::real
     assume "e > 0"
     obtain d where "d>0" and d: "\<And>x'. \<lbrakk>x' \<in> {a..b} - S; \<bar>x' - x\<bar> < d\<rbrakk> \<Longrightarrow> norm(f x' - f x) \<le> e"
       using \<open>e>0\<close> fx by (auto simp: continuous_within_eps_delta dist_norm less_imp_le)
     have "norm (integral {a..y} f - integral {a..x} f - (y-x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>" (is "?lhs \<le> ?rhs")
            if y: "y \<in> {a..b} - S" and yx: "\<bar>y - x\<bar> < d" for y
     proof (cases "y < x")
       case False
       have "f integrable_on {a..y}"
         using f y by (simp add: integrable_subinterval_real)
       then have Idiff: "?I a y - ?I a x = ?I x y"
         using False x by (simp add: algebra_simps integral_combine)
       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {x..y} f - (y-x) *\<^sub>R f x) {x..y}"
       proof (rule has_integral_diff)
         show "(f has_integral integral {x..y} f) {x..y}"
           using x y by (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]])
         show "((\<lambda>u. f x) has_integral (y - x) *\<^sub>R f x) {x..y}"
           using has_integral_const_real [of "f x" x y] False by simp
       qed
       have "?lhs \<le> e * content {x..y}"
         using yx False d x y \<open>e>0\<close> assms 
         by (intro has_integral_bound_real[where f="(\<lambda>u. f u - f x)"]) (auto simp: Idiff fux_int)
       also have "... \<le> ?rhs"
         using False by auto
       finally show ?thesis .
     next
       case True
       have "f integrable_on {a..x}"
         using f x by (simp add: integrable_subinterval_real)
       then have Idiff: "?I a x - ?I a y = ?I y x"
         using True x y by (simp add: algebra_simps integral_combine)
       have fux_int: "((\<lambda>u. f u - f x) has_integral integral {y..x} f - (x - y) *\<^sub>R f x) {y..x}"
       proof (rule has_integral_diff)
         show "(f has_integral integral {y..x} f) {y..x}"
           using x y by (auto intro: integrable_integral [OF integrable_subinterval_real [OF f]])
         show "((\<lambda>u. f x) has_integral (x - y) *\<^sub>R f x) {y..x}"
           using has_integral_const_real [of "f x" y x] True by simp
       qed
       have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \<le> e * content {y..x}"
         using yx True d x y \<open>e>0\<close> assms 
         by (intro has_integral_bound_real[where f="(\<lambda>u. f u - f x)"]) (auto simp: Idiff fux_int)
       also have "... \<le> e * \<bar>y - x\<bar>"
         using True by auto
       finally have "norm (integral {a..x} f - integral {a..y} f - (x - y) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>" .
       then show ?thesis
         by (simp add: algebra_simps norm_minus_commute)
     qed
     then have "\<exists>d>0. \<forall>y\<in>{a..b} - S. \<bar>y - x\<bar> < d \<longrightarrow> norm (integral {a..y} f - integral {a..x} f - (y-x) *\<^sub>R f x) \<le> e * \<bar>y - x\<bar>"
       using \<open>d>0\<close> by blast
   }
   then show ?thesis
     by (simp add: has_vector_derivative_def has_derivative_within_alt bounded_linear_scaleR_left)
 qed
 
 
 lemma integral_has_vector_derivative:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "continuous_on {a..b} f"
     and "x \<in> {a..b}"
   shows "((\<lambda>u. integral {a..u} f) has_vector_derivative f(x)) (at x within {a..b})"
 using assms integral_has_vector_derivative_continuous_at [OF integrable_continuous_real]
   by (fastforce simp: continuous_on_eq_continuous_within)
 
 lemma integral_has_real_derivative:
   assumes "continuous_on {a..b} g"
   assumes "t \<in> {a..b}"
   shows "((\<lambda>x. integral {a..x} g) has_real_derivative g t) (at t within {a..b})"
   using integral_has_vector_derivative[of a b g t] assms
   by (auto simp: has_field_derivative_iff_has_vector_derivative)
 
 lemma antiderivative_continuous:
   fixes q b :: real
   assumes "continuous_on {a..b} f"
   obtains g where "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative (f x::_::banach)) (at x within {a..b})"
   using integral_has_vector_derivative[OF assms] by auto
 
 subsection \<open>Combined fundamental theorem of calculus\<close>
 
 lemma antiderivative_integral_continuous:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "continuous_on {a..b} f"
   obtains g where "\<forall>u\<in>{a..b}. \<forall>v \<in> {a..b}. u \<le> v \<longrightarrow> (f has_integral (g v - g u)) {u..v}"
 proof -
   obtain g 
     where g: "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative f x) (at x within {a..b})" 
     using  antiderivative_continuous[OF assms] by metis
   have "(f has_integral g v - g u) {u..v}" if "u \<in> {a..b}" "v \<in> {a..b}" "u \<le> v" for u v
   proof -
     have "\<And>x. x \<in> cbox u v \<Longrightarrow> (g has_vector_derivative f x) (at x within cbox u v)"
       by (metis atLeastAtMost_iff atLeastatMost_subset_iff box_real(2) g
           has_vector_derivative_within_subset subsetCE that(1,2))
     then show ?thesis
       by (metis box_real(2) that(3) fundamental_theorem_of_calculus)
   qed
   then show ?thesis
     using that by blast
 qed
 
 
 subsection \<open>General "twiddling" for interval-to-interval function image\<close>
 
 lemma has_integral_twiddle:
   assumes "0 < r"
     and hg: "\<And>x. h(g x) = x"
     and gh: "\<And>x. g(h x) = x"
     and contg: "\<And>x. continuous (at x) g"
     and g: "\<And>u v. \<exists>w z. g ` cbox u v = cbox w z"
     and h: "\<And>u v. \<exists>w z. h ` cbox u v = cbox w z"
     and r: "\<And>u v. content(g ` cbox u v) = r * content (cbox u v)"
     and intfi: "(f has_integral i) (cbox a b)"
   shows "((\<lambda>x. f(g x)) has_integral (1 / r) *\<^sub>R i) (h ` cbox a b)"
 proof (cases "cbox a b = {}")
   case True
   then show ?thesis 
     using intfi by auto
 next
   case False
   obtain w z where wz: "h ` cbox a b = cbox w z"
     using h by blast
   have inj: "inj g" "inj h"
     using hg gh injI by metis+
   from h obtain ha hb where h_eq: "h ` cbox a b = cbox ha hb" by blast
   have "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of h ` cbox a b \<and> d fine p 
               \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e)"
     if "e > 0" for e
   proof -
     have "e * r > 0" using that \<open>0 < r\<close> by simp
     with intfi[unfolded has_integral]
     obtain d where "gauge d"
                and d: "\<And>p. p tagged_division_of cbox a b \<and> d fine p 
                         \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - i) < e * r" 
       by metis
     define d' where "d' x = g -` d (g x)" for x
     show ?thesis
     proof (rule_tac x=d' in exI, safe)
       show "gauge d'"
         using \<open>gauge d\<close> continuous_open_vimage[OF _ contg] by (auto simp: gauge_def d'_def)
     next
       fix p
       assume ptag: "p tagged_division_of h ` cbox a b" and finep: "d' fine p"
       note p = tagged_division_ofD[OF ptag]
       have gab: "g y \<in> cbox a b" if "y \<in> K" "(x, K) \<in> p" for x y K
         by (metis hg inj(2) inj_image_mem_iff p(3) subsetCE that that)
       have gimp: "(\<lambda>(x,K). (g x, g ` K)) ` p tagged_division_of (cbox a b) \<and> 
                   d fine (\<lambda>(x, k). (g x, g ` k)) ` p"
         unfolding tagged_division_of
       proof safe
         show "finite ((\<lambda>(x, k). (g x, g ` k)) ` p)"
           using ptag by auto
         show "d fine (\<lambda>(x, k). (g x, g ` k)) ` p"
           using finep unfolding fine_def d'_def by auto
       next
         fix x k
         assume xk: "(x, k) \<in> p"
         show "g x \<in> g ` k"
           using p(2)[OF xk] by auto
         show "\<exists>u v. g ` k = cbox u v"
           using p(4)[OF xk] using assms(5-6) by auto
         fix x' K' u
         assume xk': "(x', K') \<in> p" and u: "u \<in> interior (g ` k)" "u \<in> interior (g ` K')"
         have "interior k \<inter> interior K' \<noteq> {}"
         proof 
           assume "interior k \<inter> interior K' = {}"
           moreover have "u \<in> g ` (interior k \<inter> interior K')"
             using interior_image_subset[OF \<open>inj g\<close> contg] u
             unfolding image_Int[OF inj(1)] by blast
           ultimately show False by blast
         qed
         then have same: "(x, k) = (x', K')"
           using ptag xk' xk by blast
         then show "g x = g x'"
           by auto
         show "g u \<in> g ` K'"if "u \<in> k" for u
           using that same by auto
         show "g u \<in> g ` k"if "u \<in> K'" for u
           using that same by auto
       next
         fix x
         assume "x \<in> cbox a b"
         then have "h x \<in>  \<Union>{k. \<exists>x. (x, k) \<in> p}"
           using p(6) by auto
         then obtain X y where "h x \<in> X" "(y, X) \<in> p" by blast
         then show "x \<in> \<Union>{k. \<exists>x. (x, k) \<in> (\<lambda>(x, k). (g x, g ` k)) ` p}"
           by clarsimp (metis (no_types, lifting) gh image_eqI pair_imageI)
       qed (use gab in auto)
       have *: "inj_on (\<lambda>(x, k). (g x, g ` k)) p"
         using inj(1) unfolding inj_on_def by fastforce
       have "(\<Sum>(x,K)\<in>(\<lambda>(y,L). (g y, g ` L)) ` p. content K *\<^sub>R f x) 
           = (\<Sum>u\<in>p. case case u of (x,K) \<Rightarrow> (g x, g ` K) of (y,L) \<Rightarrow> content L *\<^sub>R f y)"
         by (metis (mono_tags, lifting) "*" sum.reindex_cong)
       also have "... = (\<Sum>(x,K)\<in>p. r *\<^sub>R content K *\<^sub>R f (g x))"
         using r by (auto intro!: * sum.cong simp: bij_betw_def dest!: p(4))
       finally
       have "(\<Sum>(x, K)\<in>(\<lambda>(x,K). (g x, g ` K)) ` p. content K *\<^sub>R f x) - i = r *\<^sub>R (\<Sum>(x,K)\<in>p. content K *\<^sub>R f (g x)) - i" 
         by (simp add: scaleR_right.sum split_def)
       also have "\<dots> = r *\<^sub>R ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i)" 
         using \<open>0 < r\<close> by (auto simp: scaleR_diff_right)
       finally show "norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f (g x)) - (1 / r) *\<^sub>R i) < e"
         using d[OF gimp] \<open>0 < r\<close> by auto
     qed
   qed
   then show ?thesis
     by (auto simp: h_eq has_integral)
 qed
 
 
 subsection \<open>Special case of a basic affine transformation\<close>
 
 lemma AE_lborel_inner_neq:
   assumes k: "k \<in> Basis"
   shows "AE x in lborel. x \<bullet> k \<noteq> c"
 proof -
   interpret finite_product_sigma_finite "\<lambda>_. lborel" Basis
     proof qed simp
     have "emeasure lborel {x\<in>space lborel. x \<bullet> k = c} 
         = emeasure (\<Pi>\<^sub>M j::'a\<in>Basis. lborel) (\<Pi>\<^sub>E j\<in>Basis. if j = k then {c} else UNIV)"
     using k
     by (auto simp add: lborel_eq[where 'a='a] emeasure_distr intro!: arg_cong2[where f=emeasure])
        (auto simp: space_PiM PiE_iff extensional_def split: if_split_asm)
   also have "\<dots> = (\<Prod>j\<in>Basis. emeasure lborel (if j = k then {c} else UNIV))"
     by (intro measure_times) auto
   also have "\<dots> = 0"
     by (intro prod_zero bexI[OF _ k]) auto
   finally show ?thesis
     by (subst AE_iff_measurable[OF _ refl]) auto
 qed
 
 lemma content_image_stretch_interval:
   fixes m :: "'a::euclidean_space \<Rightarrow> real"
   defines "s f x \<equiv> (\<Sum>k::'a\<in>Basis. (f k * (x\<bullet>k)) *\<^sub>R k)"
   shows "content (s m ` cbox a b) = \<bar>\<Prod>k\<in>Basis. m k\<bar> * content (cbox a b)"
 proof cases
   have s[measurable]: "s f \<in> borel \<rightarrow>\<^sub>M borel" for f
     by (auto simp: s_def[abs_def])
   assume m: "\<forall>k\<in>Basis. m k \<noteq> 0"
   then have s_comp_s: "s (\<lambda>k. 1 / m k) \<circ> s m = id" "s m \<circ> s (\<lambda>k. 1 / m k) = id"
     by (auto simp: s_def[abs_def] fun_eq_iff euclidean_representation)
   then have "inv (s (\<lambda>k. 1 / m k)) = s m" "bij (s (\<lambda>k. 1 / m k))"
     by (auto intro: inv_unique_comp o_bij)
   then have eq: "s m ` cbox a b = s (\<lambda>k. 1 / m k) -` cbox a b"
     using bij_vimage_eq_inv_image[OF \<open>bij (s (\<lambda>k. 1 / m k))\<close>, of "cbox a b"] by auto
   show ?thesis
     using m unfolding eq measure_def
     by (subst lborel_affine_euclidean[where c=m and t=0])
        (simp_all add: emeasure_density measurable_sets_borel[OF s] abs_prod nn_integral_cmult
                       s_def[symmetric] emeasure_distr vimage_comp s_comp_s enn2real_mult prod_nonneg)
 next
   assume "\<not> (\<forall>k\<in>Basis. m k \<noteq> 0)"
   then obtain k where k: "k \<in> Basis" "m k = 0" by auto
   then have [simp]: "(\<Prod>k\<in>Basis. m k) = 0"
     by (intro prod_zero) auto
   have "emeasure lborel {x\<in>space lborel. x \<in> s m ` cbox a b} = 0"
   proof (rule emeasure_eq_0_AE)
     show "AE x in lborel. x \<notin> s m ` cbox a b"
       using AE_lborel_inner_neq[OF \<open>k\<in>Basis\<close>]
     proof eventually_elim
       show "x \<bullet> k \<noteq> 0 \<Longrightarrow> x \<notin> s m ` cbox a b " for x
         using k by (auto simp: s_def[abs_def] cbox_def)
     qed
   qed
   then show ?thesis
     by (simp add: measure_def)
 qed
 
 lemma interval_image_affinity_interval:
   "\<exists>u v. (\<lambda>x. m *\<^sub>R (x::'a::euclidean_space) + c) ` cbox a b = cbox u v"
   unfolding image_affinity_cbox
   by auto
 
 lemma content_image_affinity_cbox:
   "content((\<lambda>x::'a::euclidean_space. m *\<^sub>R x + c) ` cbox a b) =
     \<bar>m\<bar> ^ DIM('a) * content (cbox a b)" (is "?l = ?r")
 proof (cases "cbox a b = {}")
   case True then show ?thesis by simp
 next
   case False
   show ?thesis
   proof (cases "m \<ge> 0")
     case True
     with \<open>cbox a b \<noteq> {}\<close> have "cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c) \<noteq> {}"
       by (simp add: box_ne_empty inner_left_distrib mult_left_mono)
     moreover from True have *: "\<And>i. (m *\<^sub>R b + c) \<bullet> i - (m *\<^sub>R a + c) \<bullet> i = m *\<^sub>R (b-a) \<bullet> i"
       by (simp add: inner_simps field_simps)
     ultimately show ?thesis
       by (simp add: image_affinity_cbox True content_cbox'
         prod.distrib inner_diff_left)
   next
     case False
     with \<open>cbox a b \<noteq> {}\<close> have "cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c) \<noteq> {}"
       by (simp add: box_ne_empty inner_left_distrib mult_left_mono)
     moreover from False have *: "\<And>i. (m *\<^sub>R a + c) \<bullet> i - (m *\<^sub>R b + c) \<bullet> i = (-m) *\<^sub>R (b-a) \<bullet> i"
       by (simp add: inner_simps field_simps)
     ultimately show ?thesis using False
       by (simp add: image_affinity_cbox content_cbox'
         prod.distrib[symmetric] inner_diff_left flip: prod_constant)
   qed
 qed
 
 lemma has_integral_affinity:
   fixes a :: "'a::euclidean_space"
   assumes "(f has_integral i) (cbox a b)"
       and "m \<noteq> 0"
   shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral (1 / (\<bar>m\<bar> ^ DIM('a))) *\<^sub>R i) ((\<lambda>x. (1 / m) *\<^sub>R x + -((1 / m) *\<^sub>R c)) ` cbox a b)"
 proof (rule has_integral_twiddle)
   show "\<exists>w z. (\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox u v = cbox w z" 
        "\<exists>w z. (\<lambda>x. m *\<^sub>R x + c) ` cbox u v = cbox w z" for u v
     using interval_image_affinity_interval by blast+
   show "content ((\<lambda>x. m *\<^sub>R x + c) ` cbox u v) = \<bar>m\<bar> ^ DIM('a) * content (cbox u v)" for u v
     using content_image_affinity_cbox by blast
 qed (use assms zero_less_power in \<open>auto simp: field_simps\<close>)
 
 lemma integrable_affinity:
   assumes "f integrable_on cbox a b"
     and "m \<noteq> 0"
   shows "(\<lambda>x. f(m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x + -((1/m) *\<^sub>R c)) ` cbox a b)"
   using has_integral_affinity assms
   unfolding integrable_on_def by blast
 
 lemmas has_integral_affinity01 = has_integral_affinity [of _ _ 0 "1::real", simplified]
 
 lemma integrable_on_affinity:
   assumes "m \<noteq> 0" "f integrable_on (cbox a b)"
   shows   "(\<lambda>x. f (m *\<^sub>R x + c)) integrable_on ((\<lambda>x. (1 / m) *\<^sub>R x - ((1 / m) *\<^sub>R c)) ` cbox a b)"
 proof -
   from assms obtain I where "(f has_integral I) (cbox a b)"
     by (auto simp: integrable_on_def)
   from has_integral_affinity[OF this assms(1), of c] show ?thesis
     by (auto simp: integrable_on_def)
 qed
 
 lemma has_integral_cmul_iff:
   assumes "c \<noteq> 0"
   shows   "((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R I)) A \<longleftrightarrow> (f has_integral I) A"
   using assms has_integral_cmul[of f I A c]
         has_integral_cmul[of "\<lambda>x. c *\<^sub>R f x" "c *\<^sub>R I" A "inverse c"] by (auto simp: field_simps)
 
 lemma has_integral_affinity':
   fixes a :: "'a::euclidean_space"
   assumes "(f has_integral i) (cbox a b)" and "m > 0"
   shows "((\<lambda>x. f(m *\<^sub>R x + c)) has_integral (i /\<^sub>R m ^ DIM('a)))
            (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m))"
 proof (cases "cbox a b = {}")
   case True
   hence "(cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) = {}"
     using \<open>m > 0\<close> unfolding box_eq_empty by (auto simp: algebra_simps)
   with True and assms show ?thesis by simp
 next
   case False
   have "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral (1 / \<bar>m\<bar> ^ DIM('a)) *\<^sub>R i)
           ((\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b)"
     using assms by (intro has_integral_affinity) auto
   also have "((\<lambda>x. (1 / m) *\<^sub>R x + - ((1 / m) *\<^sub>R c)) ` cbox a b) =
                ((\<lambda>x.  - ((1 / m) *\<^sub>R c) + x) ` (\<lambda>x. (1 / m) *\<^sub>R x) ` cbox a b)"
     by (simp add: image_image algebra_simps)
   also have "(\<lambda>x. (1 / m) *\<^sub>R x) ` cbox a b = cbox ((1 / m) *\<^sub>R a) ((1 / m) *\<^sub>R b)" using \<open>m > 0\<close> False
     by (subst image_smult_cbox) simp_all
   also have "(\<lambda>x. - ((1 / m) *\<^sub>R c) + x) ` \<dots> = cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)"
     by (subst cbox_translation [symmetric]) (simp add: field_simps vector_add_divide_simps)
   finally show ?thesis using \<open>m > 0\<close> by (simp add: field_simps)
 qed
 
 lemma has_integral_affinity_iff:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: real_normed_vector"
   assumes "m > 0"
   shows   "((\<lambda>x. f (m *\<^sub>R x + c)) has_integral (I /\<^sub>R m ^ DIM('a)))
                (cbox ((a - c) /\<^sub>R m) ((b - c) /\<^sub>R m)) \<longleftrightarrow>
            (f has_integral I) (cbox a b)" (is "?lhs = ?rhs")
 proof
   assume ?lhs
   from has_integral_affinity'[OF this, of "1 / m" "-c /\<^sub>R m"] and \<open>m > 0\<close>
     show ?rhs by (simp add: vector_add_divide_simps) (simp add: field_simps)
 next
   assume ?rhs
   from has_integral_affinity'[OF this, of m c] and \<open>m > 0\<close>
   show ?lhs by simp
 qed
 
 
 subsection \<open>Special case of stretching coordinate axes separately\<close>
 
 lemma has_integral_stretch:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "(f has_integral i) (cbox a b)"
     and "\<forall>k\<in>Basis. m k \<noteq> 0"
   shows "((\<lambda>x. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) has_integral
          ((1/ \<bar>prod m Basis\<bar>) *\<^sub>R i)) ((\<lambda>x. (\<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k)) ` cbox a b)"
   apply (rule has_integral_twiddle[where f=f])
   unfolding zero_less_abs_iff content_image_stretch_interval
   unfolding image_stretch_interval empty_as_interval euclidean_eq_iff[where 'a='a]
   using assms
   by auto
 
 lemma has_integral_stretch_real:
   fixes f :: "real \<Rightarrow> 'b::real_normed_vector"
   assumes "(f has_integral i) {a..b}" and "m \<noteq> 0"
   shows "((\<lambda>x. f (m * x)) has_integral (1 / \<bar>m\<bar>) *\<^sub>R i) ((\<lambda>x. x / m) ` {a..b})"
   using has_integral_stretch [of f i a b "\<lambda>b. m"] assms by simp
 
 lemma integrable_stretch:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "f integrable_on cbox a b"
     and "\<forall>k\<in>Basis. m k \<noteq> 0"
   shows "(\<lambda>x::'a. f (\<Sum>k\<in>Basis. (m k * (x\<bullet>k))*\<^sub>R k)) integrable_on
     ((\<lambda>x. \<Sum>k\<in>Basis. (1 / m k * (x\<bullet>k))*\<^sub>R k) ` cbox a b)"
   using assms unfolding integrable_on_def
   by (force dest: has_integral_stretch)
 
 lemma vec_lambda_eq_sum:
      "(\<chi> k. f k (x $ k)) = (\<Sum>k\<in>Basis. (f (axis_index k) (x \<bullet> k)) *\<^sub>R k)" (is "?lhs = ?rhs")
 proof -
   have "?lhs = (\<chi> k. f k (x \<bullet> axis k 1))"
     by (simp add: cart_eq_inner_axis)
   also have "... = (\<Sum>u\<in>UNIV. f u (x \<bullet> axis u 1) *\<^sub>R axis u 1)"
     by (simp add: vec_eq_iff axis_def if_distrib cong: if_cong)
   also have "... = ?rhs"
     by (simp add: Basis_vec_def UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def)
   finally show ?thesis .
 qed
 
 lemma has_integral_stretch_cart:
   fixes m :: "'n::finite \<Rightarrow> real"
   assumes f: "(f has_integral i) (cbox a b)" and m: "\<And>k. m k \<noteq> 0"
   shows "((\<lambda>x. f(\<chi> k. m k * x$k)) has_integral i /\<^sub>R \<bar>prod m UNIV\<bar>)
             ((\<lambda>x. \<chi> k. x$k / m k) ` (cbox a b))"
 proof -
   have *: "\<forall>k:: real^'n \<in> Basis. m (axis_index k) \<noteq> 0"
     using axis_index by (simp add: m)
   have eqp: "(\<Prod>k:: real^'n \<in> Basis. m (axis_index k)) = prod m UNIV"
     by (simp add: Basis_vec_def UNION_singleton_eq_range prod.reindex axis_eq_axis inj_on_def)
   show ?thesis
     using has_integral_stretch [OF f *] vec_lambda_eq_sum [where f="\<lambda>i x. m i * x"] vec_lambda_eq_sum [where f="\<lambda>i x. x / m i"]
     by (simp add: field_simps eqp)
 qed
 
 lemma image_stretch_interval_cart:
   fixes m :: "'n::finite \<Rightarrow> real"
   shows "(\<lambda>x. \<chi> k. m k * x$k) ` cbox a b =
             (if cbox a b = {} then {}
             else cbox (\<chi> k. min (m k * a$k) (m k * b$k)) (\<chi> k. max (m k * a$k) (m k * b$k)))"
 proof -
   have *: "(\<Sum>k\<in>Basis. min (m (axis_index k) * (a \<bullet> k)) (m (axis_index k) * (b \<bullet> k)) *\<^sub>R k)
            = (\<chi> k. min (m k * a $ k) (m k * b $ k))"
           "(\<Sum>k\<in>Basis. max (m (axis_index k) * (a \<bullet> k)) (m (axis_index k) * (b \<bullet> k)) *\<^sub>R k)
            = (\<chi> k. max (m k * a $ k) (m k * b $ k))"
     apply (simp_all add: Basis_vec_def cart_eq_inner_axis UNION_singleton_eq_range sum.reindex axis_eq_axis inj_on_def)
     apply (simp_all add: vec_eq_iff axis_def if_distrib cong: if_cong)
     done
   show ?thesis
     by (simp add: vec_lambda_eq_sum [where f="\<lambda>i x. m i * x"] image_stretch_interval eq_cbox *)
 qed
 
 
 subsection \<open>even more special cases\<close>
 
 lemma uminus_interval_vector[simp]:
   fixes a b :: "'a::euclidean_space"
   shows "uminus ` cbox a b = cbox (-b) (-a)"
 proof -
   have "x \<in> uminus ` cbox a b" if "x \<in> cbox (- b) (- a)" for x
   proof -
     have "-x \<in> cbox a b"
       using that by (auto simp: mem_box)
     then show ?thesis
       by force
   qed
   then show ?thesis
     by (auto simp: mem_box)
 qed
 
 lemma has_integral_reflect_lemma[intro]:
   assumes "(f has_integral i) (cbox a b)"
   shows "((\<lambda>x. f(-x)) has_integral i) (cbox (-b) (-a))"
   using has_integral_affinity[OF assms, of "-1" 0]
   by auto
 
 lemma has_integral_reflect_lemma_real[intro]:
   assumes "(f has_integral i) {a..b::real}"
   shows "((\<lambda>x. f(-x)) has_integral i) {-b .. -a}"
   using assms
   unfolding box_real[symmetric]
   by (rule has_integral_reflect_lemma)
 
 lemma has_integral_reflect[simp]:
   "((\<lambda>x. f (-x)) has_integral i) (cbox (-b) (-a)) \<longleftrightarrow> (f has_integral i) (cbox a b)"
   by (auto dest: has_integral_reflect_lemma)
 
 lemma has_integral_reflect_real[simp]:
   fixes a b::real
   shows "((\<lambda>x. f (-x)) has_integral i) {-b..-a} \<longleftrightarrow> (f has_integral i) {a..b}"
   by (metis has_integral_reflect interval_cbox)
 
 lemma integrable_reflect[simp]: "(\<lambda>x. f(-x)) integrable_on cbox (-b) (-a) \<longleftrightarrow> f integrable_on cbox a b"
   unfolding integrable_on_def by auto
 
 lemma integrable_reflect_real[simp]: "(\<lambda>x. f(-x)) integrable_on {-b .. -a} \<longleftrightarrow> f integrable_on {a..b::real}"
   unfolding box_real[symmetric]
   by (rule integrable_reflect)
 
 lemma integral_reflect[simp]: "integral (cbox (-b) (-a)) (\<lambda>x. f (-x)) = integral (cbox a b) f"
   unfolding integral_def by auto
 
 lemma integral_reflect_real[simp]: "integral {-b .. -a} (\<lambda>x. f (-x)) = integral {a..b::real} f"
   unfolding box_real[symmetric]
   by (rule integral_reflect)
 
 subsection \<open>Stronger form of FCT; quite a tedious proof\<close>
 
 lemma split_minus[simp]: "(\<lambda>(x, k). f x k) x - (\<lambda>(x, k). g x k) x = (\<lambda>(x, k). f x k - g x k) x"
   by (simp add: split_def)
 
 theorem fundamental_theorem_of_calculus_interior:
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "a \<le> b"
     and contf: "continuous_on {a..b} f"
     and derf: "\<And>x. x \<in> {a <..< b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
   shows "(f' has_integral (f b - f a)) {a..b}"
 proof (cases "a = b")
   case True
   then have *: "cbox a b = {b}" "f b - f a = 0"
     by (auto simp add:  order_antisym)
   with True show ?thesis by auto
 next
   case False
   with \<open>a \<le> b\<close> have ab: "a < b" by arith
   show ?thesis
     unfolding has_integral_factor_content_real
   proof (intro allI impI)
     fix e :: real
     assume e: "e > 0"
     then have eba8: "(e * (b-a)) / 8 > 0"
       using ab by (auto simp add: field_simps)
     note derf_exp = derf[unfolded has_vector_derivative_def has_derivative_at_alt, THEN conjunct2, rule_format]
     thm derf_exp
     have bounded: "\<And>x. x \<in> {a<..<b} \<Longrightarrow> bounded_linear (\<lambda>u. u *\<^sub>R f' x)"
       by (simp add: bounded_linear_scaleR_left)
     have "\<forall>x \<in> box a b. \<exists>d>0. \<forall>y. norm (y-x) < d \<longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e/2 * norm (y-x)"
       (is "\<forall>x \<in> box a b. ?Q x") \<comment>\<open>The explicit quantifier is required by the following step\<close>
     proof
       fix x assume "x \<in> box a b"
       with e show "?Q x"
         using derf_exp [of x "e/2"] by auto
     qed
     then obtain d where d: "\<And>x. 0 < d x"
       "\<And>x y. \<lbrakk>x \<in> box a b; norm (y-x) < d x\<rbrakk> \<Longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e/2 * norm (y-x)"
       unfolding bgauge_existence_lemma by metis
     have "bounded (f ` cbox a b)"
       using compact_cbox assms by (auto simp: compact_imp_bounded compact_continuous_image)
     then obtain B 
       where "0 < B" and B: "\<And>x. x \<in> f ` cbox a b \<Longrightarrow> norm x \<le> B"
       unfolding bounded_pos by metis
     obtain da where "0 < da"
       and da: "\<And>c. \<lbrakk>a \<le> c; {a..c} \<subseteq> {a..b}; {a..c} \<subseteq> ball a da\<rbrakk>
                           \<Longrightarrow> norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> (e * (b-a)) / 4"
     proof -
       have "continuous (at a within {a..b}) f"
         using contf continuous_on_eq_continuous_within by force
       with eba8 obtain k where "0 < k"
         and k: "\<And>x. \<lbrakk>x \<in> {a..b}; 0 < norm (x-a); norm (x-a) < k\<rbrakk> \<Longrightarrow> norm (f x - f a) < e * (b-a) / 8"
         unfolding continuous_within Lim_within dist_norm by metis
       obtain l where l: "0 < l" "norm (l *\<^sub>R f' a) \<le> e * (b-a) / 8" 
       proof (cases "f' a = 0")
         case True with ab e that show ?thesis by auto
       next
         case False
         show ?thesis
         proof
           show "norm ((e * (b - a) / 8 / norm (f' a)) *\<^sub>R f' a) \<le> e * (b - a) / 8"
                "0 < e * (b - a) / 8 / norm (f' a)"
             using False ab e by (auto simp add: field_simps)
         qed 
       qed
       have "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b-a) / 4"
         if "a \<le> c" "{a..c} \<subseteq> {a..b}" and bmin: "{a..c} \<subseteq> ball a (min k l)" for c
       proof -
         have minkl: "\<bar>a - x\<bar> < min k l" if "x \<in> {a..c}" for x
           using bmin dist_real_def that by auto
         then have lel: "\<bar>c - a\<bar> \<le> \<bar>l\<bar>"
           using that by force
         have "norm ((c - a) *\<^sub>R f' a - (f c - f a)) \<le> norm ((c - a) *\<^sub>R f' a) + norm (f c - f a)"
           by (rule norm_triangle_ineq4)
         also have "\<dots> \<le> e * (b-a) / 8 + e * (b-a) / 8"
         proof (rule add_mono)
           have "norm ((c - a) *\<^sub>R f' a) \<le> norm (l *\<^sub>R f' a)"
             by (auto intro: mult_right_mono [OF lel])
           also have "... \<le> e * (b-a) / 8"
             by (rule l)
           finally show "norm ((c - a) *\<^sub>R f' a) \<le> e * (b-a) / 8" .
         next
           have "norm (f c - f a) < e * (b-a) / 8"
           proof (cases "a = c")
             case True then show ?thesis
               using eba8 by auto
           next
             case False show ?thesis
               by (rule k) (use minkl \<open>a \<le> c\<close> that False in auto)
           qed
           then show "norm (f c - f a) \<le> e * (b-a) / 8" by simp
         qed
         finally show "norm (content {a..c} *\<^sub>R f' a - (f c - f a)) \<le> e * (b-a) / 4"
           unfolding content_real[OF \<open>a \<le> c\<close>] by auto
       qed
       then show ?thesis
         by (rule_tac da="min k l" in that) (auto simp: l \<open>0 < k\<close>)
     qed
     obtain db where "0 < db"
       and db: "\<And>c. \<lbrakk>c \<le> b; {c..b} \<subseteq> {a..b}; {c..b} \<subseteq> ball b db\<rbrakk>
                           \<Longrightarrow> norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> (e * (b-a)) / 4"
     proof -
       have "continuous (at b within {a..b}) f"
         using contf continuous_on_eq_continuous_within by force
       with eba8 obtain k
         where "0 < k"
           and k: "\<And>x. \<lbrakk>x \<in> {a..b}; 0 < norm(x-b); norm(x-b) < k\<rbrakk>
                      \<Longrightarrow> norm (f b - f x) < e * (b-a) / 8"
         unfolding continuous_within Lim_within dist_norm norm_minus_commute by metis
       obtain l where l: "0 < l" "norm (l *\<^sub>R f' b) \<le> (e * (b-a)) / 8"
       proof (cases "f' b = 0")
         case True thus ?thesis 
           using ab e that by auto
       next
         case False show ?thesis
         proof
           show "norm ((e * (b - a) / 8 / norm (f' b)) *\<^sub>R f' b) \<le> e * (b - a) / 8" 
                "0 < e * (b - a) / 8 / norm (f' b)"
             using False ab e by (auto simp add: field_simps)
         qed
       qed
       have "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b-a) / 4" 
         if "c \<le> b" "{c..b} \<subseteq> {a..b}" and bmin: "{c..b} \<subseteq> ball b (min k l)" for c
       proof -
         have minkl: "\<bar>b - x\<bar> < min k l" if "x \<in> {c..b}" for x
           using bmin dist_real_def that by auto
         then have lel: "\<bar>b - c\<bar> \<le> \<bar>l\<bar>"
           using that by force
         have "norm ((b - c) *\<^sub>R f' b - (f b - f c)) \<le> norm ((b - c) *\<^sub>R f' b) + norm (f b - f c)"
           by (rule norm_triangle_ineq4)
         also have "\<dots> \<le> e * (b-a) / 8 + e * (b-a) / 8"
         proof (rule add_mono)
           have "norm ((b - c) *\<^sub>R f' b) \<le> norm (l *\<^sub>R f' b)"
             by (auto intro: mult_right_mono [OF lel])
           also have "... \<le> e * (b-a) / 8"
             by (rule l)
           finally show "norm ((b - c) *\<^sub>R f' b) \<le> e * (b-a) / 8" .
         next
           have "norm (f b - f c) < e * (b-a) / 8"
           proof (cases "b = c")
             case True with eba8 show ?thesis
               by auto
           next
             case False show ?thesis
               by (rule k) (use minkl \<open>c \<le> b\<close> that False in auto)
           qed
           then show "norm (f b - f c) \<le> e * (b-a) / 8" by simp
         qed
         finally show "norm (content {c..b} *\<^sub>R f' b - (f b - f c)) \<le> e * (b-a) / 4"
           unfolding content_real[OF \<open>c \<le> b\<close>] by auto
       qed
       then show ?thesis
         by (rule_tac db="min k l" in that) (auto simp: l \<open>0 < k\<close>)
     qed
     let ?d = "(\<lambda>x. ball x (if x=a then da else if x=b then db else d x))"
     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
               norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b})"
     proof (rule exI, safe)
       show "gauge ?d"
         using ab \<open>db > 0\<close> \<open>da > 0\<close> d(1) by (auto intro: gauge_ball_dependent)
     next
       fix p
       assume ptag: "p tagged_division_of {a..b}" and fine: "?d fine p"
       let ?A = "{t. fst t \<in> {a, b}}"
       note p = tagged_division_ofD[OF ptag]
       have pA: "p = (p \<inter> ?A) \<union> (p - ?A)" "finite (p \<inter> ?A)" "finite (p - ?A)" "(p \<inter> ?A) \<inter> (p - ?A) = {}"
         using ptag fine by auto
       have le_xz: "\<And>w x y z::real. y \<le> z/2 \<Longrightarrow> w - x \<le> z/2 \<Longrightarrow> w + y \<le> x + z"
         by arith
       have non: False if xk: "(x,K) \<in> p" and "x \<noteq> a" "x \<noteq> b"
         and less: "e * (Sup K - Inf K)/2 < norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))"
       for x K
       proof -
         obtain u v where k: "K = cbox u v"
           using p(4) xk by blast
         then have "u \<le> v" and uv: "{u, v} \<subseteq> cbox u v"
           using p(2)[OF xk] by auto
         then have result: "e * (v - u)/2 < norm ((v - u) *\<^sub>R f' x - (f v - f u))"
           using less[unfolded k box_real interval_bounds_real content_real] by auto
         then have "x \<in> box a b"
           using p(2) p(3) \<open>x \<noteq> a\<close> \<open>x \<noteq> b\<close> xk by fastforce
         with d have *: "\<And>y. norm (y-x) < d x 
                 \<Longrightarrow> norm (f y - f x - (y-x) *\<^sub>R f' x) \<le> e/2 * norm (y-x)"
           by metis
         have xd: "norm (u - x) < d x" "norm (v - x) < d x"
           using fineD[OF fine xk] \<open>x \<noteq> a\<close> \<open>x \<noteq> b\<close> uv 
           by (auto simp add: k subset_eq dist_commute dist_real_def)
         have "norm ((v - u) *\<^sub>R f' x - (f v - f u)) =
               norm ((f u - f x - (u - x) *\<^sub>R f' x) - (f v - f x - (v - x) *\<^sub>R f' x))"
           by (rule arg_cong[where f=norm]) (auto simp: scaleR_left.diff)
         also have "\<dots> \<le> e/2 * norm (u - x) + e/2 * norm (v - x)"
           by (metis norm_triangle_le_diff add_mono * xd)
         also have "\<dots> \<le> e/2 * norm (v - u)"
           using p(2)[OF xk] by (auto simp add: field_simps k)
         also have "\<dots> < norm ((v - u) *\<^sub>R f' x - (f v - f u))"
           using result by (simp add: \<open>u \<le> v\<close>)
         finally have "e * (v - u)/2 < e * (v - u)/2"
           using uv by auto
         then show False by auto
       qed
       have "norm (\<Sum>(x, K)\<in>p - ?A. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))
           \<le> (\<Sum>(x, K)\<in>p - ?A. norm (content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))))"
         by (auto intro: sum_norm_le)
       also have "... \<le> (\<Sum>n\<in>p - ?A. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k)/2)"
         using non by (fastforce intro: sum_mono)
       finally have I: "norm (\<Sum>(x, k)\<in>p - ?A.
                   content k *\<^sub>R f' x - (f (Sup k) - f (Inf k)))
              \<le> (\<Sum>n\<in>p - ?A. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))/2"
         by (simp add: sum_divide_distrib)
       have II: "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) -
              (\<Sum>n\<in>p \<inter> ?A. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))
              \<le> (\<Sum>n\<in>p - ?A. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))/2"
       proof -
         have ge0: "0 \<le> e * (Sup k - Inf k)" if xkp: "(x, k) \<in> p \<inter> ?A" for x k
         proof -
           obtain u v where uv: "k = cbox u v"
             by (meson Int_iff xkp p(4))
           with p(2) that uv have "cbox u v \<noteq> {}"
             by blast
           then show "0 \<le> e * ((Sup k) - (Inf k))"
             unfolding uv using e by (auto simp add: field_simps)
         qed
         let ?B = "\<lambda>x. {t \<in> p. fst t = x \<and> content (snd t) \<noteq> 0}"
         let ?C = "{t \<in> p. fst t \<in> {a, b} \<and> content (snd t) \<noteq> 0}"
         have "norm (\<Sum>(x, k)\<in>p \<inter> {t. fst t \<in> {a, b}}. content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \<le> e * (b-a)/2"
         proof -
           have *: "\<And>S f e. sum f S = sum f (p \<inter> ?C) \<Longrightarrow> norm (sum f (p \<inter> ?C)) \<le> e \<Longrightarrow> norm (sum f S) \<le> e"
             by auto
           have 1: "content K *\<^sub>R (f' x) - (f ((Sup K)) - f ((Inf K))) = 0"
             if "(x,K) \<in> p \<inter> {t. fst t \<in> {a, b}} - p \<inter> ?C" for x K
           proof -
             have xk: "(x,K) \<in> p" and k0: "content K = 0"
               using that by auto
             then obtain u v where uv: "K = cbox u v"
               using p(4) by blast
             then have "u = v"
               using xk k0 p(2) by force
             then show "content K *\<^sub>R (f' x) - (f ((Sup K)) - f ((Inf K))) = 0"
               using xk unfolding uv by auto
           qed
           have 2: "norm(\<Sum>(x,K)\<in>p \<inter> ?C. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))  \<le> e * (b-a)/2"
           proof -
             have norm_le: "norm (sum f S) \<le> e"
               if "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x = y" "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> e" "e > 0"
               for S f and e :: real
             proof (cases "S = {}")
               case True
               with that show ?thesis by auto
             next
               case False then obtain x where "x \<in> S"
                 by auto
               then have "S = {x}"
                 using that(1) by auto
               then show ?thesis
                 using \<open>x \<in> S\<close> that(2) by auto
             qed
             have *: "p \<inter> ?C = ?B a \<union> ?B b"
               by blast
             then have "norm (\<Sum>(x,K)\<in>p \<inter> ?C. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) =
                        norm (\<Sum>(x,K) \<in> ?B a \<union> ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))"
               by simp
             also have "... = norm ((\<Sum>(x,K) \<in> ?B a. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) + 
                                    (\<Sum>(x,K) \<in> ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))))"
               using p(1) ab e by (subst sum.union_disjoint) auto
             also have "... \<le> e * (b - a) / 4 + e * (b - a) / 4"
             proof (rule norm_triangle_le [OF add_mono])
               have pa: "\<exists>v. k = cbox a v \<and> a \<le> v" if "(a, k) \<in> p" for k
                 using p(2) p(3) p(4) that by fastforce
               show "norm (\<Sum>(x,K) \<in> ?B a. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \<le> e * (b - a) / 4"
               proof (intro norm_le; clarsimp)
                 fix K K'
                 assume K: "(a, K) \<in> p" "(a, K') \<in> p" and ne0: "content K \<noteq> 0" "content K' \<noteq> 0"
                 with pa obtain v v' where v: "K = cbox a v" "a \<le> v" and v': "K' = cbox a v'" "a \<le> v'"
                   by blast
                 let ?v = "min v v'"
                 have "box a ?v \<subseteq> K \<inter> K'"
                   unfolding v v' by (auto simp add: mem_box)
                 then have "interior (box a (min v v')) \<subseteq> interior K \<inter> interior K'"
                   using interior_Int interior_mono by blast
                 moreover have "(a + ?v)/2 \<in> box a ?v"
                   using ne0  unfolding v v' content_eq_0 not_le
                   by (auto simp add: mem_box)
                 ultimately have "(a + ?v)/2 \<in> interior K \<inter> interior K'"
                   unfolding interior_open[OF open_box] by auto
                 then show "K = K'"
                   using p(5)[OF K] by auto
               next
                 fix K 
                 assume K: "(a, K) \<in> p" and ne0: "content K \<noteq> 0"
                 show "norm (content c *\<^sub>R f' a - (f (Sup c) - f (Inf c))) * 4 \<le> e * (b-a)"
                   if "(a, c) \<in> p" and ne0: "content c \<noteq> 0" for c
                 proof -
                   obtain v where v: "c = cbox a v" and "a \<le> v"
                     using pa[OF \<open>(a, c) \<in> p\<close>] by metis 
                   then have "a \<in> {a..v}" "v \<le> b"
                     using p(3)[OF \<open>(a, c) \<in> p\<close>] by auto
                   moreover have "{a..v} \<subseteq> ball a da"
                     using fineD[OF \<open>?d fine p\<close> \<open>(a, c) \<in> p\<close>] by (simp add: v split: if_split_asm)
                   ultimately show ?thesis
                     unfolding v interval_bounds_real[OF \<open>a \<le> v\<close>] box_real
                     using da \<open>a \<le> v\<close> by auto
                 qed
               qed (use ab e in auto)
             next
               have pb: "\<exists>v. k = cbox v b \<and> b \<ge> v" if "(b, k) \<in> p" for k
                 using p(2) p(3) p(4) that by fastforce
               show "norm (\<Sum>(x,K) \<in> ?B b. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K))) \<le> e * (b - a) / 4"
               proof (intro norm_le; clarsimp)
                 fix K K'
                 assume K: "(b, K) \<in> p" "(b, K') \<in> p" and ne0: "content K \<noteq> 0" "content K' \<noteq> 0"
                 with pb obtain v v' where v: "K = cbox v b" "v \<le> b" and v': "K' = cbox v' b" "v' \<le> b"
                   by blast
                 let ?v = "max v v'"
                 have "box ?v b \<subseteq> K \<inter> K'"
                   unfolding v v' by (auto simp: mem_box)
                 then have "interior (box (max v v') b) \<subseteq> interior K \<inter> interior K'"
                   using interior_Int interior_mono by blast
                 moreover have " ((b + ?v)/2) \<in> box ?v b"
                   using ne0 unfolding v v' content_eq_0 not_le by (auto simp: mem_box)
                 ultimately have "((b + ?v)/2) \<in> interior K \<inter> interior K'"
                   unfolding interior_open[OF open_box] by auto
                 then show "K = K'"
                   using p(5)[OF K] by auto
               next
                 fix K 
                 assume K: "(b, K) \<in> p" and ne0: "content K \<noteq> 0"
                 show "norm (content c *\<^sub>R f' b - (f (Sup c) - f (Inf c))) * 4 \<le> e * (b-a)"
                   if "(b, c) \<in> p" and ne0: "content c \<noteq> 0" for c
                 proof -
                 obtain v where v: "c = cbox v b" and "v \<le> b"
                   using \<open>(b, c) \<in> p\<close> pb by blast
                 then have "v \<ge> a""b \<in> {v.. b}"  
                   using p(3)[OF \<open>(b, c) \<in> p\<close>] by auto
                 moreover have "{v..b} \<subseteq> ball b db"
                   using fineD[OF \<open>?d fine p\<close> \<open>(b, c) \<in> p\<close>] box_real(2) v False by force
                 ultimately show ?thesis
                   using db v by auto
                 qed
               qed (use ab e in auto)
             qed
             also have "... = e * (b-a)/2"
               by simp
             finally show "norm (\<Sum>(x,k)\<in>p \<inter> ?C.
                         content k *\<^sub>R f' x - (f (Sup k) - f (Inf k))) \<le> e * (b-a)/2" .
           qed
           show "norm (\<Sum>(x, k)\<in>p \<inter> ?A. content k *\<^sub>R f' x - (f ((Sup k)) - f ((Inf k)))) \<le> e * (b-a)/2"
             apply (rule * [OF sum.mono_neutral_right[OF pA(2)]])
             using 1 2 by (auto simp: split_paired_all)
         qed
         also have "... = (\<Sum>n\<in>p. e * (case n of (x, k) \<Rightarrow> Sup k - Inf k))/2"
           unfolding sum_distrib_left[symmetric]
           by (subst additive_tagged_division_1[OF \<open>a \<le> b\<close> ptag]) auto
         finally have norm_le: "norm (\<Sum>(x,K)\<in>p \<inter> {t. fst t \<in> {a, b}}. content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))
                 \<le> (\<Sum>n\<in>p. e * (case n of (x, K) \<Rightarrow> Sup K - Inf K))/2" .
         have le2: "\<And>x s1 s2::real. 0 \<le> s1 \<Longrightarrow> x \<le> (s1 + s2)/2 \<Longrightarrow> x - s1 \<le> s2/2"
           by auto
         show ?thesis
           apply (rule le2 [OF sum_nonneg])
           using ge0 apply force
           by (metis (no_types, lifting) Diff_Diff_Int Diff_subset norm_le p(1) sum.subset_diff)
       qed
       note * = additive_tagged_division_1[OF assms(1) ptag, symmetric]
       have "norm (\<Sum>(x,K)\<in>p \<inter> ?A \<union> (p - ?A). content K *\<^sub>R f' x - (f (Sup K) - f (Inf K)))
                \<le> e * (\<Sum>(x,K)\<in>p \<inter> ?A \<union> (p - ?A). Sup K - Inf K)"
         unfolding sum_distrib_left
         unfolding sum.union_disjoint[OF pA(2-)]
         using le_xz norm_triangle_le I II by blast
       then
       show "norm ((\<Sum>(x,K)\<in>p. content K *\<^sub>R f' x) - (f b - f a)) \<le> e * content {a..b}"
         by (simp only: content_real[OF \<open>a \<le> b\<close>] *[of "\<lambda>x. x"] *[of f] sum_subtractf[symmetric] split_minus pA(1) [symmetric])
     qed
   qed
 qed
 
 
 subsection \<open>Stronger form with finite number of exceptional points\<close>
 
 lemma fundamental_theorem_of_calculus_interior_strong:
  fixes f :: "real \<Rightarrow> 'a::banach"
  assumes "finite S"
    and "a \<le> b" "\<And>x. x \<in> {a <..< b} - S \<Longrightarrow> (f has_vector_derivative f'(x)) (at x)"
    and "continuous_on {a .. b} f"
  shows "(f' has_integral (f b - f a)) {a .. b}"
   using assms
 proof (induction arbitrary: a b)
 case empty
   then show ?case
     using fundamental_theorem_of_calculus_interior by force
 next
 case (insert x S)
   show ?case
   proof (cases "x \<in> {a<..<b}")
     case False then show ?thesis
       using insert by blast
   next
     case True then have "a < x" "x < b"
       by auto
     have "(f' has_integral f x - f a) {a..x}" "(f' has_integral f b - f x) {x..b}"
       using \<open>continuous_on {a..b} f\<close> \<open>a < x\<close> \<open>x < b\<close> continuous_on_subset by (force simp: intro!: insert)+
     then have "(f' has_integral f x - f a + (f b - f x)) {a..b}"
       using \<open>a < x\<close> \<open>x < b\<close> has_integral_combine less_imp_le by blast
     then show ?thesis
       by simp
   qed
 qed
 
 corollary fundamental_theorem_of_calculus_strong:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "finite S"
     and "a \<le> b"
     and vec: "\<And>x. x \<in> {a..b} - S \<Longrightarrow> (f has_vector_derivative f'(x)) (at x)"
     and "continuous_on {a..b} f"
   shows "(f' has_integral (f b - f a)) {a..b}"
   by (rule fundamental_theorem_of_calculus_interior_strong [OF \<open>finite S\<close>]) (force simp: assms)+
 
 proposition indefinite_integral_continuous_left:
   fixes f:: "real \<Rightarrow> 'a::banach"
   assumes intf: "f integrable_on {a..b}" and "a < c" "c \<le> b" "e > 0"
   obtains d where "d > 0"
     and "\<forall>t. c - d < t \<and> t \<le> c \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e"
 proof -
   obtain w where "w > 0" and w: "\<And>t. \<lbrakk>c - w < t; t < c\<rbrakk> \<Longrightarrow> norm (f c) * norm(c - t) < e/3"
   proof (cases "f c = 0")
     case False
     hence e3: "0 < e/3 / norm (f c)" using \<open>e>0\<close> by simp
     moreover have "norm (f c) * norm (c - t) < e/3" 
       if "t < c" and "c - e/3 / norm (f c) < t" for t
     proof -
       have "norm (c - t) < e/3 / norm (f c)"
         using that by auto
       then show "norm (f c) * norm (c - t) < e/3"
         by (metis e3 mult.commute norm_not_less_zero pos_less_divide_eq zero_less_divide_iff)
     qed
     ultimately show ?thesis
       using that by auto
   next
     case True then show ?thesis
       using \<open>e > 0\<close> that by auto
   qed
 
   let ?SUM = "\<lambda>p. (\<Sum>(x,K) \<in> p. content K *\<^sub>R f x)"
   have e3: "e/3 > 0"
     using \<open>e > 0\<close> by auto
   have "f integrable_on {a..c}"
     using \<open>a < c\<close> \<open>c \<le> b\<close> by (auto intro: integrable_subinterval_real[OF intf])
   then obtain d1 where "gauge d1" and d1:
     "\<And>p. \<lbrakk>p tagged_division_of {a..c}; d1 fine p\<rbrakk> \<Longrightarrow> norm (?SUM p - integral {a..c} f) < e/3"
     using integrable_integral has_integral_real e3 by metis
   define d where [abs_def]: "d x = ball x w \<inter> d1 x" for x
   have "gauge d"
     unfolding d_def using \<open>w > 0\<close> \<open>gauge d1\<close> by auto
   then obtain k where "0 < k" and k: "ball c k \<subseteq> d c"
     by (meson gauge_def open_contains_ball)
 
   let ?d = "min k (c - a)/2"
   show thesis
   proof (intro that[of ?d] allI impI, safe)
     show "?d > 0"
       using \<open>0 < k\<close> \<open>a < c\<close> by auto
   next
     fix t
     assume t: "c - ?d < t" "t \<le> c"
     show "norm (integral ({a..c}) f - integral ({a..t}) f) < e"
     proof (cases "t < c")
       case False with \<open>t \<le> c\<close> show ?thesis
         by (simp add: \<open>e > 0\<close>)
     next
       case True
       have "f integrable_on {a..t}"
         using \<open>t < c\<close> \<open>c \<le> b\<close> by (auto intro: integrable_subinterval_real[OF intf])
       then obtain d2 where d2: "gauge d2"
         "\<And>p. p tagged_division_of {a..t} \<and> d2 fine p \<Longrightarrow> norm (?SUM p - integral {a..t} f) < e/3"
         using integrable_integral has_integral_real e3 by metis
       define d3 where "d3 x = (if x \<le> t then d1 x \<inter> d2 x else d1 x)" for x
       have "gauge d3"
         using \<open>gauge d1\<close> \<open>gauge d2\<close> unfolding d3_def gauge_def by auto
       then obtain p where ptag: "p tagged_division_of {a..t}" and pfine: "d3 fine p"
         by (metis box_real(2) fine_division_exists)
       note p' = tagged_division_ofD[OF ptag]
       have pt: "(x,K)\<in>p \<Longrightarrow> x \<le> t" for x K
         by (meson atLeastAtMost_iff p'(2) p'(3) subsetCE)
       with pfine have "d2 fine p"
         unfolding fine_def d3_def by fastforce
       then have d2_fin: "norm (?SUM p - integral {a..t} f) < e/3"
         using d2(2) ptag by auto
       have eqs: "{a..c} \<inter> {x. x \<le> t} = {a..t}" "{a..c} \<inter> {x. x \<ge> t} = {t..c}"
         using t by (auto simp add: field_simps)
       have "p \<union> {(c, {t..c})} tagged_division_of {a..c}"
       proof (intro tagged_division_Un_interval_real)
         show "{(c, {t..c})} tagged_division_of {a..c} \<inter> {x. t \<le> x \<bullet> 1}"
           using \<open>t \<le> c\<close> by (auto simp: eqs tagged_division_of_self_real)
       qed (auto simp: eqs ptag)
       moreover
       have "d1 fine p \<union> {(c, {t..c})}"
         unfolding fine_def
       proof safe
         fix x K y
         assume "(x,K) \<in> p" and "y \<in> K" then show "y \<in> d1 x"
           by (metis Int_iff d3_def subsetD fineD pfine)
       next
         fix x assume "x \<in> {t..c}"
         then have "dist c x < k"
           using t(1) by (auto simp add: field_simps dist_real_def)
         with k show "x \<in> d1 c"
           unfolding d_def by auto
       qed  
       ultimately have d1_fin: "norm (?SUM(p \<union> {(c, {t..c})}) - integral {a..c} f) < e/3"
         using d1 by metis
       have SUMEQ: "?SUM(p \<union> {(c, {t..c})}) = (c - t) *\<^sub>R f c + ?SUM p"
       proof -
         have "?SUM(p \<union> {(c, {t..c})}) = (content{t..c} *\<^sub>R f c) + ?SUM p"
         proof (subst sum.union_disjoint)
           show "p \<inter> {(c, {t..c})} = {}"
             using \<open>t < c\<close> pt by force
         qed (use p'(1) in auto)
         also have "... = (c - t) *\<^sub>R f c + ?SUM p"
           using \<open>t \<le> c\<close> by auto
         finally show ?thesis .
       qed
       have "c - k < t"
         using \<open>k>0\<close> t(1) by (auto simp add: field_simps)
       moreover have "k \<le> w"
       proof (rule ccontr)
         assume "\<not> k \<le> w"
         then have "c + (k + w) / 2 \<notin> d c"
           by (auto simp add: field_simps not_le not_less dist_real_def d_def)
         then have "c + (k + w) / 2 \<notin> ball c k"
           using k by blast
         then show False
           using \<open>0 < w\<close> \<open>\<not> k \<le> w\<close> dist_real_def by auto
       qed
       ultimately have cwt: "c - w < t"
         by (auto simp add: field_simps)
       have eq: "integral {a..c} f - integral {a..t} f = -(((c - t) *\<^sub>R f c + ?SUM p) -
              integral {a..c} f) + (?SUM p - integral {a..t} f) + (c - t) *\<^sub>R f c"
         by auto
       have "norm (integral {a..c} f - integral {a..t} f) < e/3 + e/3 + e/3"
         unfolding eq
       proof (intro norm_triangle_lt add_strict_mono)
         show "norm (- ((c - t) *\<^sub>R f c + ?SUM p - integral {a..c} f)) < e/3"
           by (metis SUMEQ d1_fin norm_minus_cancel)
         show "norm (?SUM p - integral {a..t} f) < e/3"
           using d2_fin by blast
         show "norm ((c - t) *\<^sub>R f c) < e/3"
           using w cwt \<open>t < c\<close> by simp (simp add: field_simps)
       qed
       then show ?thesis by simp
     qed
   qed
 qed
 
 lemma indefinite_integral_continuous_right:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}"
     and "a \<le> c"
     and "c < b"
     and "e > 0"
   obtains d where "0 < d"
     and "\<forall>t. c \<le> t \<and> t < c + d \<longrightarrow> norm (integral {a..c} f - integral {a..t} f) < e"
 proof -
   have intm: "(\<lambda>x. f (- x)) integrable_on {-b .. -a}" "- b < - c" "- c \<le> - a"
     using assms by auto
   from indefinite_integral_continuous_left[OF intm \<open>e>0\<close>]
   obtain d where "0 < d"
     and d: "\<And>t. \<lbrakk>- c - d < t; t \<le> -c\<rbrakk> 
              \<Longrightarrow> norm (integral {- b..- c} (\<lambda>x. f (-x)) - integral {- b..t} (\<lambda>x. f (-x))) < e"
     by metis
   let ?d = "min d (b - c)" 
   show ?thesis
   proof (intro that[of "?d"] allI impI)
     show "0 < ?d"
       using \<open>0 < d\<close> \<open>c < b\<close> by auto
     fix t :: real
     assume t: "c \<le> t \<and> t < c + ?d"
     have *: "integral {a..c} f = integral {a..b} f - integral {c..b} f"
             "integral {a..t} f = integral {a..b} f - integral {t..b} f"
       using assms t by (auto simp: algebra_simps integral_combine)
     have "(- c) - d < (- t)" "- t \<le> - c"
       using t by auto 
     from d[OF this] show "norm (integral {a..c} f - integral {a..t} f) < e"
       by (auto simp add: algebra_simps norm_minus_commute *)
   qed
 qed
 
 lemma indefinite_integral_continuous_1:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}"
   shows "continuous_on {a..b} (\<lambda>x. integral {a..x} f)"
 proof -
   have "\<exists>d>0. \<forall>x'\<in>{a..b}. dist x' x < d \<longrightarrow> dist (integral {a..x'} f) (integral {a..x} f) < e" 
        if x: "x \<in> {a..b}" and "e > 0" for x e :: real
   proof (cases "a = b")
     case True
     with that show ?thesis by force
   next
     case False
     with x have "a < b" by force
     with x consider "x = a" | "x = b" | "a < x" "x < b"
       by force
     then show ?thesis 
     proof cases
       case 1 then show ?thesis
         by (force simp: dist_norm algebra_simps intro: indefinite_integral_continuous_right [OF assms _ \<open>a < b\<close> \<open>e > 0\<close>])
     next
       case 2 then show ?thesis
         by (force simp: dist_norm norm_minus_commute algebra_simps intro: indefinite_integral_continuous_left [OF assms \<open>a < b\<close> _ \<open>e > 0\<close>])
     next
       case 3
       obtain d1 where "0 < d1" 
         and d1: "\<And>t. \<lbrakk>x - d1 < t; t \<le> x\<rbrakk> \<Longrightarrow> norm (integral {a..x} f - integral {a..t} f) < e"
         using 3 by (auto intro: indefinite_integral_continuous_left [OF assms \<open>a < x\<close> _ \<open>e > 0\<close>])
       obtain d2 where "0 < d2" 
         and d2: "\<And>t. \<lbrakk>x \<le> t; t < x + d2\<rbrakk> \<Longrightarrow> norm (integral {a..x} f - integral {a..t} f) < e"
         using 3 by (auto intro: indefinite_integral_continuous_right [OF assms _ \<open>x < b\<close> \<open>e > 0\<close>])
       show ?thesis
       proof (intro exI ballI conjI impI)
         show "0 < min d1 d2"
           using \<open>0 < d1\<close> \<open>0 < d2\<close> by simp
         show "dist (integral {a..y} f) (integral {a..x} f) < e"
              if "y \<in> {a..b}" "dist y x < min d1 d2" for y
         proof (cases "y < x")
           case True
           with that d1 show ?thesis by (auto simp: dist_commute dist_norm)
         next
           case False
           with that d2 show ?thesis
             by (auto simp: dist_commute dist_norm)
         qed
       qed
     qed
   qed
   then show ?thesis
     by (auto simp: continuous_on_iff)
 qed
 
 lemma indefinite_integral_continuous_1':
   fixes f::"real \<Rightarrow> 'a::banach"
   assumes "f integrable_on {a..b}"
   shows "continuous_on {a..b} (\<lambda>x. integral {x..b} f)"
 proof -
   have "integral {a..b} f - integral {a..x} f = integral {x..b} f" if "x \<in> {a..b}" for x
     using integral_combine[OF _ _ assms, of x] that
     by (auto simp: algebra_simps)
   with _ show ?thesis
     by (rule continuous_on_eq) (auto intro!: continuous_intros indefinite_integral_continuous_1 assms)
 qed
 
 theorem integral_has_vector_derivative':
   fixes f :: "real \<Rightarrow> 'b::banach"
   assumes "continuous_on {a..b} f"
     and "x \<in> {a..b}"
   shows "((\<lambda>u. integral {u..b} f) has_vector_derivative - f x) (at x within {a..b})"
 proof -
   have *: "integral {x..b} f = integral {a .. b} f - integral {a .. x} f" if "a \<le> x" "x \<le> b" for x
     using integral_combine[of a x b for x, OF that integrable_continuous_real[OF assms(1)]]
     by (simp add: algebra_simps)
   show ?thesis
     using \<open>x \<in> _\<close> *
     by (rule has_vector_derivative_transform)
       (auto intro!: derivative_eq_intros assms integral_has_vector_derivative)
 qed
 
 lemma integral_has_real_derivative':
   assumes "continuous_on {a..b} g"
   assumes "t \<in> {a..b}"
   shows "((\<lambda>x. integral {x..b} g) has_real_derivative -g t) (at t within {a..b})"
   using integral_has_vector_derivative'[OF assms]
   by (auto simp: has_field_derivative_iff_has_vector_derivative)
 
 
 subsection \<open>This doesn't directly involve integration, but that gives an easy proof\<close>
 
 lemma has_derivative_zero_unique_strong_interval:
   fixes f :: "real \<Rightarrow> 'a::banach"
   assumes "finite k"
     and contf: "continuous_on {a..b} f"
     and "f a = y"
     and fder: "\<And>x. x \<in> {a..b} - k \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within {a..b})"
     and x: "x \<in> {a..b}"
   shows "f x = y"
 proof -
   have "a \<le> b" "a \<le> x"
     using assms by auto
   have "((\<lambda>x. 0::'a) has_integral f x - f a) {a..x}"
   proof (rule fundamental_theorem_of_calculus_interior_strong[OF \<open>finite k\<close> \<open>a \<le> x\<close>]; clarify?)
     have "{a..x} \<subseteq> {a..b}"
       using x by auto
     then show "continuous_on {a..x} f"
       by (rule continuous_on_subset[OF contf])
     show "(f has_vector_derivative 0) (at z)" if z: "z \<in> {a<..<x}" and notin: "z \<notin> k" for z
       unfolding has_vector_derivative_def
     proof (simp add: at_within_open[OF z, symmetric])
       show "(f has_derivative (\<lambda>x. 0)) (at z within {a<..<x})"
         by (rule has_derivative_subset [OF fder]) (use x z notin in auto)
     qed
   qed
   from has_integral_unique[OF has_integral_0 this]
   show ?thesis
     unfolding assms by auto
 qed
 
 
 subsection \<open>Generalize a bit to any convex set\<close>
 
 lemma has_derivative_zero_unique_strong_convex:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "convex S" "finite K"
     and contf: "continuous_on S f"
     and "c \<in> S" "f c = y"
     and derf: "\<And>x. x \<in> S - K \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within S)"
     and "x \<in> S"
   shows "f x = y"
 proof (cases "x = c")
   case True with \<open>f c = y\<close> show ?thesis
     by blast
 next
   case False
   let ?\<phi> = "\<lambda>u. (1 - u) *\<^sub>R c + u *\<^sub>R x"
   have contf': "continuous_on {0 ..1} (f \<circ> ?\<phi>)"
   proof (rule continuous_intros continuous_on_subset[OF contf])+
     show "(\<lambda>u. (1 - u) *\<^sub>R c + u *\<^sub>R x) ` {0..1} \<subseteq> S"
       using \<open>convex S\<close> \<open>x \<in> S\<close> \<open>c \<in> S\<close> by (auto simp add: convex_alt algebra_simps)
   qed
   have "t = u" if "?\<phi> t = ?\<phi> u" for t u
   proof -
     from that have "(t - u) *\<^sub>R x = (t - u) *\<^sub>R c"
       by (auto simp add: algebra_simps)
     then show ?thesis
       using \<open>x \<noteq> c\<close> by auto
   qed
   then have eq: "(SOME t. ?\<phi> t = ?\<phi> u) = u" for u
     by blast
   then have "(?\<phi> -` K) \<subseteq> (\<lambda>z. SOME t. ?\<phi> t = z) ` K"
     by (clarsimp simp: image_iff) (metis (no_types) eq)
   then have fin: "finite (?\<phi> -` K)"
     by (rule finite_surj[OF \<open>finite K\<close>])
 
   have derf': "((\<lambda>u. f (?\<phi> u)) has_derivative (\<lambda>h. 0)) (at t within {0..1})"
                if "t \<in> {0..1} - {t. ?\<phi> t \<in> K}" for t
   proof -
     have df: "(f has_derivative (\<lambda>h. 0)) (at (?\<phi> t) within ?\<phi> ` {0..1})"
       using \<open>convex S\<close> \<open>x \<in> S\<close> \<open>c \<in> S\<close> that 
       by (auto simp add: convex_alt algebra_simps intro: has_derivative_subset [OF derf])
     have "(f \<circ> ?\<phi> has_derivative (\<lambda>x. 0) \<circ> (\<lambda>z. (0 - z *\<^sub>R c) + z *\<^sub>R x)) (at t within {0..1})"
       by (rule derivative_eq_intros df | simp)+
     then show ?thesis
       unfolding o_def .
   qed
   have "(f \<circ> ?\<phi>) 1 = y"
     apply (rule has_derivative_zero_unique_strong_interval[OF fin contf'])
     unfolding o_def using \<open>f c = y\<close> derf' by auto
   then show ?thesis
     by auto
 qed
 
 
 text \<open>Also to any open connected set with finite set of exceptions. Could
  generalize to locally convex set with limpt-free set of exceptions.\<close>
 
 lemma has_derivative_zero_unique_strong_connected:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "connected S"
     and "open S"
     and "finite K"
     and contf: "continuous_on S f"
     and "c \<in> S"
     and "f c = y"
     and derf: "\<And>x. x \<in> S - K \<Longrightarrow> (f has_derivative (\<lambda>h. 0)) (at x within S)"
     and "x \<in> S"
   shows "f x = y"
 proof -
   have "\<exists>e>0. ball x e \<subseteq> (S \<inter> f -` {f x})" if "x \<in> S" for x
   proof -
     obtain e where "0 < e" and e: "ball x e \<subseteq> S"
       using \<open>x \<in> S\<close> \<open>open S\<close> open_contains_ball by blast
     have "ball x e \<subseteq> {u \<in> S. f u \<in> {f x}}"
     proof safe
       fix y
       assume y: "y \<in> ball x e"
       then show "y \<in> S"
         using e by auto
       show "f y = f x"
       proof (rule has_derivative_zero_unique_strong_convex[OF convex_ball \<open>finite K\<close>])
         show "continuous_on (ball x e) f"
           using contf continuous_on_subset e by blast
         show "(f has_derivative (\<lambda>h. 0)) (at u within ball x e)"
              if "u \<in> ball x e - K" for u
           by (metis Diff_iff contra_subsetD derf e has_derivative_subset that)
       qed (use y e \<open>0 < e\<close> in auto)
     qed
     then show "\<exists>e>0. ball x e \<subseteq> (S \<inter> f -` {f x})"
       using \<open>0 < e\<close> by blast
   qed
   then have "openin (top_of_set S) (S \<inter> f -` {y})"
     by (auto intro!: open_openin_trans[OF \<open>open S\<close>] simp: open_contains_ball)
   moreover have "closedin (top_of_set S) (S \<inter> f -` {y})"
     by (force intro!: continuous_closedin_preimage [OF contf])
   ultimately have "(S \<inter> f -` {y}) = {} \<or> (S \<inter> f -` {y}) = S"
     using \<open>connected S\<close> by (simp add: connected_clopen)
   then show ?thesis
     using \<open>x \<in> S\<close> \<open>f c = y\<close> \<open>c \<in> S\<close> by auto
 qed
 
 lemma has_derivative_zero_connected_constant:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "connected S"
       and "open S"
       and "finite k"
       and "continuous_on S f"
       and "\<forall>x\<in>(S - k). (f has_derivative (\<lambda>h. 0)) (at x within S)"
     obtains c where "\<And>x. x \<in> S \<Longrightarrow> f(x) = c"
 proof (cases "S = {}")
   case True
   then show ?thesis
     by (metis empty_iff that)
 next
   case False
   then obtain c where "c \<in> S"
     by (metis equals0I)
   then show ?thesis
     by (metis has_derivative_zero_unique_strong_connected assms that)
 qed
 
 lemma DERIV_zero_connected_constant:
   fixes f :: "'a::{real_normed_field,euclidean_space} \<Rightarrow> 'a"
   assumes "connected S"
       and "open S"
       and "finite K"
       and "continuous_on S f"
       and "\<forall>x\<in>(S - K). DERIV f x :> 0"
     obtains c where "\<And>x. x \<in> S \<Longrightarrow> f(x) = c"
   using has_derivative_zero_connected_constant [OF assms(1-4)] assms
   by (metis DERIV_const has_derivative_const Diff_iff at_within_open frechet_derivative_at has_field_derivative_def)
 
 
 subsection \<open>Integrating characteristic function of an interval\<close>
 
 lemma has_integral_restrict_open_subinterval:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes intf: "(f has_integral i) (cbox c d)"
     and cb: "cbox c d \<subseteq> cbox a b"
   shows "((\<lambda>x. if x \<in> box c d then f x else 0) has_integral i) (cbox a b)"
 proof (cases "cbox c d = {}")
   case True
   then have "box c d = {}"
     by (metis bot.extremum_uniqueI box_subset_cbox)
   then show ?thesis
     using True intf by auto
 next
   case False
   then obtain p where pdiv: "p division_of cbox a b" and inp: "cbox c d \<in> p"
     using cb partial_division_extend_1 by blast
   define g where [abs_def]: "g x = (if x \<in>box c d then f x else 0)" for x
   interpret operative "lift_option plus" "Some (0 :: 'b)"
     "\<lambda>i. if g integrable_on i then Some (integral i g) else None"
     by (fact operative_integralI)
   note operat = division [OF pdiv, symmetric]
   show ?thesis
   proof (cases "(g has_integral i) (cbox a b)")
     case True then show ?thesis
       by (simp add: g_def)
   next
     case False
     have iterate:"F (\<lambda>i. if g integrable_on i then Some (integral i g) else None) (p - {cbox c d}) = Some 0"
     proof (intro neutral ballI)
       fix x
       assume x: "x \<in> p - {cbox c d}"
       then have "x \<in> p"
         by auto
       then obtain u v where uv: "x = cbox u v"
         using pdiv by blast
       have "interior x \<inter> interior (cbox c d) = {}"
         using pdiv inp x by blast 
       then have "(g has_integral 0) x"
         unfolding uv using has_integral_spike_interior[where f="\<lambda>x. 0"]
         by (metis (no_types, lifting) disjoint_iff_not_equal g_def has_integral_0_eq interior_cbox)
       then show "(if g integrable_on x then Some (integral x g) else None) = Some 0"
         by auto
     qed
     interpret comm_monoid_set "lift_option plus" "Some (0 :: 'b)"
       by (intro comm_monoid_set.intro comm_monoid_lift_option add.comm_monoid_axioms)
     have intg: "g integrable_on cbox c d"
       using integrable_spike_interior[where f=f]
       by (meson g_def has_integral_integrable intf)
     moreover have "integral (cbox c d) g = i"
     proof (rule has_integral_unique[OF has_integral_spike_interior intf])
       show "\<And>x. x \<in> box c d \<Longrightarrow> f x = g x"
         by (auto simp: g_def)
       show "(g has_integral integral (cbox c d) g) (cbox c d)"
         by (rule integrable_integral[OF intg])
     qed
     ultimately have "F (\<lambda>A. if g integrable_on A then Some (integral A g) else None) p = Some i"
       by (metis (full_types, lifting) division_of_finite inp iterate pdiv remove right_neutral)
     then
     have "(g has_integral i) (cbox a b)"
       by (metis integrable_on_def integral_unique operat option.inject option.simps(3))
     with False show ?thesis
       by blast
   qed
 qed
 
 
 lemma has_integral_restrict_closed_subinterval:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "(f has_integral i) (cbox c d)"
     and "cbox c d \<subseteq> cbox a b"
   shows "((\<lambda>x. if x \<in> cbox c d then f x else 0) has_integral i) (cbox a b)"
 proof -
   note has_integral_restrict_open_subinterval[OF assms]
   note * = has_integral_spike[OF negligible_frontier_interval _ this]
   show ?thesis
     by (rule *[of c d]) (use box_subset_cbox[of c d] in auto)
 qed
 
 lemma has_integral_restrict_closed_subintervals_eq:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "cbox c d \<subseteq> cbox a b"
   shows "((\<lambda>x. if x \<in> cbox c d then f x else 0) has_integral i) (cbox a b) \<longleftrightarrow> (f has_integral i) (cbox c d)"
   (is "?l = ?r")
 proof (cases "cbox c d = {}")
   case False
   let ?g = "\<lambda>x. if x \<in> cbox c d then f x else 0"
   show ?thesis
   proof 
     assume ?l
     then have "?g integrable_on cbox c d"
       using assms has_integral_integrable integrable_subinterval by blast
     then have "f integrable_on cbox c d"
       by (rule integrable_eq) auto
     moreover then have "i = integral (cbox c d) f"
       by (meson \<open>((\<lambda>x. if x \<in> cbox c d then f x else 0) has_integral i) (cbox a b)\<close> assms has_integral_restrict_closed_subinterval has_integral_unique integrable_integral)
     ultimately show ?r by auto
   next
     assume ?r then show ?l
       by (rule has_integral_restrict_closed_subinterval[OF _ assms])
   qed
 qed auto
 
 
 text \<open>Hence we can apply the limit process uniformly to all integrals.\<close>
 
 lemma has_integral':
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "(f has_integral i) S \<longleftrightarrow>
     (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       (\<exists>z. ((\<lambda>x. if x \<in> S then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - i) < e))"
   (is "?l \<longleftrightarrow> (\<forall>e>0. ?r e)")
 proof (cases "\<exists>a b. S = cbox a b")
   case False then show ?thesis 
     by (simp add: has_integral_alt)
 next
   case True
   then obtain a b where S: "S = cbox a b"
     by blast 
   obtain B where " 0 < B" and B: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm x \<le> B"
     using bounded_cbox[unfolded bounded_pos] by blast
   show ?thesis
   proof safe
     fix e :: real
     assume ?l and "e > 0"
     have "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) (cbox c d)"
       if "ball 0 (B+1) \<subseteq> cbox c d" for c d
         unfolding S using B that
         by (force intro: \<open>?l\<close>[unfolded S] has_integral_restrict_closed_subinterval)
     then show "?r e"
       by (meson \<open>0 < B\<close> \<open>0 < e\<close> add_pos_pos le_less_trans zero_less_one norm_pths(2))
   next
     assume as: "\<forall>e>0. ?r e"
     then obtain C 
       where C: "\<And>a b. ball 0 C \<subseteq> cbox a b \<Longrightarrow>
                 \<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b)"
       by (meson zero_less_one)
     define c :: 'n where "c = (\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)"
     define d :: 'n where "d = (\<Sum>i\<in>Basis. max B C *\<^sub>R i)"
     have "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" if "norm x \<le> B" "i \<in> Basis" for x i
       using that and Basis_le_norm[OF \<open>i\<in>Basis\<close>, of x]
       by (auto simp add: field_simps sum_negf c_def d_def)
     then have c_d: "cbox a b \<subseteq> cbox c d"
       by (meson B mem_box(2) subsetI)
     have "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
       if x: "norm (0 - x) < C" and i: "i \<in> Basis" for x i
         using Basis_le_norm[OF i, of x] x i by (auto simp: sum_negf c_def d_def)
       then have "ball 0 C \<subseteq> cbox c d"
         by (auto simp: mem_box dist_norm)
     with C obtain y where y: "(f has_integral y) (cbox a b)"
       using c_d has_integral_restrict_closed_subintervals_eq S by blast
     have "y = i"
     proof (rule ccontr)
       assume "y \<noteq> i"
       then have "0 < norm (y - i)"
         by auto
       from as[rule_format,OF this]
       obtain C where C: "\<And>a b. ball 0 C \<subseteq> cbox a b \<Longrightarrow> 
            \<exists>z. ((\<lambda>x. if x \<in> S then f x else 0) has_integral z) (cbox a b) \<and> norm (z-i) < norm (y-i)"
         by auto
       define c :: 'n where "c = (\<Sum>i\<in>Basis. (- max B C) *\<^sub>R i)"
       define d :: 'n where "d = (\<Sum>i\<in>Basis. max B C *\<^sub>R i)"
       have "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i"
         if "norm x \<le> B" and "i \<in> Basis" for x i
           using that Basis_le_norm[of i x] by (auto simp add: field_simps sum_negf c_def d_def)
         then have c_d: "cbox a b \<subseteq> cbox c d"
         by (simp add: B mem_box(2) subset_eq)
       have "c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" if "norm (0 - x) < C" and "i \<in> Basis" for x i
         using Basis_le_norm[of i x] that by (auto simp: sum_negf c_def d_def)
       then have "ball 0 C \<subseteq> cbox c d"
         by (auto simp: mem_box dist_norm)
       with C obtain z where z: "(f has_integral z) (cbox a b)" "norm (z-i) < norm (y-i)"
         using has_integral_restrict_closed_subintervals_eq[OF c_d] S by blast
       moreover then have "z = y" 
         by (blast intro: has_integral_unique[OF _ y])
       ultimately show False
         by auto
     qed
     then show ?l
       using y by (auto simp: S)
   qed
 qed
 
 lemma has_integral_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes fg: "(f has_integral i) S" "(g has_integral j) S"
     and le: "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x"
   shows "i \<le> j"
   using has_integral_component_le[OF _ fg, of 1] le  by auto
 
 lemma integral_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "f integrable_on S"
     and "g integrable_on S"
     and "\<And>x. x \<in> S \<Longrightarrow> f x \<le> g x"
   shows "integral S f \<le> integral S g"
   by (rule has_integral_le[OF assms(1,2)[unfolded has_integral_integral] assms(3)])
 
 lemma has_integral_nonneg:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "(f has_integral i) S"
     and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
   shows "0 \<le> i"
   using has_integral_component_nonneg[of 1 f i S]
   unfolding o_def
   using assms
   by auto
 
 lemma integral_nonneg:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes f: "f integrable_on S" and 0: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x"
   shows "0 \<le> integral S f"
   by (rule has_integral_nonneg[OF f[unfolded has_integral_integral] 0])
 
 
 text \<open>Hence a general restriction property.\<close>
 
 lemma has_integral_restrict [simp]:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   assumes "S \<subseteq> T"
   shows "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) T \<longleftrightarrow> (f has_integral i) S"
 proof -
   have *: "\<And>x. (if x \<in> T then if x \<in> S then f x else 0 else 0) =  (if x\<in>S then f x else 0)"
     using assms by auto
   show ?thesis
     apply (subst(2) has_integral')
     apply (subst has_integral')
       apply (simp add: *)
     done
 qed
 
 corollary has_integral_restrict_UNIV:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "((\<lambda>x. if x \<in> s then f x else 0) has_integral i) UNIV \<longleftrightarrow> (f has_integral i) s"
   by auto
 
 lemma has_integral_restrict_Int:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) T \<longleftrightarrow> (f has_integral i) (S \<inter> T)"
 proof -
   have "((\<lambda>x. if x \<in> T then if x \<in> S then f x else 0 else 0) has_integral i) UNIV =
         ((\<lambda>x. if x \<in> S \<inter> T then f x else 0) has_integral i) UNIV"
     by (rule has_integral_cong) auto
   then show ?thesis
     using has_integral_restrict_UNIV by fastforce
 qed
 
 lemma integral_restrict_Int:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "integral T (\<lambda>x. if x \<in> S then f x else 0) = integral (S \<inter> T) f"
   by (metis (no_types, lifting) has_integral_cong has_integral_restrict_Int integrable_integral integral_unique not_integrable_integral)
 
 lemma integrable_restrict_Int:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "(\<lambda>x. if x \<in> S then f x else 0) integrable_on T \<longleftrightarrow> f integrable_on (S \<inter> T)"
   using has_integral_restrict_Int by fastforce
 
 lemma has_integral_on_superset:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "(f has_integral i) S"
       and "\<And>x. x \<notin> S \<Longrightarrow> f x = 0"
       and "S \<subseteq> T"
     shows "(f has_integral i) T"
 proof -
   have "(\<lambda>x. if x \<in> S then f x else 0) = (\<lambda>x. if x \<in> T then f x else 0)"
     using assms by fastforce
   with f show ?thesis
     by (simp only: has_integral_restrict_UNIV [symmetric, of f])
 qed
 
 lemma integrable_on_superset:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on S"
     and "\<And>x. x \<notin> S \<Longrightarrow> f x = 0"
     and "S \<subseteq> t"
   shows "f integrable_on t"
   using assms
   unfolding integrable_on_def
   by (auto intro:has_integral_on_superset)
 
 lemma integral_restrict_UNIV:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "integral UNIV (\<lambda>x. if x \<in> S then f x else 0) = integral S f"
   by (simp add: integral_restrict_Int)
 
 lemma integrable_restrict_UNIV:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "(\<lambda>x. if x \<in> s then f x else 0) integrable_on UNIV \<longleftrightarrow> f integrable_on s"
   unfolding integrable_on_def
   by auto
 
 lemma has_integral_subset_component_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes k: "k \<in> Basis"
     and as: "S \<subseteq> T" "(f has_integral i) S" "(f has_integral j) T" "\<And>x. x\<in>T \<Longrightarrow> 0 \<le> f(x)\<bullet>k"
   shows "i\<bullet>k \<le> j\<bullet>k"
 proof -
   have \<section>: "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) UNIV"
           "((\<lambda>x. if x \<in> T then f x else 0) has_integral j) UNIV"
     by (simp_all add: assms)
   show ?thesis
     using as by (force intro!: has_integral_component_le[OF k \<section>])
 qed
 
 subsection\<open>Integrals on set differences\<close>
 
 lemma has_integral_setdiff:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes S: "(f has_integral i) S" and T: "(f has_integral j) T"
     and neg: "negligible (T - S)"
   shows "(f has_integral (i - j)) (S - T)"
 proof -
   show ?thesis
     unfolding has_integral_restrict_UNIV [symmetric, of f]
   proof (rule has_integral_spike [OF neg])
     have eq: "(\<lambda>x. (if x \<in> S then f x else 0) - (if x \<in> T then f x else 0)) =
             (\<lambda>x. if x \<in> T - S then - f x else if x \<in> S - T then f x else 0)"
       by (force simp add: )
     have "((\<lambda>x. if x \<in> S then f x else 0) has_integral i) UNIV"
       "((\<lambda>x. if x \<in> T then f x else 0) has_integral j) UNIV"
       using S T has_integral_restrict_UNIV by auto
     from has_integral_diff [OF this]
     show "((\<lambda>x. if x \<in> T - S then - f x else if x \<in> S - T then f x else 0)
                    has_integral i-j) UNIV"
       by (simp add: eq)
   qed force
 qed
 
 lemma integral_setdiff:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "f integrable_on S" "f integrable_on T" "negligible(T - S)"
  shows "integral (S - T) f = integral S f - integral T f"
   by (rule integral_unique) (simp add: assms has_integral_setdiff integrable_integral)
 
 lemma integrable_setdiff:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   assumes "(f has_integral i) S" "(f has_integral j) T" "negligible (T - S)"
   shows "f integrable_on (S - T)"
   using has_integral_setdiff [OF assms]
   by (simp add: has_integral_iff)
 
 lemma negligible_setdiff [simp]: "T \<subseteq> S \<Longrightarrow> negligible (T - S)"
   by (metis Diff_eq_empty_iff negligible_empty)
 
 lemma negligible_on_intervals: "negligible s \<longleftrightarrow> (\<forall>a b. negligible(s \<inter> cbox a b))" (is "?l \<longleftrightarrow> ?r")
 proof
   assume R: ?r
   show ?l
     unfolding negligible_def
   proof safe
     fix a b
     have "negligible (s \<inter> cbox a b)"
       by (simp add: R)
     then show "(indicator s has_integral 0) (cbox a b)"
       by (meson Diff_iff Int_iff has_integral_negligible indicator_simps(2))
   qed
 qed (simp add: negligible_Int)
 
 lemma negligible_translation:
   assumes "negligible S"
     shows "negligible ((+) c ` S)"
 proof -
   have inj: "inj ((+) c)"
     by simp
   show ?thesis
   using assms
   proof (clarsimp simp: negligible_def)
     fix a b
     assume "\<forall>x y. (indicator S has_integral 0) (cbox x y)"
     then have *: "(indicator S has_integral 0) (cbox (a-c) (b-c))"
       by (meson Diff_iff assms has_integral_negligible indicator_simps(2))
     have eq: "indicator ((+) c ` S) = (\<lambda>x. indicator S (x - c))"
       by (force simp add: indicator_def)
     show "(indicator ((+) c ` S) has_integral 0) (cbox a b)"
       using has_integral_affinity [OF *, of 1 "-c"]
             cbox_translation [of "c" "-c+a" "-c+b"]
       by (simp add: eq) (simp add: ac_simps)
   qed
 qed
 
 lemma negligible_translation_rev:
   assumes "negligible ((+) c ` S)"
     shows "negligible S"
 by (metis negligible_translation [OF assms, of "-c"] translation_galois)
 
 lemma has_integral_spike_set_eq:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
   shows "(f has_integral y) S \<longleftrightarrow> (f has_integral y) T"
 proof -
   have "((\<lambda>x. if x \<in> S then f x else 0) has_integral y) UNIV =
         ((\<lambda>x. if x \<in> T then f x else 0) has_integral y) UNIV"
   proof (rule has_integral_spike_eq)
     show "negligible ({x \<in> S - T. f x \<noteq> 0} \<union> {x \<in> T - S. f x \<noteq> 0})"
       by (rule negligible_Un [OF assms])
   qed auto
   then show ?thesis
     by (simp add: has_integral_restrict_UNIV)
 qed
 
 corollary integral_spike_set:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
   shows "integral S f = integral T f"
   using has_integral_spike_set_eq [OF assms]
   by (metis eq_integralD integral_unique)
 
 lemma integrable_spike_set:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on S" and neg: "negligible {x \<in> S - T. f x \<noteq> 0}" "negligible {x \<in> T - S. f x \<noteq> 0}"
   shows "f integrable_on T"
   using has_integral_spike_set_eq [OF neg] f by blast
 
 lemma integrable_spike_set_eq:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "negligible ((S - T) \<union> (T - S))"
   shows "f integrable_on S \<longleftrightarrow> f integrable_on T"
   by (blast intro: integrable_spike_set assms negligible_subset)
 
 lemma integrable_on_insert_iff: "f integrable_on (insert x X) \<longleftrightarrow> f integrable_on X"
   for f::"_ \<Rightarrow> 'a::banach"
   by (rule integrable_spike_set_eq) (auto simp: insert_Diff_if)
 
 lemma has_integral_interior:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "negligible(frontier S) \<Longrightarrow> (f has_integral y) (interior S) \<longleftrightarrow> (f has_integral y) S"
   by (rule has_integral_spike_set_eq [OF empty_imp_negligible negligible_subset])
      (use interior_subset in \<open>auto simp: frontier_def closure_def\<close>)
 
 lemma has_integral_closure:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "negligible(frontier S) \<Longrightarrow> (f has_integral y) (closure S) \<longleftrightarrow> (f has_integral y) S"
   by (rule has_integral_spike_set_eq [OF negligible_subset empty_imp_negligible]) (auto simp: closure_Un_frontier )
 
 lemma has_integral_open_interval:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "(f has_integral y) (box a b) \<longleftrightarrow> (f has_integral y) (cbox a b)"
   unfolding interior_cbox [symmetric]
   by (metis frontier_cbox has_integral_interior negligible_frontier_interval)
 
 lemma integrable_on_open_interval:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "f integrable_on box a b \<longleftrightarrow> f integrable_on cbox a b"
   by (simp add: has_integral_open_interval integrable_on_def)
 
 lemma integral_open_interval:
   fixes f :: "'a :: euclidean_space \<Rightarrow> 'b :: banach"
   shows "integral(box a b) f = integral(cbox a b) f"
   by (metis has_integral_integrable_integral has_integral_open_interval not_integrable_integral)
 
 
 subsection \<open>More lemmas that are useful later\<close>
 
 lemma has_integral_subset_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "s \<subseteq> t"
     and "(f has_integral i) s"
     and "(f has_integral j) t"
     and "\<forall>x\<in>t. 0 \<le> f x"
   shows "i \<le> j"
   using has_integral_subset_component_le[OF _ assms(1), of 1 f i j]
   using assms
   by auto
 
 lemma integral_subset_component_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'm::euclidean_space"
   assumes "k \<in> Basis"
     and "s \<subseteq> t"
     and "f integrable_on s"
     and "f integrable_on t"
     and "\<forall>x \<in> t. 0 \<le> f x \<bullet> k"
   shows "(integral s f)\<bullet>k \<le> (integral t f)\<bullet>k"
   by (meson assms has_integral_subset_component_le integrable_integral)
 
 lemma integral_subset_le:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "s \<subseteq> t"
     and "f integrable_on s"
     and "f integrable_on t"
     and "\<forall>x \<in> t. 0 \<le> f x"
   shows "integral s f \<le> integral t f"
   using assms has_integral_subset_le by blast
 
 lemma has_integral_alt':
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "(f has_integral i) s \<longleftrightarrow> 
           (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b) \<and>
           (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
             norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e)"
   (is "?l = ?r")
 proof
   assume rhs: ?r
   show ?l
   proof (subst has_integral', intro allI impI)
     fix e::real
     assume "e > 0"
     from rhs[THEN conjunct2,rule_format,OF this] 
     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
                    (\<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z)
                          (cbox a b) \<and> norm (z - i) < e)"
       by (simp add: has_integral_iff rhs)
   qed
 next
   let ?\<Phi> = "\<lambda>e a b. \<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b) \<and> norm (z - i) < e"
   assume ?l 
   then have lhs: "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> ?\<Phi> e a b" if "e > 0" for e
     using that has_integral'[of f] by auto
   let ?f = "\<lambda>x. if x \<in> s then f x else 0"
   show ?r
   proof (intro conjI allI impI)
     fix a b :: 'n
     from lhs[OF zero_less_one]
     obtain B where "0 < B" and B: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> ?\<Phi> 1 a b"
       by blast
     let ?a = "\<Sum>i\<in>Basis. min (a\<bullet>i) (-B) *\<^sub>R i::'n"
     let ?b = "\<Sum>i\<in>Basis. max (b\<bullet>i) B *\<^sub>R i::'n"
     show "?f integrable_on cbox a b"
     proof (rule integrable_subinterval[of _ ?a ?b])
       have "?a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> ?b \<bullet> i" if "norm (0 - x) < B" "i \<in> Basis" for x i
         using Basis_le_norm[of i x] that by (auto simp add:field_simps)
       then have "ball 0 B \<subseteq> cbox ?a ?b"
         by (auto simp: mem_box dist_norm)
       then show "?f integrable_on cbox ?a ?b"
         unfolding integrable_on_def using B by blast
       show "cbox a b \<subseteq> cbox ?a ?b"
         by (force simp: mem_box)
     qed
   
     fix e :: real
     assume "e > 0"
     with lhs show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
       norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - i) < e"
       by (metis (no_types, lifting) has_integral_integrable_integral)
   qed
 qed
 
 
 subsection \<open>Continuity of the integral (for a 1-dimensional interval)\<close>
 
 lemma integrable_alt:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   shows "f integrable_on s \<longleftrightarrow>
     (\<forall>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b) \<and>
     (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
     norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) -
       integral (cbox c d)  (\<lambda>x. if x \<in> s then f x else 0)) < e)"
   (is "?l = ?r")
 proof
   let ?F = "\<lambda>x. if x \<in> s then f x else 0"
   assume ?l
   then obtain y where intF: "\<And>a b. ?F integrable_on cbox a b"
           and y: "\<And>e. 0 < e \<Longrightarrow>
               \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) ?F - y) < e"
     unfolding integrable_on_def has_integral_alt'[of f] by auto
   show ?r
   proof (intro conjI allI impI intF)
     fix e::real
     assume "e > 0"
     then have "e/2 > 0"
       by auto
     obtain B where "0 < B" 
        and B: "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> norm (integral (cbox a b) ?F - y) < e/2"
       using \<open>0 < e/2\<close> y by blast
     show "\<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
                   norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
     proof (intro conjI exI impI allI, rule \<open>0 < B\<close>)
       fix a b c d::'n
       assume sub: "ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d"
       show "norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
         using sub by (auto intro: norm_triangle_half_l dest: B)
     qed
   qed
 next
   let ?F = "\<lambda>x. if x \<in> s then f x else 0"
   assume rhs: ?r
   let ?cube = "\<lambda>n. cbox (\<Sum>i\<in>Basis. - real n *\<^sub>R i::'n) (\<Sum>i\<in>Basis. real n *\<^sub>R i)"
   have "Cauchy (\<lambda>n. integral (?cube n) ?F)"
     unfolding Cauchy_def
   proof (intro allI impI)
     fix e::real
     assume "e > 0"
     with rhs obtain B where "0 < B" 
       and B: "\<And>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d 
                         \<Longrightarrow> norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e"
       by blast
     obtain N where N: "B \<le> real N"
       using real_arch_simple by blast
     have "ball 0 B \<subseteq> ?cube n" if n: "n \<ge> N" for n
     proof -
       have "sum ((*\<^sub>R) (- real n)) Basis \<bullet> i \<le> x \<bullet> i \<and>
             x \<bullet> i \<le> sum ((*\<^sub>R) (real n)) Basis \<bullet> i"
         if "norm x < B" "i \<in> Basis" for x i::'n
           using Basis_le_norm[of i x] n N that by (auto simp add: field_simps sum_negf)
       then show ?thesis
         by (auto simp: mem_box dist_norm)
     qed
     then show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (integral (?cube m) ?F) (integral (?cube n) ?F) < e"
       by (fastforce simp add: dist_norm intro!: B)
   qed
   then obtain i where i: "(\<lambda>n. integral (?cube n) ?F) \<longlonglongrightarrow> i"
     using convergent_eq_Cauchy by blast
   have "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) ?F - i) < e"
     if "e > 0" for e
   proof -
     have *: "e/2 > 0" using that by auto
     then obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> norm (i - integral (?cube n) ?F) < e/2"
       using i[THEN LIMSEQ_D, simplified norm_minus_commute] by meson
     obtain B where "0 < B" 
       and B: "\<And>a b c d. \<lbrakk>ball 0 B \<subseteq> cbox a b; ball 0 B \<subseteq> cbox c d\<rbrakk> \<Longrightarrow>
                   norm (integral (cbox a b) ?F - integral (cbox c d) ?F) < e/2"
       using rhs * by meson
     let ?B = "max (real N) B"
     show ?thesis
     proof (intro exI conjI allI impI)
       show "0 < ?B"
         using \<open>B > 0\<close> by auto
       fix a b :: 'n
       assume "ball 0 ?B \<subseteq> cbox a b"
       moreover obtain n where n: "max (real N) B \<le> real n"
         using real_arch_simple by blast
       moreover have "ball 0 B \<subseteq> ?cube n"
       proof 
         fix x :: 'n
         assume x: "x \<in> ball 0 B"
         have "\<lbrakk>norm (0 - x) < B; i \<in> Basis\<rbrakk>
               \<Longrightarrow> sum ((*\<^sub>R) (-n)) Basis \<bullet> i\<le> x \<bullet> i \<and> x \<bullet> i \<le> sum ((*\<^sub>R) n) Basis \<bullet> i" for i
           using Basis_le_norm[of i x] n by (auto simp add: field_simps sum_negf)
         then show "x \<in> ?cube n"
           using x by (auto simp: mem_box dist_norm)
       qed 
       ultimately show "norm (integral (cbox a b) ?F - i) < e"
         using norm_triangle_half_l [OF B N] by force
     qed
   qed
   then show ?l unfolding integrable_on_def has_integral_alt'[of f]
     using rhs by blast
 qed
 
 lemma integrable_altD:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on s"
   shows "\<And>a b. (\<lambda>x. if x \<in> s then f x else 0) integrable_on cbox a b"
     and "\<And>e. e > 0 \<Longrightarrow> \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
       norm (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)  (\<lambda>x. if x \<in> s then f x else 0)) < e"
   using assms[unfolded integrable_alt[of f]] by auto
 
 lemma integrable_alt_subset:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::banach"
   shows
      "f integrable_on S \<longleftrightarrow>
       (\<forall>a b. (\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b) \<and>
       (\<forall>e>0. \<exists>B>0. \<forall>a b c d.
                       ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d
                       \<longrightarrow> norm(integral (cbox a b) (\<lambda>x. if x \<in> S then f x else 0) -
                                integral (cbox c d) (\<lambda>x. if x \<in> S then f x else 0)) < e)"
       (is "_ = ?rhs")
 proof -
   let ?g = "\<lambda>x. if x \<in> S then f x else 0"
   have "f integrable_on S \<longleftrightarrow>
         (\<forall>a b. ?g integrable_on cbox a b) \<and>
         (\<forall>e>0. \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
            norm (integral (cbox a b) ?g - integral (cbox c d)  ?g) < e)"
     by (rule integrable_alt)
   also have "\<dots> = ?rhs"
   proof -
     { fix e :: "real"
       assume e: "\<And>e. e>0 \<Longrightarrow> \<exists>B>0. \<forall>a b c d. ball 0 B \<subseteq> cbox a b \<and> cbox a b \<subseteq> cbox c d \<longrightarrow>
                                    norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
         and "e > 0"
       obtain B where "B > 0"
         and B: "\<And>a b c d. \<lbrakk>ball 0 B \<subseteq> cbox a b; cbox a b \<subseteq> cbox c d\<rbrakk> \<Longrightarrow>
                            norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e/2"
         using \<open>e > 0\<close> e [of "e/2"] by force
       have "\<exists>B>0. \<forall>a b c d.
                ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
                norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
       proof (intro exI allI conjI impI)
         fix a b c d :: "'a"
         let ?\<alpha> = "\<Sum>i\<in>Basis. max (a \<bullet> i) (c \<bullet> i) *\<^sub>R i"
         let ?\<beta> = "\<Sum>i\<in>Basis. min (b \<bullet> i) (d \<bullet> i) *\<^sub>R i"
         show "norm (integral (cbox a b) ?g - integral (cbox c d) ?g) < e"
           if ball: "ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d"
         proof -
           have B': "norm (integral (cbox a b \<inter> cbox c d) ?g - integral (cbox x y) ?g) < e/2"
             if "cbox a b \<inter> cbox c d \<subseteq> cbox x y" for x y
             using B [of ?\<alpha> ?\<beta> x y] ball that by (simp add: Int_interval [symmetric])
           show ?thesis
             using B' [of a b] B' [of c d] norm_triangle_half_r by blast
         qed
       qed (use \<open>B > 0\<close> in auto)}
     then show ?thesis
       by force
   qed
   finally show ?thesis .
 qed
 
 lemma integrable_on_subcbox:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes intf: "f integrable_on S"
     and sub: "cbox a b \<subseteq> S"
   shows "f integrable_on cbox a b"
 proof -
   have "(\<lambda>x. if x \<in> S then f x else 0) integrable_on cbox a b"
     by (simp add: intf integrable_altD(1))
 then show ?thesis
   by (metis (mono_tags) sub integrable_restrict_Int le_inf_iff order_refl subset_antisym)
 qed
 
 
 subsection \<open>A straddling criterion for integrability\<close>
 
 lemma integrable_straddle_interval:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "\<And>e. e>0 \<Longrightarrow> \<exists>g h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and>
                             \<bar>i - j\<bar> < e \<and> (\<forall>x\<in>cbox a b. (g x) \<le> f x \<and> f x \<le> h x)"
   shows "f integrable_on cbox a b"
 proof -
   have "\<exists>d. gauge d \<and>
             (\<forall>p1 p2. p1 tagged_division_of cbox a b \<and> d fine p1 \<and>
                      p2 tagged_division_of cbox a b \<and> d fine p2 \<longrightarrow>
                      \<bar>(\<Sum>(x,K)\<in>p1. content K *\<^sub>R f x) - (\<Sum>(x,K)\<in>p2. content K *\<^sub>R f x)\<bar> < e)"
     if "e > 0" for e
   proof -
     have e: "e/3 > 0"
       using that by auto
     then obtain g h i j where ij: "\<bar>i - j\<bar> < e/3"
             and "(g has_integral i) (cbox a b)"
             and "(h has_integral j) (cbox a b)"
             and fgh: "\<And>x. x \<in> cbox a b \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
       using assms real_norm_def by metis
     then obtain d1 d2 where "gauge d1" "gauge d2"
             and d1: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; d1 fine p\<rbrakk> \<Longrightarrow>
                           \<bar>(\<Sum>(x,K)\<in>p. content K *\<^sub>R g x) - i\<bar> < e/3"
             and d2: "\<And>p. \<lbrakk>p tagged_division_of cbox a b; d2 fine p\<rbrakk> \<Longrightarrow>
                           \<bar>(\<Sum>(x,K) \<in> p. content K *\<^sub>R h x) - j\<bar> < e/3"
       by (metis e has_integral real_norm_def)
     have "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R f x) - (\<Sum>(x,K) \<in> p2. content K *\<^sub>R f x)\<bar> < e"
       if p1: "p1 tagged_division_of cbox a b" and 11: "d1 fine p1" and 21: "d2 fine p1"
        and p2: "p2 tagged_division_of cbox a b" and 12: "d1 fine p2" and 22: "d2 fine p2" for p1 p2
     proof -
       have *: "\<And>g1 g2 h1 h2 f1 f2.
                   \<lbrakk>\<bar>g2 - i\<bar> < e/3; \<bar>g1 - i\<bar> < e/3; \<bar>h2 - j\<bar> < e/3; \<bar>h1 - j\<bar> < e/3;
                    g1 - h2 \<le> f1 - f2; f1 - f2 \<le> h1 - g2\<rbrakk>
                   \<Longrightarrow> \<bar>f1 - f2\<bar> < e"
         using \<open>e > 0\<close> ij by arith
       have 0: "(\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R g x) \<ge> 0"
               "0 \<le> (\<Sum>(x, k)\<in>p2. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)"
               "(\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R g x) \<ge> 0"
               "0 \<le> (\<Sum>(x, k)\<in>p1. content k *\<^sub>R h x) - (\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x)"
         unfolding sum_subtractf[symmetric]
            apply (auto intro!: sum_nonneg)
            apply (meson fgh measure_nonneg mult_left_mono tag_in_interval that sum_nonneg)+
         done
       show ?thesis
       proof (rule *)
         show "\<bar>(\<Sum>(x,K) \<in> p2. content K *\<^sub>R g x) - i\<bar> < e/3"
           by (rule d1[OF p2 12])
         show "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R g x) - i\<bar> < e/3"
           by (rule d1[OF p1 11])
         show "\<bar>(\<Sum>(x,K) \<in> p2. content K *\<^sub>R h x) - j\<bar> < e/3"
           by (rule d2[OF p2 22])
         show "\<bar>(\<Sum>(x,K) \<in> p1. content K *\<^sub>R h x) - j\<bar> < e/3"
           by (rule d2[OF p1 21])
       qed (use 0 in auto)
     qed
     then show ?thesis
       by (rule_tac x="\<lambda>x. d1 x \<inter> d2 x" in exI)
         (auto simp: fine_Int intro: \<open>gauge d1\<close> \<open>gauge d2\<close> d1 d2)
   qed
   then show ?thesis
     by (simp add: integrable_Cauchy)
 qed
 
 lemma integrable_straddle:
   fixes f :: "'n::euclidean_space \<Rightarrow> real"
   assumes "\<And>e. e>0 \<Longrightarrow> \<exists>g h i j. (g has_integral i) s \<and> (h has_integral j) s \<and>
                      \<bar>i - j\<bar> < e \<and> (\<forall>x\<in>s. g x \<le> f x \<and> f x \<le> h x)"
   shows "f integrable_on s"
 proof -
   let ?fs = "(\<lambda>x. if x \<in> s then f x else 0)"
   have "?fs integrable_on cbox a b" for a b
   proof (rule integrable_straddle_interval)
     fix e::real
     assume "e > 0"
     then have *: "e/4 > 0"
       by auto
     with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s"
                  and ij: "\<bar>i - j\<bar> < e/4"
                  and fgh: "\<And>x. x \<in> s \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
       by metis
     let ?gs = "(\<lambda>x. if x \<in> s then g x else 0)"
     let ?hs = "(\<lambda>x. if x \<in> s then h x else 0)"
     obtain Bg where Bg: "\<And>a b. ball 0 Bg \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?gs - i\<bar> < e/4"
               and int_g: "\<And>a b. ?gs integrable_on cbox a b"
       using g * unfolding has_integral_alt' real_norm_def by meson
     obtain Bh where
           Bh: "\<And>a b. ball 0 Bh \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?hs - j\<bar> < e/4"
          and int_h: "\<And>a b. ?hs integrable_on cbox a b"
       using h * unfolding has_integral_alt' real_norm_def by meson
     define c where "c = (\<Sum>i\<in>Basis. min (a\<bullet>i) (- (max Bg Bh)) *\<^sub>R i)"
     define d where "d = (\<Sum>i\<in>Basis. max (b\<bullet>i) (max Bg Bh) *\<^sub>R i)"
     have "\<lbrakk>norm (0 - x) < Bg; i \<in> Basis\<rbrakk> \<Longrightarrow> c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" for x i
       using Basis_le_norm[of i x] unfolding c_def d_def by auto
     then have ballBg: "ball 0 Bg \<subseteq> cbox c d"
       by (auto simp: mem_box dist_norm)
     have "\<lbrakk>norm (0 - x) < Bh; i \<in> Basis\<rbrakk> \<Longrightarrow> c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i" for x i
       using Basis_le_norm[of i x] unfolding c_def d_def by auto
     then have ballBh: "ball 0 Bh \<subseteq> cbox c d"
       by (auto simp: mem_box dist_norm)
     have ab_cd: "cbox a b \<subseteq> cbox c d"
       by (auto simp: c_def d_def subset_box_imp)
     have **: "\<And>ch cg ag ah::real. \<lbrakk>\<bar>ah - ag\<bar> \<le> \<bar>ch - cg\<bar>; \<bar>cg - i\<bar> < e/4; \<bar>ch - j\<bar> < e/4\<rbrakk>
        \<Longrightarrow> \<bar>ag - ah\<bar> < e"
       using ij by arith
     show "\<exists>g h i j. (g has_integral i) (cbox a b) \<and> (h has_integral j) (cbox a b) \<and> \<bar>i - j\<bar> < e \<and>
           (\<forall>x\<in>cbox a b. g x \<le> (if x \<in> s then f x else 0) \<and>
                         (if x \<in> s then f x else 0) \<le> h x)"
     proof (intro exI ballI conjI)
       have eq: "\<And>x f g. (if x \<in> s then f x else 0) - (if x \<in> s then g x else 0) =
                        (if x \<in> s then f x - g x else (0::real))"
         by auto
       have int_hg: "(\<lambda>x. if x \<in> s then h x - g x else 0) integrable_on cbox a b"
                    "(\<lambda>x. if x \<in> s then h x - g x else 0) integrable_on cbox c d"
         by (metis (no_types) integrable_diff g h has_integral_integrable integrable_altD(1))+
       show "(?gs has_integral integral (cbox a b) ?gs) (cbox a b)"
            "(?hs has_integral integral (cbox a b) ?hs) (cbox a b)"
         by (intro integrable_integral int_g int_h)+
       then have "integral (cbox a b) ?gs \<le> integral (cbox a b) ?hs"
         using fgh by (force intro: has_integral_le)
       then have "0 \<le> integral (cbox a b) ?hs - integral (cbox a b) ?gs"
         by simp
       then have "\<bar>integral (cbox a b) ?hs - integral (cbox a b) ?gs\<bar>
               \<le> \<bar>integral (cbox c d) ?hs - integral (cbox c d) ?gs\<bar>"
         apply (simp add: integral_diff [symmetric] int_g int_h)
         apply (subst abs_of_nonneg[OF integral_nonneg[OF integrable_diff, OF int_h int_g]])
         using fgh apply (force simp: eq intro!: integral_subset_le [OF ab_cd int_hg])+
         done
       then show "\<bar>integral (cbox a b) ?gs - integral (cbox a b) ?hs\<bar> < e"
         using ** Bg ballBg Bh ballBh by blast
       show "\<And>x. x \<in> cbox a b \<Longrightarrow> ?gs x \<le> ?fs x" "\<And>x. x \<in> cbox a b \<Longrightarrow> ?fs x \<le> ?hs x"
         using fgh by auto
     qed
   qed
   then have int_f: "?fs integrable_on cbox a b" for a b
     by simp
   have "\<exists>B>0. \<forall>a b c d.
                   ball 0 B \<subseteq> cbox a b \<and> ball 0 B \<subseteq> cbox c d \<longrightarrow>
                   abs (integral (cbox a b) ?fs - integral (cbox c d) ?fs) < e"
       if "0 < e" for e
   proof -
     have *: "e/3 > 0"
       using that by auto
     with assms obtain g h i j where g: "(g has_integral i) s" and h: "(h has_integral j) s"
                  and ij: "\<bar>i - j\<bar> < e/3"
                  and fgh: "\<And>x. x \<in> s \<Longrightarrow> g x \<le> f x \<and> f x \<le> h x"
       by metis
     let ?gs = "(\<lambda>x. if x \<in> s then g x else 0)"
     let ?hs = "(\<lambda>x. if x \<in> s then h x else 0)"
     obtain Bg where "Bg > 0"
               and Bg: "\<And>a b. ball 0 Bg \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?gs - i\<bar> < e/3"
               and int_g: "\<And>a b. ?gs integrable_on cbox a b"
       using g * unfolding has_integral_alt' real_norm_def by meson
     obtain Bh where "Bh > 0"
               and Bh: "\<And>a b. ball 0 Bh \<subseteq> cbox a b \<Longrightarrow> \<bar>integral (cbox a b) ?hs - j\<bar> < e/3"
               and int_h: "\<And>a b. ?hs integrable_on cbox a b"
       using h * unfolding has_integral_alt' real_norm_def by meson
     { fix a b c d :: 'n
       assume as: "ball 0 (max Bg Bh) \<subseteq> cbox a b" "ball 0 (max Bg Bh) \<subseteq> cbox c d"
       have **: "ball 0 Bg \<subseteq> ball (0::'n) (max Bg Bh)" "ball 0 Bh \<subseteq> ball (0::'n) (max Bg Bh)"
         by auto
       have *: "\<And>ga gc ha hc fa fc. \<lbrakk>\<bar>ga - i\<bar> < e/3; \<bar>gc - i\<bar> < e/3; \<bar>ha - j\<bar> < e/3;
                      \<bar>hc - j\<bar> < e/3; ga \<le> fa; fa \<le> ha; gc \<le> fc; fc \<le> hc\<rbrakk> \<Longrightarrow>
         \<bar>fa - fc\<bar> < e"
         using ij by arith
       have "abs (integral (cbox a b) (\<lambda>x. if x \<in> s then f x else 0) - integral (cbox c d)
         (\<lambda>x. if x \<in> s then f x else 0)) < e"
       proof (rule *)
         show "\<bar>integral (cbox a b) ?gs - i\<bar> < e/3"
           using "**" Bg as by blast
         show "\<bar>integral (cbox c d) ?gs - i\<bar> < e/3"
           using "**" Bg as by blast
         show "\<bar>integral (cbox a b) ?hs - j\<bar> < e/3"
           using "**" Bh as by blast
         show "\<bar>integral (cbox c d) ?hs - j\<bar> < e/3"
           using "**" Bh as by blast
       qed (use int_f int_g int_h fgh in \<open>simp_all add: integral_le\<close>)
     }
     then show ?thesis
       apply (rule_tac x="max Bg Bh" in exI)
         using \<open>Bg > 0\<close> by auto
   qed
   then show ?thesis
     unfolding integrable_alt[of f] real_norm_def by (blast intro: int_f)
 qed
 
 
 subsection \<open>Adding integrals over several sets\<close>
 
 lemma has_integral_Un:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "(f has_integral i) S" "(f has_integral j) T"
     and neg: "negligible (S \<inter> T)"
   shows "(f has_integral (i + j)) (S \<union> T)"
   unfolding has_integral_restrict_UNIV[symmetric, of f]
 proof (rule has_integral_spike[OF neg])
   let ?f = "\<lambda>x. (if x \<in> S then f x else 0) + (if x \<in> T then f x else 0)"
   show "(?f has_integral i + j) UNIV"
     by (simp add: f has_integral_add)
 qed auto
 
 lemma integral_Un [simp]:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on S" "f integrable_on T" "negligible (S \<inter> T)"
   shows "integral (S \<union> T) f = integral S f + integral T f"
   by (simp add: has_integral_Un assms integrable_integral integral_unique)
 
 lemma integrable_Un:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: banach"
   assumes "negligible (A \<inter> B)" "f integrable_on A" "f integrable_on B"
   shows   "f integrable_on (A \<union> B)"
 proof -
   from assms obtain y z where "(f has_integral y) A" "(f has_integral z) B"
      by (auto simp: integrable_on_def)
   from has_integral_Un[OF this assms(1)] show ?thesis by (auto simp: integrable_on_def)
 qed
 
 lemma integrable_Un':
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b :: banach"
   assumes "f integrable_on A" "f integrable_on B" "negligible (A \<inter> B)" "C = A \<union> B"
   shows   "f integrable_on C"
   using integrable_Un[of A B f] assms by simp
 
 lemma has_integral_Union:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes \<T>: "finite \<T>"
     and int: "\<And>S. S \<in> \<T> \<Longrightarrow> (f has_integral (i S)) S"
     and neg: "pairwise (\<lambda>S S'. negligible (S \<inter> S')) \<T>"
   shows "(f has_integral (sum i \<T>)) (\<Union>\<T>)"
 proof -
   let ?\<U> = "((\<lambda>(a,b). a \<inter> b) ` {(a,b). a \<in> \<T> \<and> b \<in> {y. y \<in> \<T> \<and> a \<noteq> y}})"
   have "((\<lambda>x. if x \<in> \<Union>\<T> then f x else 0) has_integral sum i \<T>) UNIV"
   proof (rule has_integral_spike)
     show "negligible (\<Union>?\<U>)"
     proof (rule negligible_Union)
       have "finite (\<T> \<times> \<T>)"
         by (simp add: \<T>)
       moreover have "{(a, b). a \<in> \<T> \<and> b \<in> {y \<in> \<T>. a \<noteq> y}} \<subseteq> \<T> \<times> \<T>"
         by auto
       ultimately show "finite ?\<U>"
         by (blast intro: finite_subset[of _ "\<T> \<times> \<T>"])
       show "\<And>t. t \<in> ?\<U> \<Longrightarrow> negligible t"
         using neg unfolding pairwise_def by auto
     qed
   next
     show "(if x \<in> \<Union>\<T> then f x else 0) = (\<Sum>A\<in>\<T>. if x \<in> A then f x else 0)"
       if "x \<in> UNIV - (\<Union>?\<U>)" for x
     proof clarsimp
       fix S assume "S \<in> \<T>" "x \<in> S"
       moreover then have "\<forall>b\<in>\<T>. x \<in> b \<longleftrightarrow> b = S"
         using that by blast
       ultimately show "f x = (\<Sum>A\<in>\<T>. if x \<in> A then f x else 0)"
         by (simp add: sum.delta[OF \<T>])
     qed
   next
     show "((\<lambda>x. \<Sum>A\<in>\<T>. if x \<in> A then f x else 0) has_integral (\<Sum>A\<in>\<T>. i A)) UNIV"
       using int by (simp add: has_integral_restrict_UNIV has_integral_sum [OF \<T>])
   qed
   then show ?thesis
     using has_integral_restrict_UNIV by blast
 qed
 
 
 text \<open>In particular adding integrals over a division, maybe not of an interval.\<close>
 
 lemma has_integral_combine_division:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "\<D> division_of S"
     and "\<And>k. k \<in> \<D> \<Longrightarrow> (f has_integral (i k)) k"
   shows "(f has_integral (sum i \<D>)) S"
 proof -
   note \<D> = division_ofD[OF assms(1)]
   have neg: "negligible (S \<inter> s')" if "S \<in> \<D>" "s' \<in> \<D>" "S \<noteq> s'" for S s'
   proof -
     obtain a c b \<D> where obt: "S = cbox a b" "s' = cbox c \<D>"
       by (meson \<open>S \<in> \<D>\<close> \<open>s' \<in> \<D>\<close> \<D>(4))
     from \<D>(5)[OF that] show ?thesis
       unfolding obt interior_cbox
       by (metis (no_types, lifting) Diff_empty Int_interval box_Int_box negligible_frontier_interval)
   qed
   show ?thesis
     unfolding \<D>(6)[symmetric]
     by (auto intro: \<D> neg assms has_integral_Union pairwiseI)
 qed
 
 lemma integral_combine_division_bottomup:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "\<D> division_of S" "\<And>k. k \<in> \<D> \<Longrightarrow> f integrable_on k"
   shows "integral S f = sum (\<lambda>i. integral i f) \<D>"
   by (meson assms integral_unique has_integral_combine_division has_integral_integrable_integral)
 
 lemma has_integral_combine_division_topdown:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on S"
     and \<D>: "\<D> division_of K"
     and "K \<subseteq> S"
   shows "(f has_integral (sum (\<lambda>i. integral i f) \<D>)) K"
 proof -
   have "f integrable_on L" if "L \<in> \<D>" for L
   proof -
     have "L \<subseteq> S"
       using \<open>K \<subseteq> S\<close> \<D> that by blast
     then show "f integrable_on L"
       using that by (metis (no_types) f \<D> division_ofD(4) integrable_on_subcbox)
   qed
   then show ?thesis
     by (meson \<D> has_integral_combine_division has_integral_integrable_integral)
 qed
 
 lemma integral_combine_division_topdown:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on S"
     and "\<D> division_of S"
   shows "integral S f = sum (\<lambda>i. integral i f) \<D>"
   using assms has_integral_combine_division_topdown by blast
 
 lemma integrable_combine_division:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes \<D>: "\<D> division_of S"
     and f: "\<And>i. i \<in> \<D> \<Longrightarrow> f integrable_on i"
   shows "f integrable_on S"
   using f unfolding integrable_on_def by (metis has_integral_combine_division[OF \<D>])
 
 lemma integrable_on_subdivision:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes \<D>: "\<D> division_of i"
     and f: "f integrable_on S"
     and "i \<subseteq> S"
   shows "f integrable_on i"
 proof -
   have "f integrable_on i" if "i \<in> \<D>" for i
 proof -
   have "i \<subseteq> S"
     using assms that by auto
   then show "f integrable_on i"
     using that by (metis (no_types) \<D> f division_ofD(4) integrable_on_subcbox)
 qed
   then show ?thesis
     using \<D> integrable_combine_division by blast
 qed
 
 
 subsection \<open>Also tagged divisions\<close>
 
 lemma has_integral_combine_tagged_division:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "p tagged_division_of S"
     and "\<And>x k. (x,k) \<in> p \<Longrightarrow> (f has_integral (i k)) k"
   shows "(f has_integral (\<Sum>(x,k)\<in>p. i k)) S"
 proof -
   have *: "(f has_integral (\<Sum>k\<in>snd`p. integral k f)) S"
   proof -
     have "snd ` p division_of S"
       by (simp add: assms(1) division_of_tagged_division)
     with assms show ?thesis
       by (metis (mono_tags, lifting) has_integral_combine_division has_integral_integrable_integral imageE prod.collapse)
   qed
   also have "(\<Sum>k\<in>snd`p. integral k f) = (\<Sum>(x, k)\<in>p. integral k f)"
     by (intro sum.over_tagged_division_lemma[OF assms(1), symmetric] integral_null)
        (simp add: content_eq_0_interior)
   finally show ?thesis
     using assms by (auto simp add: has_integral_iff intro!: sum.cong)
 qed
 
 lemma integral_combine_tagged_division_bottomup:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes p: "p tagged_division_of (cbox a b)"
     and f: "\<And>x k. (x,k)\<in>p \<Longrightarrow> f integrable_on k"
   shows "integral (cbox a b) f = sum (\<lambda>(x,k). integral k f) p"
   by (simp add: has_integral_combine_tagged_division[OF p] integral_unique f integrable_integral)
 
 lemma has_integral_combine_tagged_division_topdown:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes f: "f integrable_on cbox a b"
     and p: "p tagged_division_of (cbox a b)"
   shows "(f has_integral (sum (\<lambda>(x,K). integral K f) p)) (cbox a b)"
 proof -
   have "(f has_integral integral K f) K" if "(x,K) \<in> p" for x K
     by (metis assms integrable_integral integrable_on_subcbox tagged_division_ofD(3,4) that)
   then show ?thesis
     by (simp add: has_integral_combine_tagged_division p)
 qed
 
 lemma integral_combine_tagged_division_topdown:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on cbox a b"
     and "p tagged_division_of (cbox a b)"
   shows "integral (cbox a b) f = sum (\<lambda>(x,k). integral k f) p"
   using assms by (auto intro: integral_unique [OF has_integral_combine_tagged_division_topdown])
 
 
 subsection \<open>Henstock's lemma\<close>
 
 lemma Henstock_lemma_part1:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes intf: "f integrable_on cbox a b"
     and "e > 0"
     and "gauge d"
     and less_e: "\<And>p. \<lbrakk>p tagged_division_of (cbox a b); d fine p\<rbrakk> \<Longrightarrow>
                      norm (sum (\<lambda>(x,K). content K *\<^sub>R f x) p - integral(cbox a b) f) < e"
     and p: "p tagged_partial_division_of (cbox a b)" "d fine p"
   shows "norm (sum (\<lambda>(x,K). content K *\<^sub>R f x - integral K f) p) \<le> e" (is "?lhs \<le> e")
 proof (rule field_le_epsilon)
   fix k :: real
   assume "k > 0"
   let ?SUM = "\<lambda>p. (\<Sum>(x,K) \<in> p. content K *\<^sub>R f x)"
   note p' = tagged_partial_division_ofD[OF p(1)]
   have "\<Union>(snd ` p) \<subseteq> cbox a b"
     using p'(3) by fastforce
   then obtain q where q: "snd ` p \<subseteq> q" and qdiv: "q division_of cbox a b"
     by (meson p(1) partial_division_extend_interval partial_division_of_tagged_division)
   note q' = division_ofD[OF qdiv]
   define r where "r = q - snd ` p"
   have "snd ` p \<inter> r = {}"
     unfolding r_def by auto
   have "finite r"
     using q' unfolding r_def by auto
   have "\<exists>p. p tagged_division_of i \<and> d fine p \<and>
         norm (?SUM p - integral i f) < k / (real (card r) + 1)"
     if "i\<in>r" for i
   proof -
     have gt0: "k / (real (card r) + 1) > 0" using \<open>k > 0\<close> by simp
     have i: "i \<in> q"
       using that unfolding r_def by auto
     then obtain u v where uv: "i = cbox u v"
       using q'(4) by blast
     then have "cbox u v \<subseteq> cbox a b"
       using i q'(2) by auto  
     then have "f integrable_on cbox u v"
       by (rule integrable_subinterval[OF intf])
     with integrable_integral[OF this, unfolded has_integral[of f]]
     obtain dd where "gauge dd" and dd:
       "\<And>\<D>. \<lbrakk>\<D> tagged_division_of cbox u v; dd fine \<D>\<rbrakk> \<Longrightarrow>
     norm (?SUM \<D> - integral (cbox u v) f) < k / (real (card r) + 1)"
       using gt0 by auto
     with gauge_Int[OF \<open>gauge d\<close> \<open>gauge dd\<close>]
     obtain qq where qq: "qq tagged_division_of cbox u v" "(\<lambda>x. d x \<inter> dd x) fine qq"
       using fine_division_exists by blast
     with dd[of qq]  show ?thesis
       by (auto simp: fine_Int uv)
   qed
   then obtain qq where qq: "\<And>i. i \<in> r \<Longrightarrow> qq i tagged_division_of i \<and>
       d fine qq i \<and> norm (?SUM (qq i) - integral i f) < k / (real (card r) + 1)"
     by metis
 
   let ?p = "p \<union> \<Union>(qq ` r)"
   have "norm (?SUM ?p - integral (cbox a b) f) < e"
   proof (rule less_e)
     show "d fine ?p"
       by (metis (mono_tags, hide_lams) qq fine_Un fine_Union imageE p(2))
     note ptag = tagged_partial_division_of_Union_self[OF p(1)]
     have "p \<union> \<Union>(qq ` r) tagged_division_of \<Union>(snd ` p) \<union> \<Union>r"
     proof (rule tagged_division_Un[OF ptag tagged_division_Union [OF \<open>finite r\<close>]])
       show "\<And>i. i \<in> r \<Longrightarrow> qq i tagged_division_of i"
         using qq by auto
       show "\<And>i1 i2. \<lbrakk>i1 \<in> r; i2 \<in> r; i1 \<noteq> i2\<rbrakk> \<Longrightarrow> interior i1 \<inter> interior i2 = {}"
         by (simp add: q'(5) r_def)
       show "interior (\<Union>(snd ` p)) \<inter> interior (\<Union>r) = {}"
       proof (rule Int_interior_Union_intervals [OF \<open>finite r\<close>])
         show "open (interior (\<Union>(snd ` p)))"
           by blast
         show "\<And>T. T \<in> r \<Longrightarrow> \<exists>a b. T = cbox a b"
           by (simp add: q'(4) r_def)
         have "interior T \<inter> interior (\<Union>(snd ` p)) = {}" if "T \<in> r" for T
         proof (rule Int_interior_Union_intervals)
           show "\<And>U. U \<in> snd ` p \<Longrightarrow> \<exists>a b. U = cbox a b"
             using q q'(4) by blast
           show "\<And>U. U \<in> snd ` p \<Longrightarrow> interior T \<inter> interior U = {}"
             by (metis DiffE q q'(5) r_def subsetD that)
         qed (use p' in auto)
         then show "\<And>T. T \<in> r \<Longrightarrow> interior (\<Union>(snd ` p)) \<inter> interior T = {}"
           by (metis Int_commute)
       qed
     qed
     moreover have "\<Union>(snd ` p) \<union> \<Union>r = cbox a b" and "{qq i |i. i \<in> r} = qq ` r"
       using qdiv q unfolding Union_Un_distrib[symmetric] r_def by auto
     ultimately show "?p tagged_division_of (cbox a b)"
       by fastforce
   qed
   then have "norm (?SUM p + (?SUM (\<Union>(qq ` r))) - integral (cbox a b) f) < e"
   proof (subst sum.union_inter_neutral[symmetric, OF \<open>finite p\<close>], safe)
     show "content L *\<^sub>R f x = 0" if "(x, L) \<in> p" "(x, L) \<in> qq K" "K \<in> r" for x K L 
     proof -
       obtain u v where uv: "L = cbox u v"
         using \<open>(x,L) \<in> p\<close> p'(4) by blast
       have "L \<subseteq> K"
         using  qq[OF that(3)] tagged_division_ofD(3) \<open>(x,L) \<in> qq K\<close> by metis
       have "L \<in> snd ` p" 
         using \<open>(x,L) \<in> p\<close> image_iff by fastforce 
       then have "L \<in> q" "K \<in> q" "L \<noteq> K"
         using that(1,3) q(1) unfolding r_def by auto
       with q'(5) have "interior L = {}"
         using interior_mono[OF \<open>L \<subseteq> K\<close>] by blast
       then show "content L *\<^sub>R f x = 0"
         unfolding uv content_eq_0_interior[symmetric] by auto
     qed
     show "finite (\<Union>(qq ` r))"
       by (meson finite_UN qq \<open>finite r\<close> tagged_division_of_finite)
   qed
   moreover have "content M *\<^sub>R f x = 0" 
       if x: "(x,M) \<in> qq K" "(x,M) \<in> qq L" and KL: "qq K \<noteq> qq L" and r: "K \<in> r" "L \<in> r"
     for x M K L
   proof -
     note kl = tagged_division_ofD(3,4)[OF qq[THEN conjunct1]]
     obtain u v where uv: "M = cbox u v"
       using \<open>(x, M) \<in> qq L\<close> \<open>L \<in> r\<close> kl(2) by blast
     have empty: "interior (K \<inter> L) = {}"
       by (metis DiffD1 interior_Int q'(5) r_def KL r)
     have "interior M = {}"
       by (metis (no_types, lifting) Int_assoc empty inf.absorb_iff2 interior_Int kl(1) subset_empty x r)
     then show "content M *\<^sub>R f x = 0"
       unfolding uv content_eq_0_interior[symmetric]
       by auto
   qed 
   ultimately have "norm (?SUM p + sum ?SUM (qq ` r) - integral (cbox a b) f) < e"
     apply (subst (asm) sum.Union_comp)
     using qq by (force simp: split_paired_all)+
   moreover have "content M *\<^sub>R f x = 0" 
        if "K \<in> r" "L \<in> r" "K \<noteq> L" "qq K = qq L" "(x, M) \<in> qq K" for K L x M
     using tagged_division_ofD(6) qq that by (metis (no_types, lifting)) 
   ultimately have less_e: "norm (?SUM p + sum (?SUM \<circ> qq) r - integral (cbox a b) f) < e"
   proof (subst (asm) sum.reindex_nontrivial [OF \<open>finite r\<close>])
     qed (auto simp: split_paired_all sum.neutral)
   have norm_le: "norm (cp - ip) \<le> e + k"
                   if "norm ((cp + cr) - i) < e" "norm (cr - ir) < k" "ip + ir = i"
                   for ir ip i cr cp::'a
   proof -
     from that show ?thesis
       using norm_triangle_le[of "cp + cr - i" "- (cr - ir)"]
       unfolding that(3)[symmetric] norm_minus_cancel
       by (auto simp add: algebra_simps)
   qed
 
   have "?lhs =  norm (?SUM p - (\<Sum>(x, k)\<in>p. integral k f))"
     unfolding split_def sum_subtractf ..
   also have "\<dots> \<le> e + k"
   proof (rule norm_le[OF less_e])
     have lessk: "k * real (card r) / (1 + real (card r)) < k"
       using \<open>k>0\<close> by (auto simp add: field_simps)
     have "norm (sum (?SUM \<circ> qq) r - (\<Sum>k\<in>r. integral k f)) \<le> (\<Sum>x\<in>r. k / (real (card r) + 1))"
       unfolding sum_subtractf[symmetric] by (force dest: qq intro!: sum_norm_le)
     also have "... < k"
       by (simp add: lessk add.commute mult.commute)
     finally show "norm (sum (?SUM \<circ> qq) r - (\<Sum>k\<in>r. integral k f)) < k" .
   next
     from q(1) have [simp]: "snd ` p \<union> q = q" by auto
     have "integral l f = 0"
       if inp: "(x, l) \<in> p" "(y, m) \<in> p" and ne: "(x, l) \<noteq> (y, m)" and "l = m" for x l y m
     proof -
       obtain u v where uv: "l = cbox u v"
         using inp p'(4) by blast
       have "content (cbox u v) = 0"
-        unfolding content_eq_0_interior using that p(1) uv by auto
+        unfolding content_eq_0_interior using that p(1) uv
+        by (auto dest: tagged_partial_division_ofD)
       then show ?thesis
         using uv by blast
     qed
     then have "(\<Sum>(x, K)\<in>p. integral K f) = (\<Sum>K\<in>snd ` p. integral K f)"
       apply (subst sum.reindex_nontrivial [OF \<open>finite p\<close>])
       unfolding split_paired_all split_def by auto
     then show "(\<Sum>(x, k)\<in>p. integral k f) + (\<Sum>k\<in>r. integral k f) = integral (cbox a b) f"
       unfolding integral_combine_division_topdown[OF intf qdiv] r_def
       using q'(1) p'(1) sum.union_disjoint [of "snd ` p" "q - snd ` p", symmetric]
         by simp
   qed
   finally show "?lhs \<le> e + k" .
 qed
 
 lemma Henstock_lemma_part2:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes fed: "f integrable_on cbox a b" "e > 0" "gauge d"
     and less_e: "\<And>\<D>. \<lbrakk>\<D> tagged_division_of (cbox a b); d fine \<D>\<rbrakk> \<Longrightarrow>
                      norm (sum (\<lambda>(x,k). content k *\<^sub>R f x) \<D> - integral (cbox a b) f) < e"
     and tag: "p tagged_partial_division_of (cbox a b)"
     and "d fine p"
   shows "sum (\<lambda>(x,k). norm (content k *\<^sub>R f x - integral k f)) p \<le> 2 * real (DIM('n)) * e"
 proof -
   have "finite p"
-    using tag by blast
+    using tag tagged_partial_division_ofD by blast
   then show ?thesis
     unfolding split_def
   proof (rule sum_norm_allsubsets_bound)
     fix Q
     assume Q: "Q \<subseteq> p"
     then have fine: "d fine Q"
       by (simp add: \<open>d fine p\<close> fine_subset)
     show "norm (\<Sum>x\<in>Q. content (snd x) *\<^sub>R f (fst x) - integral (snd x) f) \<le> e"
       apply (rule Henstock_lemma_part1[OF fed less_e, unfolded split_def])
       using Q tag tagged_partial_division_subset by (force simp add: fine)+
   qed
 qed
 
 lemma Henstock_lemma:
   fixes f :: "'m::euclidean_space \<Rightarrow> 'n::euclidean_space"
   assumes intf: "f integrable_on cbox a b"
     and "e > 0"
   obtains \<gamma> where "gauge \<gamma>"
     and "\<And>p. \<lbrakk>p tagged_partial_division_of (cbox a b); \<gamma> fine p\<rbrakk> \<Longrightarrow>
              sum (\<lambda>(x,k). norm(content k *\<^sub>R f x - integral k f)) p < e"
 proof -
   have *: "e/(2 * (real DIM('n) + 1)) > 0" using \<open>e > 0\<close> by simp
   with integrable_integral[OF intf, unfolded has_integral]
   obtain \<gamma> where "gauge \<gamma>"
     and \<gamma>: "\<And>\<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 x) - integral (cbox a b) f)
          < e/(2 * (real DIM('n) + 1))"
     by metis
   show thesis
   proof (rule that [OF \<open>gauge \<gamma>\<close>])
     fix p
     assume p: "p tagged_partial_division_of cbox a b" "\<gamma> fine p"
     have "(\<Sum>(x,K)\<in>p. norm (content K *\<^sub>R f x - integral K f)) 
           \<le> 2 * real DIM('n) * (e/(2 * (real DIM('n) + 1)))"
       using Henstock_lemma_part2[OF intf * \<open>gauge \<gamma>\<close> \<gamma> p] by metis
     also have "... < e"
       using \<open>e > 0\<close> by (auto simp add: field_simps)
     finally
     show "(\<Sum>(x,K)\<in>p. norm (content K *\<^sub>R f x - integral K f)) < e" .
   qed
 qed
 
 
 subsection \<open>Monotone convergence (bounded interval first)\<close>
 
 lemma bounded_increasing_convergent:
   fixes f :: "nat \<Rightarrow> real"
   shows "\<lbrakk>bounded (range f); \<And>n. f n \<le> f (Suc n)\<rbrakk> \<Longrightarrow> \<exists>l. f \<longlonglongrightarrow> l"
   using Bseq_mono_convergent[of f] incseq_Suc_iff[of f]
   by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)
 
 lemma monotone_convergence_interval:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
   assumes intf: "\<And>k. (f k) integrable_on cbox a b"
     and le: "\<And>k x. x \<in> cbox a b \<Longrightarrow> (f k x) \<le> f (Suc k) x"
     and fg: "\<And>x. x \<in> cbox a b \<Longrightarrow> ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
     and bou: "bounded (range (\<lambda>k. integral (cbox a b) (f k)))"
   shows "g integrable_on cbox a b \<and> ((\<lambda>k. integral (cbox a b) (f k)) \<longlongrightarrow> integral (cbox a b) g) sequentially"
 proof (cases "content (cbox a b) = 0")
   case True then show ?thesis
     by auto
 next
   case False
   have fg1: "(f k x) \<le> (g x)" if x: "x \<in> cbox a b" for x k
   proof -
     have "\<forall>\<^sub>F j in sequentially. f k x \<le> f j x"
     proof (rule eventually_sequentiallyI [of k])
       show "\<And>j. k \<le> j \<Longrightarrow> f k x \<le> f j x"
         using le x by (force intro: transitive_stepwise_le)
     qed
     then show "f k x \<le> g x"
       using tendsto_lowerbound [OF fg] x trivial_limit_sequentially by blast
   qed
   have int_inc: "\<And>n. integral (cbox a b) (f n) \<le> integral (cbox a b) (f (Suc n))"
     by (metis integral_le intf le)
   then obtain i where i: "(\<lambda>k. integral (cbox a b) (f k)) \<longlonglongrightarrow> i"
     using bounded_increasing_convergent bou by blast
   have "\<And>k. \<forall>\<^sub>F x in sequentially. integral (cbox a b) (f k) \<le> integral (cbox a b) (f x)"
     unfolding eventually_sequentially
     by (force intro: transitive_stepwise_le int_inc)
   then have i': "\<And>k. (integral(cbox a b) (f k)) \<le> i"
     using tendsto_le [OF trivial_limit_sequentially i] by blast
   have "(g has_integral i) (cbox a b)"
     unfolding has_integral real_norm_def
   proof clarify
     fix e::real
     assume e: "e > 0"
     have "\<And>k. (\<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>\<D>. \<D> tagged_division_of (cbox a b) \<and> \<gamma> fine \<D> \<longrightarrow>
       abs ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f k x) - integral (cbox a b) (f k)) < e/2 ^ (k + 2)))"
       using intf e by (auto simp: has_integral_integral has_integral)
     then obtain c where c: "\<And>x. gauge (c x)"
           "\<And>x \<D>. \<lbrakk>\<D> tagged_division_of cbox a b; c x fine \<D>\<rbrakk> \<Longrightarrow>
               abs ((\<Sum>(u,K)\<in>\<D>. content K *\<^sub>R f x u) - integral (cbox a b) (f x))
               < e/2 ^ (x + 2)"
       by metis
 
     have "\<exists>r. \<forall>k\<ge>r. 0 \<le> i - (integral (cbox a b) (f k)) \<and> i - (integral (cbox a b) (f k)) < e/4"
     proof -
       have "e/4 > 0"
         using e by auto
       show ?thesis
         using LIMSEQ_D [OF i \<open>e/4 > 0\<close>] i' by auto
     qed
     then obtain r where r: "\<And>k. r \<le> k \<Longrightarrow> 0 \<le> i - integral (cbox a b) (f k)"
                        "\<And>k. r \<le> k \<Longrightarrow> i - integral (cbox a b) (f k) < e/4" 
       by metis
     have "\<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> (g x) - (f k x) \<and> (g x) - (f k x) < e/(4 * content(cbox a b))"
       if "x \<in> cbox a b" for x
     proof -
       have "e/(4 * content (cbox a b)) > 0"
         by (simp add: False content_lt_nz e)
       with fg that LIMSEQ_D
       obtain N where "\<forall>n\<ge>N. norm (f n x - g x) < e/(4 * content (cbox a b))"
         by metis
       then show "\<exists>n\<ge>r. \<forall>k\<ge>n. 0 \<le> g x - f k x \<and> g x - f k x < e/(4 * content (cbox a b))"
         apply (rule_tac x="N + r" in exI)
         using fg1[OF that] by (auto simp add: field_simps)
     qed
     then obtain m where r_le_m: "\<And>x. x \<in> cbox a b \<Longrightarrow> r \<le> m x"
        and m: "\<And>x k. \<lbrakk>x \<in> cbox a b; m x \<le> k\<rbrakk>
                      \<Longrightarrow> 0 \<le> g x - f k x \<and> g x - f k x < e/(4 * content (cbox a b))"
       by metis
     define d where "d x = c (m x) x" for x
     show "\<exists>\<gamma>. gauge \<gamma> \<and>
              (\<forall>\<D>. \<D> tagged_division_of cbox a b \<and>
                   \<gamma> fine \<D> \<longrightarrow> abs ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - i) < e)"
     proof (rule exI, safe)
       show "gauge d"
         using c(1) unfolding gauge_def d_def by auto
     next
       fix \<D>
       assume ptag: "\<D> tagged_division_of (cbox a b)" and "d fine \<D>"
       note p'=tagged_division_ofD[OF ptag]
       obtain s where s: "\<And>x. x \<in> \<D> \<Longrightarrow> m (fst x) \<le> s"
         by (metis finite_imageI finite_nat_set_iff_bounded_le p'(1) rev_image_eqI)
       have *: "\<bar>a - d\<bar> < e" if "\<bar>a - b\<bar> \<le> e/4" "\<bar>b - c\<bar> < e/2" "\<bar>c - d\<bar> < e/4" for a b c d
         using that norm_triangle_lt[of "a - b" "b - c" "3* e/4"]
           norm_triangle_lt[of "a - b + (b - c)" "c - d" e]
         by (auto simp add: algebra_simps)
       show "\<bar>(\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - i\<bar> < e"
       proof (rule *)
         have "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x)\<bar> 
               \<le> (\<Sum>i\<in>\<D>. \<bar>(case i of (x, K) \<Rightarrow> content K *\<^sub>R g x) - (case i of (x, K) \<Rightarrow> content K *\<^sub>R f (m x) x)\<bar>)"
           by (metis (mono_tags) sum_subtractf sum_abs) 
         also have "... \<le> (\<Sum>(x, k)\<in>\<D>. content k * (e/(4 * content (cbox a b))))"
         proof (rule sum_mono, simp add: split_paired_all)
           fix x K
           assume xk: "(x,K) \<in> \<D>"
           with ptag have x: "x \<in> cbox a b"
             by blast
           then have "abs (content K * (g x - f (m x) x)) \<le> content K * (e/(4 * content (cbox a b)))"
             by (metis m[OF x] mult_nonneg_nonneg abs_of_nonneg less_eq_real_def measure_nonneg mult_left_mono order_refl)
           then show "\<bar>content K * g x - content K * f (m x) x\<bar> \<le> content K * e/(4 * content (cbox a b))"
             by (simp add: algebra_simps)
         qed
         also have "... = (e/(4 * content (cbox a b))) * (\<Sum>(x, k)\<in>\<D>. content k)"
           by (simp add: sum_distrib_left sum_divide_distrib split_def mult.commute)
         also have "... \<le> e/4"
           by (metis False additive_content_tagged_division [OF ptag] nonzero_mult_divide_mult_cancel_right order_refl times_divide_eq_left)
         finally show "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R g x) - (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x)\<bar> \<le> e/4" .
 
       next
         have "norm ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>\<D>. integral K (f (m x))))
               \<le> norm (\<Sum>j = 0..s. \<Sum>(x,K)\<in>{xk \<in> \<D>. m (fst xk) = j}. content K *\<^sub>R f (m x) x - integral K (f (m x)))"
           apply (subst sum.group)
           using s by (auto simp: sum_subtractf split_def p'(1))
         also have "\<dots> < e/2"
         proof -
           have "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> \<D>. m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x)))
                 \<le> (\<Sum>i = 0..s. e/2 ^ (i + 2))"
           proof (rule sum_norm_le)
             fix t
             assume "t \<in> {0..s}"
             have "norm (\<Sum>(x,k)\<in>{xk \<in> \<D>. m (fst xk) = t}. content k *\<^sub>R f (m x) x - integral k (f (m x))) =
                   norm (\<Sum>(x,k)\<in>{xk \<in> \<D>. m (fst xk) = t}. content k *\<^sub>R f t x - integral k (f t))"
               by (force intro!: sum.cong arg_cong[where f=norm])
             also have "... \<le> e/2 ^ (t + 2)"
             proof (rule Henstock_lemma_part1 [OF intf])
               show "{xk \<in> \<D>. m (fst xk) = t} tagged_partial_division_of cbox a b"
               proof (rule tagged_partial_division_subset[of \<D>])
                 show "\<D> tagged_partial_division_of cbox a b"
                   using ptag tagged_division_of_def by blast
               qed auto
               show "c t fine {xk \<in> \<D>. m (fst xk) = t}"
                 using \<open>d fine \<D>\<close> by (auto simp: fine_def d_def)
             qed (use c e in auto)
             finally show "norm (\<Sum>(x,K)\<in>{xk \<in> \<D>. m (fst xk) = t}. content K *\<^sub>R f (m x) x -
                                 integral K (f (m x))) \<le> e/2 ^ (t + 2)" .
           qed
           also have "... = (e/2/2) * (\<Sum>i = 0..s. (1/2) ^ i)"
             by (simp add: sum_distrib_left field_simps)
           also have "\<dots> < e/2"
             by (simp add: sum_gp mult_strict_left_mono[OF _ e])
           finally show "norm (\<Sum>j = 0..s. \<Sum>(x, k)\<in>{xk \<in> \<D>.
             m (fst xk) = j}. content k *\<^sub>R f (m x) x - integral k (f (m x))) < e/2" .
         qed 
         finally show "\<bar>(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f (m x) x) - (\<Sum>(x,K)\<in>\<D>. integral K (f (m x)))\<bar> < e/2"
           by simp
       next
         have comb: "integral (cbox a b) (f y) = (\<Sum>(x, k)\<in>\<D>. integral k (f y))" for y
           using integral_combine_tagged_division_topdown[OF intf ptag] by metis
         have f_le: "\<And>y m n. \<lbrakk>y \<in> cbox a b; n\<ge>m\<rbrakk> \<Longrightarrow> f m y \<le> f n y"
           using le by (auto intro: transitive_stepwise_le)        
         have "(\<Sum>(x, k)\<in>\<D>. integral k (f r)) \<le> (\<Sum>(x, K)\<in>\<D>. integral K (f (m x)))"
         proof (rule sum_mono, simp add: split_paired_all)
           fix x K
           assume xK: "(x, K) \<in> \<D>"
           show "integral K (f r) \<le> integral K (f (m x))"
           proof (rule integral_le)
             show "f r integrable_on K"
               by (metis integrable_on_subcbox intf p'(3) p'(4) xK)
             show "f (m x) integrable_on K"
               by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK)
             show "f r y \<le> f (m x) y" if "y \<in> K" for y
               using that r_le_m[of x] p'(2-3)[OF xK] f_le by auto
           qed
         qed
         moreover have "(\<Sum>(x, K)\<in>\<D>. integral K (f (m x))) \<le> (\<Sum>(x, k)\<in>\<D>. integral k (f s))"
         proof (rule sum_mono, simp add: split_paired_all)
           fix x K
           assume xK: "(x, K) \<in> \<D>"
           show "integral K (f (m x)) \<le> integral K (f s)"
           proof (rule integral_le)
             show "f (m x) integrable_on K"
               by (metis elementary_interval integrable_on_subdivision intf p'(3) p'(4) xK)
             show "f s integrable_on K"
               by (metis integrable_on_subcbox intf p'(3) p'(4) xK)
             show "f (m x) y \<le> f s y" if "y \<in> K" for y
               using that s xK f_le p'(3) by fastforce
           qed
         qed
         moreover have "0 \<le> i - integral (cbox a b) (f r)" "i - integral (cbox a b) (f r) < e/4"
           using r by auto
         ultimately show "\<bar>(\<Sum>(x,K)\<in>\<D>. integral K (f (m x))) - i\<bar> < e/4"
           using comb i'[of s] by auto
       qed
     qed
   qed 
   with i integral_unique show ?thesis
     by blast
 qed
 
 lemma monotone_convergence_increasing:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
   assumes int_f: "\<And>k. (f k) integrable_on S"
     and "\<And>k x. x \<in> S \<Longrightarrow> (f k x) \<le> (f (Suc k) x)"
     and fg: "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
     and bou: "bounded (range (\<lambda>k. integral S (f k)))"
   shows "g integrable_on S \<and> ((\<lambda>k. integral S (f k)) \<longlongrightarrow> integral S g) sequentially"
 proof -
   have lem: "g integrable_on S \<and> ((\<lambda>k. integral S (f k)) \<longlongrightarrow> integral S g) sequentially"
     if f0: "\<And>k x. x \<in> S \<Longrightarrow> 0 \<le> f k x"
     and int_f: "\<And>k. (f k) integrable_on S"
     and le: "\<And>k x. x \<in> S \<Longrightarrow> f k x \<le> f (Suc k) x"
     and lim: "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
     and bou: "bounded (range(\<lambda>k. integral S (f k)))"
     for f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real" and g S
   proof -
     have fg: "(f k x) \<le> (g x)" if "x \<in> S" for x k
     proof -
       have "\<And>xa. k \<le> xa \<Longrightarrow> f k x \<le> f xa x"
         using le  by (force intro: transitive_stepwise_le that)
       then show ?thesis
         using tendsto_lowerbound [OF lim [OF that]] eventually_sequentiallyI by force
     qed
     obtain i where i: "(\<lambda>k. integral S (f k)) \<longlonglongrightarrow> i"
       using bounded_increasing_convergent [OF bou] le int_f integral_le by blast
     have i': "(integral S (f k)) \<le> i" for k
     proof -
       have "\<And>k. \<And>x. x \<in> S \<Longrightarrow> \<forall>n\<ge>k. f k x \<le> f n x"
         using le  by (force intro: transitive_stepwise_le)
       then show ?thesis
         using tendsto_lowerbound [OF i eventually_sequentiallyI trivial_limit_sequentially]
         by (meson int_f integral_le)
     qed
     let ?f = "(\<lambda>k x. if x \<in> S then f k x else 0)"
     let ?g = "(\<lambda>x. if x \<in> S then g x else 0)"
     have int: "?f k integrable_on cbox a b" for a b k
       by (simp add: int_f integrable_altD(1))
     have int': "\<And>k a b. f k integrable_on cbox a b \<inter> S"
       using int by (simp add: Int_commute integrable_restrict_Int) 
     have g: "?g integrable_on cbox a b \<and>
              (\<lambda>k. integral (cbox a b) (?f k)) \<longlonglongrightarrow> integral (cbox a b) ?g" for a b
     proof (rule monotone_convergence_interval)
       have "norm (integral (cbox a b) (?f k)) \<le> norm (integral S (f k))" for k
       proof -
         have "0 \<le> integral (cbox a b) (?f k)"
           by (metis (no_types) integral_nonneg Int_iff f0 inf_commute integral_restrict_Int int')
         moreover have "0 \<le> integral S (f k)"
           by (simp add: integral_nonneg f0 int_f)
         moreover have "integral (S \<inter> cbox a b) (f k) \<le> integral S (f k)"
           by (metis f0 inf_commute int' int_f integral_subset_le le_inf_iff order_refl)
         ultimately show ?thesis
           by (simp add: integral_restrict_Int)
       qed
       moreover obtain B where "\<And>x. x \<in> range (\<lambda>k. integral S (f k)) \<Longrightarrow> norm x \<le> B"
         using bou unfolding bounded_iff by blast
       ultimately show "bounded (range (\<lambda>k. integral (cbox a b) (?f k)))"
         unfolding bounded_iff by (blast intro: order_trans)
     qed (use int le lim in auto)
     moreover have "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow> norm (integral (cbox a b) ?g - i) < e"
       if "0 < e" for e
     proof -
       have "e/4>0"
         using that by auto
       with LIMSEQ_D [OF i] obtain N where N: "\<And>n. n \<ge> N \<Longrightarrow> norm (integral S (f n) - i) < e/4"
         by metis
       with int_f[of N, unfolded has_integral_integral has_integral_alt'[of "f N"]] 
       obtain B where "0 < B" and B: 
         "\<And>a b. ball 0 B \<subseteq> cbox a b \<Longrightarrow> norm (integral (cbox a b) (?f N) - integral S (f N)) < e/4"
         by (meson \<open>0 < e/4\<close>)
       have "norm (integral (cbox a b) ?g - i) < e" if  ab: "ball 0 B \<subseteq> cbox a b" for a b
       proof -
         obtain M where M: "\<And>n. n \<ge> M \<Longrightarrow> abs (integral (cbox a b) (?f n) - integral (cbox a b) ?g) < e/2"
           using \<open>e > 0\<close> g by (fastforce simp add: dest!: LIMSEQ_D [where r = "e/2"])
         have *: "\<And>\<alpha> \<beta> g. \<lbrakk>\<bar>\<alpha> - i\<bar> < e/2; \<bar>\<beta> - g\<bar> < e/2; \<alpha> \<le> \<beta>; \<beta> \<le> i\<rbrakk> \<Longrightarrow> \<bar>g - i\<bar> < e"
           unfolding real_inner_1_right by arith
         show "norm (integral (cbox a b) ?g - i) < e"
           unfolding real_norm_def
         proof (rule *)
           show "\<bar>integral (cbox a b) (?f N) - i\<bar> < e/2"
           proof (rule abs_triangle_half_l)
             show "\<bar>integral (cbox a b) (?f N) - integral S (f N)\<bar> < e/2/2"
               using B[OF ab] by simp
             show "abs (i - integral S (f N)) < e/2/2"
               using N by (simp add: abs_minus_commute)
           qed
           show "\<bar>integral (cbox a b) (?f (M + N)) - integral (cbox a b) ?g\<bar> < e/2"
             by (metis le_add1 M[of "M + N"])
           show "integral (cbox a b) (?f N) \<le> integral (cbox a b) (?f (M + N))"
           proof (intro ballI integral_le[OF int int])
             fix x assume "x \<in> cbox a b"
             have "(f m x) \<le> (f n x)" if "x \<in> S" "n \<ge> m" for m n
             proof (rule transitive_stepwise_le [OF \<open>n \<ge> m\<close> order_refl])
               show "\<And>u y z. \<lbrakk>f u x \<le> f y x; f y x \<le> f z x\<rbrakk> \<Longrightarrow> f u x \<le> f z x"
                 using dual_order.trans by blast
             qed (simp add: le \<open>x \<in> S\<close>)
             then show "(?f N)x \<le> (?f (M+N))x"
               by auto
           qed
           have "integral (cbox a b \<inter> S) (f (M + N)) \<le> integral S (f (M + N))"
             by (metis Int_lower1 f0 inf_commute int' int_f integral_subset_le)
           then have "integral (cbox a b) (?f (M + N)) \<le> integral S (f (M + N))"
             by (metis (no_types) inf_commute integral_restrict_Int)
           also have "... \<le> i"
             using i'[of "M + N"] by auto
           finally show "integral (cbox a b) (?f (M + N)) \<le> i" .
         qed
       qed
       then show ?thesis
         using \<open>0 < B\<close> by blast
     qed
     ultimately have "(g has_integral i) S"
       unfolding has_integral_alt' by auto
     then show ?thesis
       using has_integral_integrable_integral i integral_unique by metis
   qed
 
   have sub: "\<And>k. integral S (\<lambda>x. f k x - f 0 x) = integral S (f k) - integral S (f 0)"
     by (simp add: integral_diff int_f)
   have *: "\<And>x m n. x \<in> S \<Longrightarrow> n\<ge>m \<Longrightarrow> f m x \<le> f n x"
     using assms(2) by (force intro: transitive_stepwise_le)
   have gf: "(\<lambda>x. g x - f 0 x) integrable_on S \<and> ((\<lambda>k. integral S (\<lambda>x. f (Suc k) x - f 0 x)) \<longlongrightarrow>
     integral S (\<lambda>x. g x - f 0 x)) sequentially"
   proof (rule lem)
     show "\<And>k. (\<lambda>x. f (Suc k) x - f 0 x) integrable_on S"
       by (simp add: integrable_diff int_f)
     show "(\<lambda>k. f (Suc k) x - f 0 x) \<longlonglongrightarrow> g x - f 0 x" if "x \<in> S" for x
     proof -
       have "(\<lambda>n. f (Suc n) x) \<longlonglongrightarrow> g x"
         using LIMSEQ_ignore_initial_segment[OF fg[OF \<open>x \<in> S\<close>], of 1] by simp
       then show ?thesis
         by (simp add: tendsto_diff)
     qed
     show "bounded (range (\<lambda>k. integral S (\<lambda>x. f (Suc k) x - f 0 x)))"
     proof -
       obtain B where B: "\<And>k. norm (integral S (f k)) \<le> B"
         using  bou by (auto simp: bounded_iff)
       then have "norm (integral S (\<lambda>x. f (Suc k) x - f 0 x))
               \<le> B + norm (integral S (f 0))" for k
         unfolding sub by (meson add_le_cancel_right norm_triangle_le_diff)
       then show ?thesis
         unfolding bounded_iff by blast
     qed
   qed (use * in auto)
   then have "(\<lambda>x. integral S (\<lambda>xa. f (Suc x) xa - f 0 xa) + integral S (f 0))
              \<longlonglongrightarrow> integral S (\<lambda>x. g x - f 0 x) + integral S (f 0)"
     by (auto simp add: tendsto_add)
   moreover have "(\<lambda>x. g x - f 0 x + f 0 x) integrable_on S"
     using gf integrable_add int_f [of 0] by metis
   ultimately show ?thesis
     by (simp add: integral_diff int_f LIMSEQ_imp_Suc sub)
 qed
 
 lemma has_integral_monotone_convergence_increasing:
   fixes f :: "nat \<Rightarrow> 'a::euclidean_space \<Rightarrow> real"
   assumes f: "\<And>k. (f k has_integral x k) s"
   assumes "\<And>k x. x \<in> s \<Longrightarrow> f k x \<le> f (Suc k) x"
   assumes "\<And>x. x \<in> s \<Longrightarrow> (\<lambda>k. f k x) \<longlonglongrightarrow> g x"
   assumes "x \<longlonglongrightarrow> x'"
   shows "(g has_integral x') s"
 proof -
   have x_eq: "x = (\<lambda>i. integral s (f i))"
     by (simp add: integral_unique[OF f])
   then have x: "range(\<lambda>k. integral s (f k)) = range x"
     by auto
   have *: "g integrable_on s \<and> (\<lambda>k. integral s (f k)) \<longlonglongrightarrow> integral s g"
   proof (intro monotone_convergence_increasing allI ballI assms)
     show "bounded (range(\<lambda>k. integral s (f k)))"
       using x convergent_imp_bounded assms by metis
   qed (use f in auto)
   then have "integral s g = x'"
     by (intro LIMSEQ_unique[OF _ \<open>x \<longlonglongrightarrow> x'\<close>]) (simp add: x_eq)
   with * show ?thesis
     by (simp add: has_integral_integral)
 qed
 
 lemma monotone_convergence_decreasing:
   fixes f :: "nat \<Rightarrow> 'n::euclidean_space \<Rightarrow> real"
   assumes intf: "\<And>k. (f k) integrable_on S"
     and le: "\<And>k x. x \<in> S \<Longrightarrow> f (Suc k) x \<le> f k x"
     and fg: "\<And>x. x \<in> S \<Longrightarrow> ((\<lambda>k. f k x) \<longlongrightarrow> g x) sequentially"
     and bou: "bounded (range(\<lambda>k. integral S (f k)))"
   shows "g integrable_on S \<and> (\<lambda>k. integral S (f k)) \<longlonglongrightarrow> integral S g"
 proof -
   have *: "range(\<lambda>k. integral S (\<lambda>x. - f k x)) = (*\<^sub>R) (- 1) ` (range(\<lambda>k. integral S (f k)))"
     by force
   have "(\<lambda>x. - g x) integrable_on S \<and> (\<lambda>k. integral S (\<lambda>x. - f k x)) \<longlonglongrightarrow> integral S (\<lambda>x. - g x)"
   proof (rule monotone_convergence_increasing)
     show "\<And>k. (\<lambda>x. - f k x) integrable_on S"
       by (blast intro: integrable_neg intf)
     show "\<And>k x. x \<in> S \<Longrightarrow> - f k x \<le> - f (Suc k) x"
       by (simp add: le)
     show "\<And>x. x \<in> S \<Longrightarrow> (\<lambda>k. - f k x) \<longlonglongrightarrow> - g x"
       by (simp add: fg tendsto_minus)
     show "bounded (range(\<lambda>k. integral S (\<lambda>x. - f k x)))"
       using "*" bou bounded_scaling by auto
   qed
   then show ?thesis
     by (force dest: integrable_neg tendsto_minus)
 qed
 
 lemma integral_norm_bound_integral:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes int_f: "f integrable_on S"
     and int_g: "g integrable_on S"
     and le_g: "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"
   shows "norm (integral S f) \<le> integral S g"
 proof -
   have norm: "norm \<eta> \<le> y + e"
     if "norm \<zeta> \<le> x" and "\<bar>x - y\<bar> < e/2" and "norm (\<zeta> - \<eta>) < e/2"
     for e x y and \<zeta> \<eta> :: 'a
   proof -
     have "norm (\<eta> - \<zeta>) < e/2"
       by (metis norm_minus_commute that(3))
     moreover have "x \<le> y + e/2"
       using that(2) by linarith
     ultimately show ?thesis
       using that(1) le_less_trans[OF norm_triangle_sub[of \<eta> \<zeta>]] by (auto simp: less_imp_le)
   qed
   have lem: "norm (integral(cbox a b) f) \<le> integral (cbox a b) g"
     if f: "f integrable_on cbox a b"
     and g: "g integrable_on cbox a b"
     and nle: "\<And>x. x \<in> cbox a b \<Longrightarrow> norm (f x) \<le> g x"
     for f :: "'n \<Rightarrow> 'a" and g a b
   proof (rule field_le_epsilon)
     fix e :: real
     assume "e > 0"
     then have e: "e/2 > 0"
       by auto
     with integrable_integral[OF f,unfolded has_integral[of f]]
     obtain \<gamma> where \<gamma>: "gauge \<gamma>"
               "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D> 
            \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2"
       by meson 
     moreover
     from integrable_integral[OF g,unfolded has_integral[of g]] e
     obtain \<delta> where \<delta>: "gauge \<delta>"
               "\<And>\<D>. \<D> tagged_division_of cbox a b \<and> \<delta> fine \<D> 
            \<Longrightarrow> norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - integral (cbox a b) g) < e/2"
       by meson
     ultimately have "gauge (\<lambda>x. \<gamma> x \<inter> \<delta> x)"
       using gauge_Int by blast
     with fine_division_exists obtain \<D> 
       where p: "\<D> tagged_division_of cbox a b" "(\<lambda>x. \<gamma> x \<inter> \<delta> x) fine \<D>" 
       by metis
     have "\<gamma> fine \<D>" "\<delta> fine \<D>"
       using fine_Int p(2) by blast+
     show "norm (integral (cbox a b) f) \<le> integral (cbox a b) g + e"
     proof (rule norm)
       have "norm (content K *\<^sub>R f x) \<le> content K *\<^sub>R g x" if  "(x, K) \<in> \<D>" for x K
       proof-
         have K: "x \<in> K" "K \<subseteq> cbox a b"
           using \<open>(x, K) \<in> \<D>\<close> p(1) by blast+
         obtain u v where  "K = cbox u v"
           using \<open>(x, K) \<in> \<D>\<close> p(1) by blast
         moreover have "content K * norm (f x) \<le> content K * g x"
           by (metis K subsetD dual_order.antisym measure_nonneg mult_zero_left nle not_le real_mult_le_cancel_iff2)
         then show ?thesis
           by simp
       qed
       then show "norm (\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) \<le> (\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x)"
         by (simp add: sum_norm_le split_def)
       show "norm ((\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R f x) - integral (cbox a b) f) < e/2"
         using \<open>\<gamma> fine \<D>\<close> \<gamma> p(1) by simp
       show "\<bar>(\<Sum>(x, k)\<in>\<D>. content k *\<^sub>R g x) - integral (cbox a b) g\<bar> < e/2"
         using \<open>\<delta> fine \<D>\<close> \<delta> p(1) by simp
     qed
   qed
   show ?thesis
   proof (rule field_le_epsilon)
     fix e :: real
     assume "e > 0"
     then have e: "e/2 > 0"
       by auto
     let ?f = "(\<lambda>x. if x \<in> S then f x else 0)"
     let ?g = "(\<lambda>x. if x \<in> S then g x else 0)"
     have f: "?f integrable_on cbox a b" and g: "?g integrable_on cbox a b" for a b
       using int_f int_g integrable_altD by auto
     obtain Bf where "0 < Bf"
       and Bf: "\<And>a b. ball 0 Bf \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?f has_integral z) (cbox a b) \<and> norm (z - integral S f) < e/2"
       using integrable_integral [OF int_f,unfolded has_integral'[of f]] e that by blast
     obtain Bg where "0 < Bg"
       and Bg: "\<And>a b. ball 0 Bg \<subseteq> cbox a b \<Longrightarrow>
         \<exists>z. (?g has_integral z) (cbox a b) \<and> norm (z - integral S g) < e/2"
       using integrable_integral [OF int_g,unfolded has_integral'[of g]] e that by blast
     obtain a b::'n where ab: "ball 0 Bf \<union> ball 0 Bg \<subseteq> cbox a b"
       using ball_max_Un  by (metis bounded_ball bounded_subset_cbox_symmetric)
     have "ball 0 Bf \<subseteq> cbox a b"
       using ab by auto
     with Bf obtain z where int_fz: "(?f has_integral z) (cbox a b)" and z: "norm (z - integral S f) < e/2"
       by meson
     have "ball 0 Bg \<subseteq> cbox a b"
       using ab by auto
     with Bg obtain w where int_gw: "(?g has_integral w) (cbox a b)" and w: "norm (w - integral S g) < e/2"
       by meson
     show "norm (integral S f) \<le> integral S g + e"
     proof (rule norm)
       show "norm (integral (cbox a b) ?f) \<le> integral (cbox a b) ?g"
         by (simp add: le_g lem[OF f g, of a b])
       show "\<bar>integral (cbox a b) ?g - integral S g\<bar> < e/2"
         using int_gw integral_unique w by auto
       show "norm (integral (cbox a b) ?f - integral S f) < e/2"
         using int_fz integral_unique z by blast
     qed
   qed
 qed
 
 lemma continuous_on_imp_absolutely_integrable_on:
   fixes f::"real \<Rightarrow> 'a::banach"
   shows "continuous_on {a..b} f \<Longrightarrow>
     norm (integral {a..b} f) \<le> integral {a..b} (\<lambda>x. norm (f x))"
   by (intro integral_norm_bound_integral integrable_continuous_real continuous_on_norm) auto
 
 lemma integral_bound:
   fixes f::"real \<Rightarrow> 'a::banach"
   assumes "a \<le> b"
   assumes "continuous_on {a .. b} f"
   assumes "\<And>t. t \<in> {a .. b} \<Longrightarrow> norm (f t) \<le> B"
   shows "norm (integral {a .. b} f) \<le> B * (b - a)"
 proof -
   note continuous_on_imp_absolutely_integrable_on[OF assms(2)]
   also have "integral {a..b} (\<lambda>x. norm (f x)) \<le> integral {a..b} (\<lambda>_. B)"
     by (rule integral_le)
       (auto intro!: integrable_continuous_real continuous_intros assms)
   also have "\<dots> = B * (b - a)" using assms by simp
   finally show ?thesis .
 qed
 
 lemma integral_norm_bound_integral_component:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   fixes g :: "'n \<Rightarrow> 'b::euclidean_space"
   assumes f: "f integrable_on S" and g: "g integrable_on S"
     and fg: "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> (g x)\<bullet>k"
   shows "norm (integral S f) \<le> (integral S g)\<bullet>k"
 proof -
   have "norm (integral S f) \<le> integral S ((\<lambda>x. x \<bullet> k) \<circ> g)"
     using integral_norm_bound_integral[OF f integrable_linear[OF g]]
     by (simp add: bounded_linear_inner_left fg)
   then show ?thesis
     unfolding o_def integral_component_eq[OF g] .
 qed
 
 lemma has_integral_norm_bound_integral_component:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   fixes g :: "'n \<Rightarrow> 'b::euclidean_space"
   assumes f: "(f has_integral i) S"
     and g: "(g has_integral j) S"
     and "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> (g x)\<bullet>k"
   shows "norm i \<le> j\<bullet>k"
   using integral_norm_bound_integral_component[of f S g k] 
   unfolding integral_unique[OF f] integral_unique[OF g]
   using assms
   by auto
 
 
 lemma uniformly_convergent_improper_integral:
   fixes f :: "'b \<Rightarrow> real \<Rightarrow> 'a :: {banach}"
   assumes deriv: "\<And>x. x \<ge> a \<Longrightarrow> (G has_field_derivative g x) (at x within {a..})"
   assumes integrable: "\<And>a' b x. x \<in> A \<Longrightarrow> a' \<ge> a \<Longrightarrow> b \<ge> a' \<Longrightarrow> f x integrable_on {a'..b}"
   assumes G: "convergent G"
   assumes le: "\<And>y x. y \<in> A \<Longrightarrow> x \<ge> a \<Longrightarrow> norm (f y x) \<le> g x"
   shows   "uniformly_convergent_on A (\<lambda>b x. integral {a..b} (f x))"
 proof (intro Cauchy_uniformly_convergent uniformly_Cauchy_onI', goal_cases)
   case (1 \<epsilon>)
   from G have "Cauchy G"
     by (auto intro!: convergent_Cauchy)
   with 1 obtain M where M: "dist (G (real m)) (G (real n)) < \<epsilon>" if "m \<ge> M" "n \<ge> M" for m n
     by (force simp: Cauchy_def)
   define M' where "M' = max (nat \<lceil>a\<rceil>) M"
 
   show ?case
   proof (rule exI[of _ M'], safe, goal_cases)
     case (1 x m n)
     have M': "M' \<ge> a" "M' \<ge> M" unfolding M'_def by linarith+
     have int_g: "(g has_integral (G (real n) - G (real m))) {real m..real n}"
       using 1 M' by (intro fundamental_theorem_of_calculus) 
                     (auto simp: has_field_derivative_iff_has_vector_derivative [symmetric] 
                           intro!: DERIV_subset[OF deriv])
     have int_f: "f x integrable_on {a'..real n}" if "a' \<ge> a" for a'
       using that 1 by (cases "a' \<le> real n") (auto intro: integrable)
 
     have "dist (integral {a..real m} (f x)) (integral {a..real n} (f x)) =
             norm (integral {a..real n} (f x) - integral {a..real m} (f x))"
       by (simp add: dist_norm norm_minus_commute)
     also have "integral {a..real m} (f x) + integral {real m..real n} (f x) = 
                  integral {a..real n} (f x)"
       using M' and 1 by (intro integral_combine int_f) auto
     hence "integral {a..real n} (f x) - integral {a..real m} (f x) = 
              integral {real m..real n} (f x)"
       by (simp add: algebra_simps)
     also have "norm \<dots> \<le> integral {real m..real n} g"
       using le 1 M' int_f int_g by (intro integral_norm_bound_integral) auto 
     also from int_g have "integral {real m..real n} g = G (real n) - G (real m)"
       by (simp add: has_integral_iff)
     also have "\<dots> \<le> dist (G m) (G n)" 
       by (simp add: dist_norm)
     also from 1 and M' have "\<dots> < \<epsilon>"
       by (intro M) auto
     finally show ?case .
   qed
 qed
 
 lemma uniformly_convergent_improper_integral':
   fixes f :: "'b \<Rightarrow> real \<Rightarrow> 'a :: {banach, real_normed_algebra}"
   assumes deriv: "\<And>x. x \<ge> a \<Longrightarrow> (G has_field_derivative g x) (at x within {a..})"
   assumes integrable: "\<And>a' b x. x \<in> A \<Longrightarrow> a' \<ge> a \<Longrightarrow> b \<ge> a' \<Longrightarrow> f x integrable_on {a'..b}"
   assumes G: "convergent G"
   assumes le: "eventually (\<lambda>x. \<forall>y\<in>A. norm (f y x) \<le> g x) at_top"
   shows   "uniformly_convergent_on A (\<lambda>b x. integral {a..b} (f x))"
 proof -
   from le obtain a'' where le: "\<And>y x. y \<in> A \<Longrightarrow> x \<ge> a'' \<Longrightarrow> norm (f y x) \<le> g x"
     by (auto simp: eventually_at_top_linorder)
   define a' where "a' = max a a''"
 
   have "uniformly_convergent_on A (\<lambda>b x. integral {a'..real b} (f x))"
   proof (rule uniformly_convergent_improper_integral)
     fix t assume t: "t \<ge> a'"
     hence "(G has_field_derivative g t) (at t within {a..})"
       by (intro deriv) (auto simp: a'_def)
     moreover have "{a'..} \<subseteq> {a..}" unfolding a'_def by auto
     ultimately show "(G has_field_derivative g t) (at t within {a'..})"
       by (rule DERIV_subset)
   qed (insert le, auto simp: a'_def intro: integrable G)
   hence "uniformly_convergent_on A (\<lambda>b x. integral {a..a'} (f x) + integral {a'..real b} (f x))" 
     (is ?P) by (intro uniformly_convergent_add) auto
   also have "eventually (\<lambda>x. \<forall>y\<in>A. integral {a..a'} (f y) + integral {a'..x} (f y) =
                    integral {a..x} (f y)) sequentially"
     by (intro eventually_mono [OF eventually_ge_at_top[of "nat \<lceil>a'\<rceil>"]] ballI integral_combine)
        (auto simp: a'_def intro: integrable)
   hence "?P \<longleftrightarrow> ?thesis"
     by (intro uniformly_convergent_cong) simp_all
   finally show ?thesis .
 qed
 
 subsection \<open>differentiation under the integral sign\<close>
 
 lemma integral_continuous_on_param:
   fixes f::"'a::topological_space \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
   assumes cont_fx: "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). f x t)"
   shows "continuous_on U (\<lambda>x. integral (cbox a b) (f x))"
 proof cases
   assume "content (cbox a b) \<noteq> 0"
   then have ne: "cbox a b \<noteq> {}" by auto
 
   note [continuous_intros] =
     continuous_on_compose2[OF cont_fx, where f="\<lambda>y. Pair x y" for x,
       unfolded split_beta fst_conv snd_conv]
 
   show ?thesis
     unfolding continuous_on_def
   proof (safe intro!: tendstoI)
     fix e'::real and x
     assume "e' > 0"
     define e where "e = e' / (content (cbox a b) + 1)"
     have "e > 0" using \<open>e' > 0\<close> by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos)
     assume "x \<in> U"
     from continuous_on_prod_compactE[OF cont_fx compact_cbox \<open>x \<in> U\<close> \<open>0 < e\<close>]
     obtain X0 where X0: "x \<in> X0" "open X0"
       and fx_bound: "\<And>y t. y \<in> X0 \<inter> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> norm (f y t - f x t) \<le> e"
       unfolding split_beta fst_conv snd_conv dist_norm
       by metis
     have "\<forall>\<^sub>F y in at x within U. y \<in> X0 \<inter> U"
       using X0(1) X0(2) eventually_at_topological by auto
     then show "\<forall>\<^sub>F y in at x within U. dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'"
     proof eventually_elim
       case (elim y)
       have "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) =
         norm (integral (cbox a b) (\<lambda>t. f y t - f x t))"
         using elim \<open>x \<in> U\<close>
         unfolding dist_norm
         by (subst integral_diff)
            (auto intro!: integrable_continuous continuous_intros)
       also have "\<dots> \<le> e * content (cbox a b)"
         using elim \<open>x \<in> U\<close>
         by (intro integrable_bound)
            (auto intro!: fx_bound \<open>x \<in> U \<close> less_imp_le[OF \<open>0 < e\<close>]
               integrable_continuous continuous_intros)
       also have "\<dots> < e'"
         using \<open>0 < e'\<close> \<open>e > 0\<close>
         by (auto simp: e_def field_split_simps)
       finally show "dist (integral (cbox a b) (f y)) (integral (cbox a b) (f x)) < e'" .
     qed
   qed
 qed (auto intro!: continuous_on_const)
 
 lemma leibniz_rule:
   fixes f::"'a::banach \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
   assumes fx: "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow>
     ((\<lambda>x. f x t) has_derivative blinfun_apply (fx x t)) (at x within U)"
   assumes integrable_f2: "\<And>x. x \<in> U \<Longrightarrow> f x integrable_on cbox a b"
   assumes cont_fx: "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
   assumes [intro]: "x0 \<in> U"
   assumes "convex U"
   shows
     "((\<lambda>x. integral (cbox a b) (f x)) has_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
     (is "(?F has_derivative ?dF) _")
 proof cases
   assume "content (cbox a b) \<noteq> 0"
   then have ne: "cbox a b \<noteq> {}" by auto
   note [continuous_intros] =
     continuous_on_compose2[OF cont_fx, where f="\<lambda>y. Pair x y" for x,
       unfolded split_beta fst_conv snd_conv]
   show ?thesis
   proof (intro has_derivativeI bounded_linear_scaleR_left tendstoI, fold norm_conv_dist)
     have cont_f1: "\<And>t. t \<in> cbox a b \<Longrightarrow> continuous_on U (\<lambda>x. f x t)"
       by (auto simp: continuous_on_eq_continuous_within intro!: has_derivative_continuous fx)
     note [continuous_intros] = continuous_on_compose2[OF cont_f1]
     fix e'::real
     assume "e' > 0"
     define e where "e = e' / (content (cbox a b) + 1)"
     have "e > 0" using \<open>e' > 0\<close> by (auto simp: e_def intro!: divide_pos_pos add_nonneg_pos)
     from continuous_on_prod_compactE[OF cont_fx compact_cbox \<open>x0 \<in> U\<close> \<open>e > 0\<close>]
     obtain X0 where X0: "x0 \<in> X0" "open X0"
       and fx_bound: "\<And>x t. x \<in> X0 \<inter> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> norm (fx x t - fx x0 t) \<le> e"
       unfolding split_beta fst_conv snd_conv
       by (metis dist_norm)
 
     note eventually_closed_segment[OF \<open>open X0\<close> \<open>x0 \<in> X0\<close>, of U]
     moreover
     have "\<forall>\<^sub>F x in at x0 within U. x \<in> X0"
       using \<open>open X0\<close> \<open>x0 \<in> X0\<close> eventually_at_topological by blast
     moreover have "\<forall>\<^sub>F x in at x0 within U. x \<noteq> x0"
       by (auto simp: eventually_at_filter)
     moreover have "\<forall>\<^sub>F x in at x0 within U. x \<in> U"
       by (auto simp: eventually_at_filter)
     ultimately
     show "\<forall>\<^sub>F x in at x0 within U. norm ((?F x - ?F x0 - ?dF (x - x0)) /\<^sub>R norm (x - x0)) < e'"
     proof eventually_elim
       case (elim x)
       from elim have "0 < norm (x - x0)" by simp
       have "closed_segment x0 x \<subseteq> U"
         by (rule \<open>convex U\<close>[unfolded convex_contains_segment, rule_format, OF \<open>x0 \<in> U\<close> \<open>x \<in> U\<close>])
       from elim have [intro]: "x \<in> U" by auto
       have "?F x - ?F x0 - ?dF (x - x0) =
         integral (cbox a b) (\<lambda>y. f x y - f x0 y - fx x0 y (x - x0))"
         (is "_ = ?id")
         using \<open>x \<noteq> x0\<close>
         by (subst blinfun_apply_integral integral_diff,
             auto intro!: integrable_diff integrable_f2 continuous_intros
               intro: integrable_continuous)+
       also
       {
         fix t assume t: "t \<in> (cbox a b)"
         have seg: "\<And>t. t \<in> {0..1} \<Longrightarrow> x0 + t *\<^sub>R (x - x0) \<in> X0 \<inter> U"
           using \<open>closed_segment x0 x \<subseteq> U\<close>
             \<open>closed_segment x0 x \<subseteq> X0\<close>
           by (force simp: closed_segment_def algebra_simps)
         from t have deriv:
           "((\<lambda>x. f x t) has_derivative (fx y t)) (at y within X0 \<inter> U)"
           if "y \<in> X0 \<inter> U" for y
           unfolding has_vector_derivative_def[symmetric]
           using that \<open>x \<in> X0\<close>
           by (intro has_derivative_subset[OF fx]) auto
         have "\<And>x. x \<in> X0 \<inter> U \<Longrightarrow> onorm (blinfun_apply (fx x t) - (fx x0 t)) \<le> e"
           using fx_bound t
           by (auto simp add: norm_blinfun_def fun_diff_def blinfun.bilinear_simps[symmetric])
         from differentiable_bound_linearization[OF seg deriv this] X0
         have "norm (f x t - f x0 t - fx x0 t (x - x0)) \<le> e * norm (x - x0)"
           by (auto simp add: ac_simps)
       }
       then have "norm ?id \<le> integral (cbox a b) (\<lambda>_. e * norm (x - x0))"
         by (intro integral_norm_bound_integral)
           (auto intro!: continuous_intros integrable_diff integrable_f2
             intro: integrable_continuous)
       also have "\<dots> = content (cbox a b) * e * norm (x - x0)"
         by simp
       also have "\<dots> < e' * norm (x - x0)"
       proof (intro mult_strict_right_mono[OF _ \<open>0 < norm (x - x0)\<close>])
         show "content (cbox a b) * e < e'"
           using \<open>e' > 0\<close> by (simp add: divide_simps e_def not_less)
       qed
       finally have "norm (?F x - ?F x0 - ?dF (x - x0)) < e' * norm (x - x0)" .
       then show ?case
         by (auto simp: divide_simps)
     qed
   qed (rule blinfun.bounded_linear_right)
 qed (auto intro!: derivative_eq_intros simp: blinfun.bilinear_simps)
 
 lemma has_vector_derivative_eq_has_derivative_blinfun:
   "(f has_vector_derivative f') (at x within U) \<longleftrightarrow>
     (f has_derivative blinfun_scaleR_left f') (at x within U)"
   by (simp add: has_vector_derivative_def)
 
 lemma leibniz_rule_vector_derivative:
   fixes f::"real \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
   assumes fx: "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow>
       ((\<lambda>x. f x t) has_vector_derivative (fx x t)) (at x within U)"
   assumes integrable_f2: "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
   assumes cont_fx: "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). fx x t)"
   assumes U: "x0 \<in> U" "convex U"
   shows "((\<lambda>x. integral (cbox a b) (f x)) has_vector_derivative integral (cbox a b) (fx x0))
       (at x0 within U)"
 proof -
   note [continuous_intros] =
     continuous_on_compose2[OF cont_fx, where f="\<lambda>y. Pair x y" for x,
       unfolded split_beta fst_conv snd_conv]
   show ?thesis
     unfolding has_vector_derivative_eq_has_derivative_blinfun
   proof (rule has_derivative_eq_rhs [OF leibniz_rule[OF _ integrable_f2 _ U]])
     show "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). blinfun_scaleR_left (fx x t))"
       using cont_fx by (auto simp: split_beta intro!: continuous_intros)
     show "blinfun_apply (integral (cbox a b) (\<lambda>t. blinfun_scaleR_left (fx x0 t))) =
           blinfun_apply (blinfun_scaleR_left (integral (cbox a b) (fx x0)))"
     by (subst integral_linear[symmetric])
        (auto simp: has_vector_derivative_def o_def
          intro!: integrable_continuous U continuous_intros bounded_linear_intros)
     qed (use fx in \<open>auto simp: has_vector_derivative_def\<close>)
 qed
 
 lemma has_field_derivative_eq_has_derivative_blinfun:
   "(f has_field_derivative f') (at x within U) \<longleftrightarrow> (f has_derivative blinfun_mult_right f') (at x within U)"
   by (simp add: has_field_derivative_def)
 
 lemma leibniz_rule_field_derivative:
   fixes f::"'a::{real_normed_field, banach} \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'a"
   assumes fx: "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
   assumes integrable_f2: "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
   assumes cont_fx: "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
   assumes U: "x0 \<in> U" "convex U"
   shows "((\<lambda>x. integral (cbox a b) (f x)) has_field_derivative integral (cbox a b) (fx x0)) (at x0 within U)"
 proof -
   note [continuous_intros] =
     continuous_on_compose2[OF cont_fx, where f="\<lambda>y. Pair x y" for x,
       unfolded split_beta fst_conv snd_conv]
   have *: "blinfun_mult_right (integral (cbox a b) (fx x0)) =
     integral (cbox a b) (\<lambda>t. blinfun_mult_right (fx x0 t))"
     by (subst integral_linear[symmetric])
       (auto simp: has_vector_derivative_def o_def
         intro!: integrable_continuous U continuous_intros bounded_linear_intros)
   show ?thesis
     unfolding has_field_derivative_eq_has_derivative_blinfun
   proof (rule has_derivative_eq_rhs [OF leibniz_rule[OF _ integrable_f2 _ U, where fx="\<lambda>x t. blinfun_mult_right (fx x t)"]])
     show "continuous_on (U \<times> cbox a b) (\<lambda>(x, t). blinfun_mult_right (fx x t))"
       using cont_fx by (auto simp: split_beta intro!: continuous_intros)
     show "blinfun_apply (integral (cbox a b) (\<lambda>t. blinfun_mult_right (fx x0 t))) =
           blinfun_apply (blinfun_mult_right (integral (cbox a b) (fx x0)))"
       by (subst integral_linear[symmetric])
         (auto simp: has_vector_derivative_def o_def
           intro!: integrable_continuous U continuous_intros bounded_linear_intros)
   qed (use fx in \<open>auto simp: has_field_derivative_def\<close>)
 qed
 
 lemma leibniz_rule_field_differentiable:
   fixes f::"'a::{real_normed_field, banach} \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'a"
   assumes "\<And>x t. x \<in> U \<Longrightarrow> t \<in> cbox a b \<Longrightarrow> ((\<lambda>x. f x t) has_field_derivative fx x t) (at x within U)"
   assumes "\<And>x. x \<in> U \<Longrightarrow> (f x) integrable_on cbox a b"
   assumes "continuous_on (U \<times> (cbox a b)) (\<lambda>(x, t). fx x t)"
   assumes "x0 \<in> U" "convex U"
   shows "(\<lambda>x. integral (cbox a b) (f x)) field_differentiable at x0 within U"
   using leibniz_rule_field_derivative[OF assms] by (auto simp: field_differentiable_def)
 
 
 subsection \<open>Exchange uniform limit and integral\<close>
 
 lemma uniform_limit_integral_cbox:
   fixes f::"'a \<Rightarrow> 'b::euclidean_space \<Rightarrow> 'c::banach"
   assumes u: "uniform_limit (cbox a b) f g F"
   assumes c: "\<And>n. continuous_on (cbox a b) (f n)"
   assumes [simp]: "F \<noteq> bot"
   obtains I J where
     "\<And>n. (f n has_integral I n) (cbox a b)"
     "(g has_integral J) (cbox a b)"
     "(I \<longlongrightarrow> J) F"
 proof -
   have fi[simp]: "f n integrable_on (cbox a b)" for n
     by (auto intro!: integrable_continuous assms)
   then obtain I where I: "\<And>n. (f n has_integral I n) (cbox a b)"
     by atomize_elim (auto simp: integrable_on_def intro!: choice)
 
   moreover
   have gi[simp]: "g integrable_on (cbox a b)"
     by (auto intro!: integrable_continuous uniform_limit_theorem[OF _ u] eventuallyI c)
   then obtain J where J: "(g has_integral J) (cbox a b)"
     by blast
 
   moreover
   have "(I \<longlongrightarrow> J) F"
   proof cases
     assume "content (cbox a b) = 0"
     hence "I = (\<lambda>_. 0)" "J = 0"
       by (auto intro!: has_integral_unique I J)
     thus ?thesis by simp
   next
     assume content_nonzero: "content (cbox a b) \<noteq> 0"
     show ?thesis
     proof (rule tendstoI)
       fix e::real
       assume "e > 0"
       define e' where "e' = e/2"
       with \<open>e > 0\<close> have "e' > 0" by simp
       then have "\<forall>\<^sub>F n in F. \<forall>x\<in>cbox a b. norm (f n x - g x) < e' / content (cbox a b)"
         using u content_nonzero by (auto simp: uniform_limit_iff dist_norm zero_less_measure_iff)
       then show "\<forall>\<^sub>F n in F. dist (I n) J < e"
       proof eventually_elim
         case (elim n)
         have "I n = integral (cbox a b) (f n)"
             "J = integral (cbox a b) g"
           using I[of n] J by (simp_all add: integral_unique)
         then have "dist (I n) J = norm (integral (cbox a b) (\<lambda>x. f n x - g x))"
           by (simp add: integral_diff dist_norm)
         also have "\<dots> \<le> integral (cbox a b) (\<lambda>x. (e' / content (cbox a b)))"
           using elim
           by (intro integral_norm_bound_integral) (auto intro!: integrable_diff)
         also have "\<dots> < e"
           using \<open>0 < e\<close>
           by (simp add: e'_def)
         finally show ?case .
       qed
     qed
   qed
   ultimately show ?thesis ..
 qed
 
 lemma uniform_limit_integral:
   fixes f::"'a \<Rightarrow> 'b::ordered_euclidean_space \<Rightarrow> 'c::banach"
   assumes u: "uniform_limit {a..b} f g F"
   assumes c: "\<And>n. continuous_on {a..b} (f n)"
   assumes [simp]: "F \<noteq> bot"
   obtains I J where
     "\<And>n. (f n has_integral I n) {a..b}"
     "(g has_integral J) {a..b}"
     "(I \<longlongrightarrow> J) F"
   by (metis interval_cbox assms uniform_limit_integral_cbox)
 
 
 subsection \<open>Integration by parts\<close>
 
 lemma integration_by_parts_interior_strong:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes bilinear: "bounded_bilinear (prod)"
   assumes s: "finite s" and le: "a \<le> b"
   assumes cont [continuous_intros]: "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes deriv: "\<And>x. x\<in>{a<..<b} - s \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
                  "\<And>x. x\<in>{a<..<b} - s \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes int: "((\<lambda>x. prod (f x) (g' x)) has_integral
                   (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
   shows   "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
 proof -
   interpret bounded_bilinear prod by fact
   have "((\<lambda>x. prod (f x) (g' x) + prod (f' x) (g x)) has_integral
           (prod (f b) (g b) - prod (f a) (g a))) {a..b}"
     using deriv by (intro fundamental_theorem_of_calculus_interior_strong[OF s le])
                    (auto intro!: continuous_intros continuous_on has_vector_derivative)
   from has_integral_diff[OF this int] show ?thesis by (simp add: algebra_simps)
 qed
 
 lemma integration_by_parts_interior:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes "bounded_bilinear (prod)" "a \<le> b"
           "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes "\<And>x. x\<in>{a<..<b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
           "\<And>x. x\<in>{a<..<b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes "((\<lambda>x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
   shows   "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
   by (rule integration_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all)
 
 lemma integration_by_parts:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes "bounded_bilinear (prod)" "a \<le> b"
           "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes "\<And>x. x\<in>{a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
           "\<And>x. x\<in>{a..b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes "((\<lambda>x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) - y)) {a..b}"
   shows   "((\<lambda>x. prod (f' x) (g x)) has_integral y) {a..b}"
   by (rule integration_by_parts_interior[of _ _ _ f g f' g']) (insert assms, simp_all)
 
 lemma integrable_by_parts_interior_strong:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes bilinear: "bounded_bilinear (prod)"
   assumes s: "finite s" and le: "a \<le> b"
   assumes cont [continuous_intros]: "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes deriv: "\<And>x. x\<in>{a<..<b} - s \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
                  "\<And>x. x\<in>{a<..<b} - s \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes int: "(\<lambda>x. prod (f x) (g' x)) integrable_on {a..b}"
   shows   "(\<lambda>x. prod (f' x) (g x)) integrable_on {a..b}"
 proof -
   from int obtain I where "((\<lambda>x. prod (f x) (g' x)) has_integral I) {a..b}"
     unfolding integrable_on_def by blast
   hence "((\<lambda>x. prod (f x) (g' x)) has_integral (prod (f b) (g b) - prod (f a) (g a) -
            (prod (f b) (g b) - prod (f a) (g a) - I))) {a..b}" by simp
   from integration_by_parts_interior_strong[OF assms(1-7) this]
     show ?thesis unfolding integrable_on_def by blast
 qed
 
 lemma integrable_by_parts_interior:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes "bounded_bilinear (prod)" "a \<le> b"
           "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes "\<And>x. x\<in>{a<..<b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
           "\<And>x. x\<in>{a<..<b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes "(\<lambda>x. prod (f x) (g' x)) integrable_on {a..b}"
   shows   "(\<lambda>x. prod (f' x) (g x)) integrable_on {a..b}"
   by (rule integrable_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all)
 
 lemma integrable_by_parts:
   fixes prod :: "_ \<Rightarrow> _ \<Rightarrow> 'b :: banach"
   assumes "bounded_bilinear (prod)" "a \<le> b"
           "continuous_on {a..b} f" "continuous_on {a..b} g"
   assumes "\<And>x. x\<in>{a..b} \<Longrightarrow> (f has_vector_derivative f' x) (at x)"
           "\<And>x. x\<in>{a..b} \<Longrightarrow> (g has_vector_derivative g' x) (at x)"
   assumes "(\<lambda>x. prod (f x) (g' x)) integrable_on {a..b}"
   shows   "(\<lambda>x. prod (f' x) (g x)) integrable_on {a..b}"
   by (rule integrable_by_parts_interior_strong[of _ "{}" _ _ f g f' g']) (insert assms, simp_all)
 
 
 subsection \<open>Integration by substitution\<close>
 
 lemma has_integral_substitution_general:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> real"
   assumes s: "finite s" and le: "a \<le> b"
       and subset: "g ` {a..b} \<subseteq> {c..d}"
       and f [continuous_intros]: "continuous_on {c..d} f"
       and g [continuous_intros]: "continuous_on {a..b} g"
       and deriv [derivative_intros]:
               "\<And>x. x \<in> {a..b} - s \<Longrightarrow> (g has_field_derivative g' x) (at x within {a..b})"
     shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f - integral {g b..g a} f)) {a..b}"
 proof -
   let ?F = "\<lambda>x. integral {c..g x} f"
   have cont_int: "continuous_on {a..b} ?F"
     by (rule continuous_on_compose2[OF _ g subset] indefinite_integral_continuous_1
           f integrable_continuous_real)+
   have deriv: "(((\<lambda>x. integral {c..x} f) \<circ> g) has_vector_derivative g' x *\<^sub>R f (g x))
                  (at x within {a..b})" if "x \<in> {a..b} - s" for x
   proof (rule has_vector_derivative_eq_rhs [OF vector_diff_chain_within refl])
     show "(g has_vector_derivative g' x) (at x within {a..b})"
       using deriv has_field_derivative_iff_has_vector_derivative that by blast
     show "((\<lambda>x. integral {c..x} f) has_vector_derivative f (g x)) 
           (at (g x) within g ` {a..b})"
       using that le subset
       by (blast intro: has_vector_derivative_within_subset integral_has_vector_derivative f)
   qed
   have deriv: "(?F has_vector_derivative g' x *\<^sub>R f (g x))
                   (at x)" if "x \<in> {a..b} - (s \<union> {a,b})" for x
     using deriv[of x] that by (simp add: at_within_Icc_at o_def)
   have "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral (?F b - ?F a)) {a..b}"
     using le cont_int s deriv cont_int
     by (intro fundamental_theorem_of_calculus_interior_strong[of "s \<union> {a,b}"]) simp_all
   also
   from subset have "g x \<in> {c..d}" if "x \<in> {a..b}" for x using that by blast
   from this[of a] this[of b] le have cd: "c \<le> g a" "g b \<le> d" "c \<le> g b" "g a \<le> d" by auto
   have "integral {c..g b} f - integral {c..g a} f = integral {g a..g b} f - integral {g b..g a} f"
   proof cases
     assume "g a \<le> g b"
     note le = le this
     from cd have "integral {c..g a} f + integral {g a..g b} f = integral {c..g b} f"
       by (intro integral_combine integrable_continuous_real continuous_on_subset[OF f] le) simp_all
     with le show ?thesis
       by (cases "g a = g b") (simp_all add: algebra_simps)
   next
     assume less: "\<not>g a \<le> g b"
     then have "g a \<ge> g b" by simp
     note le = le this
     from cd have "integral {c..g b} f + integral {g b..g a} f = integral {c..g a} f"
       by (intro integral_combine integrable_continuous_real continuous_on_subset[OF f] le) simp_all
     with less show ?thesis
       by (simp_all add: algebra_simps)
   qed
   finally show ?thesis .
 qed
 
 lemma has_integral_substitution_strong:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> real"
   assumes s: "finite s" and le: "a \<le> b" "g a \<le> g b"
     and subset: "g ` {a..b} \<subseteq> {c..d}"
     and f [continuous_intros]: "continuous_on {c..d} f"
     and g [continuous_intros]: "continuous_on {a..b} g"
     and deriv [derivative_intros]:
     "\<And>x. x \<in> {a..b} - s \<Longrightarrow> (g has_field_derivative g' x) (at x within {a..b})"
   shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f)) {a..b}"
   using has_integral_substitution_general[OF s le(1) subset f g deriv] le(2)
   by (cases "g a = g b") auto
 
 lemma has_integral_substitution:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space" and g :: "real \<Rightarrow> real"
   assumes "a \<le> b" "g a \<le> g b" "g ` {a..b} \<subseteq> {c..d}"
       and "continuous_on {c..d} f"
       and "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_field_derivative g' x) (at x within {a..b})"
     shows "((\<lambda>x. g' x *\<^sub>R f (g x)) has_integral (integral {g a..g b} f)) {a..b}"
   by (intro has_integral_substitution_strong[of "{}" a b g c d] assms)
      (auto intro: DERIV_continuous_on assms)
 
 lemma integral_shift:
   fixes f :: "real \<Rightarrow> 'a::euclidean_space"
   assumes cont: "continuous_on {a + c..b + c} f"
   shows "integral {a..b} (f \<circ> (\<lambda>x. x + c)) = integral {a + c..b + c} f"
 proof (cases "a \<le> b")
   case True
   have "((\<lambda>x. 1 *\<^sub>R f (x + c)) has_integral integral {a+c..b+c} f) {a..b}"
     using True cont
     by (intro has_integral_substitution[where c = "a + c" and d = "b + c"])
        (auto intro!: derivative_eq_intros)
   thus ?thesis by (simp add: has_integral_iff o_def)
 qed auto
 
 
 subsection \<open>Compute a double integral using iterated integrals and switching the order of integration\<close>
 
 lemma continuous_on_imp_integrable_on_Pair1:
   fixes f :: "_ \<Rightarrow> 'b::banach"
   assumes con: "continuous_on (cbox (a,c) (b,d)) f" and x: "x \<in> cbox a b"
   shows "(\<lambda>y. f (x, y)) integrable_on (cbox c d)"
 proof -
   have "f \<circ> (\<lambda>y. (x, y)) integrable_on (cbox c d)"
   proof (intro integrable_continuous continuous_on_compose [OF _ continuous_on_subset [OF con]])
     show "continuous_on (cbox c d) (Pair x)"
       by (simp add: continuous_on_Pair)
     show "Pair x ` cbox c d \<subseteq> cbox (a,c) (b,d)"
       using x by blast
   qed
   then show ?thesis
     by (simp add: o_def)
 qed
 
 lemma integral_integrable_2dim:
   fixes f :: "('a::euclidean_space * 'b::euclidean_space) \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a,c) (b,d)) f"
     shows "(\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y))) integrable_on cbox a b"
 proof (cases "content(cbox c d) = 0")
 case True
   then show ?thesis
     by (simp add: True integrable_const)
 next
   case False
   have uc: "uniformly_continuous_on (cbox (a,c) (b,d)) f"
     by (simp add: assms compact_cbox compact_uniformly_continuous)
   { fix x::'a and e::real
     assume x: "x \<in> cbox a b" and e: "0 < e"
     then have e2_gt: "0 < e/2 / content (cbox c d)" and e2_less: "e/2 / content (cbox c d) * content (cbox c d) < e"
       by (auto simp: False content_lt_nz e)
     then obtain dd
     where dd: "\<And>x x'. \<lbrakk>x\<in>cbox (a, c) (b, d); x'\<in>cbox (a, c) (b, d); norm (x' - x) < dd\<rbrakk>
                        \<Longrightarrow> norm (f x' - f x) \<le> e/(2 * content (cbox c d))"  "dd>0"
       using uc [unfolded uniformly_continuous_on_def, THEN spec, of "e/(2 * content (cbox c d))"]
       by (auto simp: dist_norm intro: less_imp_le)
     have "\<exists>delta>0. \<forall>x'\<in>cbox a b. norm (x' - x) < delta \<longrightarrow> norm (integral (cbox c d) (\<lambda>u. f (x', u) - f (x, u))) < e"
       using dd e2_gt assms x
       apply (rule_tac x=dd in exI)
       apply clarify
       apply (rule le_less_trans [OF integrable_bound e2_less])
       apply (auto intro: integrable_diff continuous_on_imp_integrable_on_Pair1)
       done
   } note * = this
   show ?thesis
   proof (rule integrable_continuous)
     show "continuous_on (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y)))"
       by (simp add: * continuous_on_iff dist_norm integral_diff [symmetric] continuous_on_imp_integrable_on_Pair1 [OF assms])
   qed
 qed
 
 lemma integral_split:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
   assumes f: "f integrable_on (cbox a b)"
       and k: "k \<in> Basis"
   shows "integral (cbox a b) f =
            integral (cbox a b \<inter> {x. x\<bullet>k \<le> c}) f +
            integral (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) f"
   using k f
   by (auto simp: has_integral_integral intro: integral_unique [OF has_integral_split])
 
 lemma integral_swap_operativeI:
   fixes f :: "('a::euclidean_space * 'b::euclidean_space) \<Rightarrow> 'c::banach"
   assumes f: "continuous_on s f" and e: "0 < e"
     shows "operative conj True
            (\<lambda>k. \<forall>a b c d.
                 cbox (a,c) (b,d) \<subseteq> k \<and> cbox (a,c) (b,d) \<subseteq> s
                 \<longrightarrow> norm(integral (cbox (a,c) (b,d)) f -
                          integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f((x,y)))))
                     \<le> e * content (cbox (a,c) (b,d)))"
 proof (standard, auto)
   fix a::'a and c::'b and b::'a and d::'b and u::'a and v::'a and w::'b and z::'b
   assume *: "box (a, c) (b, d) = {}"
      and cb1: "cbox (u, w) (v, z) \<subseteq> cbox (a, c) (b, d)"
      and cb2: "cbox (u, w) (v, z) \<subseteq> s"
   then have c0: "content (cbox (a, c) (b, d)) = 0"
     using * unfolding content_eq_0_interior by simp
   have c0': "content (cbox (u, w) (v, z)) = 0"
     by (fact content_0_subset [OF c0 cb1])
   show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))))
           \<le> e * content (cbox (u,w) (v,z))"
     using content_cbox_pair_eq0_D [OF c0']
     by (force simp add: c0')
 next
   fix a::'a and c::'b and b::'a and d::'b
   and M::real and i::'a and j::'b
   and u::'a and v::'a and w::'b and z::'b
   assume ij: "(i,j) \<in> Basis"
      and n1: "\<forall>a' b' c' d'.
                 cbox (a',c') (b',d') \<subseteq> cbox (a,c) (b,d) \<and>
                 cbox (a',c') (b',d') \<subseteq> {x. x \<bullet> (i,j) \<le> M} \<and> cbox (a',c') (b',d') \<subseteq> s \<longrightarrow>
                 norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (\<lambda>x. integral (cbox c' d') (\<lambda>y. f (x,y))))
                 \<le> e * content (cbox (a',c') (b',d'))"
      and n2: "\<forall>a' b' c' d'.
                 cbox (a',c') (b',d') \<subseteq> cbox (a,c) (b,d) \<and>
                 cbox (a',c') (b',d') \<subseteq> {x. M \<le> x \<bullet> (i,j)} \<and> cbox (a',c') (b',d') \<subseteq> s \<longrightarrow>
                 norm (integral (cbox (a',c') (b',d')) f - integral (cbox a' b') (\<lambda>x. integral (cbox c' d') (\<lambda>y. f (x,y))))
                 \<le> e * content (cbox (a',c') (b',d'))"
      and subs: "cbox (u,w) (v,z) \<subseteq> cbox (a,c) (b,d)"  "cbox (u,w) (v,z) \<subseteq> s"
   have fcont: "continuous_on (cbox (u, w) (v, z)) f"
     using assms(1) continuous_on_subset  subs(2) by blast
   then have fint: "f integrable_on cbox (u, w) (v, z)"
     by (metis integrable_continuous)
   consider "i \<in> Basis" "j=0" | "j \<in> Basis" "i=0"  using ij
     by (auto simp: Euclidean_Space.Basis_prod_def)
   then show "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y))))
              \<le> e * content (cbox (u,w) (v,z))" (is ?normle)
   proof cases
     case 1
     then have e: "e * content (cbox (u, w) (v, z)) =
                   e * (content (cbox u v \<inter> {x. x \<bullet> i \<le> M}) * content (cbox w z)) +
                   e * (content (cbox u v \<inter> {x. M \<le> x \<bullet> i}) * content (cbox w z))"
       by (simp add: content_split [where c=M] content_Pair algebra_simps)
     have *: "integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))) =
                 integral (cbox u v \<inter> {x. x \<bullet> i \<le> M}) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))) +
                 integral (cbox u v \<inter> {x. M \<le> x \<bullet> i}) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y)))"
       using 1 f subs integral_integrable_2dim continuous_on_subset
       by (blast intro: integral_split)
     show ?normle
       apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e])
       using 1 subs
       apply (simp_all add: cbox_Pair_eq setcomp_dot1 [of "\<lambda>u. M\<le>u"] setcomp_dot1 [of "\<lambda>u. u\<le>M"] Sigma_Int_Paircomp1)
       apply (simp_all add: interval_split ij flip: cbox_Pair_eq content_Pair)
       apply (force simp flip: interval_split intro!: n1 [rule_format])
       apply (force simp flip: interval_split intro!: n2 [rule_format])
       done
   next
     case 2
     then have e: "e * content (cbox (u, w) (v, z)) =
                   e * (content (cbox u v) * content (cbox w z \<inter> {x. x \<bullet> j \<le> M})) +
                   e * (content (cbox u v) * content (cbox w z \<inter> {x. M \<le> x \<bullet> j}))"
       by (simp add: content_split [where c=M] content_Pair algebra_simps)
     have "(\<lambda>x. integral (cbox w z \<inter> {x. x \<bullet> j \<le> M}) (\<lambda>y. f (x, y))) integrable_on cbox u v"
          "(\<lambda>x. integral (cbox w z \<inter> {x. M \<le> x \<bullet> j}) (\<lambda>y. f (x, y))) integrable_on cbox u v"
       using 2 subs
       apply (simp_all add: interval_split)
       apply (rule integral_integrable_2dim [OF continuous_on_subset [OF f]]; auto simp: cbox_Pair_eq interval_split [symmetric])+
       done
     with 2 have *: "integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))) =
                    integral (cbox u v) (\<lambda>x. integral (cbox w z \<inter> {x. x \<bullet> j \<le> M}) (\<lambda>y. f (x, y))) +
                    integral (cbox u v) (\<lambda>x. integral (cbox w z \<inter> {x. M \<le> x \<bullet> j}) (\<lambda>y. f (x, y)))"
       by (simp add: integral_add [symmetric] integral_split [symmetric]
                     continuous_on_imp_integrable_on_Pair1 [OF fcont] cong: integral_cong)
     show ?normle
       apply (rule norm_diff2 [OF integral_split [where c=M, OF fint ij] * e])
       using 2 subs
       apply (simp_all add: cbox_Pair_eq setcomp_dot2 [of "\<lambda>u. M\<le>u"] setcomp_dot2 [of "\<lambda>u. u\<le>M"] Sigma_Int_Paircomp2)
       apply (simp_all add: interval_split ij flip: cbox_Pair_eq content_Pair)
       apply (force simp flip: interval_split intro!: n1 [rule_format])
       apply (force simp flip: interval_split intro!: n2 [rule_format])
       done
   qed
 qed
 
 lemma integral_Pair_const:
     "integral (cbox (a,c) (b,d)) (\<lambda>x. k) =
      integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. k))"
   by (simp add: content_Pair)
 
 lemma integral_prod_continuous:
   fixes f :: "('a::euclidean_space * 'b::euclidean_space) \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a, c) (b, d)) f" (is "continuous_on ?CBOX f")
     shows "integral (cbox (a, c) (b, d)) f = integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y)))"
 proof (cases "content ?CBOX = 0")
   case True
   then show ?thesis
     by (auto simp: content_Pair)
 next
   case False
   then have cbp: "content ?CBOX > 0"
     using content_lt_nz by blast
   have "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y)))) = 0"
   proof (rule dense_eq0_I, simp)
     fix e :: real 
     assume "0 < e"
     with \<open>content ?CBOX > 0\<close> have "0 < e/content ?CBOX"
       by simp
     have f_int_acbd: "f integrable_on ?CBOX"
       by (rule integrable_continuous [OF assms])
     { fix p
       assume p: "p division_of ?CBOX"
       then have "finite p"
         by blast
       define e' where "e' = e/content ?CBOX"
       with \<open>0 < e\<close> \<open>0 < e/content ?CBOX\<close>
       have "0 < e'"
         by simp
       interpret operative conj True
            "\<lambda>k. \<forall>a' b' c' d'.
                 cbox (a', c') (b', d') \<subseteq> k \<and> cbox (a', c') (b', d') \<subseteq> ?CBOX
                 \<longrightarrow> norm (integral (cbox (a', c') (b', d')) f -
                          integral (cbox a' b') (\<lambda>x. integral (cbox c' d') (\<lambda>y. f ((x, y)))))
                     \<le> e' * content (cbox (a', c') (b', d'))"
         using assms \<open>0 < e'\<close> by (rule integral_swap_operativeI)
       have "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y))))
           \<le> e' * content ?CBOX"
         if "\<And>t u v w z. t \<in> p \<Longrightarrow> cbox (u, w) (v, z) \<subseteq> t \<Longrightarrow> cbox (u, w) (v, z) \<subseteq> ?CBOX
           \<Longrightarrow> norm (integral (cbox (u, w) (v, z)) f -
               integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))))
               \<le> e' * content (cbox (u, w) (v, z))"
         using that division [of p "(a, c)" "(b, d)"] p \<open>finite p\<close> by (auto simp add: comm_monoid_set_F_and)
       with False have "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x, y))))
           \<le> e"
         if "\<And>t u v w z. t \<in> p \<Longrightarrow> cbox (u, w) (v, z) \<subseteq> t \<Longrightarrow> cbox (u, w) (v, z) \<subseteq> ?CBOX
           \<Longrightarrow> norm (integral (cbox (u, w) (v, z)) f -
               integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))))
               \<le> e * content (cbox (u, w) (v, z)) / content ?CBOX"
         using that by (simp add: e'_def)
     } note op_acbd = this
     { fix k::real and \<D> and u::'a and v w and z::'b and t1 t2 l
       assume k: "0 < k"
          and nf: "\<And>x y u v.
                   \<lbrakk>x \<in> cbox a b; y \<in> cbox c d; u \<in> cbox a b; v\<in>cbox c d; norm (u-x, v-y) < k\<rbrakk>
                   \<Longrightarrow> norm (f(u,v) - f(x,y)) < e/(2 * (content ?CBOX))"
          and p_acbd: "\<D> tagged_division_of cbox (a,c) (b,d)"
          and fine: "(\<lambda>x. ball x k) fine \<D>"  "((t1,t2), l) \<in> \<D>"
          and uwvz_sub: "cbox (u,w) (v,z) \<subseteq> l" "cbox (u,w) (v,z) \<subseteq> cbox (a,c) (b,d)"
       have t: "t1 \<in> cbox a b" "t2 \<in> cbox c d"
         by (meson fine p_acbd cbox_Pair_iff tag_in_interval)+
       have ls: "l \<subseteq> ball (t1,t2) k"
         using fine by (simp add: fine_def Ball_def)
       { fix x1 x2
         assume xuvwz: "x1 \<in> cbox u v" "x2 \<in> cbox w z"
         then have x: "x1 \<in> cbox a b" "x2 \<in> cbox c d"
           using uwvz_sub by auto
         have "norm (x1 - t1, x2 - t2) = norm (t1 - x1, t2 - x2)"
           by (simp add: norm_Pair norm_minus_commute)
         also have "norm (t1 - x1, t2 - x2) < k"
           using xuvwz ls uwvz_sub unfolding ball_def
           by (force simp add: cbox_Pair_eq dist_norm )
         finally have "norm (f (x1,x2) - f (t1,t2)) \<le> e/content ?CBOX/2"
           using nf [OF t x]  by simp
       } note nf' = this
       have f_int_uwvz: "f integrable_on cbox (u,w) (v,z)"
         using f_int_acbd uwvz_sub integrable_on_subcbox by blast
       have f_int_uv: "\<And>x. x \<in> cbox u v \<Longrightarrow> (\<lambda>y. f (x,y)) integrable_on cbox w z"
         using assms continuous_on_subset uwvz_sub
         by (blast intro: continuous_on_imp_integrable_on_Pair1)
       have 1: "norm (integral (cbox (u,w) (v,z)) f - integral (cbox (u,w) (v,z)) (\<lambda>x. f (t1,t2)))
              \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
         using cbp \<open>0 < e/content ?CBOX\<close> nf'
         apply (simp only: integral_diff [symmetric] f_int_uwvz integrable_const)
         apply (auto simp: integrable_diff f_int_uwvz integrable_const intro: order_trans [OF integrable_bound [of "e/content ?CBOX/2"]])
         done
       have int_integrable: "(\<lambda>x. integral (cbox w z) (\<lambda>y. f (x, y))) integrable_on cbox u v"
         using assms integral_integrable_2dim continuous_on_subset uwvz_sub(2) by blast
       have normint_wz:
          "\<And>x. x \<in> cbox u v \<Longrightarrow>
                norm (integral (cbox w z) (\<lambda>y. f (x, y)) - integral (cbox w z) (\<lambda>y. f (t1, t2)))
                \<le> e * content (cbox w z) / content (cbox (a, c) (b, d))/2"
         using cbp \<open>0 < e/content ?CBOX\<close> nf'
         apply (simp only: integral_diff [symmetric] f_int_uv integrable_const)
         apply (auto simp: integrable_diff f_int_uv integrable_const intro: order_trans [OF integrable_bound [of "e/content ?CBOX/2"]])
         done
       have "norm (integral (cbox u v)
                (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y)) - integral (cbox w z) (\<lambda>y. f (t1,t2))))
             \<le> e * content (cbox w z) / content ?CBOX/2 * content (cbox u v)"
         using cbp \<open>0 < e/content ?CBOX\<close>
         apply (intro integrable_bound [OF _ _ normint_wz])
         apply (auto simp: field_split_simps integrable_diff int_integrable integrable_const)
         done
       also have "... \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
         by (simp add: content_Pair field_split_simps)
       finally have 2: "norm (integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y))) -
                       integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (t1,t2))))
                 \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX/2"
         by (simp only: integral_diff [symmetric] int_integrable integrable_const)
       have norm_xx: "\<lbrakk>x' = y'; norm(x - x') \<le> e/2; norm(y - y') \<le> e/2\<rbrakk> \<Longrightarrow> norm(x - y) \<le> e" for x::'c and y x' y' e
         using norm_triangle_mono [of "x-y'" "e/2" "y'-y" "e/2"] field_sum_of_halves
         by (simp add: norm_minus_commute)
       have "norm (integral (cbox (u,w) (v,z)) f - integral (cbox u v) (\<lambda>x. integral (cbox w z) (\<lambda>y. f (x,y))))
             \<le> e * content (cbox (u,w) (v,z)) / content ?CBOX"
         by (rule norm_xx [OF integral_Pair_const 1 2])
     } note * = this
     have "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y)))) \<le> e" 
       if "\<forall>x\<in>?CBOX. \<forall>x'\<in>?CBOX. norm (x' - x) < k \<longrightarrow> norm (f x' - f x) < e /(2 * content (?CBOX))" "0 < k" for k
     proof -
       obtain p where ptag: "p tagged_division_of cbox (a, c) (b, d)" 
                  and fine: "(\<lambda>x. ball x k) fine p"
         using fine_division_exists \<open>0 < k\<close> by blast
       show ?thesis
         using that fine ptag \<open>0 < k\<close> 
         by (auto simp: * intro: op_acbd [OF division_of_tagged_division [OF ptag]])
     qed
     then show "norm (integral ?CBOX f - integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. f (x,y)))) \<le> e"
       using compact_uniformly_continuous [OF assms compact_cbox]
       apply (simp add: uniformly_continuous_on_def dist_norm)
       apply (drule_tac x="e/2 / content?CBOX" in spec)
       using cbp \<open>0 < e\<close> by (auto simp: zero_less_mult_iff)
   qed
   then show ?thesis
     by simp
 qed
 
 lemma integral_swap_2dim:
   fixes f :: "['a::euclidean_space, 'b::euclidean_space] \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
     shows "integral (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y) = integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)"
 proof -
   have "((\<lambda>(x, y). f x y) has_integral integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)) (prod.swap ` (cbox (c, a) (d, b)))"
   proof (rule has_integral_twiddle [of 1 prod.swap prod.swap "\<lambda>(x,y). f y x" "integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)", simplified])
     show "\<And>u v. content (prod.swap ` cbox u v) = content (cbox u v)"
       by (metis content_Pair mult.commute old.prod.exhaust swap_cbox_Pair)
     show "((\<lambda>(x, y). f y x) has_integral integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)) (cbox (c, a) (d, b))"
       by (simp add: assms integrable_continuous integrable_integral swap_continuous)
   qed (use isCont_swap in \<open>fastforce+\<close>)
  then show ?thesis
    by force
 qed
 
 theorem integral_swap_continuous:
   fixes f :: "['a::euclidean_space, 'b::euclidean_space] \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
     shows "integral (cbox a b) (\<lambda>x. integral (cbox c d) (f x)) =
            integral (cbox c d) (\<lambda>y. integral (cbox a b) (\<lambda>x. f x y))"
 proof -
   have "integral (cbox a b) (\<lambda>x. integral (cbox c d) (f x)) = integral (cbox (a, c) (b, d)) (\<lambda>(x, y). f x y)"
     using integral_prod_continuous [OF assms] by auto
   also have "... = integral (cbox (c, a) (d, b)) (\<lambda>(x, y). f y x)"
     by (rule integral_swap_2dim [OF assms])
   also have "... = integral (cbox c d) (\<lambda>y. integral (cbox a b) (\<lambda>x. f x y))"
     using integral_prod_continuous [OF swap_continuous] assms
     by auto
   finally show ?thesis .
 qed
 
 
 subsection \<open>Definite integrals for exponential and power function\<close>
 
 lemma has_integral_exp_minus_to_infinity:
   assumes a: "a > 0"
   shows   "((\<lambda>x::real. exp (-a*x)) has_integral exp (-a*c)/a) {c..}"
 proof -
   define f where "f = (\<lambda>k x. if x \<in> {c..real k} then exp (-a*x) else 0)"
   {
     fix k :: nat assume k: "of_nat k \<ge> c"
     from k a
       have "((\<lambda>x. exp (-a*x)) has_integral (-exp (-a*real k)/a - (-exp (-a*c)/a))) {c..real k}"
       by (intro fundamental_theorem_of_calculus)
          (auto intro!: derivative_eq_intros
                simp: has_field_derivative_iff_has_vector_derivative [symmetric])
     hence "(f k has_integral (exp (-a*c)/a - exp (-a*real k)/a)) {c..}" unfolding f_def
       by (subst has_integral_restrict) simp_all
   } note has_integral_f = this
 
   have [simp]: "f k = (\<lambda>_. 0)" if "of_nat k < c" for k using that by (auto simp: fun_eq_iff f_def)
   have integral_f: "integral {c..} (f k) =
                       (if real k \<ge> c then exp (-a*c)/a - exp (-a*real k)/a else 0)"
     for k using integral_unique[OF has_integral_f[of k]] by simp
 
   have A: "(\<lambda>x. exp (-a*x)) integrable_on {c..} \<and>
              (\<lambda>k. integral {c..} (f k)) \<longlonglongrightarrow> integral {c..} (\<lambda>x. exp (-a*x))"
   proof (intro monotone_convergence_increasing allI ballI)
     fix k ::nat
     have "(\<lambda>x. exp (-a*x)) integrable_on {c..of_real k}" (is ?P)
       unfolding f_def by (auto intro!: continuous_intros integrable_continuous_real)
     hence  "(f k) integrable_on {c..of_real k}"
       by (rule integrable_eq) (simp add: f_def)
     then show "f k integrable_on {c..}"
       by (rule integrable_on_superset) (auto simp: f_def)
   next
     fix x assume x: "x \<in> {c..}"
     have "sequentially \<le> principal {nat \<lceil>x\<rceil>..}" unfolding at_top_def by (simp add: Inf_lower)
     also have "{nat \<lceil>x\<rceil>..} \<subseteq> {k. x \<le> real k}" by auto
     also have "inf (principal \<dots>) (principal {k. \<not>x \<le> real k}) =
                  principal ({k. x \<le> real k} \<inter> {k. \<not>x \<le> real k})" by simp
     also have "{k. x \<le> real k} \<inter> {k. \<not>x \<le> real k} = {}" by blast
     finally have "inf sequentially (principal {k. \<not>x \<le> real k}) = bot"
       by (simp add: inf.coboundedI1 bot_unique)
     with x show "(\<lambda>k. f k x) \<longlonglongrightarrow> exp (-a*x)" unfolding f_def
       by (intro filterlim_If) auto
   next
     have "\<bar>integral {c..} (f k)\<bar> \<le> exp (-a*c)/a" for k
     proof (cases "c > of_nat k")
       case False
       hence "abs (integral {c..} (f k)) = abs (exp (- (a * c)) / a - exp (- (a * real k)) / a)"
         by (simp add: integral_f)
       also have "abs (exp (- (a * c)) / a - exp (- (a * real k)) / a) =
                    exp (- (a * c)) / a - exp (- (a * real k)) / a"
         using False a by (intro abs_of_nonneg) (simp_all add: field_simps)
       also have "\<dots> \<le> exp (- a * c) / a" using a by simp
       finally show ?thesis .
     qed (insert a, simp_all add: integral_f)
     thus "bounded (range(\<lambda>k. integral {c..} (f k)))"
       by (intro boundedI[of _ "exp (-a*c)/a"]) auto
   qed (auto simp: f_def)
   have "(\<lambda>k. exp (-a*c)/a - exp (-a * of_nat k)/a) \<longlonglongrightarrow> exp (-a*c)/a - 0/a"
     by (intro tendsto_intros filterlim_compose[OF exp_at_bot]
         filterlim_tendsto_neg_mult_at_bot[OF tendsto_const] filterlim_real_sequentially)+
       (insert a, simp_all)
   moreover
   from eventually_gt_at_top[of "nat \<lceil>c\<rceil>"] have "eventually (\<lambda>k. of_nat k > c) sequentially"
     by eventually_elim linarith
   hence "eventually (\<lambda>k. exp (-a*c)/a - exp (-a * of_nat k)/a = integral {c..} (f k)) sequentially"
     by eventually_elim (simp add: integral_f) 
   ultimately have "(\<lambda>k. integral {c..} (f k)) \<longlonglongrightarrow> exp (-a*c)/a - 0/a"
     by (rule Lim_transform_eventually)
   from LIMSEQ_unique[OF conjunct2[OF A] this]
   have "integral {c..} (\<lambda>x. exp (-a*x)) = exp (-a*c)/a" by simp
   with conjunct1[OF A] show ?thesis
     by (simp add: has_integral_integral)
 qed
 
 lemma integrable_on_exp_minus_to_infinity: "a > 0 \<Longrightarrow> (\<lambda>x. exp (-a*x) :: real) integrable_on {c..}"
   using has_integral_exp_minus_to_infinity[of a c] unfolding integrable_on_def by blast
 
 lemma has_integral_powr_from_0:
   assumes a: "a > (-1::real)" and c: "c \<ge> 0"
   shows   "((\<lambda>x. x powr a) has_integral (c powr (a+1) / (a+1))) {0..c}"
 proof (cases "c = 0")
   case False
   define f where "f = (\<lambda>k x. if x \<in> {inverse (of_nat (Suc k))..c} then x powr a else 0)"
   define F where "F = (\<lambda>k. if inverse (of_nat (Suc k)) \<le> c then
                              c powr (a+1)/(a+1) - inverse (real (Suc k)) powr (a+1)/(a+1) else 0)"
   {
     fix k :: nat
     have "(f k has_integral F k) {0..c}"
     proof (cases "inverse (of_nat (Suc k)) \<le> c")
       case True
       {
         fix x assume x: "x \<ge> inverse (1 + real k)"
         have "0 < inverse (1 + real k)" by simp
         also note x
         finally have "x > 0" .
       } note x = this
       hence "((\<lambda>x. x powr a) has_integral c powr (a + 1) / (a + 1) -
                inverse (real (Suc k)) powr (a + 1) / (a + 1)) {inverse (real (Suc k))..c}"
         using True a by (intro fundamental_theorem_of_calculus)
            (auto intro!: derivative_eq_intros continuous_on_powr' continuous_on_const
               simp: has_field_derivative_iff_has_vector_derivative [symmetric])
       with True show ?thesis unfolding f_def F_def by (subst has_integral_restrict) simp_all
     next
       case False
       thus ?thesis unfolding f_def F_def by (subst has_integral_restrict) auto
     qed
   } note has_integral_f = this
   have integral_f: "integral {0..c} (f k) = F k" for k
     using has_integral_f[of k] by (rule integral_unique)
 
   have A: "(\<lambda>x. x powr a) integrable_on {0..c} \<and>
            (\<lambda>k. integral {0..c} (f k)) \<longlonglongrightarrow> integral {0..c} (\<lambda>x. x powr a)"
   proof (intro monotone_convergence_increasing ballI allI)
     fix k from has_integral_f[of k] show "f k integrable_on {0..c}"
       by (auto simp: integrable_on_def)
   next
     fix k :: nat and x :: real
     {
       assume x: "inverse (real (Suc k)) \<le> x"
       have "inverse (real (Suc (Suc k))) \<le> inverse (real (Suc k))" by (simp add: field_simps)
       also note x
       finally have "inverse (real (Suc (Suc k))) \<le> x" .
     }
     thus "f k x \<le> f (Suc k) x" by (auto simp: f_def simp del: of_nat_Suc)
   next
     fix x assume x: "x \<in> {0..c}"
     show "(\<lambda>k. f k x) \<longlonglongrightarrow> x powr a"
     proof (cases "x = 0")
       case False
       with x have "x > 0" by simp
       from order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this]
         have "eventually (\<lambda>k. x powr a = f k x) sequentially"
         by eventually_elim (insert x, simp add: f_def)
       moreover have "(\<lambda>_. x powr a) \<longlonglongrightarrow> x powr a" by simp
       ultimately show ?thesis by (blast intro: Lim_transform_eventually)
     qed (simp_all add: f_def)
   next
     {
       fix k
       from a have "F k \<le> c powr (a + 1) / (a + 1)"
         by (auto simp add: F_def divide_simps)
       also from a have "F k \<ge> 0"
         by (auto simp: F_def divide_simps simp del: of_nat_Suc intro!: powr_mono2)
       hence "F k = abs (F k)" by simp
       finally have "abs (F k) \<le>  c powr (a + 1) / (a + 1)" .
     }
     thus "bounded (range(\<lambda>k. integral {0..c} (f k)))"
       by (intro boundedI[of _ "c powr (a+1) / (a+1)"]) (auto simp: integral_f)
   qed
 
   from False c have "c > 0" by simp
   from order_tendstoD(2)[OF LIMSEQ_inverse_real_of_nat this]
     have "eventually (\<lambda>k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a+1) / (a+1) =
             integral {0..c} (f k)) sequentially"
     by eventually_elim (simp add: integral_f F_def)
   moreover have "(\<lambda>k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1))
                    \<longlonglongrightarrow> c powr (a + 1) / (a + 1) - 0 powr (a + 1) / (a + 1)"
     using a by (intro tendsto_intros LIMSEQ_inverse_real_of_nat) auto
   hence "(\<lambda>k. c powr (a + 1) / (a + 1) - inverse (real (Suc k)) powr (a + 1) / (a + 1))
           \<longlonglongrightarrow> c powr (a + 1) / (a + 1)" by simp
   ultimately have "(\<lambda>k. integral {0..c} (f k)) \<longlonglongrightarrow> c powr (a+1) / (a+1)"
     by (blast intro: Lim_transform_eventually)
   with A have "integral {0..c} (\<lambda>x. x powr a) = c powr (a+1) / (a+1)"
     by (blast intro: LIMSEQ_unique)
   with A show ?thesis by (simp add: has_integral_integral)
 qed (simp_all add: has_integral_refl)
 
 lemma integrable_on_powr_from_0:
   assumes a: "a > (-1::real)" and c: "c \<ge> 0"
   shows   "(\<lambda>x. x powr a) integrable_on {0..c}"
   using has_integral_powr_from_0[OF assms] unfolding integrable_on_def by blast
 
 lemma has_integral_powr_to_inf:
   fixes a e :: real
   assumes "e < -1" "a > 0"
   shows   "((\<lambda>x. x powr e) has_integral -(a powr (e + 1)) / (e + 1)) {a..}"
 proof -
   define f :: "nat \<Rightarrow> real \<Rightarrow> real" where "f = (\<lambda>n x. if x \<in> {a..n} then x powr e else 0)"
   define F where "F = (\<lambda>x. x powr (e + 1) / (e + 1))"
 
   have has_integral_f: "(f n has_integral (F n - F a)) {a..}"
     if n: "n \<ge> a" for n :: nat
   proof -
     from n assms have "((\<lambda>x. x powr e) has_integral (F n - F a)) {a..n}"
       by (intro fundamental_theorem_of_calculus) (auto intro!: derivative_eq_intros
             simp: has_field_derivative_iff_has_vector_derivative [symmetric] F_def)
     hence "(f n has_integral (F n - F a)) {a..n}"
       by (rule has_integral_eq [rotated]) (simp add: f_def)
     thus "(f n has_integral (F n - F a)) {a..}"
       by (rule has_integral_on_superset) (auto simp: f_def)
   qed
   have integral_f: "integral {a..} (f n) = (if n \<ge> a then F n - F a else 0)" for n :: nat
   proof (cases "n \<ge> a")
     case True
     with has_integral_f[OF this] show ?thesis by (simp add: integral_unique)
   next
     case False
     have "(f n has_integral 0) {a}" by (rule has_integral_refl)
     hence "(f n has_integral 0) {a..}"
       by (rule has_integral_on_superset) (insert False, simp_all add: f_def)
     with False show ?thesis by (simp add: integral_unique)
   qed
 
   have *: "(\<lambda>x. x powr e) integrable_on {a..} \<and>
            (\<lambda>n. integral {a..} (f n)) \<longlonglongrightarrow> integral {a..} (\<lambda>x. x powr e)"
   proof (intro monotone_convergence_increasing allI ballI)
     fix n :: nat
     from assms have "(\<lambda>x. x powr e) integrable_on {a..n}"
       by (auto intro!: integrable_continuous_real continuous_intros)
     hence "f n integrable_on {a..n}"
       by (rule integrable_eq) (auto simp: f_def)
     thus "f n integrable_on {a..}"
       by (rule integrable_on_superset) (auto simp: f_def)
   next
     fix n :: nat and x :: real
     show "f n x \<le> f (Suc n) x" by (auto simp: f_def)
   next
     fix x :: real assume x: "x \<in> {a..}"
     from filterlim_real_sequentially
       have "eventually (\<lambda>n. real n \<ge> x) at_top"
       by (simp add: filterlim_at_top)
     with x have "eventually (\<lambda>n. f n x = x powr e) at_top"
       by (auto elim!: eventually_mono simp: f_def)
     thus "(\<lambda>n. f n x) \<longlonglongrightarrow> x powr e" by (simp add: tendsto_eventually)
   next
     have "norm (integral {a..} (f n)) \<le> -F a" for n :: nat
     proof (cases "n \<ge> a")
       case True
       with assms have "a powr (e + 1) \<ge> n powr (e + 1)"
         by (intro powr_mono2') simp_all
       with assms show ?thesis by (auto simp: divide_simps F_def integral_f)
     qed (insert assms, simp add: integral_f F_def field_split_simps)
     thus "bounded (range(\<lambda>k. integral {a..} (f k)))"
       unfolding bounded_iff by (intro exI[of _ "-F a"]) auto
   qed
 
   from filterlim_real_sequentially
     have "eventually (\<lambda>n. real n \<ge> a) at_top"
     by (simp add: filterlim_at_top)
   hence "eventually (\<lambda>n. F n - F a = integral {a..} (f n)) at_top"
     by eventually_elim (simp add: integral_f)
   moreover have "(\<lambda>n. F n - F a) \<longlonglongrightarrow> 0 / (e + 1) - F a" unfolding F_def
     by (insert assms, (rule tendsto_intros filterlim_compose[OF tendsto_neg_powr]
           filterlim_ident filterlim_real_sequentially | simp)+)
   hence "(\<lambda>n. F n - F a) \<longlonglongrightarrow> -F a" by simp
   ultimately have "(\<lambda>n. integral {a..} (f n)) \<longlonglongrightarrow> -F a" by (blast intro: Lim_transform_eventually)
   from conjunct2[OF *] and this
     have "integral {a..} (\<lambda>x. x powr e) = -F a" by (rule LIMSEQ_unique)
   with conjunct1[OF *] show ?thesis
     by (simp add: has_integral_integral F_def)
 qed
 
 lemma has_integral_inverse_power_to_inf:
   fixes a :: real and n :: nat
   assumes "n > 1" "a > 0"
   shows   "((\<lambda>x. 1 / x ^ n) has_integral 1 / (real (n - 1) * a ^ (n - 1))) {a..}"
 proof -
   from assms have "real_of_int (-int n) < -1" by simp
   note has_integral_powr_to_inf[OF this \<open>a > 0\<close>]
   also have "- (a powr (real_of_int (- int n) + 1)) / (real_of_int (- int n) + 1) =
                  1 / (real (n - 1) * a powr (real (n - 1)))" using assms
     by (simp add: field_split_simps powr_add [symmetric] of_nat_diff)
   also from assms have "a powr (real (n - 1)) = a ^ (n - 1)"
     by (intro powr_realpow)
   finally show ?thesis
     by (rule has_integral_eq [rotated])
        (insert assms, simp_all add: powr_minus powr_realpow field_split_simps)
 qed
 
 subsubsection \<open>Adaption to ordered Euclidean spaces and the Cartesian Euclidean space\<close>
 
 lemma integral_component_eq_cart[simp]:
   fixes f :: "'n::euclidean_space \<Rightarrow> real^'m"
   assumes "f integrable_on s"
   shows "integral s (\<lambda>x. f x $ k) = integral s f $ k"
   using integral_linear[OF assms(1) bounded_linear_vec_nth,unfolded o_def] .
 
 lemma content_closed_interval:
   fixes a :: "'a::ordered_euclidean_space"
   assumes "a \<le> b"
   shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
   using content_cbox[of a b] assms by (simp add: cbox_interval eucl_le[where 'a='a])
 
 lemma integrable_const_ivl[intro]:
   fixes a::"'a::ordered_euclidean_space"
   shows "(\<lambda>x. c) integrable_on {a..b}"
   unfolding cbox_interval[symmetric] by (rule integrable_const)
 
 lemma integrable_on_subinterval:
   fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::banach"
   assumes "f integrable_on S" "{a..b} \<subseteq> S"
   shows "f integrable_on {a..b}"
   using integrable_on_subcbox[of f S a b] assms by (simp add: cbox_interval)
 
 end
diff --git a/src/HOL/Analysis/Improper_Integral.thy b/src/HOL/Analysis/Improper_Integral.thy
--- a/src/HOL/Analysis/Improper_Integral.thy
+++ b/src/HOL/Analysis/Improper_Integral.thy
@@ -1,2268 +1,2273 @@
 (*  Title:      HOL/Analysis/Improper_Integral.thy
     Author: LC Paulson (ported from HOL Light)
 *)
 
 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\<^marker>\<open>tag important\<close> 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)
 
 
 lemma equiintegrable_cmul:
   assumes F: "F equiintegrable_on I"
   shows "(\<Union>c \<in> {-k..k}. \<Union>f \<in> F. {(\<lambda>x. c *\<^sub>R f x)}) equiintegrable_on I"
   unfolding equiintegrable_on_def
   proof (intro conjI impI allI ballI)
   show "f integrable_on I"
     if "f \<in> (\<Union>c\<in>{- k..k}. \<Union>f\<in>F. {\<lambda>x. c *\<^sub>R f x})"
     for f :: "'a \<Rightarrow> 'b"
     using that assms equiintegrable_on_integrable integrable_cmul by blast
   show "\<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>f \<D>. f \<in> (\<Union>c\<in>{- k..k}. \<Union>f\<in>F. {\<lambda>x. c *\<^sub>R f x}) \<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> where "gauge \<gamma>"
       and \<gamma>: "\<And>f \<D>. \<lbrakk>f \<in> F; \<D> tagged_division_of I; \<gamma> fine \<D>\<rbrakk>
                     \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon> / (\<bar>k\<bar> + 1)"
       using assms \<open>\<epsilon> > 0\<close> unfolding equiintegrable_on_def
       by (metis add.commute add.right_neutral add_strict_mono divide_pos_pos norm_eq_zero real_norm_def zero_less_norm_iff zero_less_one)
     moreover have "norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R c *\<^sub>R (f x)) - integral I (\<lambda>x. c *\<^sub>R f x)) < \<epsilon>"
       if c: "c \<in> {- k..k}"
         and "f \<in> F" "\<D> tagged_division_of I" "\<gamma> fine \<D>"
       for \<D> c f
     proof -
       have "norm ((\<Sum>x\<in>\<D>. case x of (x, K) \<Rightarrow> content K *\<^sub>R c *\<^sub>R f x) - integral I (\<lambda>x. c *\<^sub>R f x))
           = \<bar>c\<bar> * norm ((\<Sum>x\<in>\<D>. case x of (x, K) \<Rightarrow> content K *\<^sub>R f x) - integral I f)"
         by (simp add: algebra_simps scale_sum_right case_prod_unfold flip: norm_scaleR)
       also have "\<dots> \<le> (\<bar>k\<bar> + 1) * norm ((\<Sum>x\<in>\<D>. case x of (x, K) \<Rightarrow> content K *\<^sub>R f x) - integral I f)"
         using c by (auto simp: mult_right_mono)
       also have "\<dots> < (\<bar>k\<bar> + 1) * (\<epsilon> / (\<bar>k\<bar> + 1))"
         by (rule mult_strict_left_mono) (use \<gamma> less_eq_real_def that in auto)
       also have "\<dots> = \<epsilon>"
         by auto
       finally show ?thesis .
     qed
     ultimately show ?thesis
       by (rule_tac x="\<gamma>" in exI) auto
   qed
 qed
 
 
 lemma equiintegrable_add:
   assumes F: "F equiintegrable_on I" and G: "G equiintegrable_on I"
   shows "(\<Union>f \<in> F. \<Union>g \<in> G. {(\<lambda>x. f x + g x)}) equiintegrable_on I"
   unfolding equiintegrable_on_def
 proof (intro conjI impI allI ballI)
   show "f integrable_on I"
     if "f \<in> (\<Union>f\<in>F. \<Union>g\<in>G. {\<lambda>x. f x + g x})" for f
     using that equiintegrable_on_integrable assms  by (auto intro: integrable_add)
   show "\<exists>\<gamma>. gauge \<gamma> \<and> (\<forall>f \<D>. f \<in> (\<Union>f\<in>F. \<Union>g\<in>G. {\<lambda>x. f x + g x}) \<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>. \<lbrakk>f \<in> F; \<D> tagged_division_of I; \<gamma>1 fine \<D>\<rbrakk>
                     \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) < \<epsilon>/2"
       using assms \<open>\<epsilon> > 0\<close> unfolding equiintegrable_on_def by (meson half_gt_zero_iff)
     obtain \<gamma>2 where  "gauge \<gamma>2"
       and \<gamma>2: "\<And>g \<D>. \<lbrakk>g \<in> G; \<D> tagged_division_of I; \<gamma>2 fine \<D>\<rbrakk>
                     \<Longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g x) - integral I g) < \<epsilon>/2"
       using assms \<open>\<epsilon> > 0\<close> unfolding equiintegrable_on_def by (meson half_gt_zero_iff)
     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 "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R h x) - integral I h) < \<epsilon>"
       if h: "h \<in> (\<Union>f\<in>F. \<Union>g\<in>G. {\<lambda>x. f x + g x})"
         and \<D>: "\<D> tagged_division_of I" and fine: "(\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x) fine \<D>"
       for h \<D>
     proof -
       obtain f g where "f \<in> F" "g \<in> G" and heq: "h = (\<lambda>x. f x + g x)"
         using h by blast
       then have int: "f integrable_on I" "g integrable_on I"
         using F G equiintegrable_on_def by blast+
       have "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R h x) - integral I h)
           = norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x + content K *\<^sub>R g x) - (integral I f + integral I g))"
         by (simp add: heq algebra_simps integral_add int)
       also have "\<dots> = norm (((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f + (\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g x) - integral I g))"
         by (simp add: sum.distrib algebra_simps case_prod_unfold)
       also have "\<dots> \<le> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R f x) - integral I f) + norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g x) - integral I g)"
         by (metis (mono_tags) add_diff_eq norm_triangle_ineq)
       also have "\<dots> < \<epsilon>/2 + \<epsilon>/2"
         using \<gamma>1 [OF \<open>f \<in> F\<close> \<D>] \<gamma>2 [OF \<open>g \<in> G\<close> \<D>] fine  by (simp add: fine_Int)
       finally show ?thesis by simp
     qed
     ultimately show ?thesis
       by meson
   qed
 qed
 
 lemma equiintegrable_minus:
   assumes "F equiintegrable_on I"
   shows "(\<Union>f \<in> F. {(\<lambda>x. - f x)}) equiintegrable_on I"
   by (force intro: equiintegrable_on_subset [OF equiintegrable_cmul [OF assms, of 1]])
 
 lemma equiintegrable_diff:
   assumes F: "F equiintegrable_on I" and G: "G equiintegrable_on I"
   shows "(\<Union>f \<in> F. \<Union>g \<in> G. {(\<lambda>x. f x - g x)}) equiintegrable_on I"
   by (rule equiintegrable_on_subset [OF equiintegrable_add [OF F equiintegrable_minus [OF G]]]) auto
 
 
 lemma equiintegrable_sum:
   fixes F :: "('a::euclidean_space \<Rightarrow> 'b::euclidean_space) set"
   assumes "F equiintegrable_on cbox a b"
   shows "(\<Union>I \<in> Collect finite. \<Union>c \<in> {c. (\<forall>i \<in> I. c i \<ge> 0) \<and> sum c I = 1}.
           \<Union>f \<in> I \<rightarrow> F. {(\<lambda>x. sum (\<lambda>i::'j. c i *\<^sub>R f i x) I)}) equiintegrable_on cbox a b"
     (is "?G equiintegrable_on _")
   unfolding equiintegrable_on_def
 proof (intro conjI impI allI ballI)
   show "f integrable_on cbox a b" if "f \<in> ?G" for f
     using that assms by (auto simp: equiintegrable_on_def intro!: integrable_sum integrable_cmul)
   show "\<exists>\<gamma>. gauge \<gamma>
            \<and> (\<forall>g \<D>. g \<in> ?G \<and> \<D> tagged_division_of cbox a b \<and> \<gamma> fine \<D>
               \<longrightarrow> norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g x) - integral (cbox a b) g) < \<epsilon>)"
     if "\<epsilon> > 0" for \<epsilon>
   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> / 2"
       using assms \<open>\<epsilon> > 0\<close> unfolding equiintegrable_on_def by (meson half_gt_zero_iff)
     moreover have "norm ((\<Sum>(x,K) \<in> \<D>. content K *\<^sub>R g x) - integral (cbox a b) g) < \<epsilon>"
       if g: "g \<in> ?G"
         and \<D>: "\<D> tagged_division_of cbox a b"
         and fine: "\<gamma> fine \<D>"
       for \<D> g
     proof -
       obtain I c f where "finite I" and 0: "\<And>i::'j. i \<in> I \<Longrightarrow> 0 \<le> c i"
         and 1: "sum c I = 1" and f: "f \<in> I \<rightarrow> F" and geq: "g = (\<lambda>x. \<Sum>i\<in>I. c i *\<^sub>R f i x)"
         using g by auto
       have fi_int: "f i integrable_on cbox a b" if "i \<in> I" for i
         by (metis Pi_iff assms equiintegrable_on_def f that)
       have *: "integral (cbox a b) (\<lambda>x. c i *\<^sub>R f i x) = (\<Sum>(x, K)\<in>\<D>. integral K (\<lambda>x. c i *\<^sub>R f i x))"
         if "i \<in> I" for i
       proof -
         have "f i integrable_on cbox a b"
           by (metis Pi_iff assms equiintegrable_on_def f that)
         then show ?thesis
           by (intro \<D> integrable_cmul integral_combine_tagged_division_topdown)
       qed
       have "finite \<D>"
         using \<D> by blast
       have swap: "(\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R (\<Sum>i\<in>I. c i *\<^sub>R f i x))
             = (\<Sum>i\<in>I. c i *\<^sub>R (\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f i x))"
         by (simp add: scale_sum_right case_prod_unfold algebra_simps) (rule sum.swap)
       have "norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R g x) - integral (cbox a b) g)
           = norm ((\<Sum>i\<in>I. c i *\<^sub>R ((\<Sum>(x,K)\<in>\<D>. content K *\<^sub>R f i x) - integral (cbox a b) (f i))))"
         unfolding geq swap
         by (simp add: scaleR_right.sum algebra_simps integral_sum fi_int integrable_cmul \<open>finite I\<close> sum_subtractf flip: sum_diff)
       also have "\<dots> \<le> (\<Sum>i\<in>I. c i * \<epsilon> / 2)"
       proof (rule sum_norm_le)
         show "norm (c i *\<^sub>R ((\<Sum>(xa, K)\<in>\<D>. content K *\<^sub>R f i xa) - integral (cbox a b) (f i))) \<le> c i * \<epsilon> / 2"
           if "i \<in> I" for i
         proof -
           have "norm ((\<Sum>(x, K)\<in>\<D>. content K *\<^sub>R f i x) - integral (cbox a b) (f i)) \<le> \<epsilon>/2"
             using \<gamma> [OF _ \<D> fine, of "f i"] funcset_mem [OF f] that  by auto
           then show ?thesis
             using that by (auto simp: 0 mult.assoc intro: mult_left_mono)
         qed
       qed
       also have "\<dots> < \<epsilon>"
         using 1 \<open>\<epsilon> > 0\<close> by (simp add: flip: sum_divide_distrib sum_distrib_right)
       finally show ?thesis .
     qed
     ultimately show ?thesis
       by (rule_tac x="\<gamma>" in exI) auto
   qed
 qed
 
 corollary equiintegrable_sum_real:
   fixes F :: "(real \<Rightarrow> 'b::euclidean_space) set"
   assumes "F equiintegrable_on {a..b}"
   shows "(\<Union>I \<in> Collect finite. \<Union>c \<in> {c. (\<forall>i \<in> I. c i \<ge> 0) \<and> sum c I = 1}.
           \<Union>f \<in> I \<rightarrow> F. {(\<lambda>x. sum (\<lambda>i. c i *\<^sub>R f i x) I)})
          equiintegrable_on {a..b}"
   using equiintegrable_sum [of F a b] assms by auto
 
 
 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)
+  unfolding equiintegrable_on_def 
+  by (metis diff_zero gauge_trivial integrable_on_null integral_null norm_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>
+  have \<section>: "\<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
+        by (auto simp add: eq sum.reindex [OF inj] intro!: sum.cong)
       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
+   show ?thesis
     using assms
     apply (auto simp: equiintegrable_on_def)
-    apply (rule integrable_eq)
-    by auto
+    subgoal for f
+      by (metis (mono_tags, lifting) comp_apply integrable_eq integrable_reflect)
+    using \<section> by fastforce
 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_cbox: "\<And>K.  K \<in> \<D> \<Longrightarrow> \<exists>a b. extend K = cbox a b"
+    using extend_def 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)
-      by fastforce
+    with i show "extend K \<subseteq> cbox a b"
+      by (auto simp: extend_def subset_box box_ne_empty)
+    have "a \<bullet> i \<le> b \<bullet> i"
+      using K by (metis bot.extremum_uniqueI box_ne_empty(1) i)
+    with K show "extend K \<noteq> {}"
+      by (simp add: extend_def i box_ne_empty)
   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 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)"
+        define \<xi> where "\<xi> \<equiv> (\<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)"
+        have [simp]: "\<xi> \<bullet> j = (if j = i then min (q \<bullet> j) (s \<bullet> j) else x \<bullet> j)" if "j \<in> Basis" for j
+          unfolding \<xi>_def
+          by (intro sum_if_inner that \<open>i \<in> Basis\<close>)
         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
+          have "min (q \<bullet> i) (s \<bullet> i) < v \<bullet> i"
+            using i s by fastforce
+          with \<open>i \<in> Basis\<close> s u ux xv
+          show "\<xi> \<in> box u v" 
+            by (force simp: mem_box)
+          have "min (q \<bullet> i) (s \<bullet> i) < z \<bullet> i"
+            using i q by force
+          with \<open>i \<in> Basis\<close> q w wx xz
+          show "\<xi> \<in> box w z"
+            by (force simp: mem_box)
         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)"
+        define \<xi> where "\<xi> \<equiv> (\<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)"
+        have [simp]: "\<xi> \<bullet> j = (if j = i then max (q \<bullet> j) (s \<bullet> j) else x \<bullet> j)" if "j \<in> Basis" for j
+          unfolding \<xi>_def
+          by (intro sum_if_inner that \<open>i \<in> Basis\<close>)
         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
+          show "\<xi> \<in> box u v"
+            using \<open>i \<in> Basis\<close> s by (force simp: mem_box ux v xv)
+          show "\<xi> \<in> box w z"
+            using \<open>i \<in> Basis\<close> q by (force simp: mem_box wx xz z)
         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)
+  define interv_diff where "interv_diff \<equiv> \<lambda>K. \<lambda>i::'a. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i"
+  have "?lhs = (\<Sum>K\<in>\<D>. (b \<bullet> i - a \<bullet> i) * content K / (interv_diff K i))"
+    by (simp add: sum_distrib_left interv_diff_def)
   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)"
+    then have "(b \<bullet> i - a \<bullet> i) * content K / (interv_diff K i)
+             = (b \<bullet> i - a \<bullet> i) * (\<Prod>i \<in> insert i (Basis \<inter> -{i}). v \<bullet> i - u \<bullet> i) / (interv_diff (cbox u v) 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)
+      using \<open>i \<in> Basis\<close> K uv by (simp add: prod.If_cases interv_diff_def) (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)"
+                        (\<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
+      by (intro prod.cong [OF refl]) (subst sum_if_inner; simp add: algebra_simps)+
     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" .
+      using \<open>i \<in> Basis\<close> K box_ne_empty \<open>K \<in> \<D>\<close> extend(1) 
+      by (auto simp add: extend_def content_cbox_if)
+    finally show "(b \<bullet> i - a \<bullet> i) * content K / (interv_diff K 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> (extend ` \<D>)"
-      using int_extend_disjoint apply (auto simp: division_of_def \<open>finite \<D>\<close> extend)
-      using extend_def apply blast
-      done
+      using int_extend_disjoint  by (auto simp: division_of_def \<open>finite \<D>\<close> extend extend_cbox)
     show "\<Union> (extend ` \<D>) \<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)"
+  define interv_diff where "interv_diff \<equiv> \<lambda>K. \<lambda>i::'a. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i"
   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')"
+
+  have zero_left: "\<And>x y. \<lbrakk>x \<in> \<D>; y \<in> \<D>; x \<noteq> y; x \<inter> {x. x \<bullet> i \<le> c} = y \<inter> {x. x \<bullet> i \<le> c}\<rbrakk>
+           \<Longrightarrow> content (y \<inter> {x. x \<bullet> i \<le> c}) = 0"
+    by (metis division_split_left_inj [OF div] lec_is_cbox content_eq_0_interior)
+  have zero_right: "\<And>x y. \<lbrakk>x \<in> \<D>; y \<in> \<D>; x \<noteq> y; x \<inter> {x. c \<le> x \<bullet> i} = y \<inter> {x. c \<le> x \<bullet> i}\<rbrakk>
+           \<Longrightarrow> content (y \<inter> {x. c \<le> x \<bullet> i}) = 0"
+    by (metis division_split_right_inj [OF div] gec_is_cbox content_eq_0_interior)
+
+  have "(b' \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>Dlec. content K / interv_diff K i) \<le> content(cbox a b')"
+    unfolding interv_diff_def
   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 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
+  have "(\<Sum>K\<in>Dlec. content K / (interv_diff K i)) = (\<Sum>K\<in>(\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}) ` \<D>. content K / interv_diff K i)"
+    unfolding Dlec_def using \<open>finite \<D>\<close> by (auto simp: sum.mono_neutral_left)
+  moreover have "... =
+        (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K)"
+    by (simp add: zero_left sum.reindex_nontrivial [OF \<open>finite \<D>\<close>])
   moreover have "(b' \<bullet> i - a \<bullet> i) = (c - a \<bullet> i)"
     by (simp add: b'_def 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)
+  have lec: "(c - a \<bullet> i) * (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K 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)"
+  have "(b \<bullet> i - a' \<bullet> i) * (\<Sum>K\<in>Dgec. content K / (interv_diff K i)) \<le> content(cbox a' b)"
+    unfolding interv_diff_def
   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 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
+  have "(\<Sum>K\<in>Dgec. content K / (interv_diff K i)) = (\<Sum>K\<in>(\<lambda>K. K \<inter> {x. c \<le> x \<bullet> i}) ` \<D>.
+       content K / interv_diff K i)"
+    unfolding Dgec_def using \<open>finite \<D>\<close> by (auto simp: sum.mono_neutral_left)
+  moreover have "\<dots> =
+        (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<ge> c}))) K)"
+    by (simp add: zero_right sum.reindex_nontrivial [OF \<open>finite \<D>\<close>])
   moreover have "(b \<bullet> i - a' \<bullet> i) = (b \<bullet> i - c)"
     by (simp add: a'_def 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)
+  have gec: "(b \<bullet> i - c) * (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K 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: i a'_def cong: if_cong)
+      moreover
+      have "(\<Sum>j\<in>Basis. (if j = i then a \<bullet> i else a \<bullet> j) *\<^sub>R j) = a"
         using euclidean_representation [of a] sum.cong [OF refl, of Basis "\<lambda>i. (a \<bullet> i) *\<^sub>R i"] by presburger
+      ultimately have "a' = a"
+        by (simp add: i a'_def cong: if_cong)
       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 _")
+      have eq: "(\<Sum>K\<in>\<D>. content (K \<inter> {x. a \<bullet> i \<le> x \<bullet> i}) / interv_diff (K \<inter> {x. a \<bullet> i \<le> x \<bullet> i}) i)
+              = (\<Sum>K\<in>\<D>. content K / interv_diff K 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
+        using gec c eq interv_diff_def by auto
     next
       assume c: "c = b \<bullet> i"
-      then have "b' = b"
-        apply (simp add: i b'_def cong: if_cong)
+      moreover have "(\<Sum>j\<in>Basis. (if j = i then b \<bullet> i else b \<bullet> j) *\<^sub>R j) = b"
         using euclidean_representation [of b] sum.cong [OF refl, of Basis "\<lambda>i. (b \<bullet> i) *\<^sub>R i"] by presburger
+      ultimately have "b' = b"
+        by (simp add: i b'_def cong: if_cong)
       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))"
+      have eq: "(\<Sum>K\<in>\<D>. content (K \<inter> {x. x \<bullet> i \<le> b \<bullet> i}) / interv_diff (K \<inter> {x. x \<bullet> i \<le> b \<bullet> i}) i)
+              = (\<Sum>K\<in>\<D>. content K / interv_diff K 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
+        using lec c eq interv_diff_def 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: field_split_simps)
-      by (metis Int_insert_left_if0 finite_Basis finite_insert le_iff_inf order_refl prod.insert subset_Compl_singleton)
+    proof -
+      have "f i * prod f (Basis \<inter> - {i}) = prod f Basis"
+        using that mk_disjoint_insert [OF i]
+        by (metis Int_insert_left_if0 finite_Basis finite_insert le_iff_inf order_refl prod.insert subset_Compl_singleton)
+      then show ?thesis
+        by (metis nonzero_mult_div_cancel_left that)
+    qed
     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)
+    then have "(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K 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)
+              "(\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K 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: field_split_simps)
     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)"
+    have "(\<Sum>K\<in>\<D>. content K / (interv_diff K i))
+          \<le> (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K) +
+            (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K 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)"
+            (\<Sum>K\<in>\<D>. ((\<lambda>K. content K / (interv_diff K i)) \<circ> ((\<lambda>K. K \<inter> {x. x \<bullet> i \<le> c}))) K +
+                    ((\<lambda>K. content K / (interv_diff K 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)
+        have uniq: "\<And>j. \<lbrakk>j \<in> Basis; \<not> u \<bullet> j \<le> v \<bullet> j\<rbrakk> \<Longrightarrow> j = i"
+          by (metis \<open>K \<in> \<D>\<close> box_ne_empty(1) div division_of_def uv)
         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)
+          using i uv uv'  by (auto simp add: interv_diff_def lemma0 dest: uniq * intro!: prod_nonneg)
       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)"
+    have "(\<Sum>K\<in>\<D>. content K / (interv_diff K i)) \<le> 2 * content (cbox a b) / (b \<bullet> i - a \<bullet> i)"
       by linarith
     then show ?thesis
-      using abc by (simp add: field_split_simps)
+      using abc interv_diff_def by (simp add: field_split_simps)
   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\<in>Basis. b \<bullet> m - a \<bullet> m) / content(cbox a b))"
-  have "gauge (\<lambda>x. ball x
+  define interv_diff where "interv_diff \<equiv> \<lambda>K. \<lambda>i::'a. interval_upperbound K \<bullet> i - interval_lowerbound K \<bullet> i"
+  have "8 * content (cbox a b) + norm (f x) * (8 * content (cbox a b)) > 0" for x
+    by (metis add.right_neutral add_pos_pos contab_gt0 mult_pos_pos mult_zero_left norm_eq_zero zero_less_norm_iff zero_less_numeral)
+  then have "gauge (\<lambda>x. ball x
                     (\<epsilon> * (INF m\<in>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 field_split_simps add_nonneg_eq_0_iff finite_less_Inf_iff)
-    apply (meson add_increasing measure_nonneg mult_nonneg_nonneg norm_ge_zero not_le zero_le_numeral)
-    done
+    using \<open>0 < content (cbox a b)\<close> \<open>0 < \<epsilon>\<close> a_less_b 
+    by (auto simp add: gauge_def field_split_simps add_nonneg_eq_0_iff finite_less_Inf_iff)
   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
+      using S unfolding tagged_partial_division_of_def by blast
     have "\<gamma>0 fine S" and fineS:
          "(\<lambda>x. ball x (\<epsilon> * (INF m\<in>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))"
+      also have "... \<le> (\<Sum>(x,K) \<in> S. \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) * content K / interv_diff K 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>"
+        show "norm (content K *\<^sub>R h x) \<le> \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) * content K / interv_diff K i"
         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
+            using S \<open>(x,K) \<in> S\<close> unfolding tagged_partial_division_of_def 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>)"
+          ultimately have "norm (h x) \<le> (\<epsilon> * (b \<bullet> i - a \<bullet> i)) / (4 * content (cbox a b) * interv_diff K i)"
           proof -
             have "dist x u < \<epsilon> * (INF m\<in>Basis. b \<bullet> m - a \<bullet> m) / (4 * (norm (f x) + 1) * content (cbox a b)) / 2"
                  "dist x v < \<epsilon> * (INF m\<in>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\<in>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
+            proof (intro mult_left_mono divide_right_mono)
+              show "(INF m\<in>Basis. b \<bullet> m - a \<bullet> m) \<le> b \<bullet> i - a \<bullet> i"
+                using \<open>i \<in> Basis\<close> by (auto intro!: cInf_le_finite)
+            qed (use \<open>0 < \<epsilon>\<close> in auto)
             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: field_split_simps split: if_split_asm)
-              by (meson add_nonneg_nonneg linorder_not_le measure_nonneg mult_nonneg_nonneg norm_ge_zero zero_le_numeral)
+            proof -
+              have "0 < norm (f x) + 1"
+                by (simp add: add.commute add_pos_nonneg)
+              then show ?thesis
+                using duv dist_uv contab_gt0
+                by (simp only: mult_ac divide_simps) auto
+            qed
             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
+            proof (intro mult_right_mono divide_left_mono divide_right_mono uvi)
+              show "norm (v - u) * \<bar>v \<bullet> i - u \<bullet> i\<bar> > 0"
+                using u_less_v [OF \<open>i \<in> Basis\<close>] 
+                by (auto simp: less_eq_real_def zero_less_mult_iff that)
+              show "\<epsilon> * (b \<bullet> i - a \<bullet> i) \<ge> 0"
+                using a_less_b \<open>0 < \<epsilon>\<close> \<open>i \<in> Basis\<close> by force
+            qed auto
+            also have "... = \<epsilon> * (b \<bullet> i - a \<bullet> i) / (4 * content (cbox a b) * interv_diff K i)"
+              using uv False that(2) u_less_v interv_diff_def 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>)"
+          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)) / interv_diff K i)"
             using mult_left_mono by fastforce
-          also have "... = \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) *
-                           content K / ?\<Delta>"
+          also have "... = \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) * content K / interv_diff K i"
             by (simp add: field_split_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))"
+      also have "... = (\<Sum>K\<in>snd ` S. \<epsilon>/4 * (b \<bullet> i - a \<bullet> i) / content (cbox a b) * content K / interv_diff K i)"
+        unfolding interv_diff_def
         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 "... = \<epsilon>/2 * ((b \<bullet> i - a \<bullet> i) / (2 * content (cbox a b)) * (\<Sum>K\<in>snd ` S. content K / interv_diff K i))"
+        by (simp add: interv_diff_def 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)"
+        have "(b \<bullet> i - a \<bullet> i) * (\<Sum>K\<in>snd ` S. content K / interv_diff K i) \<le> 2 * content (cbox a b)"
+          unfolding interv_diff_def
         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>(snd ` S) \<subseteq> cbox a b"
-            using S by force
+            using S unfolding tagged_partial_division_of_def 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"
+        then show "(b \<bullet> i - a \<bullet> i) / (2 * content (cbox a b)) * (\<Sum>K\<in>snd ` S. content K / interv_diff K 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)
+      by (simp add: inf_commute int_F integrable_split(1))
     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)
+      proof (rule bounded_equiintegral_over_thin_tagged_partial_division [OF F f, of \<open>\<epsilon>/12\<close>])
+        show "\<And>h x. \<lbrakk>h \<in> F; x \<in> cbox a b\<rbrakk> \<Longrightarrow> norm (h x) \<le> norm (f x)"
+          by (auto simp: norm_f)
+      qed (use \<open>\<epsilon> > 0\<close> in auto)
       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)
+        qed (use that \<open>\<epsilon> > 0\<close> \<open>gauge \<gamma>1\<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
+        proof (rule sum.mono_neutral_right [OF \<open>finite T\<close> \<open>T' \<subseteq> T\<close>])
+          show "\<forall>i \<in> T - T'. (case i of (x, K) \<Rightarrow> integral K f) = 0"
+            using f \<open>finite T\<close> \<open>T' \<subseteq> T\<close> integral_restrict_Int [of _ "{x. x \<bullet> i \<le> c}" h]
+            by (auto simp: T'_def Int_commute)
+        qed
         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
+        proof (rule sum.mono_neutral_right [OF \<open>finite T\<close> \<open>T' \<subseteq> T\<close>])
+          show "\<forall>i \<in> T - T'. (case i of (x, K) \<Rightarrow> content K *\<^sub>R f x) = 0"
+            using T f \<open>finite T\<close> \<open>T' \<subseteq> T\<close> by (force simp: T'_def)
+        qed
         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'. \<not> (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+
+                using T''_def T'_tagged tagged_partial_division_of_def 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= "(+)"])
               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>(snd ` A)"
-                      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
+                      using T'_tagged \<open>(x, L) \<in> A\<close> \<open>A \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by (blast dest: tagged_partial_division_ofD)
+                    have "interior (K \<inter> {x. x \<bullet> i \<le> c}) = {}"
+                    proof (rule tagged_division_split_left_inj [OF _ \<open>(x,K) \<in> A\<close> \<open>(x,L) \<in> A\<close>])
+                      show "A tagged_division_of \<Union>(snd ` A)"
+                        using A_tagged tagged_partial_division_of_Union_self by auto
+                      show "K \<inter> {x. x \<bullet> i \<le> c} = L \<inter> {x. x \<bullet> i \<le> c}"
+                        using eq \<open>i \<in> Basis\<close> by auto
+                    qed (use that in 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>(snd ` B)"
-                      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
+                      using T'_tagged \<open>(x, L) \<in> B\<close> \<open>B \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by (blast dest: tagged_partial_division_ofD)
+                    have "interior (K \<inter> {x. c \<le> x \<bullet> i}) = {}"
+                    proof (rule tagged_division_split_right_inj [OF _ \<open>(x,K) \<in> B\<close> \<open>(x,L) \<in> B\<close>])
+                      show "B tagged_division_of \<Union>(snd ` B)"
+                        using B_tagged tagged_partial_division_of_Union_self by auto
+                      show "K \<inter> {x. c \<le> x \<bullet> i} = L \<inter> {x. c \<le> x \<bullet> i}"
+                        using eq \<open>i \<in> Basis\<close> by auto
+                    qed (use that in 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
+                      using T'_tagged \<open>(x,K) \<in> B\<close> \<open>B \<subseteq> T''\<close> \<open>T'' \<subseteq> T'\<close> by (blast dest: tagged_partial_division_ofD)
+                    have huv: "h integrable_on cbox u v"
+                    proof (rule integrable_on_subcbox)
+                      show "cbox u v \<subseteq> cbox a b"
+                        using B_tagged \<open>(x,K) \<in> B\<close> uv by (blast dest: tagged_partial_division_ofD)
+                      show "h integrable_on cbox a b"
+                        by (simp add: int_F \<open>h \<in> F\<close>)
+                    qed
                     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
+                  using T''_tagged \<open>(x,K) \<in> T''\<close> by (blast dest: tagged_partial_division_ofD)
                 then have "connected K"
                   by (simp add: 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
+                  by (force simp add: mem_box sum_if_inner [where f = "\<lambda>j. c"])
                 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
+                  by (force simp add: sum_if_inner [where f = "\<lambda>j. c"] 
+                       intro!: \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> A_tagged fineA(1) \<open>h \<in> F\<close>])
+                let ?F = "\<lambda>(x,K). (x, K \<inter> {x. x \<bullet> i \<le> c})"
+                have 1: "?F ` A tagged_partial_division_of cbox a b"
+                  unfolding tagged_partial_division_of_def
+                proof (intro conjI strip)
+                  show "\<And>x K. (x, K) \<in> ?F ` A \<Longrightarrow> \<exists>a b. K = cbox a b"
+                    using A_tagged interval_split(1) [OF \<open>i \<in> Basis\<close>, of _ _ c]
+                    by (force dest: tagged_partial_division_ofD(4))
+                  show "\<And>x K. (x, K) \<in> ?F ` A \<Longrightarrow> x \<in> K"
+                    using A_def A_tagged by (fastforce dest: tagged_partial_division_ofD)
+                qed (use A_tagged in \<open>fastforce dest: tagged_partial_division_ofD\<close>)+
                 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
+                  using \<open>i \<in> Basis\<close> \<open>A \<subseteq> T''\<close> overlap
+                  by (force simp add: sum_if_inner [where f = "\<lambda>j. c"] 
+                       intro!: \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> 1 2 \<open>h \<in> F\<close>])
                 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
+                  using \<open>i \<in> Basis\<close> \<open>B \<subseteq> T''\<close> overlap
+                  by (force simp add: sum_if_inner [where f = "\<lambda>j. c"] 
+                       intro!: \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> B_tagged fineB(1) \<open>h \<in> F\<close>])
+                  let ?F = "\<lambda>(x,K). (x, K \<inter> {x. c \<le> x \<bullet> i})"
+                  have 1: "?F ` B tagged_partial_division_of cbox a b"
+                    unfolding tagged_partial_division_of_def
+                  proof (intro conjI strip)
+                    show "\<And>x K. (x, K) \<in> ?F ` B \<Longrightarrow> \<exists>a b. K = cbox a b"
+                      using B_tagged interval_split(2) [OF \<open>i \<in> Basis\<close>, of _ _ c]
+                      by (force dest: tagged_partial_division_ofD(4))
+                    show "\<And>x K. (x, K) \<in> ?F ` B \<Longrightarrow> x \<in> K"
+                      using B_def B_tagged by (fastforce dest: tagged_partial_division_ofD)
+                  qed (use B_tagged in \<open>fastforce dest: tagged_partial_division_ofD\<close>)+
                   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
+                  using \<open>i \<in> Basis\<close> \<open>A \<subseteq> T''\<close> overlap
+                  by (force simp add: B_def sum_if_inner [where f = "\<lambda>j. c"] 
+                       intro!: \<gamma>0 [OF cb_ab \<open>i \<in> Basis\<close> 1 2 \<open>h \<in> F\<close>])
                 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)
+            using T f0 that by (meson tag_in_interval)
           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)"
+            using T fh that by (meson tag_in_interval)
+          then have fh: "(\<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
+          show ?thesis
+            unfolding fh int_f
+          proof (rule less_trans [OF \<gamma>1])
+            show "\<gamma>1 fine T"
+              by (meson fine fine_Int)
+            show "\<epsilon> / (7 * Suc DIM('b)) < \<epsilon>"
+              using \<open>0 < \<epsilon>\<close> by (force simp: divide_simps)+
+          qed (use that in auto)
         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)
+      using f unfolding comp_def image_iff
+      by (metis (no_types, lifting) equation_minus_iff imageE 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
+            (\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if c \<le> x \<bullet> i then h x else 0})"   (is "?lhs = ?rhs")
+  proof
+    show "?lhs \<subseteq> ?rhs"
+      using minus_le_iff by fastforce
+    show "?rhs \<subseteq> ?lhs"
+    apply clarsimp
     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
+      using le_minus_iff by fastforce+
+  qed
   show ?thesis
     using equiintegrable_reflect [OF *] by (auto simp: eq)
 qed
 
 corollary equiintegrable_halfspace_restrictions_lt:
   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 < c then h x else 0)}) equiintegrable_on cbox a b"
          (is "?G equiintegrable_on cbox a b")
 proof -
   have *: "(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if c \<le> x \<bullet> i then h x else 0}) equiintegrable_on cbox a b"
     using equiintegrable_halfspace_restrictions_ge [OF F f] norm_f by auto
   have "(\<lambda>x. if x \<bullet> i < c then h x else 0) = (\<lambda>x. h x - (if c \<le> x \<bullet> i then h x else 0))"
     if "i \<in> Basis" "h \<in> F" for i c h
     using that by force
   then show ?thesis
     by (blast intro: equiintegrable_on_subset [OF equiintegrable_diff [OF F *]])
 qed
 
 corollary equiintegrable_halfspace_restrictions_gt:
   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 > c then h x else 0)}) equiintegrable_on cbox a b"
          (is "?G equiintegrable_on cbox a b")
 proof -
   have *: "(\<Union>i\<in>Basis. \<Union>c. \<Union>h\<in>F. {\<lambda>x. if c \<ge> x \<bullet> i then h x else 0}) equiintegrable_on cbox a b"
     using equiintegrable_halfspace_restrictions_le [OF F f] norm_f by auto
   have "(\<lambda>x. if x \<bullet> i > c then h x else 0) = (\<lambda>x. h x - (if c \<ge> x \<bullet> i then h x else 0))"
     if "i \<in> Basis" "h \<in> F" for i c h
     using that by force
   then show ?thesis
     by (blast intro: equiintegrable_on_subset [OF equiintegrable_diff [OF F *]])
 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"
+      then have "i \<in> Basis" "B \<subseteq> 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.
+      define F where "F \<equiv> (\<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})
+                {\<lambda>x. if \<xi> \<le> x \<bullet> i then h x else 0})"
+      show ?case
+      proof (rule equiintegrable_on_subset)
+        have "F equiintegrable_on cbox a b"
+          unfolding F_def
+        proof (rule equiintegrable_halfspace_restrictions_ge)
+          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"
+            by (intro * f equiintegrable_on_insert equiintegrable_halfspace_restrictions_le [OF insert.IH insertI1] \<open>B \<subseteq> Basis\<close>)
+          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 show "insert f F
              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
+          by (blast intro: f equiintegrable_on_insert)
+        show "insert f (\<Union>c d. {\<lambda>x. if \<forall>j\<in>insert i B. c \<bullet> j \<le> x \<bullet> j \<and> x \<bullet> j \<le> d \<bullet> j then f x else 0})
+            \<subseteq> insert f F"
+          using \<open>i \<in> Basis\<close> 
+          apply clarify
+          apply (simp add: F_def)
+          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: fun_eq_iff)
+          apply (drule_tac x=c in spec)
+          apply (drule_tac x=d in spec)
+          apply (simp split: if_split_asm)
           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
   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 (force intro: integrable_spike 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+
+          unfolding \<epsilon>_def 
+          by (meson Min.coboundedI euclidean_space_class.finite_Basis finite_imageI image_iff that)
         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: field_split_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: tendsto_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)
+    then have "\<exists>x \<in> cbox (a,a) (b,b). integral (cbox c d) f = integral (cbox (fst x) (snd x)) f"
+      using that by (metis cbox_Pair_iff interval_subset_is_interval is_interval_cbox prod.sel)
+    then show ?thesis
+      using B that(1) by blast
   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)"
+    define \<alpha> where "\<alpha> \<equiv> \<lambda>k. a + (b - a) /\<^sub>R real k"
+    define \<beta> where "\<beta> \<equiv> \<lambda>k. b - (b - a) /\<^sub>R real k"
+    define I where "I \<equiv> \<lambda>k. cbox (\<alpha> k) (\<beta> k)"
+    have ISuc_box: "I (Suc n) \<subseteq> box a b" for n
+      using pos unfolding I_def
+      by (intro subset_box_imp) (auto simp: \<alpha>_def \<beta>_def content_pos_lt_eq algebra_simps)
+    have ISucSuc: "I (Suc n) \<subseteq> I (Suc (Suc n))" for n
+    proof -
+      have "\<And>i. i \<in> Basis
+                 \<Longrightarrow> a \<bullet> i / Suc n + b \<bullet> i / (real n + 2)  \<le> b \<bullet> i / Suc n + a \<bullet> i / (real n + 2)"
+        using pos 
+        by (simp add: content_pos_lt_eq divide_simps) (auto simp: algebra_simps)
+      then show ?thesis
+        unfolding I_def
+        by (intro subset_box_imp) (auto simp: algebra_simps inverse_eq_divide \<alpha>_def \<beta>_def)
+    qed
+    have getN: "\<exists>N::nat. \<forall>k. k \<ge> N \<longrightarrow> x \<in> I 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>"
+      define \<Delta> where "\<Delta> \<equiv> (\<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)
+      moreover have \<Delta>: "Inf \<Delta> > 0"
+        using that by (auto simp: \<Delta>_def 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)"
+          have *: "Inf \<Delta> \<le> ((x - a) \<bullet> i) / ((b - a) \<bullet> i)"
+            unfolding \<Delta>_def 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 field_split_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)"
+          have *: "Inf \<Delta> \<le> ((b - x) \<bullet> i) / ((b - a) \<bullet> i)"
+            using that unfolding \<Delta>_def 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 field_split_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)
+        show "x \<in> I k"
+          using that \<Delta> \<open>k > 0\<close> unfolding I_def
+          by (auto simp: \<alpha>_def \<beta>_def 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
+    obtain Bf where  Bf: "\<And>c d. cbox c d \<subseteq> box a b \<Longrightarrow> norm (integral (cbox c d) f) \<le> Bf"
+      using bo unfolding bounded_iff by blast
+    obtain Bg where 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_iff real_norm_def by blast
     show "(\<lambda>x. f x \<bullet> j) absolutely_integrable_on cbox a b"
       using g
     proof     \<comment> \<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"
+        define \<phi> where "\<phi> \<equiv> \<lambda>k x. if x \<in> I (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])
-          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 monotone_convergence_increasing [of \<phi>, THEN conjunct1])
+          have *: "I (Suc k) \<inter> box a b = I (Suc k)" for k
+            using box_subset_cbox ISuc_box by fastforce
+          show "\<phi> k integrable_on box a b" for k
+          proof -
+            have "I (Suc k) \<subseteq> cbox a b"
+              using "*" box_subset_cbox by blast
+            moreover have "(\<lambda>m. f m \<bullet> j) integrable_on I (Suc k)"
+              by (metis ISuc_box I_def int_f integrable_component)
+            ultimately have "(\<lambda>m. g m - f m \<bullet> j) integrable_on I (Suc k)"
+              by (metis Henstock_Kurzweil_Integration.integrable_diff I_def int_gab integrable_on_subcbox)
+            then show ?thesis
+              by (simp add: "*" \<phi>_def integrable_restrict_Int)
+          qed
+          show "\<phi> k x \<le> \<phi> (Suc k) x" if "x \<in> box a b" for k x
+            using ISucSuc box_subset_cbox that by (force simp: \<phi>_def 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 tendsto_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)"
+            obtain N::nat where N: "\<And>k. k \<ge> N \<Longrightarrow> x \<in> I k"
               using getN [OF x] by blast
-            show "\<forall>\<^sub>F k in sequentially. ?\<phi> k x = g x - f x \<bullet> j"
+            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))"
+              have "x \<in> I (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
+              then show "\<phi> k x = g x - f x \<bullet> j"
+                by (simp add: \<phi>_def)
             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
+          have "\<bar>integral (box a b) (\<lambda>x. if x \<in> I (Suc 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"
-              by (metis I_int inf_le2 int_f)
-            then have "(\<lambda>x. f x \<bullet> j) integrable_on ?I"
-              by (simp add: integrable_component)
-            moreover have "g integrable_on ?I"
-              by (metis I_int inf.orderI int_gab integrable_on_open_interval integrable_on_subcbox)
+            have ABK_def [simp]: "I (Suc k) \<inter> box a b = I (Suc k)"
+              using ISuc_box by (simp add: Int_absorb2)
+            have int_fI: "f integrable_on I (Suc k)"
+              using ISuc_box I_def int_f by auto
             moreover
-            have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ?I f)"
+            have "\<bar>integral (I (Suc k)) (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral (I (Suc k)) 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])
+            with ISuc_box ABK_def have "\<bar>integral (I (Suc k)) (\<lambda>x. f x \<bullet> j)\<bar> \<le> Bf"
+              by (metis Bf I_def \<open>j \<in> Basis\<close> int_fI integral_component_eq norm_bound_Basis_le) 
+            ultimately 
+            have "\<bar>integral (I (Suc k)) g - integral (I (Suc k)) (\<lambda>x. f x \<bullet> j)\<bar>  \<le> Bg + Bf"
+              using "*" box_subset_cbox unfolding I_def
+              by (blast intro: Bg add_mono order_trans [OF abs_triangle_ineq4])
+            moreover have "g integrable_on I (Suc k)"
+              by (metis ISuc_box I_def int_gab integrable_on_open_interval integrable_on_subcbox)
+            moreover have "(\<lambda>x. f x \<bullet> j) integrable_on I (Suc k)"
+              using int_fI by (simp add: integrable_component)
+           ultimately show ?thesis
+              by (simp add: integral_restrict_Int integral_diff)
           qed
-          then show "bounded (range (\<lambda>k. integral (box a b) (?\<phi> k)))"
-            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
+          then show "bounded (range (\<lambda>k. integral (box a b) (\<phi> k)))"
+            by (auto simp add: bounded_iff \<phi>_def)
         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)
+        using absolutely_integrable_component_ubound [OF _ absint_g] fg by force
     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"
+        let ?\<phi> = "\<lambda>k x. if x \<in> I(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])
-          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
+          have *: "I (Suc k) \<inter> box a b = I (Suc k)" for k
+            using box_subset_cbox ISuc_box 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
+          proof (simp add: integrable_restrict_Int integral_restrict_Int *)
+            show "(\<lambda>x. f x \<bullet> j - g x) integrable_on I (Suc k)"
+              by (metis ISuc_box Henstock_Kurzweil_Integration.integrable_diff I_def int_f int_gab integrable_component integrable_on_open_interval integrable_on_subcbox)
+          qed
           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)
+            using ISucSuc box_subset_cbox that by (force simp: I_def 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 tendsto_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)"
+            obtain N::nat where N: "\<And>k. k \<ge> N \<Longrightarrow> x \<in> I 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
+            then show "\<forall>\<^sub>F k in sequentially. ?\<phi> k x = f x \<bullet> j - g x"
+              by (metis (no_types, lifting) eventually_at_top_linorderI le_Suc_eq)
           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
+                      (\<lambda>x. if x \<in> I (Suc 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"
-              by (simp add: inf.orderI int_f)
-            then have "(\<lambda>x. f x \<bullet> j) integrable_on ?I"
+            define ABK where "ABK \<equiv> cbox (a + (b - a) /\<^sub>R (1 + real k)) (b - (b - a) /\<^sub>R (1 + real k))"
+            have ABK_eq [simp]: "ABK \<inter> box a b = ABK"
+              using "*" I_def \<alpha>_def \<beta>_def ABK_def by auto
+            have int_fI: "f integrable_on ABK"
+              unfolding ABK_def
+              using ISuc_box I_def \<alpha>_def \<beta>_def int_f by force
+            then have "(\<lambda>x. f x \<bullet> j) integrable_on ABK"
               by (simp add: integrable_component)
-            moreover have "g integrable_on ?I"
-              by (metis I_int inf.orderI int_gab integrable_on_open_interval integrable_on_subcbox)
+            moreover have "g integrable_on ABK"
+              by (metis ABK_def ABK_eq IntE box_subset_cbox int_gab integrable_on_subcbox subset_eq)
             moreover
-            have "\<bar>integral ?I (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ?I f)"
+            have "\<bar>integral ABK (\<lambda>x. f x \<bullet> j)\<bar> \<le> norm (integral ABK 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])
+            then have "\<bar>integral ABK (\<lambda>x. f x \<bullet> j)\<bar> \<le> Bf"
+              by (metis ABK_eq  ABK_def Bf IntE dual_order.trans subset_eq)
             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])
+              using "*" box_subset_cbox
+              apply (simp add: integral_restrict_Int integral_diff ABK_def I_def \<alpha>_def \<beta>_def)
+               by (blast intro: Bg add_mono order_trans [OF abs_triangle_ineq4])
           qed
           then show "bounded (range (\<lambda>k. integral (box a b) (?\<phi> k)))"
-            apply (simp add: bounded_pos)
-            apply (rule_tac x="Bf+Bg" in exI)
-            using \<open>0 < Bf\<close> \<open>0 < Bg\<close>  by auto
+            by (auto simp add: bounded_iff)
         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)
+        using absint_g absolutely_integrable_absolutely_integrable_lbound gf by blast
     qed
   qed
 qed
 
-subsection\<open>Second Mean Value Theorem\<close>
-
-
-
 subsection\<open>Second mean value theorem and corollaries\<close>
 
 lemma level_approx:
   fixes f :: "real \<Rightarrow> real" and n::nat
   assumes f: "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x \<and> f x \<le> 1" and "x \<in> S" "n \<noteq> 0"
   shows "\<bar>f x - (\<Sum>k = Suc 0..n. if k / n \<le> f x then inverse n else 0)\<bar> < inverse n"
         (is "?lhs < _")
 proof -
   have "n * f x \<ge> 0"
     using assms by auto
   then obtain m::nat where m: "floor(n * f x) = int m"
     using nonneg_int_cases zero_le_floor by blast
   then have kn: "real k / real n \<le> f x \<longleftrightarrow> k \<le> m" for k
     using \<open>n \<noteq> 0\<close> by (simp add: field_split_simps) linarith
   then have "Suc n / real n \<le> f x \<longleftrightarrow> Suc n \<le> m"
     by blast
   have "real n * f x \<le> real n"
     by (simp add: \<open>x \<in> S\<close> f mult_left_le)
   then have "m \<le> n"
     using m by linarith
   have "?lhs = \<bar>f x - (\<Sum>k \<in> {Suc 0..n} \<inter> {..m}. inverse n)\<bar>"
     by (subst sum.inter_restrict) (auto simp: kn)
   also have "\<dots> < inverse n"
     using \<open>m \<le> n\<close> \<open>n \<noteq> 0\<close> m
     by (simp add: min_absorb2 field_split_simps) linarith
   finally show ?thesis .
 qed
 
 
 lemma SMVT_lemma2:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f integrable_on {a..b}"
     and g: "\<And>x y. x \<le> y \<Longrightarrow> g x \<le> g y"
   shows "(\<Union>y::real. {\<lambda>x. if g x \<ge> y then f x else 0}) equiintegrable_on {a..b}"
 proof -
   have ffab: "{f} equiintegrable_on {a..b}"
     by (metis equiintegrable_on_sing f interval_cbox)
   then have ff: "{f} equiintegrable_on (cbox a b)"
     by simp
   have ge: "(\<Union>c. {\<lambda>x. if x \<ge> c then f x else 0}) equiintegrable_on {a..b}"
     using equiintegrable_halfspace_restrictions_ge [OF ff] by auto
   have gt: "(\<Union>c. {\<lambda>x. if x > c then f x else 0}) equiintegrable_on {a..b}"
     using equiintegrable_halfspace_restrictions_gt [OF ff] by auto
   have 0: "{(\<lambda>x. 0)} equiintegrable_on {a..b}"
     by (metis box_real(2) equiintegrable_on_sing integrable_0)
   have \<dagger>: "(\<lambda>x. if g x \<ge> y then f x else 0) \<in> {(\<lambda>x. 0), f} \<union> (\<Union>z. {\<lambda>x. if z < x then f x else 0}) \<union> (\<Union>z. {\<lambda>x. if z \<le> x then f x else 0})"
     for y
   proof (cases "(\<forall>x. g x \<ge> y) \<or> (\<forall>x. \<not> (g x \<ge> y))")
     let ?\<mu> = "Inf {x. g x \<ge> y}"
     case False
     have lower: "?\<mu> \<le> x" if "g x \<ge> y" for x
-      apply (rule cInf_lower)
-      using False
-       apply (auto simp: that bdd_below_def)
-      by (meson dual_order.trans g linear)
+    proof (rule cInf_lower)
+      show "x \<in> {x. y \<le> g x}"
+        using False by (auto simp: that)
+      show "bdd_below {x. y \<le> g x}"
+        by (metis False bdd_belowI dual_order.trans g linear mem_Collect_eq)
+    qed
     have greatest: "?\<mu> \<ge> z" if  "(\<And>x. g x \<ge> y \<Longrightarrow> z \<le> x)" for z
       by (metis False cInf_greatest empty_iff mem_Collect_eq that)
     show ?thesis
     proof (cases "g ?\<mu> \<ge> y")
       case True
       then obtain \<zeta> where \<zeta>: "\<And>x. g x \<ge> y \<longleftrightarrow> x \<ge> \<zeta>"
         by (metis g lower order.trans)  \<comment> \<open>in fact y is @{term ?\<mu>}\<close>
       then show ?thesis
         by (force simp: \<zeta>)
     next
       case False
-      have "(y \<le> g x) = (?\<mu> < x)" for x
+      have "(y \<le> g x) \<longleftrightarrow> (?\<mu> < x)" for x
       proof
         show "?\<mu> < x" if "y \<le> g x"
           using that False less_eq_real_def lower by blast
         show "y \<le> g x" if "?\<mu> < x"
           by (metis g greatest le_less_trans that less_le_trans linear not_less)
       qed
       then obtain \<zeta> where \<zeta>: "\<And>x. g x \<ge> y \<longleftrightarrow> x > \<zeta>" ..
       then show ?thesis
         by (force simp: \<zeta>)
     qed
   qed auto
   show ?thesis
-    apply (rule equiintegrable_on_subset [OF equiintegrable_on_Un [OF gt equiintegrable_on_Un [OF ge equiintegrable_on_Un [OF ffab 0]]]])
-    using \<dagger> apply (simp add: UN_subset_iff)
-    done
+    using \<dagger> by (simp add: UN_subset_iff equiintegrable_on_subset [OF equiintegrable_on_Un [OF gt equiintegrable_on_Un [OF ge equiintegrable_on_Un [OF ffab 0]]]])
 qed
 
 
 lemma SMVT_lemma4:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f integrable_on {a..b}"
     and "a \<le> b"
     and g: "\<And>x y. x \<le> y \<Longrightarrow> g x \<le> g y"
     and 01: "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> 0 \<le> g x \<and> g x \<le> 1"
   obtains c where "a \<le> c" "c \<le> b" "((\<lambda>x. g x *\<^sub>R f x) has_integral integral {c..b} f) {a..b}"
 proof -
   have "connected ((\<lambda>x. integral {x..b} f) ` {a..b})"
     by (simp add: f indefinite_integral_continuous_1' connected_continuous_image)
   moreover have "compact ((\<lambda>x. integral {x..b} f) ` {a..b})"
     by (simp add: compact_continuous_image f indefinite_integral_continuous_1')
   ultimately obtain m M where int_fab: "(\<lambda>x. integral {x..b} f) ` {a..b} = {m..M}"
     using connected_compact_interval_1 by meson
   have "\<exists>c. c \<in> {a..b} \<and>
               integral {c..b} f =
               integral {a..b} (\<lambda>x. (\<Sum>k = 1..n. if g x \<ge> real k / real n then inverse n *\<^sub>R f x else 0))" for n
   proof (cases "n=0")
     case True
     then show ?thesis
       using \<open>a \<le> b\<close> by auto
   next
     case False
     have "(\<Union>c::real. {\<lambda>x. if g x \<ge> c then f x else 0}) equiintegrable_on {a..b}"
       using SMVT_lemma2 [OF f g] .
     then have int: "(\<lambda>x. if g x \<ge> c then f x else 0) integrable_on {a..b}" for c
       by (simp add: equiintegrable_on_def)
     have int': "(\<lambda>x. if g x \<ge> c then u * f x else 0) integrable_on {a..b}" for c u
     proof -
       have "(\<lambda>x. if g x \<ge> c then u * f x else 0) = (\<lambda>x. u * (if g x \<ge> c then f x else 0))"
         by (force simp: if_distrib)
       then show ?thesis
         using integrable_on_cmult_left [OF int] by simp
     qed
     have "\<exists>d. d \<in> {a..b} \<and> integral {a..b} (\<lambda>x. if g x \<ge> y then f x else 0) = integral {d..b} f" for y
     proof -
       let ?X = "{x. g x \<ge> y}"
-      have *: "\<exists>a. ?X = {x. a \<le> x} \<or> ?X = {x. a < x}"
+      have *: "\<exists>a. ?X = {a..} \<or> ?X = {a<..}"
         if 1: "?X \<noteq> {}" and 2: "?X \<noteq> UNIV"
       proof -
         let ?\<mu> = "Inf{x. g x \<ge> y}"
         have lower: "?\<mu> \<le> x" if "g x \<ge> y" for x
-          apply (rule cInf_lower)
-          using 1 2 apply (auto simp: that bdd_below_def)
-          by (meson dual_order.trans g linear)
-        have greatest: "?\<mu> \<ge> z" if  "(\<And>x. g x \<ge> y \<Longrightarrow> z \<le> x)" for z
+        proof (rule cInf_lower)
+          show "x \<in> {x. y \<le> g x}"
+            using 1 2 by (auto simp: that)
+          show "bdd_below {x. y \<le> g x}"
+            unfolding bdd_below_def
+            by (metis "2" UNIV_eq_I dual_order.trans g less_eq_real_def mem_Collect_eq not_le)
+        qed
+        have greatest: "?\<mu> \<ge> z" if "\<And>x. g x \<ge> y \<Longrightarrow> z \<le> x" for z
           by (metis cInf_greatest mem_Collect_eq that 1)
         show ?thesis
         proof (cases "g ?\<mu> \<ge> y")
           case True
           then obtain \<zeta> where \<zeta>: "\<And>x. g x \<ge> y \<longleftrightarrow> x \<ge> \<zeta>"
             by (metis g lower order.trans)  \<comment> \<open>in fact y is @{term ?\<mu>}\<close>
           then show ?thesis
             by (force simp: \<zeta>)
         next
           case False
           have "(y \<le> g x) = (?\<mu> < x)" for x
           proof
             show "?\<mu> < x" if "y \<le> g x"
               using that False less_eq_real_def lower by blast
             show "y \<le> g x" if "?\<mu> < x"
               by (metis g greatest le_less_trans that less_le_trans linear not_less)
           qed
           then obtain \<zeta> where \<zeta>: "\<And>x. g x \<ge> y \<longleftrightarrow> x > \<zeta>" ..
           then show ?thesis
             by (force simp: \<zeta>)
         qed
       qed
-      then consider "?X = {}" | "?X = UNIV" | d where "?X = {x. d \<le> x} \<or>?X = {x. d < x}"
+      then consider "?X = {}" | "?X = UNIV" | (intv) d where "?X = {d..} \<or> ?X = {d<..}"
         by metis
       then have "\<exists>d. d \<in> {a..b} \<and> integral {a..b} (\<lambda>x. if x \<in> ?X then f x else 0) = integral {d..b} f"
       proof cases
-        case 1
-        then show ?thesis
-          using \<open>a \<le> b\<close> by auto
-      next
-        case 2
-        then show ?thesis
-          using \<open>a \<le> b\<close> by auto
-      next
-        case (3 d)
+        case (intv d)
         show ?thesis
         proof (cases "d < a")
           case True
-          with 3 show ?thesis
-            apply (rule_tac x=a in exI)
-            apply (auto simp: \<open>a \<le> b\<close> iff del: mem_Collect_eq intro!: Henstock_Kurzweil_Integration.integral_cong, simp_all)
-            done
+          with intv have "integral {a..b} (\<lambda>x. if y \<le> g x then f x else 0) = integral {a..b} f"
+            by (intro Henstock_Kurzweil_Integration.integral_cong) force
+          then show ?thesis
+            by (rule_tac x=a in exI) (simp add: \<open>a \<le> b\<close>)
         next
           case False
           show ?thesis
           proof (cases "b < d")
             case True
             have "integral {a..b} (\<lambda>x. if x \<in> {x. y \<le> g x} then f x else 0) = integral {a..b} (\<lambda>x. 0)"
-              by (rule Henstock_Kurzweil_Integration.integral_cong) (use 3 True in fastforce)
+              by (rule Henstock_Kurzweil_Integration.integral_cong) (use intv True in fastforce)
             then show ?thesis
               using \<open>a \<le> b\<close> by auto
           next
             case False
-            with \<open>\<not> d < a\<close> have eq: "Collect ((\<le>) d) \<inter> {a..b} = {d..b}" "Collect ((<) d) \<inter> {a..b} = {d<..b}"
+            with \<open>\<not> d < a\<close> have eq: "{d..} \<inter> {a..b} = {d..b}" "{d<..} \<inter> {a..b} = {d<..b}"
               by force+
             moreover have "integral {d<..b} f = integral {d..b} f"
               by (rule integral_spike_set [OF empty_imp_negligible negligible_subset [OF negligible_sing [of d]]]) auto
-            ultimately
-            show ?thesis
-              using \<open>\<not> d < a\<close> False 3
-              apply (rule_tac x=d in exI)
-              apply (auto simp: \<open>a \<le> b\<close> iff del: mem_Collect_eq; subst Henstock_Kurzweil_Integration.integral_restrict_Int)
-               apply (simp_all add: \<open>a \<le> b\<close> not_less eq)
-              done
+            ultimately 
+            have "integral {a..b} (\<lambda>x. if x \<in> {x. y \<le> g x} then f x else 0) =  integral {d..b} f"
+              unfolding integral_restrict_Int using intv by presburger
+            moreover have "d \<in> {a..b}"
+              using \<open>\<not> d < a\<close> \<open>a \<le> b\<close> False by auto
+            ultimately show ?thesis
+              by auto
           qed
         qed
-      qed
+      qed (use \<open>a \<le> b\<close> in auto)
       then show ?thesis
         by auto
     qed
     then have "\<forall>k. \<exists>d. d \<in> {a..b} \<and> integral {a..b} (\<lambda>x. if real k / real n \<le> g x then f x else 0) = integral {d..b} f"
       by meson
     then obtain d where dab: "\<And>k. d k \<in> {a..b}"
       and deq: "\<And>k::nat. integral {a..b} (\<lambda>x. if k/n \<le> g x then f x else 0) = integral {d k..b} f"
       by metis
     have "(\<Sum>k = 1..n. integral {a..b} (\<lambda>x. if real k / real n \<le> g x then f x else 0)) /\<^sub>R n \<in> {m..M}"
       unfolding scaleR_right.sum
     proof (intro conjI allI impI convex [THEN iffD1, rule_format])
       show "integral {a..b} (\<lambda>xa. if real k / real n \<le> g xa then f xa else 0) \<in> {m..M}" for k
         by (metis (no_types, lifting) deq image_eqI int_fab dab)
     qed (use \<open>n \<noteq> 0\<close> in auto)
     then have "\<exists>c. c \<in> {a..b} \<and>
               integral {c..b} f = inverse n *\<^sub>R (\<Sum>k = 1..n. integral {a..b} (\<lambda>x. if g x \<ge> real k / real n then f x else 0))"
       by (metis (no_types, lifting) int_fab imageE)
     then show ?thesis
       by (simp add: sum_distrib_left if_distrib integral_sum int' flip: integral_mult_right cong: if_cong)
   qed
   then obtain c where cab: "\<And>n. c n \<in> {a..b}"
     and c: "\<And>n. integral {c n..b} f = integral {a..b} (\<lambda>x. (\<Sum>k = 1..n. if g x \<ge> real k / real n then f x /\<^sub>R n else 0))"
     by metis
   obtain d and \<sigma> :: "nat\<Rightarrow>nat"
     where "d \<in> {a..b}" and \<sigma>: "strict_mono \<sigma>" and d: "(c \<circ> \<sigma>) \<longlonglongrightarrow> d" and non0: "\<And>n. \<sigma> n \<ge> Suc 0"
   proof -
     have "compact{a..b}"
       by auto
     with cab obtain d and s0
       where "d \<in> {a..b}" and s0: "strict_mono s0" and tends: "(c \<circ> s0) \<longlonglongrightarrow> d"
       unfolding compact_def
       using that by blast
     show thesis
     proof
       show "d \<in> {a..b}"
         by fact
       show "strict_mono (s0 \<circ> Suc)"
         using s0 by (auto simp: strict_mono_def)
       show "(c \<circ> (s0 \<circ> Suc)) \<longlonglongrightarrow> d"
         by (metis tends LIMSEQ_subseq_LIMSEQ Suc_less_eq comp_assoc strict_mono_def)
       show "\<And>n. (s0 \<circ> Suc) n \<ge> Suc 0"
         by (metis comp_apply le0 not_less_eq_eq old.nat.exhaust s0 seq_suble)
     qed
   qed
   define \<phi> where "\<phi> \<equiv> \<lambda>n x. \<Sum>k = Suc 0..\<sigma> n. if k/(\<sigma> n) \<le> g x then f x /\<^sub>R (\<sigma> n) else 0"
   define \<psi> where "\<psi> \<equiv> \<lambda>n x. \<Sum>k = Suc 0..\<sigma> n. if k/(\<sigma> n) \<le> g x then inverse (\<sigma> n) else 0"
   have **: "(\<lambda>x. g x *\<^sub>R f x) integrable_on cbox a b \<and>
             (\<lambda>n. integral (cbox a b) (\<phi> n)) \<longlonglongrightarrow> integral (cbox a b) (\<lambda>x. g x *\<^sub>R f x)"
   proof (rule equiintegrable_limit)
     have \<dagger>: "((\<lambda>n. \<lambda>x. (\<Sum>k = Suc 0..n. if k / n \<le> g x then inverse n *\<^sub>R f x else 0)) ` {Suc 0..}) equiintegrable_on {a..b}"
     proof -
       have *: "(\<Union>c::real. {\<lambda>x. if g x \<ge> c then f x else 0}) equiintegrable_on {a..b}"
         using SMVT_lemma2 [OF f g] .
       show ?thesis
         apply (rule equiintegrable_on_subset [OF equiintegrable_sum_real [OF *]], clarify)
         apply (rule_tac a="{Suc 0..n}" in UN_I, force)
         apply (rule_tac a="\<lambda>k. inverse n" in UN_I, auto)
         apply (rule_tac x="\<lambda>k x. if real k / real n \<le> g x then f x else 0" in bexI)
          apply (force intro: sum.cong)+
         done
     qed
     show "range \<phi> equiintegrable_on cbox a b"
       unfolding \<phi>_def
       by (auto simp: non0 intro: equiintegrable_on_subset [OF \<dagger>])
     show "(\<lambda>n. \<phi> n x) \<longlonglongrightarrow> g x *\<^sub>R f x"
       if x: "x \<in> cbox a b" for x
     proof -
       have eq: "\<phi> n x = \<psi> n x *\<^sub>R f x"  for n
         by (auto simp: \<phi>_def \<psi>_def sum_distrib_right if_distrib intro: sum.cong)
       show ?thesis
         unfolding eq
       proof (rule tendsto_scaleR [OF _ tendsto_const])
         show "(\<lambda>n. \<psi> n x) \<longlonglongrightarrow> g x"
           unfolding lim_sequentially dist_real_def
         proof (intro allI impI)
           fix e :: real
           assume "e > 0"
           then obtain N where "N \<noteq> 0" "0 < inverse (real N)" and N: "inverse (real N) < e"
             using real_arch_inverse by metis
           moreover have "\<bar>\<psi> n x - g x\<bar> < inverse (real N)" if "n\<ge>N" for n
           proof -
             have "\<bar>g x - \<psi> n x\<bar> < inverse (real (\<sigma> n))"
               unfolding \<psi>_def
             proof (rule level_approx [of "{a..b}" g])
               show "\<sigma> n \<noteq> 0"
                 by (metis Suc_n_not_le_n non0)
             qed (use x 01 non0 in auto)
             also have "\<dots> \<le> inverse N"
               using seq_suble [OF \<sigma>] \<open>N \<noteq> 0\<close> non0 that by (auto intro: order_trans simp: field_split_simps)
             finally show ?thesis
               by linarith
           qed
           ultimately show "\<exists>N. \<forall>n\<ge>N. \<bar>\<psi> n x - g x\<bar> < e"
             using less_trans by blast
         qed
       qed
     qed
   qed
   show thesis
   proof
     show "a \<le> d" "d \<le> b"
       using \<open>d \<in> {a..b}\<close> atLeastAtMost_iff by blast+
     show "((\<lambda>x. g x *\<^sub>R f x) has_integral integral {d..b} f) {a..b}"
       unfolding has_integral_iff
     proof
       show "(\<lambda>x. g x *\<^sub>R f x) integrable_on {a..b}"
         using ** by simp
       show "integral {a..b} (\<lambda>x. g x *\<^sub>R f x) = integral {d..b} f"
       proof (rule tendsto_unique)
         show "(\<lambda>n. integral {c(\<sigma> n)..b} f) \<longlonglongrightarrow> integral {a..b} (\<lambda>x. g x *\<^sub>R f x)"
           using ** by (simp add: c \<phi>_def)
-        show "(\<lambda>n. integral {c(\<sigma> n)..b} f) \<longlonglongrightarrow> integral {d..b} f"
-          using indefinite_integral_continuous_1' [OF f]
-          using d unfolding o_def
-          apply (simp add: continuous_on_eq_continuous_within)
-          apply (drule_tac x=d in bspec)
-          using \<open>d \<in> {a..b}\<close> apply blast
-          apply (simp add: continuous_within_sequentially o_def)
-          using cab by auto
+        have "continuous (at d within {a..b}) (\<lambda>x. integral {x..b} f)"
+          using indefinite_integral_continuous_1' [OF f] \<open>d \<in> {a..b}\<close> 
+          by (simp add: continuous_on_eq_continuous_within)
+        then show "(\<lambda>n. integral {c(\<sigma> n)..b} f) \<longlonglongrightarrow> integral {d..b} f"
+          using d cab unfolding o_def
+          by (simp add: continuous_within_sequentially o_def)
       qed auto
     qed
   qed
 qed
 
 
 theorem second_mean_value_theorem_full:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f integrable_on {a..b}" and "a \<le> b"
     and g: "\<And>x y. \<lbrakk>a \<le> x; x \<le> y; y \<le> b\<rbrakk> \<Longrightarrow> g x \<le> g y"
   obtains c where "c \<in> {a..b}"
     and "((\<lambda>x. g x * f x) has_integral (g a * integral {a..c} f + g b * integral {c..b} f)) {a..b}"
 proof -
   have gab: "g a \<le> g b"
     using \<open>a \<le> b\<close> g by blast
   then consider "g a < g b" | "g a = g b"
     by linarith
   then show thesis
   proof cases
     case 1
     define h where "h \<equiv> \<lambda>x. if x < a then 0 else if b < x then 1
                              else (g x - g a) / (g b - g a)"
     obtain c where "a \<le> c" "c \<le> b" and c: "((\<lambda>x. h x *\<^sub>R f x) has_integral integral {c..b} f) {a..b}"
     proof (rule SMVT_lemma4 [OF f \<open>a \<le> b\<close>, of h])
       show "h x \<le> h y" "0 \<le> h x \<and> h x \<le> 1"  if "x \<le> y" for x y
         using that gab by (auto simp: divide_simps g h_def)
     qed
     show ?thesis
     proof
       show "c \<in> {a..b}"
         using \<open>a \<le> c\<close> \<open>c \<le> b\<close> by auto
       have I: "((\<lambda>x. g x * f x - g a * f x) has_integral (g b - g a) * integral {c..b} f) {a..b}"
       proof (subst has_integral_cong)
         show "g x * f x - g a * f x = (g b - g a) * h x *\<^sub>R f x"
           if "x \<in> {a..b}" for x
           using 1 that by (simp add: h_def field_split_simps)
         show "((\<lambda>x. (g b - g a) * h x *\<^sub>R f x) has_integral (g b - g a) * integral {c..b} f) {a..b}"
           using has_integral_mult_right [OF c, of "g b - g a"] .
       qed
       have II: "((\<lambda>x. g a * f x) has_integral g a * integral {a..b} f) {a..b}"
         using has_integral_mult_right [where c = "g a", OF integrable_integral [OF f]] .
       have "((\<lambda>x. g x * f x) has_integral (g b - g a) * integral {c..b} f + g a * integral {a..b} f) {a..b}"
         using has_integral_add [OF I II] by simp
       then show "((\<lambda>x. g x * f x) has_integral g a * integral {a..c} f + g b * integral {c..b} f) {a..b}"
         by (simp add: algebra_simps flip: integral_combine [OF \<open>a \<le> c\<close> \<open>c \<le> b\<close> f])
     qed
   next
     case 2
     show ?thesis
     proof
       show "a \<in> {a..b}"
         by (simp add: \<open>a \<le> b\<close>)
       have "((\<lambda>x. g x * f x) has_integral g a * integral {a..b} f) {a..b}"
       proof (rule has_integral_eq)
         show "((\<lambda>x. g a * f x) has_integral g a * integral {a..b} f) {a..b}"
           using f has_integral_mult_right by blast
         show "g a * f x = g x * f x"
           if "x \<in> {a..b}" for x
           by (metis atLeastAtMost_iff g less_eq_real_def not_le that 2)
       qed
       then show "((\<lambda>x. g x * f x) has_integral g a * integral {a..a} f + g b * integral {a..b} f) {a..b}"
         by (simp add: 2)
     qed
   qed
 qed
 
 
 corollary second_mean_value_theorem:
   fixes f :: "real \<Rightarrow> real"
   assumes f: "f integrable_on {a..b}" and "a \<le> b"
    and g: "\<And>x y. \<lbrakk>a \<le> x; x \<le> y; y \<le> b\<rbrakk> \<Longrightarrow> g x \<le> g y"
  obtains c where "c \<in> {a..b}"
-                 "integral {a..b} (\<lambda>x. g x * f x)
-                = g a * integral {a..c} f + g b * integral {c..b} f"
+                 "integral {a..b} (\<lambda>x. g x * f x) = g a * integral {a..c} f + g b * integral {c..b} f"
     using second_mean_value_theorem_full [where g=g, OF assms]
     by (metis (full_types) integral_unique)
 
 end
 
diff --git a/src/HOL/Analysis/Tagged_Division.thy b/src/HOL/Analysis/Tagged_Division.thy
--- a/src/HOL/Analysis/Tagged_Division.thy
+++ b/src/HOL/Analysis/Tagged_Division.thy
@@ -1,2602 +1,2602 @@
 (*  Title:      HOL/Analysis/Tagged_Division.thy
     Author:     John Harrison
     Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light); proofs reworked by LCP
 *)
 
 section \<open>Tagged Divisions for Henstock-Kurzweil Integration\<close>
 
 theory Tagged_Division
   imports Topology_Euclidean_Space
 begin
 
 lemma sum_Sigma_product:
   assumes "finite S"
     and "\<And>i. i \<in> S \<Longrightarrow> finite (T i)"
   shows "(\<Sum>i\<in>S. sum (x i) (T i)) = (\<Sum>(i, j)\<in>Sigma S T. x i j)"
   using assms
 proof induct
   case empty
   then show ?case
     by simp
 next
   case (insert a S)
   show ?case
     by (simp add: insert.hyps(1) insert.prems sum.Sigma)
 qed
 
 lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
   scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
   scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one
 
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>Sundries\<close>
 
 
 text\<open>A transitive relation is well-founded if all initial segments are finite.\<close>
 lemma wf_finite_segments:
   assumes "irrefl r" and "trans r" and "\<And>x. finite {y. (y, x) \<in> r}"
   shows "wf (r)"
   apply (simp add: trans_wf_iff wf_iff_acyclic_if_finite converse_def assms)
   using acyclic_def assms irrefl_def trans_Restr by fastforce
 
 text\<open>For creating values between \<^term>\<open>u\<close> and \<^term>\<open>v\<close>.\<close>
 lemma scaling_mono:
   fixes u::"'a::linordered_field"
   assumes "u \<le> v" "0 \<le> r" "r \<le> s"
     shows "u + r * (v - u) / s \<le> v"
 proof -
   have "r/s \<le> 1" using assms
     using divide_le_eq_1 by fastforce
   then have "(r/s) * (v - u) \<le> 1 * (v - u)"
     by (meson diff_ge_0_iff_ge mult_right_mono \<open>u \<le> v\<close>)
   then show ?thesis
     by (simp add: field_simps)
 qed
 
 subsection \<open>Some useful lemmas about intervals\<close>
 
 lemma interior_subset_union_intervals:
   assumes "i = cbox a b"
     and "j = cbox c d"
     and "interior j \<noteq> {}"
     and "i \<subseteq> j \<union> S"
     and "interior i \<inter> interior j = {}"
   shows "interior i \<subseteq> interior S"
 proof -
   have "box a b \<inter> cbox c d = {}"
      using Int_interval_mixed_eq_empty[of c d a b] assms
      unfolding interior_cbox by auto
   moreover
   have "box a b \<subseteq> cbox c d \<union> S"
     apply (rule order_trans,rule box_subset_cbox)
     using assms by auto
   ultimately
   show ?thesis
     unfolding assms interior_cbox
       by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI)
 qed
 
 lemma interior_Union_subset_cbox:
   assumes "finite f"
   assumes f: "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a b" "\<And>s. s \<in> f \<Longrightarrow> interior s \<subseteq> t"
     and t: "closed t"
   shows "interior (\<Union>f) \<subseteq> t"
 proof -
   have [simp]: "s \<in> f \<Longrightarrow> closed s" for s
     using f by auto
   define E where "E = {s\<in>f. interior s = {}}"
   then have "finite E" "E \<subseteq> {s\<in>f. interior s = {}}"
     using \<open>finite f\<close> by auto
   then have "interior (\<Union>f) = interior (\<Union>(f - E))"
   proof (induction E rule: finite_subset_induct')
     case (insert s f')
     have "interior (\<Union>(f - insert s f') \<union> s) = interior (\<Union>(f - insert s f'))"
       using insert.hyps \<open>finite f\<close> by (intro interior_closed_Un_empty_interior) auto
     also have "\<Union>(f - insert s f') \<union> s = \<Union>(f - f')"
       using insert.hyps by auto
     finally show ?case
       by (simp add: insert.IH)
   qed simp
   also have "\<dots> \<subseteq> \<Union>(f - E)"
     by (rule interior_subset)
   also have "\<dots> \<subseteq> t"
   proof (rule Union_least)
     fix s assume "s \<in> f - E"
     with f[of s] obtain a b where s: "s \<in> f" "s = cbox a b" "box a b \<noteq> {}"
       by (fastforce simp: E_def)
     have "closure (interior s) \<subseteq> closure t"
       by (intro closure_mono f \<open>s \<in> f\<close>)
     with s \<open>closed t\<close> show "s \<subseteq> t"
       by simp
   qed
   finally show ?thesis .
 qed
 
 lemma Int_interior_Union_intervals:
     "\<lbrakk>finite \<F>; open S; \<And>T. T\<in>\<F> \<Longrightarrow> \<exists>a b. T = cbox a b; \<And>T. T\<in>\<F> \<Longrightarrow> S \<inter> (interior T) = {}\<rbrakk> 
     \<Longrightarrow> S \<inter> interior (\<Union>\<F>) = {}"
   using interior_Union_subset_cbox[of \<F> "UNIV - S"] by auto
 
 lemma interval_split:
   fixes a :: "'a::euclidean_space"
   assumes "k \<in> Basis"
   shows
     "cbox a b \<inter> {x. x\<bullet>k \<le> c} = cbox a (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) c else b\<bullet>i) *\<^sub>R i)"
     "cbox a b \<inter> {x. x\<bullet>k \<ge> c} = cbox (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) c else a\<bullet>i) *\<^sub>R i) b"
   using assms by (rule_tac set_eqI; auto simp: mem_box)+
 
 lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> cbox a b \<noteq> {}"
   by (simp add: box_ne_empty)
 
 subsection \<open>Bounds on intervals where they exist\<close>
 
 definition\<^marker>\<open>tag important\<close> interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
   where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x\<in>s. x\<bullet>i) *\<^sub>R i)"
 
 definition\<^marker>\<open>tag important\<close> interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
   where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x\<in>s. x\<bullet>i) *\<^sub>R i)"
 
 lemma interval_upperbound[simp]:
   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
     interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
   unfolding interval_upperbound_def euclidean_representation_sum cbox_def
   by (safe intro!: cSup_eq) auto
 
 lemma interval_lowerbound[simp]:
   "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
     interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
   unfolding interval_lowerbound_def euclidean_representation_sum cbox_def
   by (safe intro!: cInf_eq) auto
 
 lemmas interval_bounds = interval_upperbound interval_lowerbound
 
 lemma
   fixes X::"real set"
   shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
     and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
   by (auto simp: interval_upperbound_def interval_lowerbound_def)
 
 lemma interval_bounds'[simp]:
   assumes "cbox a b \<noteq> {}"
   shows "interval_upperbound (cbox a b) = b"
     and "interval_lowerbound (cbox a b) = a"
   using assms unfolding box_ne_empty by auto
 
 lemma interval_upperbound_Times:
   assumes "A \<noteq> {}" and "B \<noteq> {}"
   shows "interval_upperbound (A \<times> B) = (interval_upperbound A, interval_upperbound B)"
 proof-
   from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
   have "(\<Sum>i\<in>Basis. (SUP x\<in>A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x\<in>A. x \<bullet> i) *\<^sub>R i)"
       by (subst (2) fst_image_times') (simp del: fst_image_times add: image_comp inner_Pair_0)
   moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
   have "(\<Sum>i\<in>Basis. (SUP x\<in>A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x\<in>B. x \<bullet> i) *\<^sub>R i)"
       by (subst (2) snd_image_times') (simp del: snd_image_times add: image_comp inner_Pair_0)
   ultimately show ?thesis unfolding interval_upperbound_def
       by (subst sum_Basis_prod_eq) (auto simp add: sum_prod)
 qed
 
 lemma interval_lowerbound_Times:
   assumes "A \<noteq> {}" and "B \<noteq> {}"
   shows "interval_lowerbound (A \<times> B) = (interval_lowerbound A, interval_lowerbound B)"
 proof-
   from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
   have "(\<Sum>i\<in>Basis. (INF x\<in>A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x\<in>A. x \<bullet> i) *\<^sub>R i)"
       by (subst (2) fst_image_times') (simp del: fst_image_times add: image_comp inner_Pair_0)
   moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
   have "(\<Sum>i\<in>Basis. (INF x\<in>A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x\<in>B. x \<bullet> i) *\<^sub>R i)"
       by (subst (2) snd_image_times') (simp del: snd_image_times add: image_comp inner_Pair_0)
   ultimately show ?thesis unfolding interval_lowerbound_def
       by (subst sum_Basis_prod_eq) (auto simp add: sum_prod)
 qed
 
 subsection \<open>The notion of a gauge --- simply an open set containing the point\<close>
 
 definition\<^marker>\<open>tag important\<close> "gauge \<gamma> \<longleftrightarrow> (\<forall>x. x \<in> \<gamma> x \<and> open (\<gamma> x))"
 
 lemma gaugeI:
   assumes "\<And>x. x \<in> \<gamma> x"
     and "\<And>x. open (\<gamma> x)"
   shows "gauge \<gamma>"
   using assms unfolding gauge_def by auto
 
 lemma gaugeD[dest]:
   assumes "gauge \<gamma>"
   shows "x \<in> \<gamma> x"
     and "open (\<gamma> x)"
   using assms unfolding gauge_def by auto
 
 lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
   unfolding gauge_def by auto
 
 lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
   unfolding gauge_def by auto
 
 lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)"
   by (rule gauge_ball) auto
 
 lemma gauge_Int[intro]: "gauge \<gamma>1 \<Longrightarrow> gauge \<gamma>2 \<Longrightarrow> gauge (\<lambda>x. \<gamma>1 x \<inter> \<gamma>2 x)"
   unfolding gauge_def by auto
 
 lemma gauge_reflect:
   fixes \<gamma> :: "'a::euclidean_space \<Rightarrow> 'a set"
   shows "gauge \<gamma> \<Longrightarrow> gauge (\<lambda>x. uminus ` \<gamma> (- x))"
   using equation_minus_iff
   by (auto simp: gauge_def surj_def intro!: open_surjective_linear_image linear_uminus)
 
 lemma gauge_Inter:
   assumes "finite S"
     and "\<And>u. u\<in>S \<Longrightarrow> gauge (f u)"
   shows "gauge (\<lambda>x. \<Inter>{f u x | u. u \<in> S})"
 proof -
   have *: "\<And>x. {f u x |u. u \<in> S} = (\<lambda>u. f u x) ` S"
     by auto
   show ?thesis
     unfolding gauge_def unfolding *
     using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
 qed
 
 lemma gauge_existence_lemma:
   "(\<forall>x. \<exists>d :: real. p x \<longrightarrow> 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. p x \<longrightarrow> q d x)"
   by (metis zero_less_one)
 
 subsection \<open>Attempt a systematic general set of "offset" results for components\<close>
 (*FIXME Restructure, the subsection consists only of 1 lemma *)
 lemma gauge_modify:
   assumes "(\<forall>S. open S \<longrightarrow> open {x. f(x) \<in> S})" "gauge d"
   shows "gauge (\<lambda>x. {y. f y \<in> d (f x)})"
   using assms unfolding gauge_def by force
 
 subsection \<open>Divisions\<close>
 
 definition\<^marker>\<open>tag important\<close> division_of (infixl "division'_of" 40)
 where
   "s division_of i \<longleftrightarrow>
     finite s \<and>
     (\<forall>K\<in>s. K \<subseteq> i \<and> K \<noteq> {} \<and> (\<exists>a b. K = cbox a b)) \<and>
     (\<forall>K1\<in>s. \<forall>K2\<in>s. K1 \<noteq> K2 \<longrightarrow> interior(K1) \<inter> interior(K2) = {}) \<and>
     (\<Union>s = i)"
 
 lemma division_ofD[dest]:
   assumes "s division_of i"
   shows "finite s"
     and "\<And>K. K \<in> s \<Longrightarrow> K \<subseteq> i"
     and "\<And>K. K \<in> s \<Longrightarrow> K \<noteq> {}"
     and "\<And>K. K \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b"
     and "\<And>K1 K2. K1 \<in> s \<Longrightarrow> K2 \<in> s \<Longrightarrow> K1 \<noteq> K2 \<Longrightarrow> interior(K1) \<inter> interior(K2) = {}"
     and "\<Union>s = i"
   using assms unfolding division_of_def by auto
 
 lemma division_ofI:
   assumes "finite s"
     and "\<And>K. K \<in> s \<Longrightarrow> K \<subseteq> i"
     and "\<And>K. K \<in> s \<Longrightarrow> K \<noteq> {}"
     and "\<And>K. K \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b"
     and "\<And>K1 K2. K1 \<in> s \<Longrightarrow> K2 \<in> s \<Longrightarrow> K1 \<noteq> K2 \<Longrightarrow> interior K1 \<inter> interior K2 = {}"
     and "\<Union>s = i"
   shows "s division_of i"
   using assms unfolding division_of_def by auto
 
 lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
   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
 
 lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
   unfolding division_of_def by auto
 
 lemma division_of_sing[simp]:
   "s division_of cbox a (a::'a::euclidean_space) \<longleftrightarrow> s = {cbox a a}"
   (is "?l = ?r")
 proof
   assume ?r
   moreover
   { fix K
     assume "s = {{a}}" "K\<in>s"
     then have "\<exists>x y. K = cbox x y"
       apply (rule_tac x=a in exI)+
       apply force
       done
   }
   ultimately show ?l
     unfolding division_of_def cbox_sing by auto
 next
   assume ?l
   have "x = {a}" if  "x \<in> s" for x
     by (metis \<open>s division_of cbox a a\<close> cbox_sing division_ofD(2) division_ofD(3) subset_singletonD that)
   moreover have "s \<noteq> {}"
     using \<open>s division_of cbox a a\<close> by auto
   ultimately show ?r
     unfolding cbox_sing by auto
 qed
 
 lemma elementary_empty: obtains p where "p division_of {}"
   unfolding division_of_trivial by auto
 
 lemma elementary_interval: obtains p where "p division_of (cbox a b)"
   by (metis division_of_trivial division_of_self)
 
 lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
   unfolding division_of_def by auto
 
 lemma forall_in_division:
   "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"
   shows "q division_of (\<Union>q)"
 proof (rule division_ofI)
   note * = division_ofD[OF assms(1)]
   show "finite q"
     using "*"(1) assms(2) infinite_super by auto
   {
     fix k
     assume "k \<in> q"
     then have kp: "k \<in> p"
       using assms(2) by auto
     show "k \<subseteq> \<Union>q"
       using \<open>k \<in> q\<close> by auto
     show "\<exists>a b. k = cbox a b"
       using *(4)[OF kp] by auto
     show "k \<noteq> {}"
       using *(3)[OF kp] by auto
   }
   fix k1 k2
   assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
   then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
     using assms(2) by auto
   show "interior k1 \<inter> interior k2 = {}"
     using *(5)[OF **] by auto
 qed auto
 
 lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
   unfolding division_of_def by auto
 
 lemma division_inter:
   fixes s1 s2 :: "'a::euclidean_space set"
   assumes "p1 division_of s1"
     and "p2 division_of s2"
   shows "{k1 \<inter> k2 | k1 k2. k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)"
   (is "?A' division_of _")
 proof -
   let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
   have *: "?A' = ?A" by auto
   show ?thesis
     unfolding *
   proof (rule division_ofI)
     have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
       by auto
     moreover have "finite (p1 \<times> p2)"
       using assms unfolding division_of_def by auto
     ultimately show "finite ?A" by auto
     have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
       by auto
     show "\<Union>?A = s1 \<inter> s2"
       unfolding * 
       using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto
     {
       fix k
       assume "k \<in> ?A"
       then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
         by auto
       then show "k \<noteq> {}"
         by auto
       show "k \<subseteq> s1 \<inter> s2"
         using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
         unfolding k by auto
       obtain a1 b1 where k1: "k1 = cbox a1 b1"
         using division_ofD(4)[OF assms(1) k(2)] by blast
       obtain a2 b2 where k2: "k2 = cbox a2 b2"
         using division_ofD(4)[OF assms(2) k(3)] by blast
       show "\<exists>a b. k = cbox a b"
         unfolding k k1 k2 unfolding Int_interval by auto
     }
     fix k1 k2
     assume "k1 \<in> ?A"
     then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
       by auto
     assume "k2 \<in> ?A"
     then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
       by auto
     assume "k1 \<noteq> k2"
     then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
       unfolding k1 k2 by auto
     have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
       interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
       interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
       interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
     show "interior k1 \<inter> interior k2 = {}"
       unfolding k1 k2
       apply (rule *)
       using assms division_ofD(5) k1 k2(2) k2(3) th apply auto
       done
   qed
 qed
 
 lemma division_inter_1:
   assumes "d division_of i"
     and "cbox a (b::'a::euclidean_space) \<subseteq> i"
   shows "{cbox a b \<inter> k | k. k \<in> d \<and> cbox a b \<inter> k \<noteq> {}} division_of (cbox a b)"
 proof (cases "cbox a b = {}")
   case True
   show ?thesis
     unfolding True and division_of_trivial by auto
 next
   case False
   have *: "cbox a b \<inter> i = cbox a b" using assms(2) by auto
   show ?thesis
     using division_inter[OF division_of_self[OF False] assms(1)]
     unfolding * by auto
 qed
 
 lemma elementary_Int:
   fixes s t :: "'a::euclidean_space set"
   assumes "p1 division_of s"
     and "p2 division_of t"
   shows "\<exists>p. p division_of (s \<inter> t)"
 using assms division_inter by blast
 
 lemma elementary_Inter:
   assumes "finite f"
     and "f \<noteq> {}"
     and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)"
   shows "\<exists>p. p division_of (\<Inter>f)"
   using assms
 proof (induct f rule: finite_induct)
   case (insert x f)
   show ?case
   proof (cases "f = {}")
     case True
     then show ?thesis
       unfolding True using insert by auto
   next
     case False
     obtain p where "p division_of \<Inter>f"
       using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
     moreover obtain px where "px division_of x"
       using insert(5)[rule_format,OF insertI1] ..
     ultimately show ?thesis
       by (simp add: elementary_Int Inter_insert)
   qed
 qed auto
 
 lemma division_disjoint_union:
   assumes "p1 division_of s1"
     and "p2 division_of s2"
     and "interior s1 \<inter> interior s2 = {}"
   shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
 proof (rule division_ofI)
   note d1 = division_ofD[OF assms(1)]
   note d2 = division_ofD[OF assms(2)]
   show "finite (p1 \<union> p2)"
     using d1(1) d2(1) by auto
   show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
     using d1(6) d2(6) by auto
   {
     fix k1 k2
     assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
     moreover
     let ?g="interior k1 \<inter> interior k2 = {}"
     {
       assume as: "k1\<in>p1" "k2\<in>p2"
       have ?g
         using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
         using assms(3) by blast
     }
     moreover
     {
       assume as: "k1\<in>p2" "k2\<in>p1"
       have ?g
         using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
         using assms(3) by blast
     }
     ultimately show ?g
       using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
   }
   fix k
   assume k: "k \<in> p1 \<union> p2"
   show "k \<subseteq> s1 \<union> s2"
     using k d1(2) d2(2) by auto
   show "k \<noteq> {}"
     using k d1(3) d2(3) by auto
   show "\<exists>a b. k = cbox a b"
     using k d1(4) d2(4) by auto
 qed
 
 lemma partial_division_extend_1:
   fixes a b c d :: "'a::euclidean_space"
   assumes incl: "cbox c d \<subseteq> cbox a b"
     and nonempty: "cbox c d \<noteq> {}"
   obtains p where "p division_of (cbox a b)" "cbox c d \<in> p"
 proof
   let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
     cbox (\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)"
   define p where "p = ?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
 
   show "cbox c d \<in> p"
     unfolding p_def
     by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
   have ord: "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i" if "i \<in> Basis" for i
     using incl nonempty that
     unfolding box_eq_empty subset_box by (auto simp: not_le)
 
   show "p division_of (cbox a b)"
   proof (rule division_ofI)
     show "finite p"
       unfolding p_def by (auto intro!: finite_PiE)
     {
       fix k
       assume "k \<in> p"
       then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
         by (auto simp: p_def)
       then show "\<exists>a b. k = cbox a b"
         by auto
       have "k \<subseteq> cbox a b \<and> k \<noteq> {}"
       proof (simp add: k box_eq_empty subset_box not_less, safe)
         fix i :: 'a
         assume i: "i \<in> Basis"
         with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
           by (auto simp: PiE_iff)
         with i ord[of i]
         show "a \<bullet> i \<le> fst (f i) \<bullet> i" "snd (f i) \<bullet> i \<le> b \<bullet> i" "fst (f i) \<bullet> i \<le> snd (f i) \<bullet> i"
           by auto
       qed
       then show "k \<noteq> {}" "k \<subseteq> cbox a b"
         by auto
       {
         fix l
         assume "l \<in> p"
         then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
           by (auto simp: p_def)
         assume "l \<noteq> k"
         have "\<exists>i\<in>Basis. f i \<noteq> g i"
         proof (rule ccontr)
           assume "\<not> ?thesis"
           with f g have "f = g"
             by (auto simp: PiE_iff extensional_def fun_eq_iff)
           with \<open>l \<noteq> k\<close> show False
             by (simp add: l k)
         qed
         then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
         then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
                   "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
           using f g by (auto simp: PiE_iff)
         with * ord[of i] show "interior l \<inter> interior k = {}"
           by (auto simp add: l k disjoint_interval intro!: bexI[of _ i])
       }
       note \<open>k \<subseteq> cbox a b\<close>
     }
     moreover
     {
       fix x assume x: "x \<in> cbox a b"
       have "\<forall>i\<in>Basis. \<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
       proof
         fix i :: 'a
         assume "i \<in> Basis"
         with x ord[of i]
         have "(a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> c \<bullet> i) \<or> (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i) \<or>
             (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
           by (auto simp: cbox_def)
         then show "\<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
           by auto
       qed
       then obtain f where
         f: "\<forall>i\<in>Basis. x \<bullet> i \<in> {fst (f i) \<bullet> i..snd (f i) \<bullet> i} \<and> f i \<in> {(a, c), (c, d), (d, b)}"
         unfolding bchoice_iff ..
       moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
         by auto
       moreover from f have "x \<in> ?B (restrict f Basis)"
         by (auto simp: mem_box)
       ultimately have "\<exists>k\<in>p. x \<in> k"
         unfolding p_def by blast
     }
     ultimately show "\<Union>p = cbox a b"
       by auto
   qed
 qed
 
 proposition partial_division_extend_interval:
   assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
   obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
 proof (cases "p = {}")
   case True
   obtain q where "q division_of (cbox a b)"
     by (rule elementary_interval)
   then show ?thesis
     using True that by blast
 next
   case False
   note p = division_ofD[OF assms(1)]
   have div_cbox: "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q"
   proof
     fix k
     assume kp: "k \<in> p"
     obtain c d where k: "k = cbox c d"
       using p(4)[OF kp] by blast
     have *: "cbox c d \<subseteq> cbox a b" "cbox c d \<noteq> {}"
       using p(2,3)[OF kp, unfolded k] using assms(2)
       by (blast intro: order.trans)+
     obtain q where "q division_of cbox a b" "cbox c d \<in> q"
       by (rule partial_division_extend_1[OF *])
     then show "\<exists>q. q division_of cbox a b \<and> k \<in> q"
       unfolding k by auto
   qed
   obtain q where q: "\<And>x. x \<in> p \<Longrightarrow> q x division_of cbox a b" "\<And>x. x \<in> p \<Longrightarrow> x \<in> q x"
     using bchoice[OF div_cbox] by blast
   have "q x division_of \<Union>(q x)" if x: "x \<in> p" for x
     apply (rule division_ofI)
     using division_ofD[OF q(1)[OF x]] by auto
   then have di: "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
     by (meson Diff_subset division_of_subset)
   have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
     apply (rule elementary_Inter [OF finite_imageI[OF p(1)]])
     apply (auto dest: di simp: False elementary_Inter [OF finite_imageI[OF p(1)]])
     done
   then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
   have "d \<union> p division_of cbox a b"
   proof -
     have te: "\<And>S f T. S \<noteq> {} \<Longrightarrow> \<forall>i\<in>S. f i \<union> i = T \<Longrightarrow> T = \<Inter>(f ` S) \<union> \<Union>S" by auto
     have cbox_eq: "cbox a b = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
     proof (rule te[OF False], clarify)
       fix i
       assume i: "i \<in> p"
       show "\<Union>(q i - {i}) \<union> i = cbox a b"
         using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
     qed
     { fix K
       assume K: "K \<in> p"
       note qk=division_ofD[OF q(1)[OF K]]
       have *: "\<And>u T S. T \<inter> S = {} \<Longrightarrow> u \<subseteq> S \<Longrightarrow> u \<inter> T = {}"
         by auto
       have "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<inter> interior K = {}"
       proof (rule *[OF Int_interior_Union_intervals])
         show "\<And>T. T\<in>q K - {K} \<Longrightarrow> interior K \<inter> interior T = {}"
           using qk(5) using q(2)[OF K] by auto
         show "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<subseteq> interior (\<Union>(q K - {K}))"
           apply (rule interior_mono)+
           using K by auto
       qed (use qk in auto)} note [simp] = this
     show "d \<union> p division_of (cbox a b)"
       unfolding cbox_eq
       apply (rule division_disjoint_union[OF d assms(1)])
       apply (rule Int_interior_Union_intervals)
       apply (rule p open_interior ballI)+
       apply simp_all
       done
   qed
   then show ?thesis
     by (meson Un_upper2 that)
 qed
 
 lemma elementary_bounded[dest]:
   fixes S :: "'a::euclidean_space set"
   shows "p division_of S \<Longrightarrow> bounded S"
   unfolding division_of_def by (metis bounded_Union bounded_cbox)
 
 lemma elementary_subset_cbox:
   "p division_of S \<Longrightarrow> \<exists>a b. S \<subseteq> cbox a (b::'a::euclidean_space)"
   by (meson bounded_subset_cbox_symmetric elementary_bounded)
 
 proposition division_union_intervals_exists:
   fixes a b :: "'a::euclidean_space"
   assumes "cbox a b \<noteq> {}"
   obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
 proof (cases "cbox c d = {}")
   case True
   with assms that show ?thesis by force
 next
   case False
   show ?thesis
   proof (cases "cbox a b \<inter> cbox c d = {}")
     case True
     then show ?thesis
       by (metis that False assms division_disjoint_union division_of_self insert_is_Un interior_Int interior_empty)
   next
     case False
     obtain u v where uv: "cbox a b \<inter> cbox c d = cbox u v"
       unfolding Int_interval by auto
     have uv_sub: "cbox u v \<subseteq> cbox c d" using uv by auto
     obtain p where pd: "p division_of cbox c d" and "cbox u v \<in> p"
       by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
     note p = this division_ofD[OF pd]
     have "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = interior(cbox u v \<inter> \<Union>(p - {cbox u v}))"
       apply (rule arg_cong[of _ _ interior])
       using p(8) uv by auto
     also have "\<dots> = {}"
       unfolding interior_Int
       apply (rule Int_interior_Union_intervals)
       using p(6) p(7)[OF p(2)] \<open>finite p\<close>
       apply auto
       done
     finally have [simp]: "interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}" by simp
     have cbe: "cbox a b \<union> cbox c d = cbox a b \<union> \<Union>(p - {cbox u v})"
       using p(8) unfolding uv[symmetric] by auto
     have "insert (cbox a b) (p - {cbox u v}) division_of cbox a b \<union> \<Union>(p - {cbox u v})"
     proof -
       have "{cbox a b} division_of cbox a b"
         by (simp add: assms division_of_self)
       then show "insert (cbox a b) (p - {cbox u v}) division_of cbox a b \<union> \<Union>(p - {cbox u v})"
         by (metis (no_types) Diff_subset \<open>interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}\<close> division_disjoint_union division_of_subset insert_is_Un p(1) p(8))
     qed
     with that[of "p - {cbox u v}"] show ?thesis by (simp add: cbe)
   qed
 qed
 
 lemma division_of_Union:
   assumes "finite f"
     and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
     and "\<And>k1 k2. k1 \<in> \<Union>f \<Longrightarrow> k2 \<in> \<Union>f \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
   shows "\<Union>f division_of \<Union>(\<Union>f)"
   using assms  by (auto intro!: division_ofI)
 
 lemma elementary_union_interval:
   fixes a b :: "'a::euclidean_space"
   assumes "p division_of \<Union>p"
   obtains q where "q division_of (cbox a b \<union> \<Union>p)"
 proof (cases "p = {} \<or> cbox a b = {}")
   case True
   obtain p where "p division_of (cbox a b)"
     by (rule elementary_interval)
   then show thesis
     using True assms that by auto
 next
   case False
   then have "p \<noteq> {}" "cbox a b \<noteq> {}"
     by auto
   note pdiv = division_ofD[OF assms]
   show ?thesis
   proof (cases "interior (cbox a b) = {}")
     case True
     show ?thesis
       apply (rule that [of "insert (cbox a b) p", OF division_ofI])
       using pdiv(1-4) True False apply auto
        apply (metis IntI equals0D pdiv(5))
       by (metis disjoint_iff_not_equal pdiv(5))
   next
     case False  
     have "\<forall>K\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> K)"
     proof
       fix K
       assume kp: "K \<in> p"
       from pdiv(4)[OF kp] obtain c d where "K = cbox c d" by blast
       then show "\<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> K)"
         by (meson \<open>cbox a b \<noteq> {}\<close> division_union_intervals_exists)
     qed
     from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert (cbox a b) (q x) division_of (cbox a b) \<union> x" ..
     note q = division_ofD[OF this[rule_format]]
     let ?D = "\<Union>{insert (cbox a b) (q K) | K. K \<in> p}"
     show thesis
     proof (rule that[OF division_ofI])
       have *: "{insert (cbox a b) (q K) |K. K \<in> p} = (\<lambda>K. insert (cbox a b) (q K)) ` p"
         by auto
       show "finite ?D"
         using "*" pdiv(1) q(1) by auto
       have "\<Union>?D = (\<Union>x\<in>p. \<Union>(insert (cbox a b) (q x)))"
         by auto
       also have "... = \<Union>{cbox a b \<union> t |t. t \<in> p}"
         using q(6) by auto
       also have "... = cbox a b \<union> \<Union>p"
         using \<open>p \<noteq> {}\<close> by auto
       finally show "\<Union>?D = cbox a b \<union> \<Union>p" .
       show "K \<subseteq> cbox a b \<union> \<Union>p" "K \<noteq> {}" if "K \<in> ?D" for K
         using q that by blast+
       show "\<exists>a b. K = cbox a b" if "K \<in> ?D" for K
         using q(4) that by auto
     next
       fix K' K
       assume K: "K \<in> ?D" and K': "K' \<in> ?D" "K \<noteq> K'"
       obtain x where x: "K \<in> insert (cbox a b) (q x)" "x\<in>p"
         using K by auto
       obtain x' where x': "K'\<in>insert (cbox a b) (q x')" "x'\<in>p"
         using K' by auto
       show "interior K \<inter> interior K' = {}"
       proof (cases "x = x' \<or> K  = cbox a b \<or> K' = cbox a b")
         case True
         then show ?thesis
           using True K' q(5) x' x by auto
       next
         case False
         then have as': "K \<noteq> cbox a b" "K' \<noteq> cbox a b"
           by auto
         obtain c d where K: "K = cbox c d"
           using q(4) x by blast
         have "interior K \<inter> interior (cbox a b) = {}"
           using as' K'(2) q(5) x by blast
         then have "interior K \<subseteq> interior x"
           by (metis \<open>interior (cbox a b) \<noteq> {}\<close> K q(2) x interior_subset_union_intervals)
         moreover
         obtain c d where c_d: "K' = cbox c d"
           using q(4)[OF x'(2,1)] by blast
         have "interior K' \<inter> interior (cbox a b) = {}"
           using as'(2) q(5) x' by blast
         then have "interior K' \<subseteq> interior x'"
           by (metis \<open>interior (cbox a b) \<noteq> {}\<close> c_d interior_subset_union_intervals q(2) x'(1) x'(2))
         moreover have "interior x \<inter> interior x' = {}"
           by (meson False assms division_ofD(5) x'(2) x(2))
         ultimately show ?thesis
           using \<open>interior K \<subseteq> interior x\<close> \<open>interior K' \<subseteq> interior x'\<close> by auto
       qed
     qed
   qed
 qed
 
 
 
 lemma elementary_unions_intervals:
   assumes fin: "finite f"
     and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a (b::'a::euclidean_space)"
   obtains p where "p division_of (\<Union>f)"
 proof -
   have "\<exists>p. p division_of (\<Union>f)"
   proof (induct_tac f rule:finite_subset_induct)
     show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
   next
     fix x F
     assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
     from this(3) obtain p where p: "p division_of \<Union>F" ..
     from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
     have *: "\<Union>F = \<Union>p"
       using division_ofD[OF p] by auto
     show "\<exists>p. p division_of \<Union>(insert x F)"
       using elementary_union_interval[OF p[unfolded *], of a b]
       unfolding Union_insert x * by metis
   qed (insert assms, auto)
   then show ?thesis
     using that by auto
 qed
 
 lemma elementary_union:
   fixes S T :: "'a::euclidean_space set"
   assumes "ps division_of S" "pt division_of T"
   obtains p where "p division_of (S \<union> T)"
 proof -
   have *: "S \<union> T = \<Union>ps \<union> \<Union>pt"
     using assms unfolding division_of_def by auto
   show ?thesis
     apply (rule elementary_unions_intervals[of "ps \<union> pt"])
     using assms apply auto
     by (simp add: * that)
 qed
 
 lemma partial_division_extend:
   fixes T :: "'a::euclidean_space set"
   assumes "p division_of S"
     and "q division_of T"
     and "S \<subseteq> T"
   obtains r where "p \<subseteq> r" and "r division_of T"
 proof -
   note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
   obtain a b where ab: "T \<subseteq> cbox a b"
     using elementary_subset_cbox[OF assms(2)] by auto
   obtain r1 where "p \<subseteq> r1" "r1 division_of (cbox a b)"
     using assms
     by (metis ab dual_order.trans partial_division_extend_interval divp(6))
   note r1 = this division_ofD[OF this(2)]
   obtain p' where "p' division_of \<Union>(r1 - p)"
     apply (rule elementary_unions_intervals[of "r1 - p"])
     using r1(3,6)
       apply auto
     done
   then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
     by (metis assms(2) divq(6) elementary_Int)
   {
     fix x
     assume x: "x \<in> T" "x \<notin> S"
     then obtain R where r: "R \<in> r1" "x \<in> R"
       unfolding r1 using ab
       by (meson division_contains r1(2) subsetCE)
     moreover have "R \<notin> p"
     proof
       assume "R \<in> p"
       then have "x \<in> S" using divp(2) r by auto
       then show False using x by auto
     qed
     ultimately have "x\<in>\<Union>(r1 - p)" by auto
   }
   then have Teq: "T = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
     unfolding divp divq using assms(3) by auto
   have "interior S \<inter> interior (\<Union>(r1-p)) = {}"
   proof (rule Int_interior_Union_intervals)
     have *: "\<And>S. (\<And>x. x \<in> S \<Longrightarrow> False) \<Longrightarrow> S = {}"
       by auto
     show "interior S \<inter> interior m = {}" if "m \<in> r1 - p" for m 
     proof -
       have "interior m \<inter> interior (\<Union>p) = {}"
       proof (rule Int_interior_Union_intervals)
         show "\<And>T. T\<in>p \<Longrightarrow> interior m \<inter> interior T = {}"
           by (metis DiffD1 DiffD2 that r1(1) r1(7) rev_subsetD)
       qed (use divp in auto)
       then show "interior S \<inter> interior m = {}"
         unfolding divp by auto
     qed 
   qed (use r1 in auto)
   then have "interior S \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
     using interior_subset by auto
   then have div: "p \<union> r2 division_of \<Union>p \<union> \<Union>(r1 - p) \<inter> \<Union>q"
     by (simp add: assms(1) division_disjoint_union divp(6) r2)
   show ?thesis
     apply (rule that[of "p \<union> r2"])
      apply (auto simp: div Teq)
     done
 qed
 
 
 lemma division_split:
   fixes a :: "'a::euclidean_space"
   assumes "p division_of (cbox a b)"
     and k: "k\<in>Basis"
   shows "{l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> p \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} division_of(cbox a b \<inter> {x. x\<bullet>k \<le> c})"
       (is "?p1 division_of ?I1")
     and "{l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> p \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}} division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
       (is "?p2 division_of ?I2")
 proof (rule_tac[!] division_ofI)
   note p = division_ofD[OF assms(1)]
   show "finite ?p1" "finite ?p2"
     using p(1) by auto
   show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2"
     unfolding p(6)[symmetric] by auto
   {
     fix K
     assume "K \<in> ?p1"
     then obtain l where l: "K = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> p" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
       by blast
     obtain u v where uv: "l = cbox u v"
       using assms(1) l(2) by blast
     show "K \<subseteq> ?I1"
       using l p(2) uv by force
     show  "K \<noteq> {}"
       by (simp add: l)
     show  "\<exists>a b. K = cbox a b"
       apply (simp add: l uv p(2-3)[OF l(2)])
       apply (subst interval_split[OF k])
       apply (auto intro: order.trans)
       done
     fix K'
     assume "K' \<in> ?p1"
     then obtain l' where l': "K' = l' \<inter> {x. x \<bullet> k \<le> c}" "l' \<in> p" "l' \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
       by blast
     assume "K \<noteq> K'"
     then show "interior K \<inter> interior K' = {}"
       unfolding l l' using p(5)[OF l(2) l'(2)] by auto
   }
   {
     fix K
     assume "K \<in> ?p2"
     then obtain l where l: "K = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> p" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}"
       by blast
     obtain u v where uv: "l = cbox u v"
       using l(2) p(4) by blast
     show "K \<subseteq> ?I2"
       using l p(2) uv by force
     show  "K \<noteq> {}"
       by (simp add: l)
     show  "\<exists>a b. K = cbox a b"
       apply (simp add: l uv p(2-3)[OF l(2)])
       apply (subst interval_split[OF k])
       apply (auto intro: order.trans)
       done
     fix K'
     assume "K' \<in> ?p2"
     then obtain l' where l': "K' = l' \<inter> {x. c \<le> x \<bullet> k}" "l' \<in> p" "l' \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}"
       by blast
     assume "K \<noteq> K'"
     then show "interior K \<inter> interior K' = {}"
       unfolding l l' using p(5)[OF l(2) l'(2)] by auto
   }
 qed
 
 subsection \<open>Tagged (partial) divisions\<close>
 
 definition\<^marker>\<open>tag important\<close> tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
   where "s tagged_partial_division_of i \<longleftrightarrow>
     finite s \<and>
     (\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and>
     (\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow>
       interior K1 \<inter> interior K2 = {})"
 
-lemma tagged_partial_division_ofD[dest]:
+lemma tagged_partial_division_ofD:
   assumes "s tagged_partial_division_of i"
   shows "finite s"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> x \<in> K"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> K \<subseteq> i"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b"
     and "\<And>x1 K1 x2 K2. (x1,K1) \<in> s \<Longrightarrow>
       (x2, K2) \<in> s \<Longrightarrow> (x1, K1) \<noteq> (x2, K2) \<Longrightarrow> interior K1 \<inter> interior K2 = {}"
   using assms unfolding tagged_partial_division_of_def by blast+
 
 definition\<^marker>\<open>tag important\<close> tagged_division_of (infixr "tagged'_division'_of" 40)
   where "s tagged_division_of i \<longleftrightarrow> s tagged_partial_division_of i \<and> (\<Union>{K. \<exists>x. (x,K) \<in> s} = i)"
 
 lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
 
 lemma tagged_division_of:
   "s tagged_division_of i \<longleftrightarrow>
     finite s \<and>
     (\<forall>x K. (x, K) \<in> s \<longrightarrow> x \<in> K \<and> K \<subseteq> i \<and> (\<exists>a b. K = cbox a b)) \<and>
     (\<forall>x1 K1 x2 K2. (x1, K1) \<in> s \<and> (x2, K2) \<in> s \<and> (x1, K1) \<noteq> (x2, K2) \<longrightarrow>
       interior K1 \<inter> interior K2 = {}) \<and>
     (\<Union>{K. \<exists>x. (x,K) \<in> s} = i)"
   unfolding tagged_division_of_def tagged_partial_division_of_def by auto
 
 lemma tagged_division_ofI:
   assumes "finite s"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> x \<in> K"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> K \<subseteq> i"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b"
     and "\<And>x1 K1 x2 K2. (x1,K1) \<in> s \<Longrightarrow> (x2, K2) \<in> s \<Longrightarrow> (x1, K1) \<noteq> (x2, K2) \<Longrightarrow>
       interior K1 \<inter> interior K2 = {}"
     and "(\<Union>{K. \<exists>x. (x,K) \<in> s} = i)"
   shows "s tagged_division_of i"
   unfolding tagged_division_of
   using assms  by fastforce
 
 lemma tagged_division_ofD[dest]:  (*FIXME USE A LOCALE*)
   assumes "s tagged_division_of i"
   shows "finite s"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> x \<in> K"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> K \<subseteq> i"
     and "\<And>x K. (x,K) \<in> s \<Longrightarrow> \<exists>a b. K = cbox a b"
     and "\<And>x1 K1 x2 K2. (x1, K1) \<in> s \<Longrightarrow> (x2, K2) \<in> s \<Longrightarrow> (x1, K1) \<noteq> (x2, K2) \<Longrightarrow>
       interior K1 \<inter> interior K2 = {}"
     and "(\<Union>{K. \<exists>x. (x,K) \<in> s} = i)"
   using assms unfolding tagged_division_of by blast+
 
 lemma division_of_tagged_division:
   assumes "s tagged_division_of i"
   shows "(snd ` s) division_of i"
 proof (rule division_ofI)
   note assm = tagged_division_ofD[OF assms]
   show "\<Union>(snd ` s) = i" "finite (snd ` s)"
     using assm by auto
   fix k
   assume k: "k \<in> snd ` s"
   then obtain xk where xk: "(xk, k) \<in> s"
     by auto
   then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = cbox a b"
     using assm by fastforce+
   fix k'
   assume k': "k' \<in> snd ` s" "k \<noteq> k'"
   from this(1) obtain xk' where xk': "(xk', k') \<in> s"
     by auto
   then show "interior k \<inter> interior k' = {}"
     using assm(5) k'(2) xk by blast
 qed
 
 lemma partial_division_of_tagged_division:
   assumes "s tagged_partial_division_of i"
   shows "(snd ` s) division_of \<Union>(snd ` s)"
 proof (rule division_ofI)
   note assm = tagged_partial_division_ofD[OF assms]
   show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
     using assm by auto
   fix k
   assume k: "k \<in> snd ` s"
   then obtain xk where xk: "(xk, k) \<in> s"
     by auto
   then show "k \<noteq> {}" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>(snd ` s)"
     using assm by auto
   fix k'
   assume k': "k' \<in> snd ` s" "k \<noteq> k'"
   from this(1) obtain xk' where xk': "(xk', k') \<in> s"
     by auto
   then show "interior k \<inter> interior k' = {}"
     using assm(5) k'(2) xk by auto
 qed
 
 lemma tagged_partial_division_subset:
   assumes "s tagged_partial_division_of i"
     and "t \<subseteq> s"
   shows "t tagged_partial_division_of i"
   using assms finite_subset[OF assms(2)]
   unfolding tagged_partial_division_of_def
   by blast
 
 lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
   by auto
 
 lemma tagged_division_of_empty: "{} tagged_division_of {}"
   unfolding tagged_division_of by auto
 
 lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
   unfolding tagged_partial_division_of_def by auto
 
 lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
   unfolding tagged_division_of by auto
 
 lemma tagged_division_of_self: "x \<in> cbox a b \<Longrightarrow> {(x,cbox a b)} tagged_division_of (cbox a b)"
   by (rule tagged_division_ofI) auto
 
 lemma tagged_division_of_self_real: "x \<in> {a .. b::real} \<Longrightarrow> {(x,{a .. b})} tagged_division_of {a .. b}"
   unfolding box_real[symmetric]
   by (rule tagged_division_of_self)
 
 lemma tagged_division_Un:
   assumes "p1 tagged_division_of s1"
     and "p2 tagged_division_of s2"
     and "interior s1 \<inter> interior s2 = {}"
   shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
 proof (rule tagged_division_ofI)
   note p1 = tagged_division_ofD[OF assms(1)]
   note p2 = tagged_division_ofD[OF assms(2)]
   show "finite (p1 \<union> p2)"
     using p1(1) p2(1) by auto
   show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
     using p1(6) p2(6) by blast
   fix x k
   assume xk: "(x, k) \<in> p1 \<union> p2"
   show "x \<in> k" "\<exists>a b. k = cbox a b"
     using xk p1(2,4) p2(2,4) by auto
   show "k \<subseteq> s1 \<union> s2"
     using xk p1(3) p2(3) by blast
   fix x' k'
   assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
   have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
     using assms(3) interior_mono by blast
   show "interior k \<inter> interior k' = {}"
     apply (cases "(x, k) \<in> p1")
     apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2))
     by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
 qed
 
 lemma tagged_division_Union:
   assumes "finite I"
     and tag: "\<And>i. i\<in>I \<Longrightarrow> pfn i tagged_division_of i"
     and disj: "\<And>i1 i2. \<lbrakk>i1 \<in> I; i2 \<in> I; i1 \<noteq> i2\<rbrakk> \<Longrightarrow> interior(i1) \<inter> interior(i2) = {}"
   shows "\<Union>(pfn ` I) tagged_division_of (\<Union>I)"
 proof (rule tagged_division_ofI)
   note assm = tagged_division_ofD[OF tag]
   show "finite (\<Union>(pfn ` I))"
     using assms by auto
   have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` I)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` I)"
     by blast
   also have "\<dots> = \<Union>I"
     using assm(6) by auto
   finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` I)} = \<Union>I" .
   fix x k
   assume xk: "(x, k) \<in> \<Union>(pfn ` I)"
   then obtain i where i: "i \<in> I" "(x, k) \<in> pfn i"
     by auto
   show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>I"
     using assm(2-4)[OF i] using i(1) by auto
   fix x' k'
   assume xk': "(x', k') \<in> \<Union>(pfn ` I)" "(x, k) \<noteq> (x', k')"
   then obtain i' where i': "i' \<in> I" "(x', k') \<in> pfn i'"
     by auto
   have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
     using i(1) i'(1) disj interior_mono by blast
   show "interior k \<inter> interior k' = {}"
   proof (cases "i = i'")
     case True then show ?thesis 
       using assm(5) i' i xk'(2) by blast
   next
     case False then show ?thesis 
     using "*" assm(3) i' i by auto
   qed
 qed
 
 lemma tagged_partial_division_of_Union_self:
   assumes "p tagged_partial_division_of s"
   shows "p tagged_division_of (\<Union>(snd ` p))"
   apply (rule tagged_division_ofI)
   using tagged_partial_division_ofD[OF assms]
   apply auto
   done
 
 lemma tagged_division_of_union_self:
   assumes "p tagged_division_of s"
   shows "p tagged_division_of (\<Union>(snd ` p))"
   apply (rule tagged_division_ofI)
   using tagged_division_ofD[OF assms]
   apply auto
   done
 
 lemma tagged_division_Un_interval:
   fixes a :: "'a::euclidean_space"
   assumes "p1 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<le> (c::real)})"
     and "p2 tagged_division_of (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
     and k: "k \<in> Basis"
   shows "(p1 \<union> p2) tagged_division_of (cbox a b)"
 proof -
   have *: "cbox a b = (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<union> (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
     by auto
   show ?thesis
     apply (subst *)
     apply (rule tagged_division_Un[OF assms(1-2)])
     unfolding interval_split[OF k] interior_cbox
     using k
     apply (auto simp add: box_def elim!: ballE[where x=k])
     done
 qed
 
 lemma tagged_division_Un_interval_real:
   fixes a :: real
   assumes "p1 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<le> (c::real)})"
     and "p2 tagged_division_of ({a .. b} \<inter> {x. x\<bullet>k \<ge> c})"
     and k: "k \<in> Basis"
   shows "(p1 \<union> p2) tagged_division_of {a .. b}"
   using assms
   unfolding box_real[symmetric]
   by (rule tagged_division_Un_interval)
 
 lemma tagged_division_split_left_inj:
   assumes d: "d tagged_division_of i"
   and tags: "(x1, K1) \<in> d" "(x2, K2) \<in> d"
   and "K1 \<noteq> K2"
   and eq: "K1 \<inter> {x. x \<bullet> k \<le> c} = K2 \<inter> {x. x \<bullet> k \<le> c}"
     shows "interior (K1 \<inter> {x. x\<bullet>k \<le> c}) = {}"
 proof -
   have "interior (K1 \<inter> K2) = {} \<or> (x2, K2) = (x1, K1)"
     using tags d by (metis (no_types) interior_Int tagged_division_ofD(5))
   then show ?thesis
     using eq \<open>K1 \<noteq> K2\<close> by (metis (no_types) inf_assoc inf_bot_left inf_left_idem interior_Int old.prod.inject)
 qed
 
 lemma tagged_division_split_right_inj:
   assumes d: "d tagged_division_of i"
   and tags: "(x1, K1) \<in> d" "(x2, K2) \<in> d"
   and "K1 \<noteq> K2"
   and eq: "K1 \<inter> {x. x\<bullet>k \<ge> c} = K2 \<inter> {x. x\<bullet>k \<ge> c}"
     shows "interior (K1 \<inter> {x. x\<bullet>k \<ge> c}) = {}"
 proof -
   have "interior (K1 \<inter> K2) = {} \<or> (x2, K2) = (x1, K1)"
     using tags d by (metis (no_types) interior_Int tagged_division_ofD(5))
   then show ?thesis
     using eq \<open>K1 \<noteq> K2\<close> by (metis (no_types) inf_assoc inf_bot_left inf_left_idem interior_Int old.prod.inject)
 qed
 
 lemma (in comm_monoid_set) over_tagged_division_lemma:
   assumes "p tagged_division_of i"
     and "\<And>u v. box u v = {} \<Longrightarrow> d (cbox u v) = \<^bold>1"
   shows "F (\<lambda>(_, k). d k) p = F d (snd ` p)"
 proof -
   have *: "(\<lambda>(_ ,k). d k) = d \<circ> snd"
     by (simp add: fun_eq_iff)
   note assm = tagged_division_ofD[OF assms(1)]
   show ?thesis
     unfolding *
   proof (rule reindex_nontrivial[symmetric])
     show "finite p"
       using assm by auto
     fix x y
     assume "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
     obtain a b where ab: "snd x = cbox a b"
       using assm(4)[of "fst x" "snd x"] \<open>x\<in>p\<close> by auto
     have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
       using \<open>x \<in> p\<close> \<open>x \<noteq> y\<close> \<open>snd x = snd y\<close> [symmetric] by auto
     with \<open>x\<in>p\<close> \<open>y\<in>p\<close> have "interior (snd x) \<inter> interior (snd y) = {}"
       by (intro assm(5)[of "fst x" _ "fst y"]) auto
     then have "box a b = {}"
       unfolding \<open>snd x = snd y\<close>[symmetric] ab by auto
     then have "d (cbox a b) = \<^bold>1"
       using assm(2)[of "fst x" "snd x"] \<open>x\<in>p\<close> ab[symmetric] by (intro assms(2)) auto
     then show "d (snd x) = \<^bold>1"
       unfolding ab by auto
   qed
 qed
 
 
 subsection \<open>Functions closed on boxes: morphisms from boxes to monoids\<close>
 
 text \<open>This auxiliary structure is used to sum up over the elements of a division. Main theorem is
   \<open>operative_division\<close>. Instances for the monoid are \<^typ>\<open>'a option\<close>, \<^typ>\<open>real\<close>, and
   \<^typ>\<open>bool\<close>.\<close>
 
 paragraph\<^marker>\<open>tag important\<close> \<open>Using additivity of lifted function to encode definedness.\<close>
 text \<open>%whitespace\<close>
 definition\<^marker>\<open>tag important\<close> lift_option :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a option \<Rightarrow> 'b option \<Rightarrow> 'c option"
 where
   "lift_option f a' b' = Option.bind a' (\<lambda>a. Option.bind b' (\<lambda>b. Some (f a b)))"
 
 lemma lift_option_simps[simp]:
   "lift_option f (Some a) (Some b) = Some (f a b)"
   "lift_option f None b' = None"
   "lift_option f a' None = None"
   by (auto simp: lift_option_def)
 
 lemma\<^marker>\<open>tag important\<close> comm_monoid_lift_option:
   assumes "comm_monoid f z"
   shows "comm_monoid (lift_option f) (Some z)"
 proof -
   from assms interpret comm_monoid f z .
   show ?thesis
     by standard (auto simp: lift_option_def ac_simps split: bind_split)
 qed
 
 lemma comm_monoid_and: "comm_monoid HOL.conj True"
   by standard auto
 
 lemma comm_monoid_set_and: "comm_monoid_set HOL.conj True"
   by (rule comm_monoid_set.intro) (fact comm_monoid_and)
 
 
 paragraph \<open>Misc\<close>
 
 lemma interval_real_split:
   "{a .. b::real} \<inter> {x. x \<le> c} = {a .. min b c}"
   "{a .. b} \<inter> {x. c \<le> x} = {max a c .. b}"
   apply (metis Int_atLeastAtMostL1 atMost_def)
   apply (metis Int_atLeastAtMostL2 atLeast_def)
   done
 
 lemma bgauge_existence_lemma: "(\<forall>x\<in>s. \<exists>d::real. 0 < d \<and> q d x) \<longleftrightarrow> (\<forall>x. \<exists>d>0. x\<in>s \<longrightarrow> q d x)"
   by (meson zero_less_one)
 
 
 paragraph \<open>Division points\<close>
 text \<open>%whitespace\<close>
 definition\<^marker>\<open>tag important\<close> "division_points (k::('a::euclidean_space) set) d =
    {(j,x). j \<in> Basis \<and> (interval_lowerbound k)\<bullet>j < x \<and> x < (interval_upperbound k)\<bullet>j \<and>
      (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
 
 lemma division_points_finite:
   fixes i :: "'a::euclidean_space set"
   assumes "d division_of i"
   shows "finite (division_points i d)"
 proof -
   note assm = division_ofD[OF assms]
   let ?M = "\<lambda>j. {(j,x)|x. (interval_lowerbound i)\<bullet>j < x \<and> x < (interval_upperbound i)\<bullet>j \<and>
     (\<exists>i\<in>d. (interval_lowerbound i)\<bullet>j = x \<or> (interval_upperbound i)\<bullet>j = x)}"
   have *: "division_points i d = \<Union>(?M ` Basis)"
     unfolding division_points_def by auto
   show ?thesis
     unfolding * using assm by auto
 qed
 
 lemma division_points_subset:
   fixes a :: "'a::euclidean_space"
   assumes "d division_of (cbox a b)"
     and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
     and k: "k \<in> Basis"
   shows "division_points (cbox a b \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l . l \<in> d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} \<subseteq>
       division_points (cbox a b) d" (is ?t1)
     and "division_points (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l . l \<in> d \<and> \<not>(l \<inter> {x. x\<bullet>k \<ge> c} = {})} \<subseteq>
       division_points (cbox a b) d" (is ?t2)
 proof -
   note assm = division_ofD[OF assms(1)]
   have *: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
     "\<forall>i\<in>Basis. a\<bullet>i \<le> (\<Sum>i\<in>Basis. (if i = k then min (b \<bullet> k) c else  b \<bullet> i) *\<^sub>R i) \<bullet> i"
     "\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (a \<bullet> k) c else a \<bullet> i) *\<^sub>R i) \<bullet> i \<le> b\<bullet>i"
     "min (b \<bullet> k) c = c" "max (a \<bullet> k) c = c"
     using assms using less_imp_le by auto
    have "\<exists>i\<in>d. interval_lowerbound i \<bullet> x = y \<or> interval_upperbound i \<bullet> x = y"
      if "a \<bullet> x < y" "y < (if x = k then c else b \<bullet> x)"
         "interval_lowerbound i \<bullet> x = y \<or> interval_upperbound i \<bullet> x = y"
         "i = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
         "x \<in> Basis" for i l x y
   proof -
     obtain u v where l: "l = cbox u v"
       using \<open>l \<in> d\<close> assms(1) by blast
     have *: "\<forall>i\<in>Basis. u \<bullet> i \<le> (\<Sum>i\<in>Basis. (if i = k then min (v \<bullet> k) c else v \<bullet> i) *\<^sub>R i) \<bullet> i"
       using that(6) unfolding l interval_split[OF k] box_ne_empty that .
     have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
       using l using that(6) unfolding box_ne_empty[symmetric] by auto
     show ?thesis
       apply (rule bexI[OF _ \<open>l \<in> d\<close>])
       using that(1-3,5) \<open>x \<in> Basis\<close>
       unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] that
       apply (auto split: if_split_asm)
       done
   qed
   moreover have "\<And>x y. \<lbrakk>y < (if x = k then c else b \<bullet> x)\<rbrakk> \<Longrightarrow> y < b \<bullet> x"
     using \<open>c < b\<bullet>k\<close> by (auto split: if_split_asm)
   ultimately show ?t1 
     unfolding division_points_def interval_split[OF k, of a b]
     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
     unfolding * by force
   have "\<And>x y i l. (if x = k then c else a \<bullet> x) < y \<Longrightarrow> a \<bullet> x < y"
     using \<open>a\<bullet>k < c\<close> by (auto split: if_split_asm)
   moreover have "\<exists>i\<in>d. interval_lowerbound i \<bullet> x = y \<or>
                        interval_upperbound i \<bullet> x = y"
     if "(if x = k then c else a \<bullet> x) < y" "y < b \<bullet> x"
       "interval_lowerbound i \<bullet> x = y \<or> interval_upperbound i \<bullet> x = y"
       "i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}"
       "x \<in> Basis" for x y i l
   proof -
     obtain u v where l: "l = cbox u v"
       using \<open>l \<in> d\<close> assm(4) by blast
     have *: "\<forall>i\<in>Basis. (\<Sum>i\<in>Basis. (if i = k then max (u \<bullet> k) c else u \<bullet> i) *\<^sub>R i) \<bullet> i \<le> v \<bullet> i"
       using that(6) unfolding l interval_split[OF k] box_ne_empty that .
     have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
       using l using that(6) unfolding box_ne_empty[symmetric] by auto
     show "\<exists>i\<in>d. interval_lowerbound i \<bullet> x = y \<or> interval_upperbound i \<bullet> x = y"
       apply (rule bexI[OF _ \<open>l \<in> d\<close>])
       using that(1-3,5) \<open>x \<in> Basis\<close>
       unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] that
       apply (auto split: if_split_asm)
       done
   qed
   ultimately show ?t2
     unfolding division_points_def interval_split[OF k, of a b]
     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
     unfolding *
     by force
 qed
 
 lemma division_points_psubset:
   fixes a :: "'a::euclidean_space"
   assumes d: "d division_of (cbox a b)"
       and altb: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
       and "l \<in> d"
       and disj: "interval_lowerbound l\<bullet>k = c \<or> interval_upperbound l\<bullet>k = c"
       and k: "k \<in> Basis"
   shows "division_points (cbox a b \<inter> {x. x\<bullet>k \<le> c}) {l \<inter> {x. x\<bullet>k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}} \<subset>
          division_points (cbox a b) d" (is "?D1 \<subset> ?D")
     and "division_points (cbox a b \<inter> {x. x\<bullet>k \<ge> c}) {l \<inter> {x. x\<bullet>k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}} \<subset>
          division_points (cbox a b) d" (is "?D2 \<subset> ?D")
 proof -
   have ab: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
     using altb by (auto intro!:less_imp_le)
   obtain u v where l: "l = cbox u v"
     using d \<open>l \<in> d\<close> by blast
   have uv: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "\<forall>i\<in>Basis. a\<bullet>i \<le> u\<bullet>i \<and> v\<bullet>i \<le> b\<bullet>i"
     apply (metis assms(5) box_ne_empty(1) cbox_division_memE d l)
     by (metis assms(5) box_ne_empty(1) cbox_division_memE d l subset_box(1))
   have *: "interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
           "interval_upperbound (cbox a b \<inter> {x. x \<bullet> k \<le> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
     unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
     using uv[rule_format, of k] ab k
     by auto
   have "\<exists>x. x \<in> ?D - ?D1"
     using assms(3-)
     unfolding division_points_def interval_bounds[OF ab]
     by (force simp add: *)
   moreover have "?D1 \<subseteq> ?D"
     by (auto simp add: assms division_points_subset)
   ultimately show "?D1 \<subset> ?D"
     by blast
   have *: "interval_lowerbound (cbox a b \<inter> {x. x \<bullet> k \<ge> interval_lowerbound l \<bullet> k}) \<bullet> k = interval_lowerbound l \<bullet> k"
     "interval_lowerbound (cbox a b \<inter> {x. x \<bullet> k \<ge> interval_upperbound l \<bullet> k}) \<bullet> k = interval_upperbound l \<bullet> k"
     unfolding l interval_split[OF k] interval_bounds[OF uv(1)]
     using uv[rule_format, of k] ab k
     by auto
   have "\<exists>x. x \<in> ?D - ?D2"
     using assms(3-)
     unfolding division_points_def interval_bounds[OF ab]
     by (force simp add: *)
   moreover have "?D2 \<subseteq> ?D"
     by (auto simp add: assms division_points_subset)
   ultimately show "?D2 \<subset> ?D"
     by blast
 qed
 
 lemma division_split_left_inj:
   fixes S :: "'a::euclidean_space set"
   assumes div: "\<D> division_of S"
     and eq: "K1 \<inter> {x::'a. x\<bullet>k \<le> c} = K2 \<inter> {x. x\<bullet>k \<le> c}"
     and "K1 \<in> \<D>" "K2 \<in> \<D>" "K1 \<noteq> K2"
   shows "interior (K1 \<inter> {x. x\<bullet>k \<le> c}) = {}"
 proof -
   have "interior K2 \<inter> interior {a. a \<bullet> k \<le> c} = interior K1 \<inter> interior {a. a \<bullet> k \<le> c}"
     by (metis (no_types) eq interior_Int)
   moreover have "\<And>A. interior A \<inter> interior K2 = {} \<or> A = K2 \<or> A \<notin> \<D>"
     by (meson div \<open>K2 \<in> \<D>\<close> division_of_def)
   ultimately show ?thesis
     using \<open>K1 \<in> \<D>\<close> \<open>K1 \<noteq> K2\<close> by auto
 qed
 
 lemma division_split_right_inj:
   fixes S :: "'a::euclidean_space set"
   assumes div: "\<D> division_of S"
     and eq: "K1 \<inter> {x::'a. x\<bullet>k \<ge> c} = K2 \<inter> {x. x\<bullet>k \<ge> c}"
     and "K1 \<in> \<D>" "K2 \<in> \<D>" "K1 \<noteq> K2"
   shows "interior (K1 \<inter> {x. x\<bullet>k \<ge> c}) = {}"
 proof -
   have "interior K2 \<inter> interior {a. a \<bullet> k \<ge> c} = interior K1 \<inter> interior {a. a \<bullet> k \<ge> c}"
     by (metis (no_types) eq interior_Int)
   moreover have "\<And>A. interior A \<inter> interior K2 = {} \<or> A = K2 \<or> A \<notin> \<D>"
     by (meson div \<open>K2 \<in> \<D>\<close> division_of_def)
   ultimately show ?thesis
     using \<open>K1 \<in> \<D>\<close> \<open>K1 \<noteq> K2\<close> by auto
 qed
 
 lemma interval_doublesplit:
   fixes a :: "'a::euclidean_space"
   assumes "k \<in> Basis"
   shows "cbox a b \<inter> {x . \<bar>x\<bullet>k - c\<bar> \<le> (e::real)} =
     cbox (\<Sum>i\<in>Basis. (if i = k then max (a\<bullet>k) (c - e) else a\<bullet>i) *\<^sub>R i)
      (\<Sum>i\<in>Basis. (if i = k then min (b\<bullet>k) (c + e) else b\<bullet>i) *\<^sub>R i)"
 proof -
   have *: "\<And>x c e::real. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e"
     by auto
   have **: "\<And>s P Q. s \<inter> {x. P x \<and> Q x} = (s \<inter> {x. Q x}) \<inter> {x. P x}"
     by blast
   show ?thesis
     unfolding * ** interval_split[OF assms] by (rule refl)
 qed
 
 lemma division_doublesplit:
   fixes a :: "'a::euclidean_space"
   assumes "p division_of (cbox a b)"
     and k: "k \<in> Basis"
   shows "(\<lambda>l. l \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e}) ` {l\<in>p. l \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e} \<noteq> {}}
          division_of  (cbox a b \<inter> {x. \<bar>x\<bullet>k - c\<bar> \<le> e})"
 proof -
   have *: "\<And>x c. \<bar>x - c\<bar> \<le> e \<longleftrightarrow> x \<ge> c - e \<and> x \<le> c + e"
     by auto
   have **: "\<And>p q p' q'. p division_of q \<Longrightarrow> p = p' \<Longrightarrow> q = q' \<Longrightarrow> p' division_of q'"
     by auto
   note division_split(1)[OF assms, where c="c+e",unfolded interval_split[OF k]]
   note division_split(2)[OF this, where c="c-e" and k=k,OF k]
   then show ?thesis
     apply (rule **)
     subgoal
       apply (simp add: abs_diff_le_iff field_simps Collect_conj_eq setcompr_eq_image [symmetric] cong: image_cong_simp)
       apply (rule equalityI)
       apply blast
       apply clarsimp
       apply (rule_tac x="xa \<inter> {x. c + e \<ge> x \<bullet> k}" in exI)
       apply auto
       done
     by (simp add: interval_split k interval_doublesplit)
 qed
               
 paragraph \<open>Operative\<close>
 
 locale operative = comm_monoid_set +
   fixes g :: "'b::euclidean_space set \<Rightarrow> 'a"
   assumes box_empty_imp: "\<And>a b. box a b = {} \<Longrightarrow> g (cbox a b) = \<^bold>1"
     and Basis_imp: "\<And>a b c k. k \<in> Basis \<Longrightarrow> g (cbox a b) = g (cbox a b \<inter> {x. x\<bullet>k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
 begin
 
 lemma empty [simp]:
   "g {} = \<^bold>1"
 proof -
   have *: "cbox One (-One) = ({}::'b set)"
     by (auto simp: box_eq_empty inner_sum_left inner_Basis sum.If_cases ex_in_conv)
   moreover have "box One (-One) = ({}::'b set)"
     using box_subset_cbox[of One "-One"] by (auto simp: *)
   ultimately show ?thesis
     using box_empty_imp [of One "-One"] by simp
 qed
        
 lemma division:
   "F g d = g (cbox a b)" if "d division_of (cbox a b)"
 proof -
   define C where [abs_def]: "C = card (division_points (cbox a b) d)"
   then show ?thesis
   using that proof (induction C arbitrary: a b d rule: less_induct)
     case (less a b d)
     show ?case
     proof cases
       assume "box a b = {}"
       { fix k assume "k\<in>d"
         then obtain a' b' where k: "k = cbox a' b'"
           using division_ofD(4)[OF less.prems] by blast
         with \<open>k\<in>d\<close> division_ofD(2)[OF less.prems] have "cbox a' b' \<subseteq> cbox a b"
           by auto
         then have "box a' b' \<subseteq> box a b"
           unfolding subset_box by auto
         then have "g k = \<^bold>1"
           using box_empty_imp [of a' b'] k by (simp add: \<open>box a b = {}\<close>) }
       then show "box a b = {} \<Longrightarrow> F g d = g (cbox a b)"
         by (auto intro!: neutral simp: box_empty_imp)
     next
       assume "box a b \<noteq> {}"
       then have ab: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i" and ab': "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
         by (auto simp: box_ne_empty)
       show "F g d = g (cbox a b)"
       proof (cases "division_points (cbox a b) d = {}")
         case True
         { fix u v and j :: 'b
           assume j: "j \<in> Basis" and as: "cbox u v \<in> d"
           then have "cbox u v \<noteq> {}"
             using less.prems by blast
           then have uv: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i" "u\<bullet>j \<le> v\<bullet>j"
             using j unfolding box_ne_empty by auto
           have *: "\<And>p r Q. \<not> j\<in>Basis \<or> p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> Q (cbox u v)"
             using as j by auto
           have "(j, u\<bullet>j) \<notin> division_points (cbox a b) d"
                "(j, v\<bullet>j) \<notin> division_points (cbox a b) d" using True by auto
           note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]
           note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]
           moreover
           have "a\<bullet>j \<le> u\<bullet>j" "v\<bullet>j \<le> b\<bullet>j"
             using division_ofD(2,2,3)[OF \<open>d division_of cbox a b\<close> as]
             apply (metis j subset_box(1) uv(1))
             by (metis \<open>cbox u v \<subseteq> cbox a b\<close> j subset_box(1) uv(1))
           ultimately have "u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j"
             unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) j by force }
         then have d': "\<forall>i\<in>d. \<exists>u v. i = cbox u v \<and>
           (\<forall>j\<in>Basis. u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = a\<bullet>j \<or> u\<bullet>j = b\<bullet>j \<and> v\<bullet>j = b\<bullet>j \<or> u\<bullet>j = a\<bullet>j \<and> v\<bullet>j = b\<bullet>j)"
           unfolding forall_in_division[OF less.prems] by blast
         have "(1/2) *\<^sub>R (a+b) \<in> cbox a b"
           unfolding mem_box using ab by (auto simp: inner_simps)
         note this[unfolded division_ofD(6)[OF \<open>d division_of cbox a b\<close>,symmetric] Union_iff]
         then obtain i where i: "i \<in> d" "(1 / 2) *\<^sub>R (a + b) \<in> i" ..
         obtain u v where uv: "i = cbox u v" 
                      "\<forall>j\<in>Basis. u \<bullet> j = a \<bullet> j \<and> v \<bullet> j = a \<bullet> j \<or>
                                 u \<bullet> j = b \<bullet> j \<and> v \<bullet> j = b \<bullet> j \<or>
                                 u \<bullet> j = a \<bullet> j \<and> v \<bullet> j = b \<bullet> j"
           using d' i(1) by auto
         have "cbox a b \<in> d"
         proof -
           have "u = a" "v = b"
             unfolding euclidean_eq_iff[where 'a='b]
           proof safe
             fix j :: 'b
             assume j: "j \<in> Basis"
             note i(2)[unfolded uv mem_box,rule_format,of j]
             then show "u \<bullet> j = a \<bullet> j" and "v \<bullet> j = b \<bullet> j"
               using uv(2)[rule_format,of j] j by (auto simp: inner_simps)
           qed
           then have "i = cbox a b" using uv by auto
           then show ?thesis using i by auto
         qed
         then have deq: "d = insert (cbox a b) (d - {cbox a b})"
           by auto
         have "F g (d - {cbox a b}) = \<^bold>1"
         proof (intro neutral ballI)
           fix x
           assume x: "x \<in> d - {cbox a b}"
           then have "x\<in>d"
             by auto note d'[rule_format,OF this]
           then obtain u v where uv: "x = cbox u v"
                       "\<forall>j\<in>Basis. u \<bullet> j = a \<bullet> j \<and> v \<bullet> j = a \<bullet> j \<or>
                                  u \<bullet> j = b \<bullet> j \<and> v \<bullet> j = b \<bullet> j \<or>
                                  u \<bullet> j = a \<bullet> j \<and> v \<bullet> j = b \<bullet> j" 
             by blast
           have "u \<noteq> a \<or> v \<noteq> b"
             using x[unfolded uv] by auto
           then obtain j where "u\<bullet>j \<noteq> a\<bullet>j \<or> v\<bullet>j \<noteq> b\<bullet>j" and j: "j \<in> Basis"
             unfolding euclidean_eq_iff[where 'a='b] by auto
           then have "u\<bullet>j = v\<bullet>j"
             using uv(2)[rule_format,OF j] by auto
           then have "box u v = {}"
             using j unfolding box_eq_empty by (auto intro!: bexI[of _ j])
           then show "g x = \<^bold>1"
             unfolding uv(1) by (rule box_empty_imp)
         qed
         then show "F g d = g (cbox a b)"
           using division_ofD[OF less.prems]
           apply (subst deq)
           apply (subst insert)
           apply auto
           done
       next
         case False
         then have "\<exists>x. x \<in> division_points (cbox a b) d"
           by auto
         then obtain k c 
           where "k \<in> Basis" "interval_lowerbound (cbox a b) \<bullet> k < c"
                 "c < interval_upperbound (cbox a b) \<bullet> k"
                 "\<exists>i\<in>d. interval_lowerbound i \<bullet> k = c \<or> interval_upperbound i \<bullet> k = c"
           unfolding division_points_def by auto
         then obtain j where "a \<bullet> k < c" "c < b \<bullet> k" 
               and "j \<in> d" and j: "interval_lowerbound j \<bullet> k = c \<or> interval_upperbound j \<bullet> k = c"
           by (metis division_of_trivial empty_iff interval_bounds' less.prems)
         let ?lec = "{x. x\<bullet>k \<le> c}" let ?gec = "{x. x\<bullet>k \<ge> c}"
         define d1 where "d1 = {l \<inter> ?lec | l. l \<in> d \<and> l \<inter> ?lec \<noteq> {}}"
         define d2 where "d2 = {l \<inter> ?gec | l. l \<in> d \<and> l \<inter> ?gec \<noteq> {}}"
         define cb where "cb = (\<Sum>i\<in>Basis. (if i = k then c else b\<bullet>i) *\<^sub>R i)"
         define ca where "ca = (\<Sum>i\<in>Basis. (if i = k then c else a\<bullet>i) *\<^sub>R i)"
         have "division_points (cbox a b \<inter> ?lec) {l \<inter> ?lec |l. l \<in> d \<and> l \<inter> ?lec \<noteq> {}} 
               \<subset> division_points (cbox a b) d"
           by (rule division_points_psubset[OF \<open>d division_of cbox a b\<close> ab \<open>a \<bullet> k < c\<close> \<open>c < b \<bullet> k\<close> \<open>j \<in> d\<close> j \<open>k \<in> Basis\<close>])
         with division_points_finite[OF \<open>d division_of cbox a b\<close>] 
         have "card
          (division_points (cbox a b \<inter> ?lec) {l \<inter> ?lec |l. l \<in> d \<and> l \<inter> ?lec \<noteq> {}})
            < card (division_points (cbox a b) d)"
           by (rule psubset_card_mono)
         moreover have "division_points (cbox a b \<inter> {x. c \<le> x \<bullet> k}) {l \<inter> {x. c \<le> x \<bullet> k} |l. l \<in> d \<and> l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}}
               \<subset> division_points (cbox a b) d"
           by (rule division_points_psubset[OF \<open>d division_of cbox a b\<close> ab \<open>a \<bullet> k < c\<close> \<open>c < b \<bullet> k\<close> \<open>j \<in> d\<close> j \<open>k \<in> Basis\<close>])
         with division_points_finite[OF \<open>d division_of cbox a b\<close>] 
         have "card (division_points (cbox a b \<inter> ?gec) {l \<inter> ?gec |l. l \<in> d \<and> l \<inter> ?gec \<noteq> {}})
               < card (division_points (cbox a b) d)"
           by (rule psubset_card_mono)
         ultimately have *: "F g d1 = g (cbox a b \<inter> ?lec)" "F g d2 = g (cbox a b \<inter> ?gec)"
           unfolding interval_split[OF \<open>k \<in> Basis\<close>]
           apply (rule_tac[!] less.hyps)
           using division_split[OF \<open>d division_of cbox a b\<close>, where k=k and c=c] \<open>k \<in> Basis\<close>
           by (simp_all add: interval_split  d1_def d2_def division_points_finite[OF \<open>d division_of cbox a b\<close>])
         have fxk_le: "g (l \<inter> ?lec) = \<^bold>1" 
           if "l \<in> d" "y \<in> d" "l \<inter> ?lec = y \<inter> ?lec" "l \<noteq> y" for l y
         proof -
           obtain u v where leq: "l = cbox u v"
             using \<open>l \<in> d\<close> less.prems by auto
           have "interior (cbox u v \<inter> ?lec) = {}"
             using that division_split_left_inj leq less.prems by blast
           then show ?thesis
             unfolding leq interval_split [OF \<open>k \<in> Basis\<close>]
             by (auto intro: box_empty_imp)
         qed
         have fxk_ge: "g (l \<inter> {x. x \<bullet> k \<ge> c}) = \<^bold>1"
           if "l \<in> d" "y \<in> d" "l \<inter> ?gec = y \<inter> ?gec" "l \<noteq> y" for l y
         proof -
           obtain u v where leq: "l = cbox u v"
             using \<open>l \<in> d\<close> less.prems by auto
           have "interior (cbox u v \<inter> ?gec) = {}"
             using that division_split_right_inj leq less.prems by blast
           then show ?thesis
             unfolding leq interval_split[OF \<open>k \<in> Basis\<close>]
             by (auto intro: box_empty_imp)
         qed
         have d1_alt: "d1 = (\<lambda>l. l \<inter> ?lec) ` {l \<in> d. l \<inter> ?lec \<noteq> {}}"
           using d1_def by auto
         have d2_alt: "d2 = (\<lambda>l. l \<inter> ?gec) ` {l \<in> d. l \<inter> ?gec \<noteq> {}}"
           using d2_def by auto
         have "g (cbox a b) = F g d1 \<^bold>* F g d2" (is "_ = ?prev")
           unfolding * using \<open>k \<in> Basis\<close>
           by (auto dest: Basis_imp)
         also have "F g d1 = F (\<lambda>l. g (l \<inter> ?lec)) d"
           unfolding d1_alt using division_of_finite[OF less.prems] fxk_le
           by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left)
         also have "F g d2 = F (\<lambda>l. g (l \<inter> ?gec)) d"
           unfolding d2_alt using division_of_finite[OF less.prems] fxk_ge
           by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left)
         also have *: "\<forall>x\<in>d. g x = g (x \<inter> ?lec) \<^bold>* g (x \<inter> ?gec)"
           unfolding forall_in_division[OF \<open>d division_of cbox a b\<close>]
           using \<open>k \<in> Basis\<close>
           by (auto dest: Basis_imp)
         have "F (\<lambda>l. g (l \<inter> ?lec)) d \<^bold>* F (\<lambda>l. g (l \<inter> ?gec)) d = F g d"
           using * by (simp add: distrib)
         finally show ?thesis by auto
       qed
     qed
   qed
 qed
 
 proposition tagged_division:
   assumes "d tagged_division_of (cbox a b)"
   shows "F (\<lambda>(_, l). g l) d = g (cbox a b)"
 proof -
   have "F (\<lambda>(_, k). g k) d = F g (snd ` d)"
     using assms box_empty_imp by (rule over_tagged_division_lemma)
   then show ?thesis
     unfolding assms [THEN division_of_tagged_division, THEN division] .
   qed
 
 end
 
 locale operative_real = comm_monoid_set +
   fixes g :: "real set \<Rightarrow> 'a"
   assumes neutral: "b \<le> a \<Longrightarrow> g {a..b} = \<^bold>1"
   assumes coalesce_less: "a < c \<Longrightarrow> c < b \<Longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
 begin
 
 sublocale operative where g = g
   rewrites "box = (greaterThanLessThan :: real \<Rightarrow> _)"
     and "cbox = (atLeastAtMost :: real \<Rightarrow> _)"
     and "\<And>x::real. x \<in> Basis \<longleftrightarrow> x = 1"
 proof -
   show "operative f z g"
   proof
     show "g (cbox a b) = \<^bold>1" if "box a b = {}" for a b
       using that by (simp add: neutral)
     show "g (cbox a b) = g (cbox a b \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. c \<le> x \<bullet> k})"
       if "k \<in> Basis" for a b c k
     proof -
       from that have [simp]: "k = 1"
         by simp
       from neutral [of 0 1] neutral [of a a for a] coalesce_less
   have [simp]: "g {} = \<^bold>1" "\<And>a. g {a} = \<^bold>1"
     "\<And>a b c. a < c \<Longrightarrow> c < b \<Longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
     by auto
       have "g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
     by (auto simp: min_def max_def le_less)
       then show "g (cbox a b) = g (cbox a b \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (cbox a b \<inter> {x. c \<le> x \<bullet> k})"
         by (simp add: atMost_def [symmetric] atLeast_def [symmetric])
 qed
   qed
   show "box = (greaterThanLessThan :: real \<Rightarrow> _)"
     and "cbox = (atLeastAtMost :: real \<Rightarrow> _)"
     and "\<And>x::real. x \<in> Basis \<longleftrightarrow> x = 1"
     by (simp_all add: fun_eq_iff)
 qed
 
 lemma coalesce_less_eq:
   "g {a..c} \<^bold>* g {c..b} = g {a..b}" if "a \<le> c" "c \<le> b"
   proof (cases "c = a \<or> c = b")
     case False
   with that have "a < c" "c < b"
     by auto
     then show ?thesis
     by (rule coalesce_less)
   next
     case True
   with that box_empty_imp [of a a] box_empty_imp [of b b] show ?thesis
     by safe simp_all
     qed
 
 end
 
 lemma operative_realI:
   "operative_real f z g" if "operative f z g"
 proof -
   interpret operative f z g
     using that .
   show ?thesis
   proof
     show "g {a..b} = z" if "b \<le> a" for a b
       using that box_empty_imp by simp
     show "f (g {a..c}) (g {c..b}) = g {a..b}" if "a < c" "c < b" for a b c
       using that
     using Basis_imp [of 1 a b c]
       by (simp_all add: atMost_def [symmetric] atLeast_def [symmetric] max_def min_def)
 qed
 qed
 
 
 subsection \<open>Special case of additivity we need for the FTC\<close>
 (*fix add explanation here *)
 lemma additive_tagged_division_1:
   fixes f :: "real \<Rightarrow> 'a::real_normed_vector"
   assumes "a \<le> b"
     and "p tagged_division_of {a..b}"
   shows "sum (\<lambda>(x,k). f(Sup k) - f(Inf k)) p = f b - f a"
 proof -
   let ?f = "(\<lambda>k::(real) set. if k = {} then 0 else f(interval_upperbound k) - f(interval_lowerbound k))"
   interpret operative_real plus 0 ?f
     rewrites "comm_monoid_set.F (+) 0 = sum"
     by standard[1] (auto simp add: sum_def)
   have p_td: "p tagged_division_of cbox a b"
     using assms(2) box_real(2) by presburger
   have **: "cbox a b \<noteq> {}"
     using assms(1) by auto
   then have "f b - f a = (\<Sum>(x, l)\<in>p. if l = {} then 0 else f (interval_upperbound l) - f (interval_lowerbound l))"
     proof -
       have "(if cbox a b = {} then 0 else f (interval_upperbound (cbox a b)) - f (interval_lowerbound (cbox a b))) = f b - f a"
         using assms by auto
       then show ?thesis
         using p_td assms by (simp add: tagged_division) 
     qed 
   then show ?thesis
     using assms by (auto intro!: sum.cong)
 qed
 
 
 subsection \<open>Fine-ness of a partition w.r.t. a gauge\<close>
 
 definition\<^marker>\<open>tag important\<close> fine  (infixr "fine" 46)
   where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
 
 lemma fineI:
   assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
   shows "d fine s"
   using assms unfolding fine_def by auto
 
 lemma fineD[dest]:
   assumes "d fine s"
   shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
   using assms unfolding fine_def by auto
 
 lemma fine_Int: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
   unfolding fine_def by auto
 
 lemma fine_Inter:
  "(\<lambda>x. \<Inter>{f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
   unfolding fine_def by blast
 
 lemma fine_Un: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
   unfolding fine_def by blast
 
 lemma fine_Union: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
   unfolding fine_def by auto
 
 lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
   unfolding fine_def by blast
 
 subsection \<open>Some basic combining lemmas\<close>
 
 lemma tagged_division_Union_exists:
   assumes "finite I"
     and "\<forall>i\<in>I. \<exists>p. p tagged_division_of i \<and> d fine p"
     and "\<forall>i1\<in>I. \<forall>i2\<in>I. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
     and "\<Union>I = i"
    obtains p where "p tagged_division_of i" and "d fine p"
 proof -
   obtain pfn where pfn:
     "\<And>x. x \<in> I \<Longrightarrow> pfn x tagged_division_of x"
     "\<And>x. x \<in> I \<Longrightarrow> d fine pfn x"
     using bchoice[OF assms(2)] by auto
   show thesis
     apply (rule_tac p="\<Union>(pfn ` I)" in that)
     using assms(1) assms(3) assms(4) pfn(1) tagged_division_Union apply force
     by (metis (mono_tags, lifting) fine_Union imageE pfn(2))
 qed
 (* FIXME structure here, do not have one lemma in each subsection *)
 
 subsection\<^marker>\<open>tag unimportant\<close> \<open>The set we're concerned with must be closed\<close>
 
 lemma division_of_closed:
   fixes i :: "'n::euclidean_space set"
   shows "s division_of i \<Longrightarrow> closed i"
   unfolding division_of_def by fastforce
 (* FIXME structure here, do not have one lemma in each subsection *)
 subsection \<open>General bisection principle for intervals; might be useful elsewhere\<close>
 (* FIXME  move? *)
 lemma interval_bisection_step:
   fixes type :: "'a::euclidean_space"
   assumes emp: "P {}"
     and Un: "\<And>S T. \<lbrakk>P S; P T; interior(S) \<inter> interior(T) = {}\<rbrakk> \<Longrightarrow> P (S \<union> T)"
     and non: "\<not> P (cbox a (b::'a))"
   obtains c d where "\<not> P (cbox c d)"
     and "\<And>i. i \<in> Basis \<Longrightarrow> a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
 proof -
   have "cbox a b \<noteq> {}"
     using emp non by metis
   then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
     by (force simp: mem_box)
   have UN_cases: "\<lbrakk>finite \<F>;
            \<And>S. S\<in>\<F> \<Longrightarrow> P S;
            \<And>S. S\<in>\<F> \<Longrightarrow> \<exists>a b. S = cbox a b;
            \<And>S T. S\<in>\<F> \<Longrightarrow> T\<in>\<F> \<Longrightarrow> S \<noteq> T \<Longrightarrow> interior S \<inter> interior T = {}\<rbrakk> \<Longrightarrow> P (\<Union>\<F>)" for \<F>
   proof (induct \<F> rule: finite_induct)
     case empty show ?case
       using emp by auto
   next
     case (insert x f)
     then show ?case
       unfolding Union_insert by (metis Int_interior_Union_intervals Un insert_iff open_interior)
   qed
   let ?ab = "\<lambda>i. (a\<bullet>i + b\<bullet>i) / 2"
   let ?A = "{cbox c d | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = ?ab i) \<or>
     (c\<bullet>i = ?ab i) \<and> (d\<bullet>i = b\<bullet>i)}"
   have "P (\<Union>?A)" 
     if "\<And>c d.  \<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i \<Longrightarrow> P (cbox c d)"
   proof (rule UN_cases)
     let ?B = "(\<lambda>S. cbox (\<Sum>i\<in>Basis. (if i \<in> S then a\<bullet>i else ?ab i) *\<^sub>R i::'a)
                         (\<Sum>i\<in>Basis. (if i \<in> S then ?ab i else b\<bullet>i) *\<^sub>R i)) ` {s. s \<subseteq> Basis}"
     have "?A \<subseteq> ?B"
     proof
       fix x
       assume "x \<in> ?A"
       then obtain c d
         where x:  "x = cbox c d"
           "\<And>i. i \<in> Basis \<Longrightarrow>
                         c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = ?ab i \<or> c \<bullet> i = ?ab i \<and> d \<bullet> i = b \<bullet> i" 
         by blast
       have "c = (\<Sum>i\<in>Basis. (if c \<bullet> i = a \<bullet> i then a \<bullet> i else ?ab i) *\<^sub>R i)"
            "d = (\<Sum>i\<in>Basis. (if c \<bullet> i = a \<bullet> i then ?ab i else b \<bullet> i) *\<^sub>R i)"
         using x(2) ab by (fastforce simp add: euclidean_eq_iff[where 'a='a])+
       then show "x \<in> ?B"
         unfolding x by (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in image_eqI) auto
     qed
     then show "finite ?A"
       by (rule finite_subset) auto
   next
     fix S
     assume "S \<in> ?A"
     then obtain c d
       where s: "S = cbox c d"
                "\<And>i. i \<in> Basis \<Longrightarrow> c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = ?ab i \<or> c \<bullet> i = ?ab i \<and> d \<bullet> i = b \<bullet> i"
       by blast
     show "P S"
       unfolding s using ab s(2) by (fastforce intro!: that)
     show "\<exists>a b. S = cbox a b"
       unfolding s by auto
     fix T
     assume "T \<in> ?A"
     then obtain e f where t:
       "T = cbox e f"
       "\<And>i. i \<in> Basis \<Longrightarrow> e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = ?ab i \<or> e \<bullet> i = ?ab i \<and> f \<bullet> i = b \<bullet> i"
       by blast
     assume "S \<noteq> T"
     then have "\<not> (c = e \<and> d = f)"
       unfolding s t by auto
     then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i': "i \<in> Basis"
       unfolding euclidean_eq_iff[where 'a='a] by auto
     then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
       using s(2) t(2) apply fastforce
       using t(2)[OF i'] \<open>c \<bullet> i \<noteq> e \<bullet> i \<or> d \<bullet> i \<noteq> f \<bullet> i\<close> i' s(2) t(2) by fastforce
     have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
       by auto
     show "interior S \<inter> interior T = {}"
       unfolding s t interior_cbox
     proof (rule *)
       fix x
       assume "x \<in> box c d" "x \<in> box e f"
       then have x: "c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i"
         unfolding mem_box using i'  by force+
       show False  using s(2)[OF i'] t(2)[OF i'] and i x  
         by auto
     qed
   qed
   also have "\<Union>?A = cbox a b"
   proof (rule set_eqI,rule)
     fix x
     assume "x \<in> \<Union>?A"
     then obtain c d where x:
       "x \<in> cbox c d"
       "\<And>i. i \<in> Basis \<Longrightarrow> c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = ?ab i \<or> c \<bullet> i = ?ab i \<and> d \<bullet> i = b \<bullet> i"
       by blast
     then show "x\<in>cbox a b"
       unfolding mem_box by force
   next
     fix x
     assume x: "x \<in> cbox a b"
     then have "\<forall>i\<in>Basis. \<exists>c d. (c = a\<bullet>i \<and> d = ?ab i \<or> c = ?ab i \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d"
       (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
       unfolding mem_box by (metis linear)
     then obtain \<alpha> \<beta> where "\<forall>i\<in>Basis. (\<alpha> \<bullet> i = a \<bullet> i \<and> \<beta> \<bullet> i = ?ab i \<or>
          \<alpha> \<bullet> i = ?ab i \<and> \<beta> \<bullet> i = b \<bullet> i) \<and> \<alpha> \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> \<beta> \<bullet> i"
       by (auto simp: choice_Basis_iff)
     then show "x\<in>\<Union>?A"
       by (force simp add: mem_box)
   qed
   finally show thesis
       by (metis (no_types, lifting) assms(3) that)
 qed
 
 lemma interval_bisection:
   fixes type :: "'a::euclidean_space"
   assumes "P {}"
     and Un: "\<And>S T. \<lbrakk>P S; P T; interior(S) \<inter> interior(T) = {}\<rbrakk> \<Longrightarrow> P (S \<union> T)"
     and "\<not> P (cbox a (b::'a))"
   obtains x where "x \<in> cbox a b"
     and "\<forall>e>0. \<exists>c d. x \<in> cbox c d \<and> cbox c d \<subseteq> ball x e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
 proof -
   have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and>
     (\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and>
        2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" (is "\<forall>x. ?P x")
   proof
     show "?P x" for x
     proof (cases "P (cbox (fst x) (snd x))")
       case True
       then show ?thesis by auto
     next
       case False
       obtain c d where "\<not> P (cbox c d)"
         "\<And>i. i \<in> Basis \<Longrightarrow>
            fst x \<bullet> i \<le> c \<bullet> i \<and>
            c \<bullet> i \<le> d \<bullet> i \<and>
            d \<bullet> i \<le> snd x \<bullet> i \<and>
            2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
         by (blast intro: interval_bisection_step[of P, OF assms(1-2) False])
       then show ?thesis
         by (rule_tac x="(c,d)" in exI) auto
     qed
   qed
   then obtain f where f:
     "\<forall>x.
       \<not> P (cbox (fst x) (snd x)) \<longrightarrow>
       \<not> P (cbox (fst (f x)) (snd (f x))) \<and>
         (\<forall>i\<in>Basis.
             fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and>
             fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and>
             snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and>
             2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)" by metis
   define AB A B where ab_def: "AB n = (f ^^ n) (a,b)" "A n = fst(AB n)" "B n = snd(AB n)" for n
   have [simp]: "A 0 = a" "B 0 = b" and ABRAW: "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and>
     (\<forall>i\<in>Basis. A(n)\<bullet>i \<le> A(Suc n)\<bullet>i \<and> A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i \<and> B(Suc n)\<bullet>i \<le> B(n)\<bullet>i \<and>
     2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i)" (is "\<And>n. ?P n")
   proof -
     show "A 0 = a" "B 0 = b"
       unfolding ab_def by auto
     note S = ab_def funpow.simps o_def id_apply
     show "?P n" for n
     proof (induct n)
       case 0
       then show ?case
         unfolding S using \<open>\<not> P (cbox a b)\<close> f by auto
     next
       case (Suc n)
       show ?case
         unfolding S
         apply (rule f[rule_format])
         using Suc
         unfolding S
         apply auto
         done
     qed
   qed
   then have AB: "A(n)\<bullet>i \<le> A(Suc n)\<bullet>i" "A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i" 
                  "B(Suc n)\<bullet>i \<le> B(n)\<bullet>i" "2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i" 
                 if "i\<in>Basis" for i n
     using that by blast+
   have notPAB: "\<not> P (cbox (A(Suc n)) (B(Suc n)))" for n
     using ABRAW by blast
   have interv: "\<exists>n. \<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e"
     if e: "0 < e" for e
   proof -
     obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
       using real_arch_pow[of 2 "(sum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] by auto
     show ?thesis
     proof (rule exI [where x=n], clarify)
       fix x y
       assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
       have "dist x y \<le> sum (\<lambda>i. \<bar>(x - y)\<bullet>i\<bar>) Basis"
         unfolding dist_norm by(rule norm_le_l1)
       also have "\<dots> \<le> sum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
       proof (rule sum_mono)
         fix i :: 'a
         assume i: "i \<in> Basis"
         show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
           using xy[unfolded mem_box,THEN bspec, OF i]
           by (auto simp: inner_diff_left)
       qed
       also have "\<dots> \<le> sum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
         unfolding sum_divide_distrib
       proof (rule sum_mono)
         show "B n \<bullet> i - A n \<bullet> i \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ n" if i: "i \<in> Basis" for i
         proof (induct n)
           case 0
           then show ?case
             unfolding AB by auto
         next
           case (Suc n)
           have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
             using AB(3) that AB(4)[of i n] using i by auto
           also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
             using Suc by (auto simp add: field_simps)
           finally show ?case .
         qed
       qed
       also have "\<dots> < e"
         using n using e by (auto simp add: field_simps)
       finally show "dist x y < e" .
     qed
   qed
   {
     fix n m :: nat
     assume "m \<le> n" then have "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)"
     proof (induction rule: inc_induct)
       case (step i)
       show ?case
         using AB by (intro order_trans[OF step.IH] subset_box_imp) auto
     qed simp
   } note ABsubset = this
   have "\<And>n. cbox (A n) (B n) \<noteq> {}"
     by (meson AB dual_order.trans interval_not_empty)
   then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
     using decreasing_closed_nest [OF closed_cbox] ABsubset interv by blast
   show thesis
   proof (rule that[rule_format, of x0])
     show "x0\<in>cbox a b"
       using \<open>A 0 = a\<close> \<open>B 0 = b\<close> x0 by blast
     fix e :: real
     assume "e > 0"
     from interv[OF this] obtain n
       where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" ..
     have "\<not> P (cbox (A n) (B n))"
     proof (cases "0 < n")
       case True then show ?thesis 
         by (metis Suc_pred' notPAB) 
     next
       case False then show ?thesis
         using \<open>A 0 = a\<close> \<open>B 0 = b\<close> \<open>\<not> P (cbox a b)\<close> by blast
     qed
     moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
       using n using x0[of n] by auto
     moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
       using ABsubset \<open>A 0 = a\<close> \<open>B 0 = b\<close> by blast
     ultimately show "\<exists>c d. x0 \<in> cbox c d \<and> cbox c d \<subseteq> ball x0 e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
       apply (rule_tac x="A n" in exI)
       apply (rule_tac x="B n" in exI)
       apply (auto simp: x0)
       done
   qed
 qed
 
 
 subsection \<open>Cousin's lemma\<close>
 
 lemma fine_division_exists: (*FIXME rename(?) *)
   fixes a b :: "'a::euclidean_space"
   assumes "gauge g"
   obtains p where "p tagged_division_of (cbox a b)" "g fine p"
 proof (cases "\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p")
   case True
   then show ?thesis
     using that by auto
 next
   case False
   assume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
   obtain x where x:
       "x \<in> (cbox a b)"
       "\<And>e. 0 < e \<Longrightarrow>
         \<exists>c d.
           x \<in> cbox c d \<and>
           cbox c d \<subseteq> ball x e \<and>
           cbox c d \<subseteq> (cbox a b) \<and>
           \<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
     apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ False])
     apply (simp add: fine_def)
     apply (metis tagged_division_Un fine_Un)
     apply auto
     done
   obtain e where e: "e > 0" "ball x e \<subseteq> g x"
     using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
   from x(2)[OF e(1)]
   obtain c d where c_d: "x \<in> cbox c d"
                         "cbox c d \<subseteq> ball x e"
                         "cbox c d \<subseteq> cbox a b"
                         "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
     by blast
   have "g fine {(x, cbox c d)}"
     unfolding fine_def using e using c_d(2) by auto
   then show ?thesis
     using tagged_division_of_self[OF c_d(1)] using c_d by auto
 qed
 
 lemma fine_division_exists_real:
   fixes a b :: real
   assumes "gauge g"
   obtains p where "p tagged_division_of {a .. b}" "g fine p"
   by (metis assms box_real(2) fine_division_exists)
 
 subsection \<open>A technical lemma about "refinement" of division\<close>
 
 lemma tagged_division_finer:
   fixes p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set"
   assumes ptag: "p tagged_division_of (cbox a b)"
     and "gauge d"
   obtains q where "q tagged_division_of (cbox a b)"
     and "d fine q"
     and "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q"
 proof -
   have p: "finite p" "p tagged_partial_division_of (cbox a b)"
-    using ptag unfolding tagged_division_of_def by auto
+    using ptag tagged_division_of_def by blast+
   have "(\<exists>q. q tagged_division_of (\<Union>{k. \<exists>x. (x,k) \<in> p}) \<and> d fine q \<and> (\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q))" 
     if "finite p" "p tagged_partial_division_of (cbox a b)" "gauge d" for p
     using that
   proof (induct p)
     case empty
     show ?case
       by (force simp add: fine_def)
   next
     case (insert xk p)
     obtain x k where xk: "xk = (x, k)"
       using surj_pair[of xk] by metis 
     obtain q1 where q1: "q1 tagged_division_of \<Union>{k. \<exists>x. (x, k) \<in> p}"
                 and "d fine q1"
                 and q1I: "\<And>x k. \<lbrakk>(x, k)\<in>p;  k \<subseteq> d x\<rbrakk> \<Longrightarrow> (x, k) \<in> q1"
       using case_prodD tagged_partial_division_subset[OF insert(4) subset_insertI]
       by (metis (mono_tags, lifting) insert.hyps(3) insert.prems(2))
     have *: "\<Union>{l. \<exists>y. (y,l) \<in> insert xk p} = k \<union> \<Union>{l. \<exists>y. (y,l) \<in> p}"
       unfolding xk by auto
     note p = tagged_partial_division_ofD[OF insert(4)]
     obtain u v where uv: "k = cbox u v"
       using p(4) xk by blast
     have "finite {k. \<exists>x. (x, k) \<in> p}"
       apply (rule finite_subset[of _ "snd ` p"])
       using image_iff apply fastforce
       using insert.hyps(1) by blast
     then have int: "interior (cbox u v) \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
     proof (rule Int_interior_Union_intervals)
       show "open (interior (cbox u v))"
         by auto
       show "\<And>T. T \<in> {k. \<exists>x. (x, k) \<in> p} \<Longrightarrow> \<exists>a b. T = cbox a b"
         using p(4) by auto
       show "\<And>T. T \<in> {k. \<exists>x. (x, k) \<in> p} \<Longrightarrow> interior (cbox u v) \<inter> interior T = {}"
         by clarify (metis insert.hyps(2) insert_iff interior_cbox p(5) uv xk)
     qed
     show ?case
     proof (cases "cbox u v \<subseteq> d x")
       case True
       have "{(x, cbox u v)} tagged_division_of cbox u v"
         by (simp add: p(2) uv xk tagged_division_of_self)
       then have "{(x, cbox u v)} \<union> q1 tagged_division_of \<Union>{k. \<exists>x. (x, k) \<in> insert xk p}"
         unfolding * uv by (metis (no_types, lifting) int q1 tagged_division_Un)
       with True show ?thesis
         apply (rule_tac x="{(x,cbox u v)} \<union> q1" in exI)
         using \<open>d fine q1\<close> fine_def q1I uv xk apply fastforce
         done
     next
       case False
       obtain q2 where q2: "q2 tagged_division_of cbox u v" "d fine q2"
         using fine_division_exists[OF assms(2)] by blast
       show ?thesis
         apply (rule_tac x="q2 \<union> q1" in exI)
         apply (intro conjI)
         unfolding * uv
         apply (rule tagged_division_Un q2 q1 int fine_Un)+
           apply (auto intro: q1 q2 fine_Un \<open>d fine q1\<close> simp add: False q1I uv xk)
         done
     qed
   qed
   with p obtain q where q: "q tagged_division_of \<Union>{k. \<exists>x. (x, k) \<in> p}" "d fine q" "\<forall>(x, k)\<in>p. k \<subseteq> d x \<longrightarrow> (x, k) \<in> q"
     by (meson \<open>gauge d\<close>)
   with ptag that show ?thesis by auto
 qed
 
 subsubsection\<^marker>\<open>tag important\<close> \<open>Covering lemma\<close>
 
 text\<open> Some technical lemmas used in the approximation results that follow. Proof of the covering
   lemma is an obvious multidimensional generalization of Lemma 3, p65 of Swartz's
   "Introduction to Gauge Integrals". \<close>
 
 proposition covering_lemma:
   assumes "S \<subseteq> cbox a b" "box a b \<noteq> {}" "gauge g"
   obtains \<D> where
     "countable \<D>"  "\<Union>\<D> \<subseteq> cbox a b"
     "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
     "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"
     "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> g x"
     "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n"
     "S \<subseteq> \<Union>\<D>"
 proof -
   have aibi: "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i" and normab: "0 < norm(b - a)"
     using \<open>box a b \<noteq> {}\<close> box_eq_empty box_sing by fastforce+
   let ?K0 = "\<lambda>(n, f::'a\<Rightarrow>nat).
                     cbox (\<Sum>i \<in> Basis. (a \<bullet> i + (f i / 2^n) * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i)
                          (\<Sum>i \<in> Basis. (a \<bullet> i + ((f i + 1) / 2^n) * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i)"
   let ?D0 = "?K0 ` (SIGMA n:UNIV. Pi\<^sub>E Basis (\<lambda>i::'a. lessThan (2^n)))"
   obtain \<D>0 where count: "countable \<D>0"
              and sub: "\<Union>\<D>0 \<subseteq> cbox a b"
              and int:  "\<And>K. K \<in> \<D>0 \<Longrightarrow> (interior K \<noteq> {}) \<and> (\<exists>c d. K = cbox c d)"
              and intdj: "\<And>A B. \<lbrakk>A \<in> \<D>0; B \<in> \<D>0\<rbrakk> \<Longrightarrow> A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}"
              and SK: "\<And>x. x \<in> S \<Longrightarrow> \<exists>K \<in> \<D>0. x \<in> K \<and> K \<subseteq> g x"
              and cbox: "\<And>u v. cbox u v \<in> \<D>0 \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n"
              and fin: "\<And>K. K \<in> \<D>0 \<Longrightarrow> finite {L \<in> \<D>0. K \<subseteq> L}"
   proof
     show "countable ?D0"
       by (simp add: countable_PiE)
   next
     show "\<Union>?D0 \<subseteq> cbox a b"
       apply (simp add: UN_subset_iff)
       apply (intro conjI allI ballI subset_box_imp)
        apply (simp add: field_simps)
       apply (auto intro: mult_right_mono aibi)
       apply (force simp: aibi scaling_mono nat_less_real_le dest: PiE_mem intro: mult_right_mono)
       done
   next
     show "\<And>K. K \<in> ?D0 \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
       using \<open>box a b \<noteq> {}\<close>
       by (clarsimp simp: box_eq_empty) (fastforce simp add: field_split_simps dest: PiE_mem)
   next
     have realff: "(real w) * 2^m < (real v) * 2^n \<longleftrightarrow> w * 2^m < v * 2^n" for m n v w
       using of_nat_less_iff less_imp_of_nat_less by fastforce
     have *: "\<forall>v w. ?K0(m,v) \<subseteq> ?K0(n,w) \<or> ?K0(n,w) \<subseteq> ?K0(m,v) \<or> interior(?K0(m,v)) \<inter> interior(?K0(n,w)) = {}"
       for m n \<comment> \<open>The symmetry argument requires a single HOL formula\<close>
     proof (rule linorder_wlog [where a=m and b=n], intro allI impI)
       fix v w m and n::nat
       assume "m \<le> n" \<comment> \<open>WLOG we can assume \<^term>\<open>m \<le> n\<close>, when the first disjunct becomes impossible\<close>
       have "?K0(n,w) \<subseteq> ?K0(m,v) \<or> interior(?K0(m,v)) \<inter> interior(?K0(n,w)) = {}"
         apply (simp add: subset_box disjoint_interval)
         apply (rule ccontr)
         apply (clarsimp simp add: aibi mult_le_cancel_right divide_le_cancel not_less not_le)
         apply (drule_tac x=i in bspec, assumption)
         using \<open>m\<le>n\<close> realff [of _ _ "1+_"] realff [of "1+_"_ "1+_"]
         apply (auto simp: divide_simps add.commute not_le nat_le_iff_add realff)
          apply (simp_all add: power_add)
         apply (metis (no_types, hide_lams) mult_Suc mult_less_cancel2 not_less_eq mult.assoc)
         apply (metis (no_types, hide_lams) mult_Suc mult_less_cancel2 not_less_eq mult.assoc)
         done
       then show "?K0(m,v) \<subseteq> ?K0(n,w) \<or> ?K0(n,w) \<subseteq> ?K0(m,v) \<or> interior(?K0(m,v)) \<inter> interior(?K0(n,w)) = {}"
         by meson
     qed auto
     show "\<And>A B. \<lbrakk>A \<in> ?D0; B \<in> ?D0\<rbrakk> \<Longrightarrow> A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}"
       apply (erule imageE SigmaE)+
       using * by simp
   next
     show "\<exists>K \<in> ?D0. x \<in> K \<and> K \<subseteq> g x" if "x \<in> S" for x
     proof (simp only: bex_simps split_paired_Bex_Sigma)
       show "\<exists>n. \<exists>f \<in> Basis \<rightarrow>\<^sub>E {..<2 ^ n}. x \<in> ?K0(n,f) \<and> ?K0(n,f) \<subseteq> g x"
       proof -
         obtain e where "0 < e"
                    and e: "\<And>y. (\<And>i. i \<in> Basis \<Longrightarrow> \<bar>x \<bullet> i - y \<bullet> i\<bar> \<le> e) \<Longrightarrow> y \<in> g x"
         proof -
           have "x \<in> g x" "open (g x)"
             using \<open>gauge g\<close> by (auto simp: gauge_def)
           then obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball x \<epsilon> \<subseteq> g x"
             using openE by blast
           have "norm (x - y) < \<epsilon>"
                if "(\<And>i. i \<in> Basis \<Longrightarrow> \<bar>x \<bullet> i - y \<bullet> i\<bar> \<le> \<epsilon> / (2 * real DIM('a)))" for y
           proof -
             have "norm (x - y) \<le> (\<Sum>i\<in>Basis. \<bar>x \<bullet> i - y \<bullet> i\<bar>)"
               by (metis (no_types, lifting) inner_diff_left norm_le_l1 sum.cong)
             also have "... \<le> DIM('a) * (\<epsilon> / (2 * real DIM('a)))"
               by (meson sum_bounded_above that)
             also have "... = \<epsilon> / 2"
               by (simp add: field_split_simps)
             also have "... < \<epsilon>"
               by (simp add: \<open>0 < \<epsilon>\<close>)
             finally show ?thesis .
           qed
           then show ?thesis
             by (rule_tac e = "\<epsilon> / 2 / DIM('a)" in that) (simp_all add:  \<open>0 < \<epsilon>\<close> dist_norm subsetD [OF \<epsilon>])
         qed
         have xab: "x \<in> cbox a b"
           using \<open>x \<in> S\<close> \<open>S \<subseteq> cbox a b\<close> by blast
         obtain n where n: "norm (b - a) / 2^n < e"
           using real_arch_pow_inv [of "e / norm(b - a)" "1/2"] normab \<open>0 < e\<close>
           by (auto simp: field_split_simps)
         then have "norm (b - a) < e * 2^n"
           by (auto simp: field_split_simps)
         then have bai: "b \<bullet> i - a \<bullet> i < e * 2 ^ n" if "i \<in> Basis" for i
         proof -
           have "b \<bullet> i - a \<bullet> i \<le> norm (b - a)"
             by (metis abs_of_nonneg dual_order.trans inner_diff_left linear norm_ge_zero Basis_le_norm that)
           also have "... < e * 2 ^ n"
             using \<open>norm (b - a) < e * 2 ^ n\<close> by blast
           finally show ?thesis .
         qed
         have D: "(a + n \<le> x \<and> x \<le> a + m) \<Longrightarrow> (a + n \<le> y \<and> y \<le> a + m) \<Longrightarrow> abs(x - y) \<le> m - n"
                  for a m n x and y::real
           by auto
         have "\<forall>i\<in>Basis. \<exists>k<2 ^ n. (a \<bullet> i + real k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le> x \<bullet> i \<and>
                x \<bullet> i \<le> a \<bullet> i + (real k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n)"
         proof
           fix i::'a assume "i \<in> Basis"
           consider "x \<bullet> i = b \<bullet> i" | "x \<bullet> i < b \<bullet> i"
             using \<open>i \<in> Basis\<close> mem_box(2) xab by force
           then show "\<exists>k<2 ^ n. (a \<bullet> i + real k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le> x \<bullet> i \<and>
                           x \<bullet> i \<le> a \<bullet> i + (real k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n)"
           proof cases
             case 1 then show ?thesis
               by (rule_tac x = "2^n - 1" in exI) (auto simp: algebra_simps field_split_simps of_nat_diff \<open>i \<in> Basis\<close> aibi)
           next
             case 2
             then have abi_less: "a \<bullet> i < b \<bullet> i"
               using \<open>i \<in> Basis\<close> xab by (auto simp: mem_box)
             let ?k = "nat \<lfloor>2 ^ n * (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)\<rfloor>"
             show ?thesis
             proof (intro exI conjI)
               show "?k < 2 ^ n"
                 using aibi xab \<open>i \<in> Basis\<close>
                 by (force simp: nat_less_iff floor_less_iff field_split_simps 2 mem_box)
             next
               have "a \<bullet> i + real ?k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le>
                     a \<bullet> i + (2 ^ n * (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (b \<bullet> i - a \<bullet> i) / 2 ^ n"
                 apply (intro add_left_mono mult_right_mono divide_right_mono of_nat_floor)
                 using aibi [OF \<open>i \<in> Basis\<close>] xab 2
                   apply (simp_all add: \<open>i \<in> Basis\<close> mem_box field_split_simps)
                 done
               also have "... = x \<bullet> i"
                 using abi_less by (simp add: field_split_simps)
               finally show "a \<bullet> i + real ?k * (b \<bullet> i - a \<bullet> i) / 2 ^ n \<le> x \<bullet> i" .
             next
               have "x \<bullet> i \<le> a \<bullet> i + (2 ^ n * (x \<bullet> i - a \<bullet> i) / (b \<bullet> i - a \<bullet> i)) * (b \<bullet> i - a \<bullet> i) / 2 ^ n"
                 using abi_less by (simp add: field_split_simps)
               also have "... \<le> a \<bullet> i + (real ?k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n"
                 apply (intro add_left_mono mult_right_mono divide_right_mono of_nat_floor)
                 using aibi [OF \<open>i \<in> Basis\<close>] xab
                   apply (auto simp: \<open>i \<in> Basis\<close> mem_box divide_simps)
                 done
               finally show "x \<bullet> i \<le> a \<bullet> i + (real ?k + 1) * (b \<bullet> i - a \<bullet> i) / 2 ^ n" .
             qed
           qed
         qed
         then have "\<exists>f\<in>Basis \<rightarrow>\<^sub>E {..<2 ^ n}. x \<in> ?K0(n,f)"
           apply (simp add: mem_box Bex_def)
           apply (clarify dest!: bchoice)
           apply (rule_tac x="restrict f Basis" in exI, simp)
           done
         moreover have "\<And>f. x \<in> ?K0(n,f) \<Longrightarrow> ?K0(n,f) \<subseteq> g x"
           apply (clarsimp simp add: mem_box)
           apply (rule e)
           apply (drule bspec D, assumption)+
           apply (erule order_trans)
           apply (simp add: divide_simps)
           using bai apply (force simp add: algebra_simps)
           done
         ultimately show ?thesis by auto
       qed
     qed
   next
     show "\<And>u v. cbox u v \<in> ?D0 \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n"
       by (force simp: eq_cbox box_eq_empty field_simps dest!: aibi)
   next
     obtain j::'a where "j \<in> Basis"
       using nonempty_Basis by blast
     have "finite {L \<in> ?D0. ?K0(n,f) \<subseteq> L}" if "f \<in> Basis \<rightarrow>\<^sub>E {..<2 ^ n}" for n f
     proof (rule finite_subset)
       let ?B = "(\<lambda>(n, f::'a\<Rightarrow>nat). cbox (\<Sum>i\<in>Basis. (a \<bullet> i + (f i) / 2^n * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i)
                                         (\<Sum>i\<in>Basis. (a \<bullet> i + ((f i) + 1) / 2^n * (b \<bullet> i - a \<bullet> i)) *\<^sub>R i))
                 ` (SIGMA m:atMost n. Pi\<^sub>E Basis (\<lambda>i::'a. lessThan (2^m)))"
       have "?K0(m,g) \<in> ?B" if "g \<in> Basis \<rightarrow>\<^sub>E {..<2 ^ m}" "?K0(n,f) \<subseteq> ?K0(m,g)" for m g
       proof -
         have dd: "w / m \<le> v / n \<and> (v+1) / n \<le> (w+1) / m
                   \<Longrightarrow> inverse n \<le> inverse m" for w m v n::real
           by (auto simp: field_split_simps)
         have bjaj: "b \<bullet> j - a \<bullet> j > 0"
           using \<open>j \<in> Basis\<close> \<open>box a b \<noteq> {}\<close> box_eq_empty(1) by fastforce
         have "((g j) / 2 ^ m) * (b \<bullet> j - a \<bullet> j) \<le> ((f j) / 2 ^ n) * (b \<bullet> j - a \<bullet> j) \<and>
               (((f j) + 1) / 2 ^ n) * (b \<bullet> j - a \<bullet> j) \<le> (((g j) + 1) / 2 ^ m) * (b \<bullet> j - a \<bullet> j)"
           using that \<open>j \<in> Basis\<close> by (simp add: subset_box field_split_simps aibi)
         then have "((g j) / 2 ^ m) \<le> ((f j) / 2 ^ n) \<and>
           ((real(f j) + 1) / 2 ^ n) \<le> ((real(g j) + 1) / 2 ^ m)"
           by (metis bjaj mult.commute of_nat_1 of_nat_add real_mult_le_cancel_iff2)
         then have "inverse (2^n) \<le> (inverse (2^m) :: real)"
           by (rule dd)
         then have "m \<le> n"
           by auto
         show ?thesis
           by (rule imageI) (simp add: \<open>m \<le> n\<close> that)
       qed
       then show "{L \<in> ?D0. ?K0(n,f) \<subseteq> L} \<subseteq> ?B"
         by auto
       show "finite ?B"
         by (intro finite_imageI finite_SigmaI finite_atMost finite_lessThan finite_PiE finite_Basis)
     qed
     then show "finite {L \<in> ?D0. K \<subseteq> L}" if "K \<in> ?D0" for K
       using that by auto
   qed
   let ?D1 = "{K \<in> \<D>0. \<exists>x \<in> S \<inter> K. K \<subseteq> g x}"
   obtain \<D> where count: "countable \<D>"
              and sub: "\<Union>\<D> \<subseteq> cbox a b"  "S \<subseteq> \<Union>\<D>"
              and int:  "\<And>K. K \<in> \<D> \<Longrightarrow> (interior K \<noteq> {}) \<and> (\<exists>c d. K = cbox c d)"
              and intdj: "\<And>A B. \<lbrakk>A \<in> \<D>; B \<in> \<D>\<rbrakk> \<Longrightarrow> A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}"
              and SK: "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x. x \<in> S \<inter> K \<and> K \<subseteq> g x"
              and cbox: "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n"
              and fin: "\<And>K. K \<in> \<D> \<Longrightarrow> finite {L. L \<in> \<D> \<and> K \<subseteq> L}"
   proof
     show "countable ?D1" using count countable_subset
       by (simp add: count countable_subset)
     show "\<Union>?D1 \<subseteq> cbox a b"
       using sub by blast
     show "S \<subseteq> \<Union>?D1"
       using SK by (force simp:)
     show "\<And>K. K \<in> ?D1 \<Longrightarrow> (interior K \<noteq> {}) \<and> (\<exists>c d. K = cbox c d)"
       using int by blast
     show "\<And>A B. \<lbrakk>A \<in> ?D1; B \<in> ?D1\<rbrakk> \<Longrightarrow> A \<subseteq> B \<or> B \<subseteq> A \<or> interior A \<inter> interior B = {}"
       using intdj by blast
     show "\<And>K. K \<in> ?D1 \<Longrightarrow> \<exists>x. x \<in> S \<inter> K \<and> K \<subseteq> g x"
       by auto
     show "\<And>u v. cbox u v \<in> ?D1 \<Longrightarrow> \<exists>n. \<forall>i \<in> Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n"
       using cbox by blast
     show "\<And>K. K \<in> ?D1 \<Longrightarrow> finite {L. L \<in> ?D1 \<and> K \<subseteq> L}"
       using fin by simp (metis (mono_tags, lifting) Collect_mono rev_finite_subset)
   qed
   let ?\<D> = "{K \<in> \<D>. \<forall>K'. K' \<in> \<D> \<and> K \<noteq> K' \<longrightarrow> \<not>(K \<subseteq> K')}"
   show ?thesis
   proof (rule that)
     show "countable ?\<D>"
       by (blast intro: countable_subset [OF _ count])
     show "\<Union>?\<D> \<subseteq> cbox a b"
       using sub by blast
     show "S \<subseteq> \<Union>?\<D>"
     proof clarsimp
       fix x
       assume "x \<in> S"
       then obtain X where "x \<in> X" "X \<in> \<D>" using \<open>S \<subseteq> \<Union>\<D>\<close> by blast
       let ?R = "{(K,L). K \<in> \<D> \<and> L \<in> \<D> \<and> L \<subset> K}"
       have irrR: "irrefl ?R" by (force simp: irrefl_def)
       have traR: "trans ?R" by (force simp: trans_def)
       have finR: "\<And>x. finite {y. (y, x) \<in> ?R}"
         by simp (metis (mono_tags, lifting) fin \<open>X \<in> \<D>\<close> finite_subset mem_Collect_eq psubset_imp_subset subsetI)
       have "{X \<in> \<D>. x \<in> X} \<noteq> {}"
         using \<open>X \<in> \<D>\<close> \<open>x \<in> X\<close> by blast
       then obtain Y where "Y \<in> {X \<in> \<D>. x \<in> X}" "\<And>Y'. (Y', Y) \<in> ?R \<Longrightarrow> Y' \<notin> {X \<in> \<D>. x \<in> X}"
         by (rule wfE_min' [OF wf_finite_segments [OF irrR traR finR]]) blast
       then show "\<exists>Y. Y \<in> \<D> \<and> (\<forall>K'. K' \<in> \<D> \<and> Y \<noteq> K' \<longrightarrow> \<not> Y \<subseteq> K') \<and> x \<in> Y"
         by blast
     qed
     show "\<And>K. K \<in> ?\<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"
       by (blast intro: dest: int)
     show "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) ?\<D>"
       using intdj by (simp add: pairwise_def) metis
     show "\<And>K. K \<in> ?\<D> \<Longrightarrow> \<exists>x \<in> S \<inter> K. K \<subseteq> g x"
       using SK by force
     show "\<And>u v. cbox u v \<in> ?\<D> \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2^n"
       using cbox by force
     qed
 qed
 
 subsection \<open>Division filter\<close>
 
 text \<open>Divisions over all gauges towards finer divisions.\<close>
 
 definition\<^marker>\<open>tag important\<close> division_filter :: "'a::euclidean_space set \<Rightarrow> ('a \<times> 'a set) set filter"
   where "division_filter s = (INF g\<in>{g. gauge g}. principal {p. p tagged_division_of s \<and> g fine p})"
 
 proposition eventually_division_filter:
   "(\<forall>\<^sub>F p in division_filter s. P p) \<longleftrightarrow>
     (\<exists>g. gauge g \<and> (\<forall>p. p tagged_division_of s \<and> g fine p \<longrightarrow> P p))"
   unfolding division_filter_def
 proof (subst eventually_INF_base; clarsimp)
   fix g1 g2 :: "'a \<Rightarrow> 'a set" show "gauge g1 \<Longrightarrow> gauge g2 \<Longrightarrow> \<exists>x. gauge x \<and>
     {p. p tagged_division_of s \<and> x fine p} \<subseteq> {p. p tagged_division_of s \<and> g1 fine p} \<and>
     {p. p tagged_division_of s \<and> x fine p} \<subseteq> {p. p tagged_division_of s \<and> g2 fine p}"
     by (intro exI[of _ "\<lambda>x. g1 x \<inter> g2 x"]) (auto simp: fine_Int)
 qed (auto simp: eventually_principal)
 
 lemma division_filter_not_empty: "division_filter (cbox a b) \<noteq> bot"
   unfolding trivial_limit_def eventually_division_filter
   by (auto elim: fine_division_exists)
 
 lemma eventually_division_filter_tagged_division:
   "eventually (\<lambda>p. p tagged_division_of s) (division_filter s)"
   unfolding eventually_division_filter by (intro exI[of _ "\<lambda>x. ball x 1"]) auto
 
 end