# HG changeset patch
# User hoelzl
# Date 1475146963 -7200
# Node ID c3da799b1b456d7b07983130082270726143e3e5
# Parent  b235e845c8e8de31d9bd93544f9f7d9fbaf6f8a3
HOL-Analysis: move gauges and (tagged) divisions to its own theory file

diff -r b235e845c8e8 -r c3da799b1b45 src/HOL/Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Analysis/Convex_Euclidean_Space.thy	Wed Sep 28 16:15:51 2016 +0200
+++ b/src/HOL/Analysis/Convex_Euclidean_Space.thy	Thu Sep 29 13:02:43 2016 +0200
@@ -6417,6 +6417,26 @@
   unfolding segment_convex_hull
   by (auto intro!: hull_subset[unfolded subset_eq, rule_format])
 
+lemma eventually_closed_segment:
+  fixes x0::"'a::real_normed_vector"
+  assumes "open X0" "x0 \<in> X0"
+  shows "\<forall>\<^sub>F x in at x0 within U. closed_segment x0 x \<subseteq> X0"
+proof -
+  from openE[OF assms]
+  obtain e where e: "0 < e" "ball x0 e \<subseteq> X0" .
+  then have "\<forall>\<^sub>F x in at x0 within U. x \<in> ball x0 e"
+    by (auto simp: dist_commute eventually_at)
+  then show ?thesis
+  proof eventually_elim
+    case (elim x)
+    have "x0 \<in> ball x0 e" using \<open>e > 0\<close> by simp
+    from convex_ball[unfolded convex_contains_segment, rule_format, OF this elim]
+    have "closed_segment x0 x \<subseteq> ball x0 e" .
+    also note \<open>\<dots> \<subseteq> X0\<close>
+    finally show ?case .
+  qed
+qed
+
 lemma segment_furthest_le:
   fixes a b x y :: "'a::euclidean_space"
   assumes "x \<in> closed_segment a b"
@@ -10500,7 +10520,7 @@
 
 lemma collinear_3_expand:
    "collinear{a,b,c} \<longleftrightarrow> a = c \<or> (\<exists>u. b = u *\<^sub>R a + (1 - u) *\<^sub>R c)"
-proof -                          
+proof -
   have "collinear{a,b,c} = collinear{a,c,b}"
     by (simp add: insert_commute)
   also have "... = collinear {0, a - c, b - c}"
diff -r b235e845c8e8 -r c3da799b1b45 src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy
--- a/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Wed Sep 28 16:15:51 2016 +0200
+++ b/src/HOL/Analysis/Equivalence_Lebesgue_Henstock_Integration.thy	Thu Sep 29 13:02:43 2016 +0200
@@ -1209,8 +1209,8 @@
       have sum_p': "(\<Sum>(x, k)\<in>p'. norm (integral k f)) = (\<Sum>k\<in>snd ` p'. norm (integral k f))"
         apply (subst setsum.over_tagged_division_lemma[OF p'',of "\<lambda>k. norm (integral k f)"])
         unfolding norm_eq_zero
-        apply (rule integral_null)
-        apply assumption
+         apply (rule integral_null)
+        apply (simp add: content_eq_0_interior)
         apply rule
         done
       note snd_p = division_ofD[OF division_of_tagged_division[OF p(1)]]
@@ -1657,7 +1657,7 @@
           show "(\<Sum>(x, k)\<in>p. norm (integral k f)) \<le> ?S"
             using partial_division_of_tagged_division[of p "cbox a b"] p(1)
             apply (subst setsum.over_tagged_division_lemma[OF p(1)])
-            apply (simp add: integral_null)
+            apply (simp add: content_eq_0_interior)
             apply (intro cSUP_upper2[OF D(2), of "snd ` p"])
             apply (auto simp: tagged_partial_division_of_def)
             done
diff -r b235e845c8e8 -r c3da799b1b45 src/HOL/Analysis/Henstock_Kurzweil_Integration.thy
--- a/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Wed Sep 28 16:15:51 2016 +0200
+++ b/src/HOL/Analysis/Henstock_Kurzweil_Integration.thy	Thu Sep 29 13:02:43 2016 +0200
@@ -6,201 +6,49 @@
 
 theory Henstock_Kurzweil_Integration
 imports
-  Lebesgue_Measure
+  Lebesgue_Measure Tagged_Division
 begin
 
-lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
-  scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
-  scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one
-
-
-subsection \<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 u} and @{term v}.\<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"
+(* BEGIN MOVE *)
+lemma swap_continuous:
+  assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
+    shows "continuous_on (cbox (c,a) (d,b)) (\<lambda>(x, y). f y x)"
 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>)
+  have "(\<lambda>(x, y). f y x) = (\<lambda>(x, y). f x y) \<circ> prod.swap"
+    by auto
   then show ?thesis
-    by (simp add: field_simps)
+    apply (rule ssubst)
+    apply (rule continuous_on_compose)
+    apply (simp add: split_def)
+    apply (rule continuous_intros | simp add: assms)+
+    done
 qed
 
-lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
-lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
-lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
-
-lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
+
+lemma norm_minus2: "norm (x1-x2, y1-y2) = norm (x2-x1, y2-y1)"
+  by (simp add: norm_minus_eqI)
+
+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
 
-declare norm_triangle_ineq4[intro]
-
-lemma transitive_stepwise_le:
-  assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" and "\<And>n. R n (Suc n)"
-  shows "\<forall>n\<ge>m. R m n"
-proof (intro allI impI)
-  show "m \<le> n \<Longrightarrow> R m n" for n
-    by (induction rule: dec_induct)
-       (use assms in blast)+
-qed
-
-subsection \<open>Some useful lemmas about intervals.\<close>
+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
 
 lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
   using nonempty_Basis
   by (fastforce simp add: set_eq_iff mem_box)
-
-lemma interior_subset_union_intervals:
-  assumes "i = cbox a b"
-    and "j = cbox c d"
-    and "interior j \<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 inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
-     unfolding assms(1,2) interior_cbox by auto
-  moreover
-  have "box a b \<subseteq> cbox c d \<union> s"
-    apply (rule order_trans,rule box_subset_cbox)
-    using assms(4) unfolding assms(1,2)
-    apply auto
-    done
-  ultimately
-  show ?thesis
-    unfolding assms interior_cbox
-      by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI)
-qed
-
-lemma 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 add: closure_box)
-  qed
-  finally show ?thesis .
-qed
-
-lemma inter_interior_unions_intervals:
-    "finite f \<Longrightarrow> open s \<Longrightarrow> \<forall>t\<in>f. \<exists>a b. t = cbox a b \<Longrightarrow> \<forall>t\<in>f. s \<inter> (interior t) = {} \<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"
-  apply (rule_tac[!] set_eqI)
-  unfolding Int_iff mem_box mem_Collect_eq
-  using assms
-  apply auto
-  done
-
-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 interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
-  where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)"
-
-definition interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
-  where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x: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_setsum 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_setsum cbox_def
-  by (safe intro!: cInf_eq) auto
-
-lemmas interval_bounds = interval_upperbound interval_lowerbound
-
-lemma
-  fixes X::"real set"
-  shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
-    and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
-  by (auto simp: interval_upperbound_def interval_lowerbound_def)
-
-lemma interval_bounds'[simp]:
-  assumes "cbox a b \<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:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:A. x \<bullet> i) *\<^sub>R i)"
-      by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
-  moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
-  have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:B. x \<bullet> i) *\<^sub>R i)"
-      by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
-  ultimately show ?thesis unfolding interval_upperbound_def
-      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
-qed
-
-lemma interval_lowerbound_Times:
-  assumes "A \<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:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:A. x \<bullet> i) *\<^sub>R i)"
-      by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
-  moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
-  have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:B. x \<bullet> i) *\<^sub>R i)"
-      by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
-  ultimately show ?thesis unfolding interval_lowerbound_def
-      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
-qed
+(* END MOVE *)
 
 subsection \<open>Content (length, area, volume...) of an interval.\<close>
 
@@ -322,1508 +170,34 @@
     by (auto simp: not_le)
 qed
 
-subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
-
-definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
-
-lemma gaugeI:
-  assumes "\<And>x. x \<in> g x"
-    and "\<And>x. open (g x)"
-  shows "gauge g"
-  using assms unfolding gauge_def by auto
-
-lemma gaugeD[dest]:
-  assumes "gauge d"
-  shows "x \<in> d x"
-    and "open (d 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_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
-  unfolding gauge_def by auto
-
-lemma gauge_inters:
-  assumes "finite s"
-    and "\<forall>d\<in>s. gauge (f d)"
-  shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> s})"
-proof -
-  have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
-    by auto
-  show ?thesis
-    unfolding gauge_def unfolding *
-    using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
-qed
-
-lemma gauge_existence_lemma:
-  "(\<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>Divisions.\<close>
-
-definition 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"
-  unfolding division_of_def 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 simp: cbox_sing)
-      done
-  }
-  ultimately show ?l
-    unfolding division_of_def cbox_sing by auto
-next
-  assume ?l
-  note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
-  {
-    fix x
-    assume x: "x \<in> s" have "x = {a}"
-      using *(2)[rule_format,OF x] by auto
-  }
-  moreover have "s \<noteq> {}"
-    using *(4) by auto
-  ultimately show ?r
-    unfolding cbox_sing by auto
-qed
-
-lemma elementary_empty: obtains p where "p division_of {}"
-  unfolding division_of_trivial by auto
-
-lemma elementary_interval: obtains p where "p division_of (cbox a b)"
-  by (metis division_of_trivial division_of_self)
-
-lemma division_contains: "s division_of i \<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 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_of_content_0:
   assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
   shows "\<forall>k\<in>d. content k = 0"
   unfolding forall_in_division[OF assms(2)]
   by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
 
-lemma division_inter:
-  fixes s1 s2 :: "'a::euclidean_space set"
-  assumes "p1 division_of s1"
-    and "p2 division_of s2"
-  shows "{k1 \<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"
-      apply (rule set_eqI)
-      unfolding * and UN_iff
-      using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
-      apply auto
-      done
-    {
-      fix k
-      assume "k \<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_inter:
-  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_inters:
-  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_inter 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)"])
-  {
-    fix i :: 'a
-    assume "i \<in> Basis"
-    with incl nonempty have "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i"
-      unfolding box_eq_empty subset_box by (auto simp: not_le)
-  }
-  note ord = this
-
-  show "p division_of (cbox a b)"
-  proof (rule division_ofI)
-    show "finite p"
-      unfolding p_def by (auto intro!: finite_PiE)
-    {
-      fix k
-      assume "k \<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 intro!: ext)
-          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 interior_cbox 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
-
-lemma 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
-  { fix x
-    assume x: "x \<in> p"
-    have "q x division_of \<Union>q x"
-      apply (rule division_ofI)
-      using division_ofD[OF q(1)[OF x]]
-      apply auto
-      done }
-  then have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
-    by (meson Diff_subset division_of_subset)
-  then have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
-    apply -
-    apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
-    apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
-    done
-  then obtain d where d: "d division_of \<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"
-      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 inter_interior_unions_intervals])
-        note qk=division_ofD[OF q(1)[OF k]]
-        show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b"
-          using qk by auto
-        show "\<forall>t\<in>q k - {k}. 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
-          apply auto
-          done
-      qed } 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 inter_interior_unions_intervals)
-      apply (rule p open_interior ballI)+
-      apply simp_all
-      done
-  qed
-  then show ?thesis
-    by (meson Un_upper2 that)
-qed
-
-lemma elementary_bounded[dest]:
-  fixes s :: "'a::euclidean_space set"
-  shows "p division_of s \<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 elementary_bounded bounded_subset_cbox)
-
-lemma 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
-  show ?thesis
-    apply (rule that[of "{}"])
-    unfolding True
-    using assms
-    apply auto
-    done
-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 "p division_of cbox c d" "cbox u v \<in> p"
-      by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
-    note p = this division_ofD[OF this(1)]
-    have "interior (cbox a b \<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 inter_interior_unions_intervals)
-      using p(6) p(7)[OF p(2)] p(3)
-      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_unions:
-  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 -
-  note assm = division_ofD[OF assms]
-  have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
-    by auto
-  have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
+lemma setsum_content_null:
+  assumes "content (cbox a b) = 0"
+    and "p tagged_division_of (cbox a b)"
+  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
+proof (rule setsum.neutral, rule)
+  fix y
+  assume y: "y \<in> p"
+  obtain x k where xk: "y = (x, k)"
+    using surj_pair[of y] by blast
+  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
+  from this(2) obtain c d where k: "k = cbox c d" by blast
+  have "(\<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 content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
+    unfolding assms(1) k
     by auto
-  {
-    presume "p = {} \<Longrightarrow> thesis"
-      "cbox a b = {} \<Longrightarrow> thesis"
-      "cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis"
-      "p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis"
-    then show thesis by auto
-  next
-    assume as: "p = {}"
-    obtain p where "p division_of (cbox a b)"
-      by (rule elementary_interval)
-    then show thesis
-      using as that by auto
-  next
-    assume as: "cbox a b = {}"
-    show thesis
-      using as assms that by auto
-  next
-    assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}"
-    show thesis
-      apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
-      unfolding finite_insert
-      apply (rule assm(1)) unfolding Union_insert
-      using assm(2-4) as
-      apply -
-      apply (fast dest: assm(5))+
-      done
-  next
-    assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}"
-    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 assm(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 as(3) 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 "*" assm(1) q(1) by auto
-      show "\<Union>?D = cbox a b \<union> \<Union>p"
-        unfolding * lem1
-        unfolding lem2[OF as(1), of "cbox a b", symmetric]
-        using q(6)
-        by auto
-      fix k
-      assume k: "k \<in> ?D"
-      then show "k \<subseteq> cbox a b \<union> \<Union>p"
-        using q(2) by auto
-      show "k \<noteq> {}"
-        using q(3) k by auto
-      show "\<exists>a b. k = cbox a b"
-        using q(4) k by auto
-      fix k'
-      assume 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'")
-        case True
-        show ?thesis
-          using True k' q(5) x' x by auto
-      next
-        case False
-        {
-          presume "k = cbox a b \<Longrightarrow> ?thesis"
-            and "k' = cbox a b \<Longrightarrow> ?thesis"
-            and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis"
-          then show ?thesis by linarith
-        next
-          assume as': "k  = cbox a b"
-          show ?thesis
-            using as' k' q(5) x' by blast
-        next
-          assume as': "k' = cbox a b"
-          show ?thesis
-            using as' k'(2) q(5) x by blast
-        }
-        assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b"
-        obtain c d where k: "k = cbox c d"
-          using q(4)[OF x(2,1)] 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"
-        using interior_subset_union_intervals
-          by (metis as(2) k q(2) x interior_subset_union_intervals)
-        moreover
-        obtain c d where c_d: "k' = cbox c d"
-          using q(4)[OF x'(2,1)] by blast
-        have "interior k' \<inter> interior (cbox a b) = {}"
-          using as'(2) q(5) x' by blast
-        then have "interior k' \<subseteq> interior x'"
-          by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
-        ultimately show ?thesis
-          using assm(5)[OF x(2) x'(2) False] by auto
-      qed
-    qed
-  }
-qed
-
-lemma elementary_unions_intervals:
-  assumes fin: "finite f"
-    and "\<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_inter)
-  {
-    fix x
-    assume x: "x \<in> t" "x \<notin> s"
-    then have "x\<in>\<Union>r1"
-      unfolding r1 using ab by auto
-    then obtain r where r: "r \<in> r1" "x \<in> r"
-      unfolding Union_iff ..
-    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 *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
-    unfolding divp divq using assms(3) by auto
-  show ?thesis
-    apply (rule that[of "p \<union> r2"])
-    unfolding *
-    defer
-    apply (rule division_disjoint_union)
-    unfolding divp(6)
-    apply(rule assms r2)+
-  proof -
-    have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
-    proof (rule inter_interior_unions_intervals)
-      show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b"
-        using r1 by auto
-      have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
-        by auto
-      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
-      proof
-        fix m x
-        assume as: "m \<in> r1 - p"
-        have "interior m \<inter> interior (\<Union>p) = {}"
-        proof (rule inter_interior_unions_intervals)
-          show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b"
-            using divp by auto
-          show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
-            by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
-        qed
-        then show "interior s \<inter> interior m = {}"
-          unfolding divp by auto
-      qed
-    qed
-    then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
-      using interior_subset by auto
-  qed auto
-qed
-
-lemma division_split_left_inj:
-  fixes type :: "'a::euclidean_space"
-  assumes "d division_of i"
-    and "k1 \<in> d"
-    and "k2 \<in> d"
-    and "k1 \<noteq> k2"
-    and "k1 \<inter> {x::'a. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
-    and k: "k\<in>Basis"
-  shows "content(k1 \<inter> {x. x\<bullet>k \<le> c}) = 0"
-proof -
-  note d=division_ofD[OF assms(1)]
-  have *: "\<And>(a::'a) b c. content (cbox a b \<inter> {x. x\<bullet>k \<le> c}) = 0 \<longleftrightarrow>
-    interior(cbox a b \<inter> {x. x\<bullet>k \<le> c}) = {}"
-    unfolding  interval_split[OF k] content_eq_0_interior by auto
-  guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
-  guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
-  have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}"
-    by auto
-  show ?thesis
-    unfolding uv1 uv2 *
-    apply (rule **[OF d(5)[OF assms(2-4)]])
-    apply (simp add: uv1)
-    using assms(5) uv1 by auto
-qed
-
-lemma division_split_right_inj:
-  fixes type :: "'a::euclidean_space"
-  assumes "d division_of i"
-    and "k1 \<in> d"
-    and "k2 \<in> d"
-    and "k1 \<noteq> k2"
-    and "k1 \<inter> {x::'a. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
-    and k: "k \<in> Basis"
-  shows "content (k1 \<inter> {x. x\<bullet>k \<ge> c}) = 0"
-proof -
-  note d=division_ofD[OF assms(1)]
-  have *: "\<And>a b::'a. \<And>c. content(cbox a b \<inter> {x. x\<bullet>k \<ge> c}) = 0 \<longleftrightarrow>
-    interior(cbox a b \<inter> {x. x\<bullet>k \<ge> c}) = {}"
-    unfolding interval_split[OF k] content_eq_0_interior by auto
-  guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
-  guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
-  have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}"
-    by auto
-  show ?thesis
-    unfolding uv1 uv2 *
-    apply (rule **[OF d(5)[OF assms(2-4)]])
-    apply (simp add: uv1)
-    using assms(5) uv1 by auto
+  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
 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 guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
-    guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
-    show "k \<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 guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
-    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 guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
-    guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
-    show "k \<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 guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
-    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 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]:
-  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 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
-  apply auto
-  apply fastforce+
-  done
-
-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
-  unfolding tagged_partial_division_of_def
-  using finite_subset[OF assms(2)]
-  by blast
-
-lemma (in comm_monoid_set) over_tagged_division_lemma:
-  assumes "p tagged_division_of i"
-    and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> content (cbox u v) = 0 \<Longrightarrow> d (cbox u v) = \<^bold>1"
-  shows "F (\<lambda>(x,k). d k) p = F d (snd ` p)"
-proof -
-  have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
-    unfolding o_def by (rule ext) auto
-  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"
-      by (metis prod.collapse \<open>x\<in>p\<close> \<open>snd x = snd y\<close> \<open>x \<noteq> y\<close>)+
-    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 "content (cbox a b) = 0"
-      unfolding \<open>snd x = snd y\<close>[symmetric] ab content_eq_0_interior 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
-
-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_union:
-  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_unions:
-  assumes "finite iset"
-    and "\<forall>i\<in>iset. pfn i tagged_division_of i"
-    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
-  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
-proof (rule tagged_division_ofI)
-  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
-  show "finite (\<Union>(pfn ` iset))"
-    using assms by auto
-  have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
-    by blast
-  also have "\<dots> = \<Union>iset"
-    using assm(6) by auto
-  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
-  fix x k
-  assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
-  then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
-    by auto
-  show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
-    using assm(2-4)[OF i] using i(1) by auto
-  fix x' k'
-  assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
-  then obtain i' where i': "i' \<in> iset" "(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)
-    using assms(3)[rule_format] interior_mono
-    by blast
-  show "interior k \<inter> interior k' = {}"
-    apply (cases "i = i'")
-    using assm(5) i' i(2) xk'(2) apply blast
-    using "*" assm(3) i' i by auto
-qed
-
-lemma tagged_partial_division_of_union_self:
-  assumes "p tagged_partial_division_of s"
-  shows "p tagged_division_of (\<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
-
-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
-  @{text operative_division}. Instances for the monoid are @{typ "'a option"}, @{typ real}, and
-  @{typ bool}.\<close>
-
-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
-
-paragraph \<open>Using additivity of lifted function to encode definedness.\<close>
-
-definition 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 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>Operative\<close>
-
-definition (in comm_monoid) operative :: "('b::euclidean_space set \<Rightarrow> 'a) \<Rightarrow> bool"
-  where "operative g \<longleftrightarrow>
-    (\<forall>a b. content (cbox a b) = 0 \<longrightarrow> g (cbox a b) = \<^bold>1) \<and>
-    (\<forall>a b c. \<forall>k\<in>Basis. 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}))"
-
-lemma (in comm_monoid) operativeD[dest]:
-  assumes "operative g"
-  shows "\<And>a b. content (cbox a b) = 0 \<Longrightarrow> g (cbox a b) = \<^bold>1"
-    and "\<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})"
-  using assms unfolding operative_def by auto
-
-lemma (in comm_monoid) operative_empty: "operative g \<Longrightarrow> g {} = \<^bold>1"
-  unfolding operative_def by (rule property_empty_interval) auto
-
 lemma operative_content[intro]: "add.operative content"
-  by (force simp add: add.operative_def content_split[symmetric])
-
-definition "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> ~(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
-  show ?t1 (*FIXME a horrible mess*)
-    unfolding division_points_def interval_split[OF k, of a b]
-    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
-    unfolding *
-    apply (rule subsetI)
-    unfolding mem_Collect_eq split_beta
-    apply (erule bexE conjE)+
-    apply (simp add: )
-    apply (erule exE conjE)+
-  proof
-    fix i l x
-    assume as:
-      "a \<bullet> fst x < snd x" "snd x < (if fst x = k then c else b \<bullet> fst x)"
-      "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
-      "i = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
-      and fstx: "fst x \<in> Basis"
-    from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
-    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 as(6) unfolding l interval_split[OF k] box_ne_empty as .
-    have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
-      using l using as(6) unfolding box_ne_empty[symmetric] by auto
-    show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
-      apply (rule bexI[OF _ \<open>l \<in> d\<close>])
-      using as(1-3,5) fstx
-      unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
-      apply (auto split: if_split_asm)
-      done
-    show "snd x < b \<bullet> fst x"
-      using as(2) \<open>c < b\<bullet>k\<close> by (auto split: if_split_asm)
-  qed
-  show ?t2
-    unfolding division_points_def interval_split[OF k, of a b]
-    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
-    unfolding *
-    unfolding subset_eq
-    apply rule
-    unfolding mem_Collect_eq split_beta
-    apply (erule bexE conjE)+
-    apply (simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
-    apply (erule exE conjE)+
-  proof
-    fix i l x
-    assume as:
-      "(if fst x = k then c else a \<bullet> fst x) < snd x" "snd x < b \<bullet> fst x"
-      "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
-      "i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}"
-      and fstx: "fst x \<in> Basis"
-    from assm(4)[OF this(5)] guess u v by (elim exE) note l=this
-    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 as(6) unfolding l interval_split[OF k] box_ne_empty as .
-    have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
-      using l using as(6) unfolding box_ne_empty[symmetric] by auto
-    show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
-      apply (rule bexI[OF _ \<open>l \<in> d\<close>])
-      using as(1-3,5) fstx
-      unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
-      apply (auto split: if_split_asm)
-      done
-    show "a \<bullet> fst x < snd x"
-      using as(1) \<open>a\<bullet>k < c\<close> by (auto split: if_split_asm)
-   qed
-qed
-
-lemma division_points_psubset:
-  fixes a :: "'a::euclidean_space"
-  assumes "d division_of (cbox a b)"
-      and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
-      and "l \<in> d"
-      and "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 assms(2) by (auto intro!:less_imp_le)
-  guess u v using division_ofD(4)[OF assms(1,5)] by (elim exE) note l=this
-  have uv: "\<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"
-    using division_ofD(2,2,3)[OF assms(1,5)] unfolding l box_ne_empty
-    using subset_box(1)
-    apply auto
-    apply blast+
-    done
-  have *: "interval_upperbound (cbox a b \<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]
-    apply -
-    apply (erule disjE)
-    apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
-    apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
-    done
-  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]
-    apply -
-    apply (erule disjE)
-    apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
-    apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
-    done
-  moreover have "?D2 \<subseteq> ?D"
-    by (auto simp add: assms division_points_subset)
-  ultimately show "?D2 \<subset> ?D"
-    by blast
-qed
-
-lemma (in comm_monoid_set) operative_division:
-  fixes g :: "'b::euclidean_space set \<Rightarrow> 'a"
-  assumes g: "operative g" and d: "d division_of (cbox a b)" shows "F g d = g (cbox a b)"
-proof -
-  define C where [abs_def]: "C = card (division_points (cbox a b) d)"
-  then show ?thesis
-    using d
-  proof (induction C arbitrary: a b d rule: less_induct)
-    case (less a b d)
-    show ?case
-    proof cases
-      show "content (cbox a b) = 0 \<Longrightarrow> F g d = g (cbox a b)"
-        using division_of_content_0[OF _ less.prems] operativeD(1)[OF  g] division_ofD(4)[OF less.prems]
-        by (fastforce intro!: neutral)
-    next
-      assume "content (cbox a b) \<noteq> 0"
-      note ab = this[unfolded content_lt_nz[symmetric] content_pos_lt_eq]
-      then have ab': "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
-        by (auto intro!: less_imp_le)
-      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 intro!: less_imp_le simp: inner_simps)
-        note this[unfolded division_ofD(6)[OF \<open>d division_of cbox a b\<close>,symmetric] Union_iff]
-        then guess i .. note i=this
-        guess u v using d'[rule_format,OF i(1)] by (elim exE conjE) note uv=this
-        have "cbox a b \<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 guess u v by (elim exE conjE) note uv=this
-          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 "content (cbox u v) = 0"
-            unfolding content_eq_0 using j
-            by force
-          then show "g x = \<^bold>1"
-            unfolding uv(1) by (rule operativeD(1)[OF g])
-        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 guess k c
-          unfolding split_paired_Ex division_points_def mem_Collect_eq split_conv
-          apply (elim exE conjE)
-          done
-        note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
-        from this(3) guess j .. note j=this
-        define d1 where "d1 = {l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
-        define d2 where "d2 = {l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<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)"
-        note division_points_psubset[OF \<open>d division_of cbox a b\<close> ab kc(1-2) j]
-        note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
-        then have *: "F g d1 = g (cbox a b \<inter> {x. x\<bullet>k \<le> c})" "F g d2 = g (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
-          unfolding interval_split[OF kc(4)]
-          apply (rule_tac[!] "less.hyps"[rule_format])
-          using division_split[OF \<open>d division_of cbox a b\<close>, where k=k and c=c]
-          apply (simp_all add: interval_split kc d1_def d2_def division_points_finite[OF \<open>d division_of cbox a b\<close>])
-          done
-        { fix l y
-          assume as: "l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y"
-          from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this
-          have "g (l \<inter> {x. x \<bullet> k \<le> c}) = \<^bold>1"
-            unfolding leq interval_split[OF kc(4)]
-            apply (rule operativeD[OF g])
-            unfolding interval_split[symmetric, OF kc(4)]
-            using division_split_left_inj less as kc leq by blast
-        } note fxk_le = this
-        { fix l y
-          assume as: "l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y"
-          from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this
-          have "g (l \<inter> {x. x \<bullet> k \<ge> c}) = \<^bold>1"
-            unfolding leq interval_split[OF kc(4)]
-            apply (rule operativeD(1)[OF g])
-            unfolding interval_split[symmetric,OF kc(4)]
-            using division_split_right_inj less leq as kc by blast
-        } note fxk_ge = this
-        have d1_alt: "d1 = (\<lambda>l. l \<inter> {x. x\<bullet>k \<le> c}) ` {l \<in> d. l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
-          using d1_def by auto
-        have d2_alt: "d2 = (\<lambda>l. l \<inter> {x. x\<bullet>k \<ge> c}) ` {l \<in> d. l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
-          using d2_def by auto
-        have "g (cbox a b) = F g d1 \<^bold>* F g d2" (is "_ = ?prev")
-          unfolding * using g kc(4) by blast
-        also have "F g d1 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d"
-          unfolding d1_alt using division_of_finite[OF less.prems] fxk_le
-          by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g])
-        also have "F g d2 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d"
-          unfolding d2_alt using division_of_finite[OF less.prems] fxk_ge
-          by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g])
-        also have *: "\<forall>x\<in>d. g x = g (x \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (x \<inter> {x. c \<le> x \<bullet> k})"
-          unfolding forall_in_division[OF \<open>d division_of cbox a b\<close>]
-          using g kc(4) by blast
-        have "F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d \<^bold>* F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d = F g d"
-          using * by (simp add: distrib)
-        finally show ?thesis by auto
-      qed
-    qed
-  qed
-qed
-
-lemma (in comm_monoid_set) operative_tagged_division:
-  assumes f: "operative g" and d: "d tagged_division_of (cbox a b)"
-  shows "F (\<lambda>(x, l). g l) d = g (cbox a b)"
-  unfolding d[THEN division_of_tagged_division, THEN operative_division[OF f], symmetric]
-  by (simp add: f[THEN operativeD(1)] over_tagged_division_lemma[OF d])
+  by (force simp add: add.operative_def content_split[symmetric] content_eq_0_interior)
 
 lemma additive_content_division: "d division_of (cbox a b) \<Longrightarrow> setsum content d = content (cbox a b)"
   by (metis operative_content setsum.operative_division)
@@ -1835,537 +209,8 @@
 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 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 (in comm_monoid) operative_1_lt:
-  "operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow>
-    ((\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1) \<and> (\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a .. c} \<^bold>* g {c .. b} = g {a .. b}))"
-  apply (simp add: operative_def content_real_eq_0 atMost_def[symmetric] atLeast_def[symmetric]
-              del: content_real_if)
-proof safe
-  fix a b c :: real
-  assume *: "\<forall>a b c. g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
-  assume "a < c" "c < b"
-  with *[rule_format, of a b c] show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
-    by (simp add: less_imp_le min.absorb2 max.absorb2)
-next
-  fix a b c :: real
-  assume as: "\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1"
-    "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
-  from as(1)[rule_format, of 0 1] as(1)[rule_format, of a a for a] as(2)
-  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
-  show "g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
-    by (auto simp: min_def max_def le_less)
-qed
-
-lemma (in comm_monoid) operative_1_le:
-  "operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow>
-    ((\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1) \<and> (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> g {a .. c} \<^bold>* g {c .. b} = g {a .. b}))"
-  unfolding operative_1_lt
-proof safe
-  fix a b c :: real
-  assume as: "\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}" "a < c" "c < b"
-  show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
-    apply (rule as(1)[rule_format])
-    using as(2-)
-    apply auto
-    done
-next
-  fix a b c :: real
-  assume "\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1"
-    and "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
-    and "a \<le> c"
-    and "c \<le> b"
-  note as = this[rule_format]
-  show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
-  proof (cases "c = a \<or> c = b")
-    case False
-    then show ?thesis
-      apply -
-      apply (subst as(2))
-      using as(3-)
-      apply auto
-      done
-  next
-    case True
-    then show ?thesis
-    proof
-      assume *: "c = a"
-      then have "g {a .. c} = \<^bold>1"
-        apply -
-        apply (rule as(1)[rule_format])
-        apply auto
-        done
-      then show ?thesis
-        unfolding * by auto
-    next
-      assume *: "c = b"
-      then have "g {c .. b} = \<^bold>1"
-        apply -
-        apply (rule as(1)[rule_format])
-        apply auto
-        done
-      then show ?thesis
-        unfolding * by auto
-    qed
-  qed
-qed
-
-subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close>
-
-definition 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_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
-  unfolding fine_def by auto
-
-lemma fine_inters:
- "(\<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_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
-  unfolding fine_def by blast
-
-lemma fine_unions: "(\<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_unions_exists:
-  assumes "finite iset"
-    and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
-    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
-    and "\<Union>iset = i"
-   obtains p where "p tagged_division_of i" and "d fine p"
-proof -
-  obtain pfn where pfn:
-    "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
-    "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
-    using bchoice[OF assms(2)] by auto
-  show thesis
-    apply (rule_tac p="\<Union>(pfn ` iset)" in that)
-    using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
-    by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
-qed
-
-
-subsection \<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
-
-subsection \<open>General bisection principle for intervals; might be useful elsewhere.\<close>
-
-lemma interval_bisection_step:
-  fixes type :: "'a::euclidean_space"
-  assumes "P {}"
-    and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
-    and "\<not> P (cbox a (b::'a))"
-  obtains c d where "\<not> P (cbox c d)"
-    and "\<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"
-proof -
-  have "cbox a b \<noteq> {}"
-    using assms(1,3) by metis
-  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
-    by (force simp: mem_box)
-  { fix f
-    have "\<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)"
-    proof (induct f rule: finite_induct)
-      case empty
-      show ?case
-        using assms(1) by auto
-    next
-      case (insert x f)
-      show ?case
-        unfolding Union_insert
-        apply (rule assms(2)[rule_format])
-        using inter_interior_unions_intervals [of f "interior x"]
-        apply (auto simp: insert)
-        by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
-    qed
-  } note UN_cases = this
-  let ?A = "{cbox c d | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or>
-    (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
-  let ?PP = "\<lambda>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"
-  {
-    presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False"
-    then show thesis
-      unfolding atomize_not not_all
-      by (blast intro: that)
-  }
-  assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)"
-  have "P (\<Union>?A)"
-  proof (rule UN_cases)
-    let ?B = "(\<lambda>s. cbox (\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i::'a)
-      (\<Sum>i\<in>Basis. (if i \<in> s then (a\<bullet>i + b\<bullet>i) / 2 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 = (a \<bullet> i + b \<bullet> i) / 2 \<or>
-                        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
-      show "x \<in> ?B"
-        unfolding image_iff x
-        apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
-        apply (rule arg_cong2 [where f = cbox])
-        using x(2) ab
-        apply (auto simp add: euclidean_eq_iff[where 'a='a])
-        by fastforce
-    qed
-    then show "finite ?A"
-      by (rule finite_subset) auto
-  next
-    fix s
-    assume "s \<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 = (a \<bullet> i + b \<bullet> i) / 2 \<or>
-                     c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
-      by blast
-    show "P s"
-      unfolding s
-      apply (rule as[rule_format])
-      using ab s(2) by force
-    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 = (a \<bullet> i + b \<bullet> i) / 2 \<or>
-        e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<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']
-      proof safe
-        assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
-        show False
-          using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
-      next
-        assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
-        show False
-          using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
-      qed
-    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 = (a \<bullet> i + b \<bullet> i) / 2 \<or>
-        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
-      by blast
-    show "x\<in>cbox a b"
-      unfolding mem_box
-    proof safe
-      fix i :: 'a
-      assume i: "i \<in> Basis"
-      then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
-        using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
-    qed
-  next
-    fix x
-    assume x: "x \<in> cbox a b"
-    have "\<forall>i\<in>Basis.
-      \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<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
-    proof
-      fix i :: 'a
-      assume i: "i \<in> Basis"
-      have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)"
-        using x[unfolded mem_box,THEN bspec, OF i] by auto
-      then show "\<exists>c d. ?P i c d"
-        by blast
-    qed
-    then show "x\<in>\<Union>?A"
-      unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
-      apply auto
-      apply (rule_tac x="cbox xa xaa" in exI)
-      unfolding mem_box
-      apply auto
-      done
-  qed
-  finally show False
-    using assms by auto
-qed
-
-lemma interval_bisection:
-  fixes type :: "'a::euclidean_space"
-  assumes "P {}"
-    and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<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 as: False
-      obtain c d where "\<not> P (cbox c d)"
-        "\<forall>i\<in>Basis.
-           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 (rule interval_bisection_step[of P, OF assms(1-2) as])
-      then show ?thesis
-        apply -
-        apply (rule_tac x="(c,d)" in exI)
-        apply auto
-        done
-    qed
-  qed
-  then obtain f where f:
-    "\<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)"
-    apply -
-    apply (drule choice)
-    apply blast
-    done
-  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 "A 0 = a" "B 0 = b" "\<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
-        apply (rule f[rule_format]) using assms(3)
-        apply auto
-        done
-    next
-      case (Suc n)
-      show ?case
-        unfolding S
-        apply (rule f[rule_format])
-        using Suc
-        unfolding S
-        apply auto
-        done
-    qed
-  qed
-  note AB = this(1-2) conjunctD2[OF this(3),rule_format]
-
-  have interv: "\<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 "(setsum (\<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> setsum (\<lambda>i. \<bar>(x - y)\<bullet>i\<bar>) Basis"
-        unfolding dist_norm by(rule norm_le_l1)
-      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
-      proof (rule setsum_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> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
-        unfolding setsum_divide_distrib
-      proof (rule setsum_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(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(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
-    qed simp
-  } note ABsubset = this
-  have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)"
-    by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
-      (metis nat.exhaust AB(1-3) assms(1,3))
-  then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
-    by blast
-  show thesis
-  proof (rule that[rule_format, of x0])
-    show "x0\<in>cbox a b"
-      using x0[of 0] unfolding AB .
-    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))"
-      apply (cases "0 < n")
-      using AB(3)[of "n - 1"] assms(3) AB(1-2)
-      apply auto
-      done
-    moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
-      using n using x0[of n] by auto
-    moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
-      unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
-    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:
-  fixes a b :: "'a::euclidean_space"
-  assumes "gauge g"
-  obtains p where "p tagged_division_of (cbox a b)" "g fine p"
-proof -
-  presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
-  then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
-    by blast
-  then show thesis ..
-next
-  assume as: "\<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 _ _ as])
-    apply (simp add: fine_def)
-    apply (metis tagged_division_union fine_union)
-    apply (auto simp: )
-    done
-  obtain e where e: "e > 0" "ball x e \<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 False
-    using tagged_division_of_self[OF c_d(1)] using c_d by auto
-qed
-
-lemma fine_division_exists_real:
-  fixes a b :: real
-  assumes "gauge g"
-  obtains p where "p tagged_division_of {a .. b}" "g fine p"
-  by (metis assms box_real(2) fine_division_exists)
-
-subsection \<open>Division filter\<close>
-
-text \<open>Divisions over all gauges towards finer divisions.\<close>
-
-definition division_filter :: "'a::euclidean_space set \<Rightarrow> ('a \<times> 'a set) set filter"
-  where "division_filter s = (INF g:{g. gauge g}. principal {p. p tagged_division_of s \<and> g fine p})"
-
-lemma 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_inter)
-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
+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
 
 subsection \<open>Gauge integral\<close>
 
@@ -2443,26 +288,6 @@
 lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
   by auto
 
-lemma setsum_content_null:
-  assumes "content (cbox a b) = 0"
-    and "p tagged_division_of (cbox a b)"
-  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
-proof (rule setsum.neutral, rule)
-  fix y
-  assume y: "y \<in> p"
-  obtain x k where xk: "y = (x, k)"
-    using surj_pair[of y] by blast
-  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
-  from this(2) obtain c d where k: "k = cbox c d" by blast
-  have "(\<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 content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
-    unfolding assms(1) k
-    by auto
-  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
-qed
-
 subsection \<open>Basic theorems about integrals.\<close>
 
 lemma has_integral_unique:
@@ -3175,38 +1000,32 @@
 
 subsection \<open>Additivity of integral on abutting intervals.\<close>
 
-lemma tagged_division_split_left_inj:
-  fixes x1 :: "'a::euclidean_space"
+lemma tagged_division_split_left_inj_content:
   assumes d: "d tagged_division_of i"
-    and k12: "(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"
+    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 -
-  have *: "\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c"
-    by force
+  from tagged_division_ofD(4)[OF d \<open>(x1, k1) \<in> d\<close>] obtain a b where k1: "k1 = cbox a b"
+    by auto
   show ?thesis
-    using k12
-    by (fastforce intro!:  division_split_left_inj[OF division_of_tagged_division[OF d]] *)
+    unfolding k1 interval_split[OF \<open>k \<in> Basis\<close>]
+    unfolding content_eq_0_interior
+    unfolding interval_split[OF \<open>k \<in> Basis\<close>, symmetric] k1[symmetric]
+    by (rule tagged_division_split_left_inj[OF assms])
 qed
 
-lemma tagged_division_split_right_inj:
-  fixes x1 :: "'a::euclidean_space"
+lemma tagged_division_split_right_inj_content:
   assumes d: "d tagged_division_of i"
-    and k12: "(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"
+    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 -
-  have *: "\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c"
-    by force
+  from tagged_division_ofD(4)[OF d \<open>(x1, k1) \<in> d\<close>] obtain a b where k1: "k1 = cbox a b"
+    by auto
   show ?thesis
-    using k12
-    by (fastforce intro!:  division_split_right_inj[OF division_of_tagged_division[OF d]] *)
+    unfolding k1 interval_split[OF \<open>k \<in> Basis\<close>]
+    unfolding content_eq_0_interior
+    unfolding interval_split[OF \<open>k \<in> Basis\<close>, symmetric] k1[symmetric]
+    by (rule tagged_division_split_right_inj[OF assms])
 qed
 
 lemma has_integral_split:
@@ -3284,7 +1103,8 @@
     have lem1: "\<And>f P Q. (\<forall>x k. (x, k) \<in> {(x, f k) | x k. P x k} \<longrightarrow> Q x k) \<longleftrightarrow>
                          (\<forall>x k. P x k \<longrightarrow> Q x (f k))"
       by auto
-    have fin_finite: "finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}" if "finite s" for f s P
+    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
@@ -3398,8 +1218,9 @@
       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) +
         (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x \<bullet> k}) *\<^sub>R f x) - (i + j)"
         unfolding lem3[OF p(3)]
-        by (subst setsum.reindex_nontrivial[OF p(3)], auto intro!: k eq0 tagged_division_split_left_inj[OF p(1)] tagged_division_split_right_inj[OF p(1)]
-              simp: cont_eq)+
+        by (subst (1 2) setsum.reindex_nontrivial[OF p(3)])
+           (auto intro!: k eq0 tagged_division_split_left_inj_content[OF p(1)] tagged_division_split_right_inj_content[OF p(1)]
+                 simp: cont_eq)+
       also note setsum.distrib[symmetric]
       also have "\<And>x. x \<in> p \<Longrightarrow>
                     (\<lambda>(x,ka). content (ka \<inter> {x. x \<bullet> k \<le> c}) *\<^sub>R f x) x +
@@ -3429,34 +1250,6 @@
 
 subsection \<open>A sort of converse, integrability on subintervals.\<close>
 
-lemma tagged_division_union_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_union[OF assms(1-2)])
-    unfolding interval_split[OF k] interior_cbox
-    using k
-    apply (auto simp add: box_def elim!: ballE[where x=k])
-    done
-qed
-
-lemma tagged_division_union_interval_real:
-  fixes a :: real
-  assumes "p1 tagged_division_of ({a .. b} \<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_union_interval)
-
 lemma has_integral_separate_sides:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
   assumes "(f has_integral i) (cbox a b)"
@@ -3621,10 +1414,10 @@
     qed
   next
     fix a b :: 'a
-    assume "content (cbox a b) = 0"
+    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)
+      by (auto simp: integrable_on_null content_eq_0_interior)
   qed
 qed
 
@@ -4069,47 +1862,6 @@
 
 subsection \<open>Negligibility of hyperplane.\<close>
 
-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])
-      apply (rule equalityI)
-      apply blast
-      apply clarsimp
-      apply (rule_tac x="l \<inter> {x. c + e \<ge> x \<bullet> k}" in exI)
-      apply auto
-      done
-    by (simp add: interval_split k interval_doublesplit)
-qed
-
 lemma content_doublesplit:
   fixes a :: "'a::euclidean_space"
   assumes "0 < e"
@@ -4262,6 +2014,8 @@
       also have "\<dots> < e"
       proof (subst setsum.over_tagged_division_lemma[OF p[THEN conjunct1]], goal_cases)
         case prems: (1 u v)
+        then have *: "content (cbox u v) = 0"
+          unfolding content_eq_0_interior by simp
         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]
           apply (rule content_subset)
@@ -4269,7 +2023,7 @@
           apply auto
           done
         then show ?case
-          unfolding prems interval_doublesplit[OF k]
+          unfolding * interval_doublesplit[OF k]
           by (blast intro: antisym)
       next
         have "(\<Sum>l\<in>snd ` p. content (l \<inter> {x. \<bar>x \<bullet> k - c\<bar> \<le> d})) =
@@ -4303,188 +2057,9 @@
 qed
 
 
-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 "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 -
-  let ?P = "\<lambda>p. p tagged_partial_division_of (cbox a b) \<longrightarrow> gauge d \<longrightarrow>
-    (\<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))"
-  {
-    have *: "finite p" "p tagged_partial_division_of (cbox a b)"
-      using assms(1)
-      unfolding tagged_division_of_def
-      by auto
-    presume "\<And>p. finite p \<Longrightarrow> ?P p"
-    from this[rule_format,OF * assms(2)] guess q .. note q=this
-    then show ?thesis
-      apply -
-      apply (rule that[of q])
-      unfolding tagged_division_ofD[OF assms(1)]
-      apply auto
-      done
-  }
-  fix p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set"
-  assume as: "finite p"
-  show "?P p"
-    apply rule
-    apply rule
-    using as
-  proof (induct p)
-    case empty
-    show ?case
-      apply (rule_tac x="{}" in exI)
-      unfolding fine_def
-      apply auto
-      done
-  next
-    case (insert xk p)
-    guess x k using surj_pair[of xk] by (elim exE) note xk=this
-    note tagged_partial_division_subset[OF insert(4) subset_insertI]
-    from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
-    have *: "\<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)]
-    from p(4)[unfolded xk, OF insertI1] guess u v by (elim exE) note uv=this
-
-    have "finite {k. \<exists>x. (x, k) \<in> p}"
-      apply (rule finite_subset[of _ "snd ` p"])
-      using p
-      apply safe
-      apply (metis image_iff snd_conv)
-      apply auto
-      done
-    then have int: "interior (cbox u v) \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
-      apply (rule inter_interior_unions_intervals)
-      apply (rule open_interior)
-      apply (rule_tac[!] ballI)
-      unfolding mem_Collect_eq
-      apply (erule_tac[!] exE)
-      apply (drule p(4)[OF insertI2])
-      apply assumption
-      apply (rule p(5))
-      unfolding uv xk
-      apply (rule insertI1)
-      apply (rule insertI2)
-      apply assumption
-      using insert(2)
-      unfolding uv xk
-      apply auto
-      done
-    show ?case
-    proof (cases "cbox u v \<subseteq> d x")
-      case True
-      then show ?thesis
-        apply (rule_tac x="{(x,cbox u v)} \<union> q1" in exI)
-        apply rule
-        unfolding * uv
-        apply (rule tagged_division_union)
-        apply (rule tagged_division_of_self)
-        apply (rule p[unfolded xk uv] insertI1)+
-        apply (rule q1)
-        apply (rule int)
-        apply rule
-        apply (rule fine_union)
-        apply (subst fine_def)
-        defer
-        apply (rule q1)
-        unfolding Ball_def split_paired_All split_conv
-        apply rule
-        apply rule
-        apply rule
-        apply rule
-        apply (erule insertE)
-        apply (simp add: uv xk)
-        apply (rule UnI2)
-        apply (drule q1(3)[rule_format])
-        unfolding xk uv
-        apply auto
-        done
-    next
-      case False
-      from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
-      show ?thesis
-        apply (rule_tac x="q2 \<union> q1" in exI)
-        apply rule
-        unfolding * uv
-        apply (rule tagged_division_union q2 q1 int fine_union)+
-        unfolding Ball_def split_paired_All split_conv
-        apply rule
-        apply (rule fine_union)
-        apply (rule q1 q2)+
-        apply rule
-        apply rule
-        apply rule
-        apply rule
-        apply (erule insertE)
-        apply (rule UnI2)
-        apply (simp add: False uv xk)
-        apply (drule q1(3)[rule_format])
-        using False
-        unfolding xk uv
-        apply auto
-        done
-    qed
-  qed
-qed
-
 
 subsection \<open>Hence the main theorem about negligible sets.\<close>
 
-lemma finite_product_dependent:
-  assumes "finite s"
-    and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
-  shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
-  using assms
-proof induct
-  case (insert x s)
-  have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
-    (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
-  show ?case
-    unfolding *
-    apply (rule finite_UnI)
-    using insert
-    apply auto
-    done
-qed auto
-
-lemma sum_sum_product:
-  assumes "finite s"
-    and "\<forall>i\<in>s. finite (t i)"
-  shows "setsum (\<lambda>i. setsum (x i) (t i)::real) s =
-    setsum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}"
-  using assms
-proof induct
-  case (insert a s)
-  have *: "{(i, j) |i j. i \<in> insert a s \<and> j \<in> t i} =
-    (\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
-  show ?case
-    unfolding *
-    apply (subst setsum.union_disjoint)
-    unfolding setsum.insert[OF insert(1-2)]
-    prefer 4
-    apply (subst insert(3))
-    unfolding add_right_cancel
-  proof -
-    show "setsum (x a) (t a) = (\<Sum>(xa, y)\<in> Pair a ` t a. x xa y)"
-      apply (subst setsum.reindex)
-      unfolding inj_on_def
-      apply auto
-      done
-    show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
-      apply (rule finite_product_dependent)
-      using insert
-      apply auto
-      done
-  qed (insert insert, auto)
-qed auto
-
 lemma has_integral_negligible:
   fixes f :: "'b::euclidean_space \<Rightarrow> 'a::real_normed_vector"
   assumes "negligible s"
@@ -4937,10 +2512,10 @@
 proof 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 "content (cbox a b) = 0"
+    if "box a b = {}"
     apply (rule_tac x=f in exI)
     using assms that
-    apply (auto intro!: integrable_on_null)
+    apply (auto simp: content_eq_0_interior)
     done
   {
     fix c g
@@ -5059,33 +2634,6 @@
 
 subsection \<open>Specialization of additivity to one dimension.\<close>
 
-subsection \<open>Special case of additivity we need for the FTC.\<close>
-
-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 "setsum (\<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))"
-  have ***: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
-    using assms by auto
-  have *: "add.operative ?f"
-    unfolding add.operative_1_lt box_eq_empty
-    by auto
-  have **: "cbox a b \<noteq> {}"
-    using assms(1) by auto
-  note setsum.operative_tagged_division[OF * assms(2)[simplified box_real[symmetric]]]
-  note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
-  show ?thesis
-    unfolding *
-    apply (rule setsum.cong)
-    unfolding split_paired_all split_conv
-    using assms(2)
-    apply auto
-    done
-qed
-
 
 subsection \<open>A useful lemma allowing us to factor out the content size.\<close>
 
@@ -5373,20 +2921,6 @@
 qed
 
 
-subsection \<open>Attempt a systematic general set of "offset" results for components.\<close>
-
-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
-  apply safe
-  defer
-  apply (erule_tac x="f x" in allE)
-  apply (erule_tac x="d (f x)" in allE)
-  apply auto
-  done
-
 
 subsection \<open>Only need trivial subintervals if the interval itself is trivial.\<close>
 
@@ -5524,10 +3058,12 @@
   shows "comm_monoid.operative op \<and> True (\<lambda>i. f integrable_on i)"
   unfolding comm_monoid.operative_def[OF comm_monoid_and]
   apply safe
-  apply (subst integrable_on_def)
-  unfolding has_integral_null_eq
-  apply (rule, rule refl)
-  apply (rule, assumption, assumption)+
+     apply (subst integrable_on_def)
+     apply rule
+     apply (rule has_integral_null_eq[where i=0, THEN iffD2])
+      apply (simp add: content_eq_0_interior)
+     apply rule
+    apply (rule, assumption, assumption)+
   unfolding integrable_on_def
   by (auto intro!: has_integral_split)
 
@@ -6121,18 +3657,6 @@
 
 subsection \<open>Stronger form of FCT; quite a tedious proof.\<close>
 
-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)
-
-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 "setsum (\<lambda>(x,k). f (Sup k) - f(Inf k)) p = f b - f a"
-  using additive_tagged_division_1[OF _ assms(2), of f]
-  using assms(1)
-  by auto
-
 lemma split_minus[simp]: "(\<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)
 
@@ -7717,39 +5241,6 @@
   shows "f integrable_on s \<longleftrightarrow> f integrable_on t"
 by (blast intro: integrable_spike_set assms negligible_subset)
 
-(*lemma integral_spike_set:
- "\<forall>f:real^M->real^N g s t.
-        negligible(s DIFF t \<union> t DIFF s)
-        \<longrightarrow> integral s f = integral t f"
-qed  REPEAT STRIP_TAC THEN REWRITE_TAC[integral] THEN
-  AP_TERM_TAC THEN ABS_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
-  ASM_MESON_TAC[]);;
-
-lemma has_integral_interior:
- "\<forall>f:real^M->real^N y s.
-        negligible(frontier s)
-        \<longrightarrow> ((f has_integral y) (interior s) \<longleftrightarrow> (f has_integral y) s)"
-qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
-  FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
-    NEGLIGIBLE_SUBSET)) THEN
-  REWRITE_TAC[frontier] THEN
-  MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
-  MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
-  SET_TAC[]);;
-
-lemma has_integral_closure:
- "\<forall>f:real^M->real^N y s.
-        negligible(frontier s)
-        \<longrightarrow> ((f has_integral y) (closure s) \<longleftrightarrow> (f has_integral y) s)"
-qed  REPEAT STRIP_TAC THEN MATCH_MP_TAC HAS_INTEGRAL_SPIKE_SET_EQ THEN
-  FIRST_X_ASSUM(MATCH_MP_TAC o MATCH_MP (REWRITE_RULE[IMP_CONJ]
-    NEGLIGIBLE_SUBSET)) THEN
-  REWRITE_TAC[frontier] THEN
-  MP_TAC(ISPEC `s:real^M->bool` INTERIOR_SUBSET) THEN
-  MP_TAC(ISPEC `s:real^M->bool` CLOSURE_SUBSET) THEN
-  SET_TAC[]);;*)
-
-
 subsection \<open>More lemmas that are useful later\<close>
 
 lemma has_integral_subset_component_le:
@@ -8434,34 +5925,26 @@
 
 subsection \<open>Also tagged divisions\<close>
 
+lemma has_integral_iff: "(f has_integral i) s \<longleftrightarrow> (f integrable_on s \<and> integral s f = i)"
+  by blast
+
 lemma has_integral_combine_tagged_division:
   fixes f :: "'n::euclidean_space \<Rightarrow> 'a::banach"
   assumes "p tagged_division_of s"
     and "\<forall>(x,k) \<in> p. (f has_integral (i k)) k"
-  shows "(f has_integral (setsum (\<lambda>(x,k). i k) p)) s"
+  shows "(f has_integral (\<Sum>(x,k)\<in>p. i k)) s"
 proof -
-  have *: "(f has_integral (setsum (\<lambda>k. integral k f) (snd ` p))) s"
-    apply (rule has_integral_combine_division)
-    apply (rule division_of_tagged_division[OF assms(1)])
+  have *: "(f has_integral (\<Sum>k\<in>snd`p. integral k f)) s"
     using assms(2)
-    unfolding has_integral_integral[symmetric]
-    apply safe
+    apply (intro has_integral_combine_division)
+    apply (auto simp: has_integral_integral[symmetric] intro: division_of_tagged_division[OF assms(1)])
     apply auto
     done
-  then show ?thesis
-    apply -
-    apply (rule subst[where P="\<lambda>i. (f has_integral i) s"])
-    defer
-    apply assumption
-    apply (rule trans[of _ "setsum (\<lambda>(x,k). integral k f) p"])
-    apply (subst eq_commute)
-    apply (rule setsum.over_tagged_division_lemma[OF assms(1)])
-    apply (rule integral_null)
-    apply assumption
-    apply (rule setsum.cong)
-    using assms(2)
-    apply auto
-    done
+  also have "(\<Sum>k\<in>snd`p. integral k f) = (\<Sum>(x, k)\<in>p. integral k f)"
+    by (intro setsum.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!: setsum.cong)
 qed
 
 lemma integral_combine_tagged_division_bottomup:
@@ -9616,84 +7099,6 @@
 
 subsection \<open>differentiation under the integral sign\<close>
 
-lemma tube_lemma:
-  assumes "compact K"
-  assumes "open W"
-  assumes "{x0} \<times> K \<subseteq> W"
-  shows "\<exists>X0. x0 \<in> X0 \<and> open X0 \<and> X0 \<times> K \<subseteq> W"
-proof -
-  {
-    fix y assume "y \<in> K"
-    then have "(x0, y) \<in> W" using assms by auto
-    with \<open>open W\<close>
-    have "\<exists>X0 Y. open X0 \<and> open Y \<and> x0 \<in> X0 \<and> y \<in> Y \<and> X0 \<times> Y \<subseteq> W"
-      by (rule open_prod_elim) blast
-  }
-  then obtain X0 Y where
-    *: "\<forall>y \<in> K. open (X0 y) \<and> open (Y y) \<and> x0 \<in> X0 y \<and> y \<in> Y y \<and> X0 y \<times> Y y \<subseteq> W"
-    by metis
-  from * have "\<forall>t\<in>Y ` K. open t" "K \<subseteq> \<Union>(Y ` K)" by auto
-  with \<open>compact K\<close> obtain CC where CC: "CC \<subseteq> Y ` K" "finite CC" "K \<subseteq> \<Union>CC"
-    by (rule compactE)
-  then obtain c where c: "\<And>C. C \<in> CC \<Longrightarrow> c C \<in> K \<and> C = Y (c C)"
-    by (force intro!: choice)
-  with * CC show ?thesis
-    by (force intro!: exI[where x="\<Inter>C\<in>CC. X0 (c C)"]) (* SLOW *)
-qed
-
-lemma continuous_on_prod_compactE:
-  fixes fx::"'a::topological_space \<times> 'b::topological_space \<Rightarrow> 'c::metric_space"
-    and e::real
-  assumes cont_fx: "continuous_on (U \<times> C) fx"
-  assumes "compact C"
-  assumes [intro]: "x0 \<in> U"
-  notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
-  assumes "e > 0"
-  obtains X0 where "x0 \<in> X0" "open X0"
-    "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
-proof -
-  define psi where "psi = (\<lambda>(x, t). dist (fx (x, t)) (fx (x0, t)))"
-  define W0 where "W0 = {(x, t) \<in> U \<times> C. psi (x, t) < e}"
-  have W0_eq: "W0 = psi -` {..<e} \<inter> U \<times> C"
-    by (auto simp: vimage_def W0_def)
-  have "open {..<e}" by simp
-  have "continuous_on (U \<times> C) psi"
-    by (auto intro!: continuous_intros simp: psi_def split_beta')
-  from this[unfolded continuous_on_open_invariant, rule_format, OF \<open>open {..<e}\<close>]
-  obtain W where W: "open W" "W \<inter> U \<times> C = W0 \<inter> U \<times> C"
-    unfolding W0_eq by blast
-  have "{x0} \<times> C \<subseteq> W \<inter> U \<times> C"
-    unfolding W
-    by (auto simp: W0_def psi_def \<open>0 < e\<close>)
-  then have "{x0} \<times> C \<subseteq> W" by blast
-  from tube_lemma[OF \<open>compact C\<close> \<open>open W\<close> this]
-  obtain X0 where X0: "x0 \<in> X0" "open X0" "X0 \<times> C \<subseteq> W"
-    by blast
-
-  have "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
-  proof safe
-    fix x assume x: "x \<in> X0" "x \<in> U"
-    fix t assume t: "t \<in> C"
-    have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
-      by (auto simp: psi_def)
-    also
-    {
-      have "(x, t) \<in> X0 \<times> C"
-        using t x
-        by auto
-      also note \<open>\<dots> \<subseteq> W\<close>
-      finally have "(x, t) \<in> W" .
-      with t x have "(x, t) \<in> W \<inter> U \<times> C"
-        by blast
-      also note \<open>W \<inter> U \<times> C = W0 \<inter> U \<times> C\<close>
-      finally  have "psi (x, t) < e"
-        by (auto simp: W0_def)
-    }
-    finally show "dist (fx (x, t)) (fx (x0, t)) \<le> e" by simp
-  qed
-  from X0(1,2) this show ?thesis ..
-qed
-
 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)"
@@ -9743,26 +7148,6 @@
   qed
 qed (auto intro!: continuous_on_const)
 
-lemma eventually_closed_segment:
-  fixes x0::"'a::real_normed_vector"
-  assumes "open X0" "x0 \<in> X0"
-  shows "\<forall>\<^sub>F x in at x0 within U. closed_segment x0 x \<subseteq> X0"
-proof -
-  from openE[OF assms]
-  obtain e where e: "0 < e" "ball x0 e \<subseteq> X0" .
-  then have "\<forall>\<^sub>F x in at x0 within U. x \<in> ball x0 e"
-    by (auto simp: dist_commute eventually_at)
-  then show ?thesis
-  proof eventually_elim
-    case (elim x)
-    have "x0 \<in> ball x0 e" using \<open>e > 0\<close> by simp
-    from convex_ball[unfolded convex_contains_segment, rule_format, OF this elim]
-    have "closed_segment x0 x \<subseteq> ball x0 e" .
-    also note \<open>\<dots> \<subseteq> X0\<close>
-    finally show ?case .
-  qed
-qed
-
 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>
@@ -9856,8 +7241,7 @@
   qed (rule blinfun.bounded_linear_right)
 qed (auto intro!: derivative_eq_intros simp: blinfun.bilinear_simps)
 
-lemma
-  has_vector_derivative_eq_has_derivative_blinfun:
+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)
@@ -9889,8 +7273,7 @@
     done
 qed
 
-lemma
-  has_field_derivative_eq_has_derivative_blinfun:
+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)
 
@@ -9922,8 +7305,7 @@
 
 subsection \<open>Exchange uniform limit and integral\<close>
 
-lemma
-  uniform_limit_integral:
+lemma uniform_limit_integral:
   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)"
@@ -10117,18 +7499,6 @@
 
 subsection \<open>Compute a double integral using iterated integrals and switching the order of integration\<close>
 
-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
-
 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"
@@ -10180,11 +7550,6 @@
     done
 qed
 
-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 integral_split:
   fixes f :: "'a::euclidean_space \<Rightarrow> 'b::{real_normed_vector,complete_space}"
   assumes f: "f integrable_on (cbox a b)"
@@ -10208,9 +7573,11 @@
                     \<le> e * content (cbox (a,c) (b,d)))"
 proof (auto simp: comm_monoid.operative_def[OF comm_monoid_and])
   fix a::'a and c::'b and b::'a and d::'b and u::'a and v::'a and w::'b and z::'b
-  assume c0: "content (cbox (a, c) (b, d)) = 0"
+  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))))
@@ -10296,9 +7663,6 @@
      integral (cbox a b) (\<lambda>x. integral (cbox c d) (\<lambda>y. k))"
   by (simp add: content_Pair)
 
-lemma norm_minus2: "norm (x1-x2, y1-y2) = norm (x2-x1, y2-y1)"
-  by (simp add: norm_minus_eqI)
-
 lemma integral_prod_continuous:
   fixes f :: "('a::euclidean_space * 'b::euclidean_space) \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a,c) (b,d)) f" (is "continuous_on ?CBOX f")
@@ -10414,20 +7778,6 @@
     by simp
 qed
 
-lemma swap_continuous:
-  assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
-    shows "continuous_on (cbox (c,a) (d,b)) (\<lambda>(x, y). f y x)"
-proof -
-  have "(\<lambda>(x, y). f y x) = (\<lambda>(x, y). f x y) \<circ> prod.swap"
-    by auto
-  then show ?thesis
-    apply (rule ssubst)
-    apply (rule continuous_on_compose)
-    apply (simp add: split_def)
-    apply (rule continuous_intros | simp add: assms)+
-    done
-qed
-
 lemma integral_swap_2dim:
   fixes f :: "['a::euclidean_space, 'b::euclidean_space] \<Rightarrow> 'c::banach"
   assumes "continuous_on (cbox (a,c) (b,d)) (\<lambda>(x,y). f x y)"
@@ -10731,277 +8081,6 @@
        (insert assms, simp_all add: powr_minus powr_realpow divide_simps)
 qed
 
-subsubsection \<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. PiE 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: divide_simps zero_le_mult_iff aibi)
-      apply (force simp: aibi scaling_mono nat_less_real_le dest: PiE_mem)
-      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: divide_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 --\<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" --\<open>WLOG we can assume @{term"m \<le> n"}, 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 add: power_add, 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 setsum.cong)
-            also have "... \<le> DIM('a) * (\<epsilon> / (2 * real DIM('a)))"
-              by (meson setsum_bounded_above that)
-            also have "... = \<epsilon> / 2"
-              by (simp add: divide_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: divide_simps)
-        then have "norm (b - a) < e * 2^n"
-          by (auto simp: divide_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 divide_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 divide_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 divide_simps)
-                done
-              also have "... = x \<bullet> i"
-                using abi_less by (simp add: divide_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: divide_simps algebra_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 by (force simp: algebra_simps)
-        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. PiE 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: divide_simps algebra_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 algebra_simps divide_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> ~(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
-
 subsubsection \<open>Adaption to ordered Euclidean spaces and the Cartesian Euclidean space\<close>
 
 lemma integral_component_eq_cart[simp]:
diff -r b235e845c8e8 -r c3da799b1b45 src/HOL/Analysis/Tagged_Division.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Tagged_Division.thy	Thu Sep 29 13:02:43 2016 +0200
@@ -0,0 +1,2817 @@
+(*  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 used for the Henstock-Kurzweil gauge integration\<close>
+
+theory Tagged_Division
+imports
+  Topology_Euclidean_Space
+begin
+
+lemma finite_product_dependent:
+  assumes "finite s"
+    and "\<And>x. x \<in> s \<Longrightarrow> finite (t x)"
+  shows "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
+  using assms
+proof induct
+  case (insert x s)
+  have *: "{(i, j) |i j. i \<in> insert x s \<and> j \<in> t i} =
+    (\<lambda>y. (x,y)) ` (t x) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
+  show ?case
+    unfolding *
+    apply (rule finite_UnI)
+    using insert
+    apply auto
+    done
+qed auto
+
+lemma sum_sum_product:
+  assumes "finite s"
+    and "\<forall>i\<in>s. finite (t i)"
+  shows "setsum (\<lambda>i. setsum (x i) (t i)::real) s =
+    setsum (\<lambda>(i,j). x i j) {(i,j) | i j. i \<in> s \<and> j \<in> t i}"
+  using assms
+proof induct
+  case (insert a s)
+  have *: "{(i, j) |i j. i \<in> insert a s \<and> j \<in> t i} =
+    (\<lambda>y. (a,y)) ` (t a) \<union> {(i, j) |i j. i \<in> s \<and> j \<in> t i}" by auto
+  show ?case
+    unfolding *
+    apply (subst setsum.union_disjoint)
+    unfolding setsum.insert[OF insert(1-2)]
+    prefer 4
+    apply (subst insert(3))
+    unfolding add_right_cancel
+  proof -
+    show "setsum (x a) (t a) = (\<Sum>(xa, y)\<in> Pair a ` t a. x xa y)"
+      apply (subst setsum.reindex)
+      unfolding inj_on_def
+      apply auto
+      done
+    show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}"
+      apply (rule finite_product_dependent)
+      using insert
+      apply auto
+      done
+  qed (insert insert, auto)
+qed auto
+
+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 \<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 u} and @{term v}.\<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
+
+lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
+lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
+lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
+
+lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
+  by auto
+
+declare norm_triangle_ineq4[intro]
+
+lemma transitive_stepwise_le:
+  assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" and "\<And>n. R n (Suc n)"
+  shows "\<forall>n\<ge>m. R m n"
+proof (intro allI impI)
+  show "m \<le> n \<Longrightarrow> R m n" for n
+    by (induction rule: dec_induct)
+       (use assms in blast)+
+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 inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
+     unfolding assms(1,2) interior_cbox by auto
+  moreover
+  have "box a b \<subseteq> cbox c d \<union> s"
+    apply (rule order_trans,rule box_subset_cbox)
+    using assms(4) unfolding assms(1,2)
+    apply auto
+    done
+  ultimately
+  show ?thesis
+    unfolding assms interior_cbox
+      by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI)
+qed
+
+lemma 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 inter_interior_unions_intervals:
+    "finite f \<Longrightarrow> open s \<Longrightarrow> \<forall>t\<in>f. \<exists>a b. t = cbox a b \<Longrightarrow> \<forall>t\<in>f. s \<inter> (interior t) = {} \<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"
+  apply (rule_tac[!] set_eqI)
+  unfolding Int_iff mem_box mem_Collect_eq
+  using assms
+  apply auto
+  done
+
+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 interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
+  where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)"
+
+definition interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
+  where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x: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_setsum 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_setsum cbox_def
+  by (safe intro!: cInf_eq) auto
+
+lemmas interval_bounds = interval_upperbound interval_lowerbound
+
+lemma
+  fixes X::"real set"
+  shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
+    and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
+  by (auto simp: interval_upperbound_def interval_lowerbound_def)
+
+lemma interval_bounds'[simp]:
+  assumes "cbox a b \<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:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:A. x \<bullet> i) *\<^sub>R i)"
+      by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
+  moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
+  have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:B. x \<bullet> i) *\<^sub>R i)"
+      by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
+  ultimately show ?thesis unfolding interval_upperbound_def
+      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
+qed
+
+lemma interval_lowerbound_Times:
+  assumes "A \<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:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:A. x \<bullet> i) *\<^sub>R i)"
+      by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
+  moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
+  have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:B. x \<bullet> i) *\<^sub>R i)"
+      by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
+  ultimately show ?thesis unfolding interval_lowerbound_def
+      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
+qed
+
+subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
+
+definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
+
+lemma gaugeI:
+  assumes "\<And>x. x \<in> g x"
+    and "\<And>x. open (g x)"
+  shows "gauge g"
+  using assms unfolding gauge_def by auto
+
+lemma gaugeD[dest]:
+  assumes "gauge d"
+  shows "x \<in> d x"
+    and "open (d 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_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
+  unfolding gauge_def by auto
+
+lemma gauge_inters:
+  assumes "finite s"
+    and "\<forall>d\<in>s. gauge (f d)"
+  shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> s})"
+proof -
+  have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
+    by auto
+  show ?thesis
+    unfolding gauge_def unfolding *
+    using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
+qed
+
+lemma gauge_existence_lemma:
+  "(\<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>
+
+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
+  apply safe
+  defer
+  apply (erule_tac x="f x" in allE)
+  apply (erule_tac x="d (f x)" in allE)
+  apply auto
+  done
+
+subsection \<open>Divisions.\<close>
+
+definition 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"
+  unfolding division_of_def 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 simp: cbox_sing)
+      done
+  }
+  ultimately show ?l
+    unfolding division_of_def cbox_sing by auto
+next
+  assume ?l
+  note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
+  {
+    fix x
+    assume x: "x \<in> s" have "x = {a}"
+      using *(2)[rule_format,OF x] by auto
+  }
+  moreover have "s \<noteq> {}"
+    using *(4) by auto
+  ultimately show ?r
+    unfolding cbox_sing by auto
+qed
+
+lemma elementary_empty: obtains p where "p division_of {}"
+  unfolding division_of_trivial by auto
+
+lemma elementary_interval: obtains p where "p division_of (cbox a b)"
+  by (metis division_of_trivial division_of_self)
+
+lemma division_contains: "s division_of i \<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 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"
+      apply (rule set_eqI)
+      unfolding * and UN_iff
+      using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
+      apply auto
+      done
+    {
+      fix k
+      assume "k \<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_inter:
+  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_inters:
+  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_inter 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)"])
+  {
+    fix i :: 'a
+    assume "i \<in> Basis"
+    with incl nonempty have "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i"
+      unfolding box_eq_empty subset_box by (auto simp: not_le)
+  }
+  note ord = this
+
+  show "p division_of (cbox a b)"
+  proof (rule division_ofI)
+    show "finite p"
+      unfolding p_def by (auto intro!: finite_PiE)
+    {
+      fix k
+      assume "k \<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 interior_cbox 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
+
+lemma 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
+  { fix x
+    assume x: "x \<in> p"
+    have "q x division_of \<Union>q x"
+      apply (rule division_ofI)
+      using division_ofD[OF q(1)[OF x]]
+      apply auto
+      done }
+  then have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
+    by (meson Diff_subset division_of_subset)
+  then have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
+    apply -
+    apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
+    apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
+    done
+  then obtain d where d: "d division_of \<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"
+      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 inter_interior_unions_intervals])
+        note qk=division_ofD[OF q(1)[OF k]]
+        show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b"
+          using qk by auto
+        show "\<forall>t\<in>q k - {k}. 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
+          apply auto
+          done
+      qed } 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 inter_interior_unions_intervals)
+      apply (rule p open_interior ballI)+
+      apply simp_all
+      done
+  qed
+  then show ?thesis
+    by (meson Un_upper2 that)
+qed
+
+lemma elementary_bounded[dest]:
+  fixes s :: "'a::euclidean_space set"
+  shows "p division_of s \<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 elementary_bounded bounded_subset_cbox)
+
+lemma 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
+  show ?thesis
+    apply (rule that[of "{}"])
+    unfolding True
+    using assms
+    apply auto
+    done
+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 "p division_of cbox c d" "cbox u v \<in> p"
+      by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
+    note p = this division_ofD[OF this(1)]
+    have "interior (cbox a b \<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 inter_interior_unions_intervals)
+      using p(6) p(7)[OF p(2)] p(3)
+      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_unions:
+  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 -
+  note assm = division_ofD[OF assms]
+  have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
+    by auto
+  have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
+    by auto
+  {
+    presume "p = {} \<Longrightarrow> thesis"
+      "cbox a b = {} \<Longrightarrow> thesis"
+      "cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis"
+      "p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis"
+    then show thesis by auto
+  next
+    assume as: "p = {}"
+    obtain p where "p division_of (cbox a b)"
+      by (rule elementary_interval)
+    then show thesis
+      using as that by auto
+  next
+    assume as: "cbox a b = {}"
+    show thesis
+      using as assms that by auto
+  next
+    assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}"
+    show thesis
+      apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
+      unfolding finite_insert
+      apply (rule assm(1)) unfolding Union_insert
+      using assm(2-4) as
+      apply -
+      apply (fast dest: assm(5))+
+      done
+  next
+    assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}"
+    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 assm(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 as(3) 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 "*" assm(1) q(1) by auto
+      show "\<Union>?D = cbox a b \<union> \<Union>p"
+        unfolding * lem1
+        unfolding lem2[OF as(1), of "cbox a b", symmetric]
+        using q(6)
+        by auto
+      fix k
+      assume k: "k \<in> ?D"
+      then show "k \<subseteq> cbox a b \<union> \<Union>p"
+        using q(2) by auto
+      show "k \<noteq> {}"
+        using q(3) k by auto
+      show "\<exists>a b. k = cbox a b"
+        using q(4) k by auto
+      fix k'
+      assume 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'")
+        case True
+        show ?thesis
+          using True k' q(5) x' x by auto
+      next
+        case False
+        {
+          presume "k = cbox a b \<Longrightarrow> ?thesis"
+            and "k' = cbox a b \<Longrightarrow> ?thesis"
+            and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis"
+          then show ?thesis by linarith
+        next
+          assume as': "k  = cbox a b"
+          show ?thesis
+            using as' k' q(5) x' by blast
+        next
+          assume as': "k' = cbox a b"
+          show ?thesis
+            using as' k'(2) q(5) x by blast
+        }
+        assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b"
+        obtain c d where k: "k = cbox c d"
+          using q(4)[OF x(2,1)] 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"
+        using interior_subset_union_intervals
+          by (metis as(2) k q(2) x interior_subset_union_intervals)
+        moreover
+        obtain c d where c_d: "k' = cbox c d"
+          using q(4)[OF x'(2,1)] by blast
+        have "interior k' \<inter> interior (cbox a b) = {}"
+          using as'(2) q(5) x' by blast
+        then have "interior k' \<subseteq> interior x'"
+          by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
+        ultimately show ?thesis
+          using assm(5)[OF x(2) x'(2) False] by auto
+      qed
+    qed
+  }
+qed
+
+lemma elementary_unions_intervals:
+  assumes fin: "finite f"
+    and "\<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_inter)
+  {
+    fix x
+    assume x: "x \<in> t" "x \<notin> s"
+    then have "x\<in>\<Union>r1"
+      unfolding r1 using ab by auto
+    then obtain r where r: "r \<in> r1" "x \<in> r"
+      unfolding Union_iff ..
+    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 *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
+    unfolding divp divq using assms(3) by auto
+  show ?thesis
+    apply (rule that[of "p \<union> r2"])
+    unfolding *
+    defer
+    apply (rule division_disjoint_union)
+    unfolding divp(6)
+    apply(rule assms r2)+
+  proof -
+    have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
+    proof (rule inter_interior_unions_intervals)
+      show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b"
+        using r1 by auto
+      have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
+        by auto
+      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
+      proof
+        fix m x
+        assume as: "m \<in> r1 - p"
+        have "interior m \<inter> interior (\<Union>p) = {}"
+        proof (rule inter_interior_unions_intervals)
+          show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b"
+            using divp by auto
+          show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
+            by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
+        qed
+        then show "interior s \<inter> interior m = {}"
+          unfolding divp by auto
+      qed
+    qed
+    then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
+      using interior_subset by auto
+  qed auto
+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 guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
+    guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
+    show "k \<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 guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
+    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 guess l unfolding mem_Collect_eq by (elim exE conjE) note l=this
+    guess u v using p(4)[OF l(2)] by (elim exE) note uv=this
+    show "k \<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 guess l' unfolding mem_Collect_eq by (elim exE conjE) note l'=this
+    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 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]:
+  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 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
+  apply auto
+  apply fastforce+
+  done
+
+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
+  unfolding tagged_partial_division_of_def
+  using finite_subset[OF assms(2)]
+  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_union:
+  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_unions:
+  assumes "finite iset"
+    and "\<forall>i\<in>iset. pfn i tagged_division_of i"
+    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
+  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
+proof (rule tagged_division_ofI)
+  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
+  show "finite (\<Union>(pfn ` iset))"
+    using assms by auto
+  have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
+    by blast
+  also have "\<dots> = \<Union>iset"
+    using assm(6) by auto
+  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
+  fix x k
+  assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
+  then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
+    by auto
+  show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
+    using assm(2-4)[OF i] using i(1) by auto
+  fix x' k'
+  assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
+  then obtain i' where i': "i' \<in> iset" "(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)
+    using assms(3)[rule_format] interior_mono
+    by blast
+  show "interior k \<inter> interior k' = {}"
+    apply (cases "i = i'")
+    using assm(5) i' i(2) xk'(2) apply blast
+    using "*" assm(3) i' i by auto
+qed
+
+lemma tagged_partial_division_of_union_self:
+  assumes "p tagged_partial_division_of s"
+  shows "p tagged_division_of (\<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
+
+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
+  @{text operative_division}. Instances for the monoid are @{typ "'a option"}, @{typ real}, and
+  @{typ bool}.\<close>
+
+paragraph \<open>Using additivity of lifted function to encode definedness.\<close>
+
+definition 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 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>Operative\<close>
+
+definition (in comm_monoid) operative :: "('b::euclidean_space set \<Rightarrow> 'a) \<Rightarrow> bool"
+  where "operative g \<longleftrightarrow>
+    (\<forall>a b. box a b = {} \<longrightarrow> g (cbox a b) = \<^bold>1) \<and>
+    (\<forall>a b c. \<forall>k\<in>Basis. 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}))"
+
+lemma (in comm_monoid) operativeD[dest]:
+  assumes "operative g"
+  shows "\<And>a b. box a b = {} \<Longrightarrow> g (cbox a b) = \<^bold>1"
+    and "\<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})"
+  using assms unfolding operative_def by auto
+
+lemma (in comm_monoid) operative_empty:
+  assumes g: "operative g" shows "g {} = \<^bold>1"
+proof -
+  have *: "cbox One (-One) = ({}::'b set)"
+    by (auto simp: box_eq_empty inner_setsum_left inner_Basis setsum.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 operativeD(1)[OF g, of One "-One"] by simp
+qed
+
+definition "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> ~(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
+  show ?t1 (*FIXME a horrible mess*)
+    unfolding division_points_def interval_split[OF k, of a b]
+    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
+    unfolding *
+    apply (rule subsetI)
+    unfolding mem_Collect_eq split_beta
+    apply (erule bexE conjE)+
+    apply (simp add: )
+    apply (erule exE conjE)+
+  proof
+    fix i l x
+    assume as:
+      "a \<bullet> fst x < snd x" "snd x < (if fst x = k then c else b \<bullet> fst x)"
+      "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
+      "i = l \<inter> {x. x \<bullet> k \<le> c}" "l \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} \<noteq> {}"
+      and fstx: "fst x \<in> Basis"
+    from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this
+    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 as(6) unfolding l interval_split[OF k] box_ne_empty as .
+    have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
+      using l using as(6) unfolding box_ne_empty[symmetric] by auto
+    show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
+      apply (rule bexI[OF _ \<open>l \<in> d\<close>])
+      using as(1-3,5) fstx
+      unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
+      apply (auto split: if_split_asm)
+      done
+    show "snd x < b \<bullet> fst x"
+      using as(2) \<open>c < b\<bullet>k\<close> by (auto split: if_split_asm)
+  qed
+  show ?t2
+    unfolding division_points_def interval_split[OF k, of a b]
+    unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)]
+    unfolding *
+    unfolding subset_eq
+    apply rule
+    unfolding mem_Collect_eq split_beta
+    apply (erule bexE conjE)+
+    apply (simp only: mem_Collect_eq inner_setsum_left_Basis simp_thms)
+    apply (erule exE conjE)+
+  proof
+    fix i l x
+    assume as:
+      "(if fst x = k then c else a \<bullet> fst x) < snd x" "snd x < b \<bullet> fst x"
+      "interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
+      "i = l \<inter> {x. c \<le> x \<bullet> k}" "l \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} \<noteq> {}"
+      and fstx: "fst x \<in> Basis"
+    from assm(4)[OF this(5)] guess u v by (elim exE) note l=this
+    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 as(6) unfolding l interval_split[OF k] box_ne_empty as .
+    have **: "\<forall>i\<in>Basis. u\<bullet>i \<le> v\<bullet>i"
+      using l using as(6) unfolding box_ne_empty[symmetric] by auto
+    show "\<exists>i\<in>d. interval_lowerbound i \<bullet> fst x = snd x \<or> interval_upperbound i \<bullet> fst x = snd x"
+      apply (rule bexI[OF _ \<open>l \<in> d\<close>])
+      using as(1-3,5) fstx
+      unfolding l interval_bounds[OF **] interval_bounds[OF *] interval_split[OF k] as
+      apply (auto split: if_split_asm)
+      done
+    show "a \<bullet> fst x < snd x"
+      using as(1) \<open>a\<bullet>k < c\<close> by (auto split: if_split_asm)
+   qed
+qed
+
+lemma division_points_psubset:
+  fixes a :: "'a::euclidean_space"
+  assumes "d division_of (cbox a b)"
+      and "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"  "a\<bullet>k < c" "c < b\<bullet>k"
+      and "l \<in> d"
+      and "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 assms(2) by (auto intro!:less_imp_le)
+  guess u v using division_ofD(4)[OF assms(1,5)] by (elim exE) note l=this
+  have uv: "\<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"
+    using division_ofD(2,2,3)[OF assms(1,5)] unfolding l box_ne_empty
+    using subset_box(1)
+    apply auto
+    apply blast+
+    done
+  have *: "interval_upperbound (cbox a b \<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]
+    apply -
+    apply (erule disjE)
+    apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
+    apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
+    done
+  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]
+    apply -
+    apply (erule disjE)
+    apply (rule_tac x="(k,(interval_lowerbound l)\<bullet>k)" in exI, force simp add: *)
+    apply (rule_tac x="(k,(interval_upperbound l)\<bullet>k)" in exI, force simp add: *)
+    done
+  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 type :: "'a::euclidean_space"
+  assumes "d division_of i"
+    and "k1 \<in> d"
+    and "k2 \<in> d"
+    and "k1 \<noteq> k2"
+    and "k1 \<inter> {x::'a. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c}"
+    and k: "k\<in>Basis"
+  shows "interior (k1 \<inter> {x. x\<bullet>k \<le> c}) = {}"
+proof -
+  note d=division_ofD[OF assms(1)]
+  guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
+  guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
+  have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}"
+    by auto
+  show ?thesis
+    unfolding uv1 uv2
+    apply (rule **[OF d(5)[OF assms(2-4)]])
+    apply (simp add: uv1)
+    using assms(5) uv1 by auto
+qed
+
+lemma division_split_right_inj:
+  fixes type :: "'a::euclidean_space"
+  assumes "d division_of i"
+    and "k1 \<in> d"
+    and "k2 \<in> d"
+    and "k1 \<noteq> k2"
+    and "k1 \<inter> {x::'a. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c}"
+    and k: "k \<in> Basis"
+  shows "interior (k1 \<inter> {x. x\<bullet>k \<ge> c}) = {}"
+proof -
+  note d=division_ofD[OF assms(1)]
+  guess u1 v1 using d(4)[OF assms(2)] by (elim exE) note uv1=this
+  guess u2 v2 using d(4)[OF assms(3)] by (elim exE) note uv2=this
+  have **: "\<And>s t u. s \<inter> t = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<subseteq> t \<Longrightarrow> u = {}"
+    by auto
+  show ?thesis
+    unfolding uv1 uv2
+    apply (rule **[OF d(5)[OF assms(2-4)]])
+    apply (simp add: uv1)
+    using assms(5) uv1 by auto
+qed
+
+lemma 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])
+      apply (rule equalityI)
+      apply blast
+      apply clarsimp
+      apply (rule_tac x="l \<inter> {x. c + e \<ge> x \<bullet> k}" in exI)
+      apply auto
+      done
+    by (simp add: interval_split k interval_doublesplit)
+qed
+
+lemma (in comm_monoid_set) operative_division:
+  fixes g :: "'b::euclidean_space set \<Rightarrow> 'a"
+  assumes g: "operative g" and d: "d division_of (cbox a b)" shows "F g d = g (cbox a b)"
+proof -
+  define C where [abs_def]: "C = card (division_points (cbox a b) d)"
+  then show ?thesis
+    using d
+  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 operativeD(1)[OF g, 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: operativeD(1)[OF g])
+    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 guess i .. note i=this
+        guess u v using d'[rule_format,OF i(1)] by (elim exE conjE) note uv=this
+        have "cbox a b \<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 guess u v by (elim exE conjE) note uv=this
+          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 operativeD(1)[OF g])
+        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 guess k c
+          unfolding split_paired_Ex division_points_def mem_Collect_eq split_conv
+          apply (elim exE conjE)
+          done
+        note this(2-4,1) note kc=this[unfolded interval_bounds[OF ab']]
+        from this(3) guess j .. note j=this
+        define d1 where "d1 = {l \<inter> {x. x\<bullet>k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
+        define d2 where "d2 = {l \<inter> {x. x\<bullet>k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x\<bullet>k \<ge> c} \<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)"
+        note division_points_psubset[OF \<open>d division_of cbox a b\<close> ab kc(1-2) j]
+        note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]
+        then have *: "F g d1 = g (cbox a b \<inter> {x. x\<bullet>k \<le> c})" "F g d2 = g (cbox a b \<inter> {x. x\<bullet>k \<ge> c})"
+          unfolding interval_split[OF kc(4)]
+          apply (rule_tac[!] "less.hyps"[rule_format])
+          using division_split[OF \<open>d division_of cbox a b\<close>, where k=k and c=c]
+          apply (simp_all add: interval_split kc d1_def d2_def division_points_finite[OF \<open>d division_of cbox a b\<close>])
+          done
+        { fix l y
+          assume as: "l \<in> d" "y \<in> d" "l \<inter> {x. x \<bullet> k \<le> c} = y \<inter> {x. x \<bullet> k \<le> c}" "l \<noteq> y"
+          from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this
+          have "g (l \<inter> {x. x \<bullet> k \<le> c}) = \<^bold>1"
+            unfolding leq interval_split[OF kc(4)]
+            apply (rule operativeD[OF g])
+            unfolding interior_cbox[symmetric] interval_split[symmetric, OF kc(4)]
+            using division_split_left_inj less as kc leq by blast
+        } note fxk_le = this
+        { fix l y
+          assume as: "l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x \<bullet> k} = y \<inter> {x. c \<le> x \<bullet> k}" "l \<noteq> y"
+          from division_ofD(4)[OF \<open>d division_of cbox a b\<close> this(1)] guess u v by (elim exE) note leq=this
+          have "g (l \<inter> {x. x \<bullet> k \<ge> c}) = \<^bold>1"
+            unfolding leq interval_split[OF kc(4)]
+            apply (rule operativeD(1)[OF g])
+            unfolding interior_cbox[symmetric] interval_split[symmetric,OF kc(4)]
+            using division_split_right_inj less leq as kc by blast
+        } note fxk_ge = this
+        have d1_alt: "d1 = (\<lambda>l. l \<inter> {x. x\<bullet>k \<le> c}) ` {l \<in> d. l \<inter> {x. x\<bullet>k \<le> c} \<noteq> {}}"
+          using d1_def by auto
+        have d2_alt: "d2 = (\<lambda>l. l \<inter> {x. x\<bullet>k \<ge> c}) ` {l \<in> d. l \<inter> {x. x\<bullet>k \<ge> c} \<noteq> {}}"
+          using d2_def by auto
+        have "g (cbox a b) = F g d1 \<^bold>* F g d2" (is "_ = ?prev")
+          unfolding * using g kc(4) by blast
+        also have "F g d1 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d"
+          unfolding d1_alt using division_of_finite[OF less.prems] fxk_le
+          by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g])
+        also have "F g d2 = F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d"
+          unfolding d2_alt using division_of_finite[OF less.prems] fxk_ge
+          by (subst reindex_nontrivial) (auto intro!: mono_neutral_cong_left simp: operative_empty[OF g])
+        also have *: "\<forall>x\<in>d. g x = g (x \<inter> {x. x \<bullet> k \<le> c}) \<^bold>* g (x \<inter> {x. c \<le> x \<bullet> k})"
+          unfolding forall_in_division[OF \<open>d division_of cbox a b\<close>]
+          using g kc(4) by blast
+        have "F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<le> c})) d \<^bold>* F (\<lambda>l. g (l \<inter> {x. x\<bullet>k \<ge> c})) d = F g d"
+          using * by (simp add: distrib)
+        finally show ?thesis by auto
+      qed
+    qed
+  qed
+qed
+
+lemma (in comm_monoid_set) over_tagged_division_lemma:
+  assumes "p tagged_division_of i"
+    and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> box u v = {} \<Longrightarrow> d (cbox u v) = \<^bold>1"
+  shows "F (\<lambda>(x,k). d k) p = F d (snd ` p)"
+proof -
+  have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
+    unfolding o_def by (rule ext) auto
+  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"
+      by (metis prod.collapse \<open>x\<in>p\<close> \<open>snd x = snd y\<close> \<open>x \<noteq> y\<close>)+
+    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
+
+lemma (in comm_monoid_set) operative_tagged_division:
+  assumes f: "operative g" and d: "d tagged_division_of (cbox a b)"
+  shows "F (\<lambda>(x, l). g l) d = g (cbox a b)"
+  unfolding d[THEN division_of_tagged_division, THEN operative_division[OF f], symmetric]
+  by (simp add: f[THEN operativeD(1)] over_tagged_division_lemma[OF d])
+
+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 (in comm_monoid) operative_1_lt:
+  "operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow>
+    ((\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1) \<and> (\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a .. c} \<^bold>* g {c .. b} = g {a .. b}))"
+  apply (simp add: operative_def atMost_def[symmetric] atLeast_def[symmetric])
+proof safe
+  fix a b c :: real
+  assume *: "\<forall>a b c. g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
+  assume "a < c" "c < b"
+  with *[rule_format, of a b c] show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
+    by (simp add: less_imp_le min.absorb2 max.absorb2)
+next
+  fix a b c :: real
+  assume as: "\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1"
+    "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
+  from as(1)[rule_format, of 0 1] as(1)[rule_format, of a a for a] as(2)
+  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
+  show "g {a..b} = g {a..min b c} \<^bold>* g {max a c..b}"
+    by (auto simp: min_def max_def le_less)
+qed
+
+lemma (in comm_monoid) operative_1_le:
+  "operative (g :: real set \<Rightarrow> 'a) \<longleftrightarrow>
+    ((\<forall>a b. b \<le> a \<longrightarrow> g {a..b} = \<^bold>1) \<and> (\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> g {a .. c} \<^bold>* g {c .. b} = g {a .. b}))"
+  unfolding operative_1_lt
+proof safe
+  fix a b c :: real
+  assume as: "\<forall>a b c. a \<le> c \<and> c \<le> b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}" "a < c" "c < b"
+  show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
+    apply (rule as(1)[rule_format])
+    using as(2-)
+    apply auto
+    done
+next
+  fix a b c :: real
+  assume "\<forall>a b. b \<le> a \<longrightarrow> g {a .. b} = \<^bold>1"
+    and "\<forall>a b c. a < c \<and> c < b \<longrightarrow> g {a..c} \<^bold>* g {c..b} = g {a..b}"
+    and "a \<le> c"
+    and "c \<le> b"
+  note as = this[rule_format]
+  show "g {a..c} \<^bold>* g {c..b} = g {a..b}"
+  proof (cases "c = a \<or> c = b")
+    case False
+    then show ?thesis
+      apply -
+      apply (subst as(2))
+      using as(3-)
+      apply auto
+      done
+  next
+    case True
+    then show ?thesis
+    proof
+      assume *: "c = a"
+      then have "g {a .. c} = \<^bold>1"
+        apply -
+        apply (rule as(1)[rule_format])
+        apply auto
+        done
+      then show ?thesis
+        unfolding * by auto
+    next
+      assume *: "c = b"
+      then have "g {c .. b} = \<^bold>1"
+        apply -
+        apply (rule as(1)[rule_format])
+        apply auto
+        done
+      then show ?thesis
+        unfolding * by auto
+    qed
+  qed
+qed
+
+lemma tagged_division_union_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_union[OF assms(1-2)])
+    unfolding interval_split[OF k] interior_cbox
+    using k
+    apply (auto simp add: box_def elim!: ballE[where x=k])
+    done
+qed
+
+lemma tagged_division_union_interval_real:
+  fixes a :: real
+  assumes "p1 tagged_division_of ({a .. b} \<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_union_interval)
+
+lemma tagged_division_split_left_inj:
+  "d tagged_division_of i \<Longrightarrow> (x1, k1) \<in> d \<Longrightarrow> (x2, k2) \<in> d \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow>
+    k1 \<inter> {x. x\<bullet>k \<le> c} = k2 \<inter> {x. x\<bullet>k \<le> c} \<Longrightarrow> k \<in> Basis \<Longrightarrow>
+    interior (k1 \<inter> {x. x\<bullet>k \<le> c}) = {}"
+  by (intro division_split_left_inj[of "snd`d" i k1 k2, OF division_of_tagged_division])
+     (auto simp add: snd_def[abs_def] image_iff split: prod.split )
+
+lemma tagged_division_split_right_inj:
+  "d tagged_division_of i \<Longrightarrow> (x1, k1) \<in> d \<Longrightarrow> (x2, k2) \<in> d \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow>
+    k1 \<inter> {x. x\<bullet>k \<ge> c} = k2 \<inter> {x. x\<bullet>k \<ge> c} \<Longrightarrow> k \<in> Basis \<Longrightarrow>
+    interior (k1 \<inter> {x. x\<bullet>k \<ge> c}) = {}"
+  by (intro division_split_right_inj[of "snd`d" i k1 k2, OF division_of_tagged_division])
+     (auto simp add: snd_def[abs_def] image_iff split: prod.split )
+
+subsection \<open>Special case of additivity we need for the FTC.\<close>
+
+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 "setsum (\<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))"
+  have ***: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
+    using assms by auto
+  have *: "add.operative ?f"
+    unfolding add.operative_1_lt box_eq_empty
+    by auto
+  have **: "cbox a b \<noteq> {}"
+    using assms(1) by auto
+  note setsum.operative_tagged_division[OF * assms(2)[simplified box_real[symmetric]]]
+  note * = this[unfolded if_not_P[OF **] interval_bounds[OF ***],symmetric]
+  show ?thesis
+    unfolding *
+    apply (rule setsum.cong)
+    unfolding split_paired_all split_conv
+    using assms(2)
+    apply auto
+    done
+qed
+
+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)
+
+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 "setsum (\<lambda>(x,k). f (Sup k) - f(Inf k)) p = f b - f a"
+  using additive_tagged_division_1[OF _ assms(2), of f]
+  using assms(1)
+  by auto
+
+subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close>
+
+definition 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_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
+  unfolding fine_def by auto
+
+lemma fine_inters:
+ "(\<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_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
+  unfolding fine_def by blast
+
+lemma fine_unions: "(\<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_unions_exists:
+  assumes "finite iset"
+    and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
+    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
+    and "\<Union>iset = i"
+   obtains p where "p tagged_division_of i" and "d fine p"
+proof -
+  obtain pfn where pfn:
+    "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
+    "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
+    using bchoice[OF assms(2)] by auto
+  show thesis
+    apply (rule_tac p="\<Union>(pfn ` iset)" in that)
+    using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
+    by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
+qed
+
+
+subsection \<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
+
+subsection \<open>General bisection principle for intervals; might be useful elsewhere.\<close>
+
+lemma interval_bisection_step:
+  fixes type :: "'a::euclidean_space"
+  assumes "P {}"
+    and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
+    and "\<not> P (cbox a (b::'a))"
+  obtains c d where "\<not> P (cbox c d)"
+    and "\<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"
+proof -
+  have "cbox a b \<noteq> {}"
+    using assms(1,3) by metis
+  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
+    by (force simp: mem_box)
+  { fix f
+    have "\<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)"
+    proof (induct f rule: finite_induct)
+      case empty
+      show ?case
+        using assms(1) by auto
+    next
+      case (insert x f)
+      show ?case
+        unfolding Union_insert
+        apply (rule assms(2)[rule_format])
+        using inter_interior_unions_intervals [of f "interior x"]
+        apply (auto simp: insert)
+        by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
+    qed
+  } note UN_cases = this
+  let ?A = "{cbox c d | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or>
+    (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
+  let ?PP = "\<lambda>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"
+  {
+    presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False"
+    then show thesis
+      unfolding atomize_not not_all
+      by (blast intro: that)
+  }
+  assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)"
+  have "P (\<Union>?A)"
+  proof (rule UN_cases)
+    let ?B = "(\<lambda>s. cbox (\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i::'a)
+      (\<Sum>i\<in>Basis. (if i \<in> s then (a\<bullet>i + b\<bullet>i) / 2 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 = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+                        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
+      show "x \<in> ?B"
+        unfolding image_iff x
+        apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
+        apply (rule arg_cong2 [where f = cbox])
+        using x(2) ab
+        apply (auto simp add: euclidean_eq_iff[where 'a='a])
+        by fastforce
+    qed
+    then show "finite ?A"
+      by (rule finite_subset) auto
+  next
+    fix s
+    assume "s \<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 = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+                     c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
+      by blast
+    show "P s"
+      unfolding s
+      apply (rule as[rule_format])
+      using ab s(2) by force
+    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 = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+        e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<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']
+      proof safe
+        assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
+        show False
+          using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
+      next
+        assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
+        show False
+          using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
+      qed
+    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 = (a \<bullet> i + b \<bullet> i) / 2 \<or>
+        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
+      by blast
+    show "x\<in>cbox a b"
+      unfolding mem_box
+    proof safe
+      fix i :: 'a
+      assume i: "i \<in> Basis"
+      then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
+        using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
+    qed
+  next
+    fix x
+    assume x: "x \<in> cbox a b"
+    have "\<forall>i\<in>Basis.
+      \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<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
+    proof
+      fix i :: 'a
+      assume i: "i \<in> Basis"
+      have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)"
+        using x[unfolded mem_box,THEN bspec, OF i] by auto
+      then show "\<exists>c d. ?P i c d"
+        by blast
+    qed
+    then show "x\<in>\<Union>?A"
+      unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
+      apply auto
+      apply (rule_tac x="cbox xa xaa" in exI)
+      unfolding mem_box
+      apply auto
+      done
+  qed
+  finally show False
+    using assms by auto
+qed
+
+lemma interval_bisection:
+  fixes type :: "'a::euclidean_space"
+  assumes "P {}"
+    and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<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 as: False
+      obtain c d where "\<not> P (cbox c d)"
+        "\<forall>i\<in>Basis.
+           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 (rule interval_bisection_step[of P, OF assms(1-2) as])
+      then show ?thesis
+        apply -
+        apply (rule_tac x="(c,d)" in exI)
+        apply auto
+        done
+    qed
+  qed
+  then obtain f where f:
+    "\<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)"
+    apply -
+    apply (drule choice)
+    apply blast
+    done
+  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 "A 0 = a" "B 0 = b" "\<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
+        apply (rule f[rule_format]) using assms(3)
+        apply auto
+        done
+    next
+      case (Suc n)
+      show ?case
+        unfolding S
+        apply (rule f[rule_format])
+        using Suc
+        unfolding S
+        apply auto
+        done
+    qed
+  qed
+  note AB = this(1-2) conjunctD2[OF this(3),rule_format]
+
+  have interv: "\<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 "(setsum (\<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> setsum (\<lambda>i. \<bar>(x - y)\<bullet>i\<bar>) Basis"
+        unfolding dist_norm by(rule norm_le_l1)
+      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
+      proof (rule setsum_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> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
+        unfolding setsum_divide_distrib
+      proof (rule setsum_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(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(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
+    qed simp
+  } note ABsubset = this
+  have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)"
+    by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
+      (metis nat.exhaust AB(1-3) assms(1,3))
+  then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
+    by blast
+  show thesis
+  proof (rule that[rule_format, of x0])
+    show "x0\<in>cbox a b"
+      using x0[of 0] unfolding AB .
+    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))"
+      apply (cases "0 < n")
+      using AB(3)[of "n - 1"] assms(3) AB(1-2)
+      apply auto
+      done
+    moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
+      using n using x0[of n] by auto
+    moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
+      unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
+    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:
+  fixes a b :: "'a::euclidean_space"
+  assumes "gauge g"
+  obtains p where "p tagged_division_of (cbox a b)" "g fine p"
+proof -
+  presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
+  then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
+    by blast
+  then show thesis ..
+next
+  assume as: "\<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 _ _ as])
+    apply (simp add: fine_def)
+    apply (metis tagged_division_union fine_union)
+    apply (auto simp: )
+    done
+  obtain e where e: "e > 0" "ball x e \<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 False
+    using tagged_division_of_self[OF c_d(1)] using c_d by auto
+qed
+
+lemma fine_division_exists_real:
+  fixes a b :: real
+  assumes "gauge g"
+  obtains p where "p tagged_division_of {a .. b}" "g fine p"
+  by (metis assms box_real(2) fine_division_exists)
+
+subsection \<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 "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 -
+  let ?P = "\<lambda>p. p tagged_partial_division_of (cbox a b) \<longrightarrow> gauge d \<longrightarrow>
+    (\<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))"
+  {
+    have *: "finite p" "p tagged_partial_division_of (cbox a b)"
+      using assms(1)
+      unfolding tagged_division_of_def
+      by auto
+    presume "\<And>p. finite p \<Longrightarrow> ?P p"
+    from this[rule_format,OF * assms(2)] guess q .. note q=this
+    then show ?thesis
+      apply -
+      apply (rule that[of q])
+      unfolding tagged_division_ofD[OF assms(1)]
+      apply auto
+      done
+  }
+  fix p :: "('a::euclidean_space \<times> ('a::euclidean_space set)) set"
+  assume as: "finite p"
+  show "?P p"
+    apply rule
+    apply rule
+    using as
+  proof (induct p)
+    case empty
+    show ?case
+      apply (rule_tac x="{}" in exI)
+      unfolding fine_def
+      apply auto
+      done
+  next
+    case (insert xk p)
+    guess x k using surj_pair[of xk] by (elim exE) note xk=this
+    note tagged_partial_division_subset[OF insert(4) subset_insertI]
+    from insert(3)[OF this insert(5)] guess q1 .. note q1 = conjunctD3[OF this]
+    have *: "\<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)]
+    from p(4)[unfolded xk, OF insertI1] guess u v by (elim exE) note uv=this
+
+    have "finite {k. \<exists>x. (x, k) \<in> p}"
+      apply (rule finite_subset[of _ "snd ` p"])
+      using p
+      apply safe
+      apply (metis image_iff snd_conv)
+      apply auto
+      done
+    then have int: "interior (cbox u v) \<inter> interior (\<Union>{k. \<exists>x. (x, k) \<in> p}) = {}"
+      apply (rule inter_interior_unions_intervals)
+      apply (rule open_interior)
+      apply (rule_tac[!] ballI)
+      unfolding mem_Collect_eq
+      apply (erule_tac[!] exE)
+      apply (drule p(4)[OF insertI2])
+      apply assumption
+      apply (rule p(5))
+      unfolding uv xk
+      apply (rule insertI1)
+      apply (rule insertI2)
+      apply assumption
+      using insert(2)
+      unfolding uv xk
+      apply auto
+      done
+    show ?case
+    proof (cases "cbox u v \<subseteq> d x")
+      case True
+      then show ?thesis
+        apply (rule_tac x="{(x,cbox u v)} \<union> q1" in exI)
+        apply rule
+        unfolding * uv
+        apply (rule tagged_division_union)
+        apply (rule tagged_division_of_self)
+        apply (rule p[unfolded xk uv] insertI1)+
+        apply (rule q1)
+        apply (rule int)
+        apply rule
+        apply (rule fine_union)
+        apply (subst fine_def)
+        defer
+        apply (rule q1)
+        unfolding Ball_def split_paired_All split_conv
+        apply rule
+        apply rule
+        apply rule
+        apply rule
+        apply (erule insertE)
+        apply (simp add: uv xk)
+        apply (rule UnI2)
+        apply (drule q1(3)[rule_format])
+        unfolding xk uv
+        apply auto
+        done
+    next
+      case False
+      from fine_division_exists[OF assms(2), of u v] guess q2 . note q2=this
+      show ?thesis
+        apply (rule_tac x="q2 \<union> q1" in exI)
+        apply rule
+        unfolding * uv
+        apply (rule tagged_division_union q2 q1 int fine_union)+
+        unfolding Ball_def split_paired_All split_conv
+        apply rule
+        apply (rule fine_union)
+        apply (rule q1 q2)+
+        apply rule
+        apply rule
+        apply rule
+        apply rule
+        apply (erule insertE)
+        apply (rule UnI2)
+        apply (simp add: False uv xk)
+        apply (drule q1(3)[rule_format])
+        using False
+        unfolding xk uv
+        apply auto
+        done
+    qed
+  qed
+qed
+
+subsubsection \<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. PiE 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: divide_simps zero_le_mult_iff aibi)
+      apply (force simp: aibi scaling_mono nat_less_real_le dest: PiE_mem)
+      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: divide_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 --\<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" --\<open>WLOG we can assume @{term"m \<le> n"}, 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 add: power_add, 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 setsum.cong)
+            also have "... \<le> DIM('a) * (\<epsilon> / (2 * real DIM('a)))"
+              by (meson setsum_bounded_above that)
+            also have "... = \<epsilon> / 2"
+              by (simp add: divide_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: divide_simps)
+        then have "norm (b - a) < e * 2^n"
+          by (auto simp: divide_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 divide_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 divide_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 divide_simps)
+                done
+              also have "... = x \<bullet> i"
+                using abi_less by (simp add: divide_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: divide_simps algebra_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 by (force simp: algebra_simps)
+        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. PiE 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: divide_simps algebra_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 algebra_simps divide_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> ~(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 division_filter :: "'a::euclidean_space set \<Rightarrow> ('a \<times> 'a set) set filter"
+  where "division_filter s = (INF g:{g. gauge g}. principal {p. p tagged_division_of s \<and> g fine p})"
+
+lemma 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_inter)
+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
\ No newline at end of file
diff -r b235e845c8e8 -r c3da799b1b45 src/HOL/Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Wed Sep 28 16:15:51 2016 +0200
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Thu Sep 29 13:02:43 2016 +0200
@@ -1414,7 +1414,7 @@
   then have "cbox a b \<subseteq> box c d" "box c d \<subseteq>cbox a b"
     by auto
   then show ?rhs
-    apply (simp add: subset_box) 
+    apply (simp add: subset_box)
     using \<open>cbox a b = box c d\<close> box_ne_empty box_sing
     apply (fastforce simp add:)
     done
@@ -1435,7 +1435,7 @@
     by auto
   then show ?rhs
     apply (simp add: subset_box)
-    using box_ne_empty(2) \<open>box a b = box c d\<close> 
+    using box_ne_empty(2) \<open>box a b = box c d\<close>
     apply auto
      apply (meson euclidean_eqI less_eq_real_def not_less)+
     done
@@ -10009,6 +10009,83 @@
     by (metis (no_types, lifting) IntD2 IntI f someI_ex)
 qed
 
+lemma tube_lemma:
+  assumes "compact K"
+  assumes "open W"
+  assumes "{x0} \<times> K \<subseteq> W"
+  shows "\<exists>X0. x0 \<in> X0 \<and> open X0 \<and> X0 \<times> K \<subseteq> W"
+proof -
+  {
+    fix y assume "y \<in> K"
+    then have "(x0, y) \<in> W" using assms by auto
+    with \<open>open W\<close>
+    have "\<exists>X0 Y. open X0 \<and> open Y \<and> x0 \<in> X0 \<and> y \<in> Y \<and> X0 \<times> Y \<subseteq> W"
+      by (rule open_prod_elim) blast
+  }
+  then obtain X0 Y where
+    *: "\<forall>y \<in> K. open (X0 y) \<and> open (Y y) \<and> x0 \<in> X0 y \<and> y \<in> Y y \<and> X0 y \<times> Y y \<subseteq> W"
+    by metis
+  from * have "\<forall>t\<in>Y ` K. open t" "K \<subseteq> \<Union>(Y ` K)" by auto
+  with \<open>compact K\<close> obtain CC where CC: "CC \<subseteq> Y ` K" "finite CC" "K \<subseteq> \<Union>CC"
+    by (rule compactE)
+  then obtain c where c: "\<And>C. C \<in> CC \<Longrightarrow> c C \<in> K \<and> C = Y (c C)"
+    by (force intro!: choice)
+  with * CC show ?thesis
+    by (force intro!: exI[where x="\<Inter>C\<in>CC. X0 (c C)"]) (* SLOW *)
+qed
+
+lemma continuous_on_prod_compactE:
+  fixes fx::"'a::topological_space \<times> 'b::topological_space \<Rightarrow> 'c::metric_space"
+    and e::real
+  assumes cont_fx: "continuous_on (U \<times> C) fx"
+  assumes "compact C"
+  assumes [intro]: "x0 \<in> U"
+  notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
+  assumes "e > 0"
+  obtains X0 where "x0 \<in> X0" "open X0"
+    "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
+proof -
+  define psi where "psi = (\<lambda>(x, t). dist (fx (x, t)) (fx (x0, t)))"
+  define W0 where "W0 = {(x, t) \<in> U \<times> C. psi (x, t) < e}"
+  have W0_eq: "W0 = psi -` {..<e} \<inter> U \<times> C"
+    by (auto simp: vimage_def W0_def)
+  have "open {..<e}" by simp
+  have "continuous_on (U \<times> C) psi"
+    by (auto intro!: continuous_intros simp: psi_def split_beta')
+  from this[unfolded continuous_on_open_invariant, rule_format, OF \<open>open {..<e}\<close>]
+  obtain W where W: "open W" "W \<inter> U \<times> C = W0 \<inter> U \<times> C"
+    unfolding W0_eq by blast
+  have "{x0} \<times> C \<subseteq> W \<inter> U \<times> C"
+    unfolding W
+    by (auto simp: W0_def psi_def \<open>0 < e\<close>)
+  then have "{x0} \<times> C \<subseteq> W" by blast
+  from tube_lemma[OF \<open>compact C\<close> \<open>open W\<close> this]
+  obtain X0 where X0: "x0 \<in> X0" "open X0" "X0 \<times> C \<subseteq> W"
+    by blast
+
+  have "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
+  proof safe
+    fix x assume x: "x \<in> X0" "x \<in> U"
+    fix t assume t: "t \<in> C"
+    have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
+      by (auto simp: psi_def)
+    also
+    {
+      have "(x, t) \<in> X0 \<times> C"
+        using t x
+        by auto
+      also note \<open>\<dots> \<subseteq> W\<close>
+      finally have "(x, t) \<in> W" .
+      with t x have "(x, t) \<in> W \<inter> U \<times> C"
+        by blast
+      also note \<open>W \<inter> U \<times> C = W0 \<inter> U \<times> C\<close>
+      finally  have "psi (x, t) < e"
+        by (auto simp: W0_def)
+    }
+    finally show "dist (fx (x, t)) (fx (x0, t)) \<le> e" by simp
+  qed
+  from X0(1,2) this show ?thesis ..
+qed
 
 no_notation
   eucl_less (infix "<e" 50)