header {* Kurzweil-Henstock gauge integration in many dimensions. *}

 (*  Author:                     John Harrison

     Translation from HOL light: Robert Himmelmann, TU Muenchen *)

 theory Integration_Aleph

   imports Derivative SMT

 begin

 declare [[smt_certificates="~~/src/HOL/Multivariate_Analysis/Integration.cert"]]

 declare [[smt_record=true]]

 declare [[z3_proofs=true]]

 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 conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto

 declare smult_conv_scaleR[simp]

 subsection {* Some useful lemmas about intervals. *}

 lemma empty_as_interval: "{} = {1..0::real^'n}"

   apply(rule set_ext,rule) defer unfolding vector_le_def mem_interval

   using UNIV_witness[where 'a='n] apply(erule_tac exE,rule_tac x=x in allE) by auto

 lemma interior_subset_union_intervals: 

   assumes "i = {a..b::real^'n}" "j = {c..d}" "interior j \<noteq> {}" "i \<subseteq> j \<union> s" "interior(i) \<inter> interior(j) = {}"

   shows "interior i \<subseteq> interior s" proof-

   have "{a<..<b} \<inter> {c..d} = {}" using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)

     unfolding assms(1,2) interior_closed_interval by auto

   moreover have "{a<..<b} \<subseteq> {c..d} \<union> s" apply(rule order_trans,rule interval_open_subset_closed)

     using assms(4) unfolding assms(1,2) by auto

   ultimately show ?thesis apply-apply(rule interior_maximal) defer apply(rule open_interior)

     unfolding assms(1,2) interior_closed_interval by auto qed

 lemma inter_interior_unions_intervals: fixes f::"(real^'n) set set"

   assumes "finite f" "open s" "\<forall>t\<in>f. \<exists>a b. t = {a..b}" "\<forall>t\<in>f. s \<inter> (interior t) = {}"

   shows "s \<inter> interior(\<Union>f) = {}" proof(rule ccontr,unfold ex_in_conv[THEN sym]) case goal1

   have lem1:"\<And>x e s U. ball x e \<subseteq> s \<inter> interior U \<longleftrightarrow> ball x e \<subseteq> s \<inter> U" apply rule  defer apply(rule_tac Int_greatest)

     unfolding open_subset_interior[OF open_ball]  using interior_subset by auto

   have lem2:"\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto

   have "\<And>f. finite f \<Longrightarrow> (\<forall>t\<in>f. \<exists>a b. t = {a..b}) \<Longrightarrow> (\<exists>x. x \<in> s \<inter> interior (\<Union>f)) \<Longrightarrow> (\<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t)" proof- case goal1

   thus ?case proof(induct rule:finite_induct) 

     case empty from this(2) guess x .. hence False unfolding Union_empty interior_empty by auto thus ?case by auto next

     case (insert i f) guess x using insert(5) .. note x = this

     then guess e unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior],rule_format] .. note e=this

     guess a using insert(4)[rule_format,OF insertI1] .. then guess b .. note ab = this

     show ?case proof(cases "x\<in>i") case False hence "x \<in> UNIV - {a..b}" unfolding ab by auto

       then guess d unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],rule_format] ..

       hence "0 < d" "ball x (min d e) \<subseteq> UNIV - i" using e unfolding ab by auto

       hence "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)" using e unfolding lem1 by auto hence "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto

       hence "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t" apply-apply(rule insert(3)) using insert(4) by auto thus ?thesis by auto next

     case True show ?thesis proof(cases "x\<in>{a<..<b}")

       case True then guess d unfolding open_contains_ball_eq[OF open_interval,rule_format] ..

       thus ?thesis apply(rule_tac x=i in bexI,rule_tac x=x in exI,rule_tac x="min d e" in exI)

 	unfolding ab using interval_open_subset_closed[of a b] and e by fastsimp+ next

     case False then obtain k where "x$k \<le> a$k \<or> x$k \<ge> b$k" unfolding mem_interval by(auto simp add:not_less) 

     hence "x$k = a$k \<or> x$k = b$k" using True unfolding ab and mem_interval apply(erule_tac x=k in allE) by auto

     hence "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)" proof(erule_tac disjE)

       let ?z = "x - (e/2) *\<^sub>R basis k" assume as:"x$k = a$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)

 	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto

 	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto

 	hence "y$k < a$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)

 	hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed

       moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof

 	fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R basis k)"

 	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R basis k"])

 	  unfolding norm_scaleR norm_basis by auto

 	also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps)

 	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed

       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto

     next let ?z = "x + (e/2) *\<^sub>R basis k" assume as:"x$k = b$k" have "ball ?z (e / 2) \<inter> i = {}" apply(rule ccontr) unfolding ex_in_conv[THEN sym] proof(erule exE)

 	fix y assume "y \<in> ball ?z (e / 2) \<inter> i" hence "dist ?z y < e/2" and yi:"y\<in>i" by auto

 	hence "\<bar>(?z - y) $ k\<bar> < e/2" using component_le_norm[of "?z - y" k] unfolding vector_dist_norm by auto

 	hence "y$k > b$k" unfolding vector_component_simps vector_scaleR_component as using e[THEN conjunct1] by(auto simp add:field_simps)

 	hence "y \<notin> i" unfolding ab mem_interval not_all by(rule_tac x=k in exI,auto) thus False using yi by auto qed

       moreover have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)" apply(rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]]) proof

 	fix y assume as:"y\<in> ball ?z (e/2)" have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R basis k)"

 	   apply-apply(rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R basis k"])

 	  unfolding norm_scaleR norm_basis by auto

 	also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2" apply(rule add_strict_left_mono) using as unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps)

 	finally show "y\<in>ball x e" unfolding mem_ball vector_dist_norm using e by(auto simp add:field_simps) qed

       ultimately show ?thesis apply(rule_tac x="?z" in exI) unfolding Union_insert by auto qed 

     then guess x .. hence "x \<in> s \<inter> interior (\<Union>f)" unfolding lem1[where U="\<Union>f",THEN sym] using centre_in_ball e[THEN conjunct1] by auto

     thus ?thesis apply-apply(rule lem2,rule insert(3)) using insert(4) by auto qed qed qed qed note * = this

   guess t using *[OF assms(1,3) goal1]  .. from this(2) guess x .. then guess e ..

   hence "x \<in> s" "x\<in>interior t" defer using open_subset_interior[OF open_ball, of x e t] by auto

   thus False using `t\<in>f` assms(4) by auto qed

 subsection {* Bounds on intervals where they exist. *}

 definition "interval_upperbound (s::(real^'n) set) = (\<chi> i. Sup {a. \<exists>x\<in>s. x$i = a})"

 definition "interval_lowerbound (s::(real^'n) set) = (\<chi> i. Inf {a. \<exists>x\<in>s. x$i = a})"

 lemma interval_upperbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_upperbound {a..b} = b"

   using assms unfolding interval_upperbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)

   apply(rule Sup_unique) unfolding setle_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer

   apply(rule,rule) apply(rule_tac x="b$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=b in bexI)

   unfolding mem_interval using assms by auto

 lemma interval_lowerbound[simp]: assumes "\<forall>i. a$i \<le> b$i" shows "interval_lowerbound {a..b} = a"

   using assms unfolding interval_lowerbound_def Cart_eq Cart_lambda_beta apply-apply(rule,erule_tac x=i in allE)

   apply(rule Inf_unique) unfolding setge_def apply rule unfolding mem_Collect_eq apply(erule bexE) unfolding mem_interval defer

   apply(rule,rule) apply(rule_tac x="a$i" in bexI) defer unfolding mem_Collect_eq apply(rule_tac x=a in bexI)

   unfolding mem_interval using assms by auto

 lemmas interval_bounds = interval_upperbound interval_lowerbound

 lemma interval_bounds'[simp]: assumes "{a..b}\<noteq>{}" shows "interval_upperbound {a..b} = b" "interval_lowerbound {a..b} = a"

   using assms unfolding interval_ne_empty by auto

 lemma interval_upperbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_upperbound {a..b} = (b::real^1)"

   apply(rule interval_upperbound) by auto

 lemma interval_lowerbound_1[simp]: "dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> interval_lowerbound {a..b} = (a::real^1)"

   apply(rule interval_lowerbound) by auto

 lemmas interval_bound_1 = interval_upperbound_1 interval_lowerbound_1

 subsection {* Content (length, area, volume...) of an interval. *}

 definition "content (s::(real^'n) set) =

        (if s = {} then 0 else (\<Prod>i\<in>UNIV. (interval_upperbound s)$i - (interval_lowerbound s)$i))"

 lemma interval_not_empty:"\<forall>i. a$i \<le> b$i \<Longrightarrow> {a..b::real^'n} \<noteq> {}"

   unfolding interval_eq_empty unfolding not_ex not_less by assumption

 lemma content_closed_interval: assumes "\<forall>i. a$i \<le> b$i"

   shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"

   using interval_not_empty[OF assms] unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms] by auto

 lemma content_closed_interval': assumes "{a..b}\<noteq>{}" shows "content {a..b} = (\<Prod>i\<in>UNIV. b$i - a$i)"

   apply(rule content_closed_interval) using assms unfolding interval_ne_empty .

 lemma content_1:"dest_vec1 a \<le> dest_vec1 b \<Longrightarrow> content {a..b} = dest_vec1 b - dest_vec1 a"

   using content_closed_interval[of a b] by auto

 lemma content_1':"a \<le> b \<Longrightarrow> content {vec1 a..vec1 b} = b - a" using content_1[of "vec a" "vec b"] by auto

 lemma content_unit[intro]: "content{0..1::real^'n} = 1" proof-

   have *:"\<forall>i. 0$i \<le> (1::real^'n::finite)$i" by auto

   have "0 \<in> {0..1::real^'n::finite}" unfolding mem_interval by auto

   thus ?thesis unfolding content_def interval_bounds[OF *] using setprod_1 by auto qed

 lemma content_pos_le[intro]: "0 \<le> content {a..b}" proof(cases "{a..b}={}")

   case False hence *:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by assumption

   have "(\<Prod>i\<in>UNIV. interval_upperbound {a..b} $ i - interval_lowerbound {a..b} $ i) \<ge> 0"

     apply(rule setprod_nonneg) unfolding interval_bounds[OF *] using * apply(erule_tac x=x in allE) by auto

   thus ?thesis unfolding content_def by(auto simp del:interval_bounds') qed(unfold content_def, auto)

 lemma content_pos_lt: assumes "\<forall>i. a$i < b$i" shows "0 < content {a..b}"

 proof- have help_lemma1: "\<forall>i. a$i < b$i \<Longrightarrow> \<forall>i. a$i \<le> ((b$i)::real)" apply(rule,erule_tac x=i in allE) by auto

   show ?thesis unfolding content_closed_interval[OF help_lemma1[OF assms]] apply(rule setprod_pos)

     using assms apply(erule_tac x=x in allE) by auto qed

 lemma content_pos_lt_1: "dest_vec1 a < dest_vec1 b \<Longrightarrow> 0 < content({a..b})"

   apply(rule content_pos_lt) by auto

 lemma content_eq_0: "content({a..b::real^'n}) = 0 \<longleftrightarrow> (\<exists>i. b$i \<le> a$i)" proof(cases "{a..b} = {}")

   case True thus ?thesis unfolding content_def if_P[OF True] unfolding interval_eq_empty apply-

     apply(rule,erule exE) apply(rule_tac x=i in exI) by auto next

   guess a using UNIV_witness[where 'a='n] .. case False note as=this[unfolded interval_eq_empty not_ex not_less]

   show ?thesis unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_UNIV]

     apply(rule) apply(erule_tac[!] exE bexE) unfolding interval_bounds[OF as] apply(rule_tac x=x in exI) defer

     apply(rule_tac x=i in bexI) using as apply(erule_tac x=i in allE) by auto qed

 lemma cond_cases:"(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)" by auto

 lemma content_closed_interval_cases:

   "content {a..b} = (if \<forall>i. a$i \<le> b$i then setprod (\<lambda>i. b$i - a$i) UNIV else 0)" apply(rule cond_cases) 

   apply(rule content_closed_interval) unfolding content_eq_0 not_all not_le defer apply(erule exE,rule_tac x=x in exI) by auto

 lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"

   unfolding content_eq_0 interior_closed_interval interval_eq_empty by auto

 lemma content_eq_0_1: "content {a..b::real^1} = 0 \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"

   unfolding content_eq_0 by auto

 lemma content_pos_lt_eq: "0 < content {a..b} \<longleftrightarrow> (\<forall>i. a$i < b$i)"

   apply(rule) defer apply(rule content_pos_lt,assumption) proof- assume "0 < content {a..b}"

   hence "content {a..b} \<noteq> 0" by auto thus "\<forall>i. a$i < b$i" unfolding content_eq_0 not_ex not_le by auto qed

 lemma content_empty[simp]: "content {} = 0" unfolding content_def by auto

 lemma content_subset: assumes "{a..b} \<subseteq> {c..d}" shows "content {a..b::real^'n} \<le> content {c..d}" proof(cases "{a..b}={}")

   case True thus ?thesis using content_pos_le[of c d] by auto next

   case False hence ab_ne:"\<forall>i. a $ i \<le> b $ i" unfolding interval_ne_empty by auto

   hence ab_ab:"a\<in>{a..b}" "b\<in>{a..b}" unfolding mem_interval by auto

   have "{c..d} \<noteq> {}" using assms False by auto

   hence cd_ne:"\<forall>i. c $ i \<le> d $ i" using assms unfolding interval_ne_empty by auto

   show ?thesis unfolding content_def unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]

     unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`] apply(rule setprod_mono,rule) proof fix i::'n

     show "0 \<le> b $ i - a $ i" using ab_ne[THEN spec[where x=i]] by auto

     show "b $ i - a $ i \<le> d $ i - c $ i"

       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]

       using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i] by auto qed qed

 lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"

   unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by auto

 subsection {* The notion of a gauge --- simply an open set containing the point. *}

 definition gauge where "gauge d \<longleftrightarrow> (\<forall>x. x\<in>(d x) \<and> open(d x))"

 lemma gaugeI:assumes "\<And>x. x\<in>g x" "\<And>x. open (g x)" shows "gauge g"

   using assms unfolding gauge_def by auto

 lemma gaugeD[dest]: assumes "gauge d" shows "x\<in>d x" "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)" apply(rule gauge_ball) by auto

 lemma gauge_inter: "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" "\<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(meson zero_less_one)

 subsection {* Divisions. *}

 definition division_of (infixl "division'_of" 40) where

   "s division_of i \<equiv>

         finite s \<and>

         (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {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>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"

   "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<Union>s = i" using assms unfolding division_of_def by auto

 lemma division_ofI:

   assumes "finite s" "\<And>k. k\<in>s \<Longrightarrow> k \<subseteq> i" "\<And>k. k\<in>s \<Longrightarrow>  k \<noteq> {}" "\<And>k. k\<in>s \<Longrightarrow> (\<exists>a b. k = {a..b})"

   "\<And>k1 k2. \<lbrakk>k1\<in>s; k2\<in>s; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}" "\<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]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {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 {a..a::real^'n} \<longleftrightarrow> s = {{a..a}}" (is "?l = ?r") proof

   assume ?r moreover { assume "s = {{a}}" moreover fix k assume "k\<in>s" 

     ultimately have"\<exists>x y. k = {x..y}" apply(rule_tac x=a in exI)+ unfolding interval_sing[THEN conjunct1] by auto }

   ultimately show ?l unfolding division_of_def interval_sing[THEN conjunct1] by auto next

   assume ?l note as=conjunctD4[OF this[unfolded division_of_def interval_sing[THEN conjunct1]]]

   { fix x assume x:"x\<in>s" have "x={a}" using as(2)[rule_format,OF x] by auto }

   moreover have "s \<noteq> {}" using as(4) by auto ultimately show ?r unfolding interval_sing[THEN conjunct1] 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 {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. {a..b} \<in> d \<longrightarrow> P {a..b}))"

   unfolding division_of_def by fastsimp

 lemma division_of_subset: assumes "p division_of (\<Union>p)" "q \<subseteq> p" shows "q division_of (\<Union>q)"

   apply(rule division_ofI) proof- note as=division_ofD[OF assms(1)]

   show "finite q" apply(rule finite_subset) using as(1) assms(2) by auto

   { fix k assume "k \<in> q" hence kp:"k\<in>p" using assms(2) by auto show "k\<subseteq>\<Union>q" using `k \<in> q` by auto

   show "\<exists>a b. k = {a..b}" using as(4)[OF kp] by auto show "k \<noteq> {}" using as(3)[OF kp] by auto }

   fix k1 k2 assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2" hence *:"k1\<in>p" "k2\<in>p" "k1\<noteq>k2" using assms(2) by auto

   show "interior k1 \<inter> interior k2 = {}" using as(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 {a..b} = 0" "d division_of {a..b}" shows "\<forall>k\<in>d. content k = 0"

   unfolding forall_in_division[OF assms(2)] apply(rule,rule,rule) apply(drule division_ofD(2)[OF assms(2)])

   apply(drule content_subset) unfolding assms(1) proof- case goal1 thus ?case using content_pos_le[of a b] by auto qed

 lemma division_inter: assumes "p1 division_of s1" "p2 division_of (s2::(real^'a) set)"

   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_ext) unfolding * and Union_image_eq UN_iff

     using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)] by auto

   { fix k assume "k\<in>?A" then obtain k1 k2 where k:"k = k1 \<inter> k2" "k1\<in>p1" "k2\<in>p2" "k\<noteq>{}" by auto thus "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

   guess a1 using division_ofD(4)[OF assms(1) k(2)] .. then guess b1 .. note ab1=this

   guess a2 using division_ofD(4)[OF assms(2) k(3)] .. then guess b2 .. note ab2=this

   show "\<exists>a b. k = {a..b}" unfolding k ab1 ab2 unfolding inter_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" hence 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 *) defer apply(rule_tac[1-4] subset_interior)

     using division_ofD(5)[OF assms(1) k1(2) k2(2)]

     using division_ofD(5)[OF assms(2) k1(3) k2(3)] using th by auto qed qed

 lemma division_inter_1: assumes "d division_of i" "{a..b::real^'n} \<subseteq> i"

   shows "{ {a..b} \<inter> k |k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {} } division_of {a..b}" proof(cases "{a..b} = {}")

   case True show ?thesis unfolding True and division_of_trivial by auto next

   have *:"{a..b} \<inter> i = {a..b}" using assms(2) by auto 

   case False show ?thesis using division_inter[OF division_of_self[OF False] assms(1)] unfolding * by auto qed

 lemma elementary_inter: assumes "p1 division_of s" "p2 division_of (t::(real^'n) set)"

   shows "\<exists>p. p division_of (s \<inter> t)"

   by(rule,rule division_inter[OF assms])

 lemma elementary_inters: assumes "finite f" "f\<noteq>{}" "\<forall>s\<in>f. \<exists>p. p division_of (s::(real^'n) set)"

   shows "\<exists>p. p division_of (\<Inter> f)" using assms apply-proof(induct f rule:finite_induct)

 case (insert x f) show ?case proof(cases "f={}")

   case True thus ?thesis unfolding True using insert by auto next

   case False guess p using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..

   moreover guess px using insert(5)[rule_format,OF insertI1] .. ultimately

   show ?thesis unfolding Inter_insert apply(rule_tac elementary_inter) by assumption+ qed qed auto

 lemma division_disjoint_union:

   assumes "p1 division_of s1" "p2 division_of s2" "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)] and 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 subset_interior[OF d1(2)[OF as(1)]] subset_interior[OF d2(2)[OF as(2)]]

       using assms(3) by blast } moreover

   { assume as:"k1\<in>p2" "k2\<in>p1" have ?g using subset_interior[OF d1(2)[OF as(2)]] subset_interior[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 = {a..b}" using k d1(4) d2(4) by auto qed

 lemma partial_division_extend_1:

   assumes "{c..d} \<subseteq> {a..b::real^'n}" "{c..d} \<noteq> {}"

   obtains p where "p division_of {a..b}" "{c..d} \<in> p"

 proof- def n \<equiv> "CARD('n)" have n:"1 \<le> n" "0 < n" "n \<noteq> 0" unfolding n_def by auto

   guess \<pi> using ex_bij_betw_nat_finite_1[OF finite_UNIV[where 'a='n]] .. note \<pi>=this

   def \<pi>' \<equiv> "inv_into {1..n} \<pi>"

   have \<pi>':"bij_betw \<pi>' UNIV {1..n}" using bij_betw_inv_into[OF \<pi>] unfolding \<pi>'_def n_def by auto

   hence \<pi>'i:"\<And>i. \<pi>' i \<in> {1..n}" unfolding bij_betw_def by auto 

   have \<pi>\<pi>'[simp]:"\<And>i. \<pi> (\<pi>' i) = i" unfolding \<pi>'_def apply(rule f_inv_into_f) unfolding n_def using \<pi> unfolding bij_betw_def by auto

   have \<pi>'\<pi>[simp]:"\<And>i. i\<in>{1..n} \<Longrightarrow> \<pi>' (\<pi> i) = i" unfolding \<pi>'_def apply(rule inv_into_f_eq) using \<pi> unfolding n_def bij_betw_def by auto

   have "{c..d} \<noteq> {}" using assms by auto

   let ?p1 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$i else a$i) .. (\<chi> i. if \<pi>' i < l then d$i else if \<pi>' i = l then c$\<pi> l else b$i)}"

   let ?p2 = "\<lambda>l. {(\<chi> i. if \<pi>' i < l then c$i else if \<pi>' i = l then d$\<pi> l else a$i) .. (\<chi> i. if \<pi>' i < l then d$i else b$i)}"

   let ?p =  "{?p1 l |l. l \<in> {1..n+1}} \<union> {?p2 l |l. l \<in> {1..n+1}}"

   have abcd:"\<And>i. a $ i \<le> c $ i \<and> c$i \<le> d$i \<and> d $ i \<le> b $ i" using assms unfolding subset_interval interval_eq_empty by(auto simp add:not_le not_less)

   show ?thesis apply(rule that[of ?p]) apply(rule division_ofI)

   proof- have "\<And>i. \<pi>' i < Suc n"

     proof(rule ccontr,unfold not_less) fix i assume "Suc n \<le> \<pi>' i"

       hence "\<pi>' i \<notin> {1..n}" by auto thus False using \<pi>' unfolding bij_betw_def by auto

     qed hence "c = (\<chi> i. if \<pi>' i < Suc n then c $ i else a $ i)"

         "d = (\<chi> i. if \<pi>' i < Suc n then d $ i else if \<pi>' i = n + 1 then c $ \<pi> (n + 1) else b $ i)"

       unfolding Cart_eq Cart_lambda_beta using \<pi>' unfolding bij_betw_def by auto

     thus cdp:"{c..d} \<in> ?p" apply-apply(rule UnI1) unfolding mem_Collect_eq apply(rule_tac x="n + 1" in exI) by auto

     have "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p1 l \<subseteq> {a..b}"  "\<And>l. l\<in>{1..n+1} \<Longrightarrow> ?p2 l \<subseteq> {a..b}"

       unfolding subset_eq apply(rule_tac[!] ballI,rule_tac[!] ccontr)

     proof- fix l assume l:"l\<in>{1..n+1}" fix x assume "x\<notin>{a..b}"

       then guess i unfolding mem_interval not_all .. note i=this

       show "x \<in> ?p1 l \<Longrightarrow> False" "x \<in> ?p2 l \<Longrightarrow> False" unfolding mem_interval apply(erule_tac[!] x=i in allE)

         apply(case_tac[!] "\<pi>' i < l", case_tac[!] "\<pi>' i = l") using abcd[of i] i by auto 

     qed moreover have "\<And>x. x \<in> {a..b} \<Longrightarrow> x \<in> \<Union>?p"

     proof- fix x assume x:"x\<in>{a..b}"

       { presume "x\<notin>{c..d} \<Longrightarrow> x \<in> \<Union>?p" thus "x \<in> \<Union>?p" using cdp by blast }

       let ?M = "{i. i\<in>{1..n+1} \<and> \<not> (c $ \<pi> i \<le> x $ \<pi> i \<and> x $ \<pi> i \<le> d $ \<pi> i)}"

       assume "x\<notin>{c..d}" then guess i0 unfolding mem_interval not_all ..

       hence "\<pi>' i0 \<in> ?M" using \<pi>' unfolding bij_betw_def by(auto intro!:le_SucI)

       hence M:"finite ?M" "?M \<noteq> {}" by auto

       def l \<equiv> "Min ?M" note l = Min_less_iff[OF M,unfolded l_def[symmetric]] Min_in[OF M,unfolded mem_Collect_eq l_def[symmetric]]

         Min_gr_iff[OF M,unfolded l_def[symmetric]]

       have "x\<in>?p1 l \<or> x\<in>?p2 l" using l(2)[THEN conjunct2] unfolding de_Morgan_conj not_le

         apply- apply(erule disjE) apply(rule disjI1) defer apply(rule disjI2)

       proof- assume as:"x $ \<pi> l < c $ \<pi> l"

         show "x \<in> ?p1 l" unfolding mem_interval Cart_lambda_beta

         proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto

           thus ?case using as x[unfolded mem_interval,rule_format,of i]

             apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])

         qed

       next assume as:"x $ \<pi> l > d $ \<pi> l"

         show "x \<in> ?p2 l" unfolding mem_interval Cart_lambda_beta

         proof case goal1 have "\<pi>' i \<in> {1..n}" using \<pi>' unfolding bij_betw_def not_le by auto

           thus ?case using as x[unfolded mem_interval,rule_format,of i]

             apply auto using l(3)[of "\<pi>' i"] by(auto elim!:ballE[where x="\<pi>' i"])

         qed qed

       thus "x \<in> \<Union>?p" using l(2) by blast 

     qed ultimately show "\<Union>?p = {a..b}" apply-apply(rule) defer apply(rule) by(assumption,blast)

     show "finite ?p" by auto

     fix k assume k:"k\<in>?p" then obtain l where l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" by auto

     show "k\<subseteq>{a..b}" apply(rule,unfold mem_interval,rule,rule) 

     proof- fix i::'n and x assume "x \<in> k" moreover have "\<pi>' i < l \<or> \<pi>' i = l \<or> \<pi>' i > l" by auto

       ultimately show "a$i \<le> x$i" "x$i \<le> b$i" using abcd[of i] using l by(auto elim:disjE elim!:allE[where x=i] simp add:vector_le_def)

     qed have "\<And>l. ?p1 l \<noteq> {}" "\<And>l. ?p2 l \<noteq> {}" unfolding interval_eq_empty not_ex apply(rule_tac[!] allI)

     proof- case goal1 thus ?case using abcd[of x] by auto

     next   case goal2 thus ?case using abcd[of x] by auto

     qed thus "k \<noteq> {}" using k by auto

     show "\<exists>a b. k = {a..b}" using k by auto

     fix k' assume k':"k' \<in> ?p" "k \<noteq> k'" then obtain l' where l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" by auto

     { fix k k' l l'

       assume k:"k\<in>?p" and l:"k = ?p1 l \<or> k = ?p2 l" "l \<in> {1..n + 1}" 

       assume k':"k' \<in> ?p" "k \<noteq> k'" and  l':"k' = ?p1 l' \<or> k' = ?p2 l'" "l' \<in> {1..n + 1}" 

       assume "l \<le> l'" fix x

       have "x \<notin> interior k \<inter> interior k'" 

       proof(rule,cases "l' = n+1") assume x:"x \<in> interior k \<inter> interior k'"

         case True hence "\<And>i. \<pi>' i < l'" using \<pi>'i by(auto simp add:less_Suc_eq_le)

         hence k':"k' = {c..d}" using l'(1) \<pi>'i by(auto simp add:Cart_nth_inverse)

         have ln:"l < n + 1" 

         proof(rule ccontr) case goal1 hence l2:"l = n+1" using l by auto

           hence "\<And>i. \<pi>' i < l" using \<pi>'i by(auto simp add:less_Suc_eq_le)

           hence "k = {c..d}" using l(1) \<pi>'i by(auto simp add:Cart_nth_inverse)

           thus False using `k\<noteq>k'` k' by auto

         qed have **:"\<pi>' (\<pi> l) = l" using \<pi>'\<pi>[of l] using l ln by auto

         have "x $ \<pi> l < c $ \<pi> l \<or> d $ \<pi> l < x $ \<pi> l" using l(1) apply-

         proof(erule disjE)

           assume as:"k = ?p1 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]

           show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto

         next assume as:"k = ?p2 l" note * = conjunct1[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]

           show ?thesis using *[of "\<pi> l"] using ln unfolding Cart_lambda_beta ** by auto

         qed thus False using x unfolding k' unfolding Int_iff interior_closed_interval mem_interval

           by(auto elim!:allE[where x="\<pi> l"])

       next case False hence "l < n + 1" using l'(2) using `l\<le>l'` by auto

         hence ln:"l \<in> {1..n}" "l' \<in> {1..n}" using l l' False by auto

         note \<pi>l = \<pi>'\<pi>[OF ln(1)] \<pi>'\<pi>[OF ln(2)]

         assume x:"x \<in> interior k \<inter> interior k'"

         show False using l(1) l'(1) apply-

         proof(erule_tac[!] disjE)+

           assume as:"k = ?p1 l" "k' = ?p1 l'"

           note * = x[unfolded as Int_iff interior_closed_interval mem_interval]

           have "l \<noteq> l'" using k'(2)[unfolded as] by auto

           thus False using * by(smt Cart_lambda_beta \<pi>l)

         next assume as:"k = ?p2 l" "k' = ?p2 l'"

           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]

           have "l \<noteq> l'" apply(rule) using k'(2)[unfolded as] by auto

           thus False using *[of "\<pi> l"] *[of "\<pi> l'"]

             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` by auto

         next assume as:"k = ?p1 l" "k' = ?p2 l'"

           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]

           show False using *[of "\<pi> l"] *[of "\<pi> l'"]

             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt 

         next assume as:"k = ?p2 l" "k' = ?p1 l'"

           note * = conjunctD2[OF x[unfolded as Int_iff interior_closed_interval mem_interval],rule_format]

           show False using *[of "\<pi> l"] *[of "\<pi> l'"]

             unfolding Cart_lambda_beta \<pi>l using `l \<le> l'` using abcd[of "\<pi> l'"] by smt

         qed qed } 

     from this[OF k l k' l'] this[OF k'(1) l' k _ l] have "\<And>x. x \<notin> interior k \<inter> interior k'"

       apply - apply(cases "l' \<le> l") using k'(2) by auto            

     thus "interior k \<inter> interior k' = {}" by auto        

 qed qed

 lemma partial_division_extend_interval: assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"

   obtains q where "p \<subseteq> q" "q division_of {a..b::real^'n}" proof(cases "p = {}")

   case True guess q apply(rule elementary_interval[of a b]) .

   thus ?thesis apply- apply(rule that[of q]) unfolding True by auto next

   case False note p = division_ofD[OF assms(1)]

   have *:"\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k\<in>q" proof case goal1

     guess c using p(4)[OF goal1] .. then guess d .. note cd_ = this

     have *:"{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}" using p(2,3)[OF goal1, unfolded cd_] using assms(2) by auto

     guess q apply(rule partial_division_extend_1[OF *]) . thus ?case unfolding cd_ by auto qed

   guess q using bchoice[OF *] .. note q = conjunctD2[OF this[rule_format]]

   have "\<And>x. x\<in>p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})" apply(rule,rule_tac p="q x" in division_of_subset) proof-

     fix x assume x:"x\<in>p" show "q x division_of \<Union>q x" apply-apply(rule division_ofI)

       using division_ofD[OF q(1)[OF x]] by auto show "q x - {x} \<subseteq> q x" by auto qed

   hence "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)" apply- apply(rule elementary_inters)

     apply(rule finite_imageI[OF p(1)]) unfolding image_is_empty apply(rule False) by auto

   then guess d .. note d = this

   show ?thesis apply(rule that[of "d \<union> p"]) proof-

     have *:"\<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 *:"{a..b} = \<Inter> (\<lambda>i. \<Union>(q i - {i})) ` p \<union> \<Union>p" apply(rule *[OF False]) proof fix i assume i:"i\<in>p"

       show "\<Union>(q i - {i}) \<union> i = {a..b}" using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto qed

     show "d \<union> p division_of {a..b}" unfolding * apply(rule division_disjoint_union[OF d assms(1)])

       apply(rule inter_interior_unions_intervals) apply(rule p open_interior ballI)+ proof(assumption,rule)

       fix k assume k:"k\<in>p" have *:"\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}" by auto

       show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<inter> interior k = {}" apply(rule *[of _ "interior (\<Union>(q k - {k}))"])

 	defer apply(subst Int_commute) apply(rule inter_interior_unions_intervals) proof- 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 = {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

 	have *:"\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x" by auto show "interior (\<Inter>(\<lambda>i. \<Union>(q i - {i})) ` p) \<subseteq> interior (\<Union>(q k - {k}))"

 	  apply(rule subset_interior *)+ using k by auto qed qed qed auto qed

 lemma elementary_bounded[dest]: "p division_of s \<Longrightarrow> bounded (s::(real^'n) set)"

   unfolding division_of_def by(metis bounded_Union bounded_interval) 

 lemma elementary_subset_interval: "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::real^'n}"

   by(meson elementary_bounded bounded_subset_closed_interval)

 lemma division_union_intervals_exists: assumes "{a..b::real^'n} \<noteq> {}"

   obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})" proof(cases "{c..d} = {}")

   case True show ?thesis apply(rule that[of "{}"]) unfolding True using assms by auto next

   case False note false=this show ?thesis proof(cases "{a..b} \<inter> {c..d} = {}")

   have *:"\<And>a b. {a,b} = {a} \<union> {b}" by auto

   case True show ?thesis apply(rule that[of "{{c..d}}"]) unfolding * apply(rule division_disjoint_union)

     using false True assms using interior_subset by auto next

   case False obtain u v where uv:"{a..b} \<inter> {c..d} = {u..v}" unfolding inter_interval by auto

   have *:"{u..v} \<subseteq> {c..d}" using uv by auto

   guess p apply(rule partial_division_extend_1[OF * False[unfolded uv]]) . note p=this division_ofD[OF this(1)]

   have *:"{a..b} \<union> {c..d} = {a..b} \<union> \<Union>(p - {{u..v}})" "\<And>x s. insert x s = {x} \<union> s" using p(8) unfolding uv[THEN sym] by auto

   show thesis apply(rule that[of "p - {{u..v}}"]) unfolding *(1) apply(subst *(2)) apply(rule division_disjoint_union)

     apply(rule,rule assms) apply(rule division_of_subset[of p]) apply(rule division_of_union_self[OF p(1)]) defer

     unfolding interior_inter[THEN sym] proof-

     have *:"\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto

     have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))" 

       apply(rule arg_cong[of _ _ interior]) apply(rule *[OF _ uv]) using p(8) by auto

     also have "\<dots> = {}" unfolding interior_inter apply(rule inter_interior_unions_intervals) using p(6) p(7)[OF p(2)] p(3) by auto

     finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" by assumption qed auto qed qed

 lemma division_of_unions: assumes "finite f"  "\<And>p. p\<in>f \<Longrightarrow> p division_of (\<Union>p)"

   "\<And>k1 k2. \<lbrakk>k1 \<in> \<Union>f; k2 \<in> \<Union>f; k1 \<noteq> k2\<rbrakk> \<Longrightarrow> interior k1 \<inter> interior k2 = {}"

   shows "\<Union>f division_of \<Union>\<Union>f" apply(rule division_ofI) prefer 5 apply(rule assms(3)|assumption)+

   apply(rule finite_Union assms(1))+ prefer 3 apply(erule UnionE) apply(rule_tac s=X in division_ofD(3)[OF assms(2)])

   using division_ofD[OF assms(2)] by auto

 lemma elementary_union_interval: assumes "p division_of \<Union>p"

   obtains q where "q division_of ({a..b::real^'n} \<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" "{a..b} = {} \<Longrightarrow> thesis" "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"

     "p\<noteq>{} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"

   thus thesis by auto

 next assume as:"p={}" guess p apply(rule elementary_interval[of a b]) .

   thus thesis apply(rule_tac that[of p]) unfolding as by auto 

 next assume as:"{a..b}={}" show thesis apply(rule that) unfolding as using assms by auto

 next assume as:"interior {a..b} = {}" "{a..b} \<noteq> {}"

   show thesis apply(rule that[of "insert {a..b} p"],rule division_ofI)

     unfolding finite_insert apply(rule assm(1)) unfolding Union_insert  

     using assm(2-4) as apply- by(fastsimp dest: assm(5))+

 next assume as:"p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b}\<noteq>{}"

   have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)" proof case goal1

     from assm(4)[OF this] guess c .. then guess d ..

     thus ?case apply-apply(rule division_union_intervals_exists[OF as(3),of c d]) by auto

   qed from bchoice[OF this] guess q .. note q=division_ofD[OF this[rule_format]]

   let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"

   show thesis apply(rule that[of "?D"]) proof(rule division_ofI)

     have *:"{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p" by auto

     show "finite ?D" apply(rule finite_Union) unfolding * apply(rule finite_imageI) using assm(1) q(1) by auto

     show "\<Union>?D = {a..b} \<union> \<Union>p" unfolding * lem1 unfolding lem2[OF as(1), of "{a..b}",THEN sym]

       using q(6) by auto

     fix k assume k:"k\<in>?D" thus " k \<subseteq> {a..b} \<union> \<Union>p" using q(2) by auto

     show "k \<noteq> {}" using q(3) k by auto show "\<exists>a b. k = {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 {a..b} (q x)"  "x\<in>p"  using k  by auto

     obtain x' where x':"k'\<in>insert {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 apply(rule q(5)) using x x' k' unfolding True by auto

     next case False 

       { presume "k = {a..b} \<Longrightarrow> ?thesis" "k' = {a..b} \<Longrightarrow> ?thesis" 

         "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"

         thus ?thesis by auto }

       { assume as':"k  = {a..b}" show ?thesis apply(rule q(5)) using x' k'(2) unfolding as' by auto }

       { assume as':"k' = {a..b}" show ?thesis apply(rule q(5)) using x  k'(2) unfolding as' by auto }

       assume as':"k \<noteq> {a..b}" "k' \<noteq> {a..b}"

       guess c using q(4)[OF x(2,1)] .. then guess d .. note c_d=this

       have "interior k  \<inter> interior {a..b} = {}" apply(rule q(5)) using x  k'(2) using as' by auto

       hence "interior k \<subseteq> interior x" apply-

         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x(2,1)]]) by auto moreover

       guess c using q(4)[OF x'(2,1)] .. then guess d .. note c_d=this

       have "interior k' \<inter> interior {a..b} = {}" apply(rule q(5)) using x' k'(2) using as' by auto

       hence "interior k' \<subseteq> interior x'" apply-

         apply(rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]]) by auto

       ultimately show ?thesis using assm(5)[OF x(2) x'(2) False] by auto

     qed qed } qed

 lemma elementary_unions_intervals:

   assumes "finite f" "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::real^'n}"

   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

     fix x F assume as:"finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"

     from this(3) guess p .. note p=this

     from assms(2)[OF as(4)] guess a .. then guess b .. note ab=this

     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 ab * by auto

   qed(insert assms,auto) thus ?thesis apply-apply(erule exE,rule that) by auto qed

 lemma elementary_union: assumes "ps division_of s" "pt division_of (t::(real^'n) set)"

   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

   hence *:"\<Union>(ps \<union> pt) = s \<union> t" by auto

   show ?thesis apply-apply(rule elementary_unions_intervals[of "ps\<union>pt"])

     unfolding * prefer 3 apply(rule_tac p=p in that)

     using assms[unfolded division_of_def] by auto qed

 lemma partial_division_extend: fixes t::"(real^'n) set"

   assumes "p division_of s" "q division_of t" "s \<subseteq> t"

   obtains r where "p \<subseteq> r" "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>{a..b}" using elementary_subset_interval[OF assms(2)] by auto

   guess r1 apply(rule partial_division_extend_interval) apply(rule assms(1)[unfolded divp(6)[THEN sym]])

     apply(rule subset_trans) by(rule ab assms[unfolded divp(6)[THEN sym]])+  note r1 = this division_ofD[OF this(2)]

   guess p' apply(rule elementary_unions_intervals[of "r1 - p"]) using r1(3,6) by auto 

   then obtain r2 where r2:"r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)" 

     apply- apply(drule elementary_inter[OF _ assms(2)[unfolded divq(6)[THEN sym]]]) by auto

   { fix x assume x:"x\<in>t" "x\<notin>s"

     hence "x\<in>\<Union>r1" unfolding r1 using ab by auto

     then guess r unfolding Union_iff .. note r=this moreover

     have "r \<notin> p" proof assume "r\<in>p" hence "x\<in>s" using divp(2) r by auto

       thus False using x by auto qed

     ultimately have "x\<in>\<Union>(r1 - p)" by auto }

   hence *:"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)" "open (interior s)" "\<forall>t\<in>r1-p. \<exists>a b. t = {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(rule)

         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" "open (interior m)" "\<forall>t\<in>p. \<exists>a b. t = {a..b}" using divp by auto

           show "\<forall>t\<in>p. interior m \<inter> interior t = {}" apply(rule, rule r1(7)) using as using r1 by auto

         qed thus "interior s \<inter> interior m = {}" unfolding divp by auto

       qed qed        

     thus "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}" using interior_subset by auto

   qed auto qed

 subsection {* Tagged (partial) divisions. *}

 definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40) where

   "(s tagged_partial_division_of i) \<equiv>

         finite s \<and>

         (\<forall>x k. (x,k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {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>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"

   "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"

   "\<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  apply- by blast+ 

 definition tagged_division_of (infixr "tagged'_division'_of" 40) where

   "(s tagged_division_of i) \<equiv>

         (s tagged_partial_division_of i) \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"

 lemma tagged_division_of_finite[dest]: "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 = {a..b})) \<and>

         (\<forall>x1 k1 x2 k2. (x1,k1) \<in> s \<and> (x2,k2) \<in> s \<and> ~((x1,k1) = (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>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"

   "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"

   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"

   shows "s tagged_division_of i"

   unfolding tagged_division_of apply(rule) defer apply rule

   apply(rule allI impI conjI assms)+ apply assumption

   apply(rule, rule assms, assumption) apply(rule assms, assumption)

   using assms(1,5-) apply- by blast+

 lemma tagged_division_ofD[dest]: assumes "s tagged_division_of i"

   shows "finite s" "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k" "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"  "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"

   "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2,k2) \<in> s \<Longrightarrow> ~((x1,k1) = (x2,k2)) \<Longrightarrow> (interior(k1) \<inter> interior(k2) = {})"

   "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)" using assms unfolding tagged_division_of apply- 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

   thus  "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}" using assm apply- by fastsimp+

   fix k' assume k':"k' \<in> snd ` s" "k \<noteq> k'" from this(1) obtain xk' where xk':"(xk', k') \<in> s" by auto

   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto

 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

   thus "k\<noteq>{}" "\<exists>a b. k = {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

   thus "interior k \<inter> interior k' = {}" apply-apply(rule assm(5)) apply(rule xk xk')+ using k' by auto

 qed

 lemma tagged_partial_division_subset: assumes "s tagged_partial_division_of i" "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 setsum_over_tagged_division_lemma: fixes d::"(real^'m) set \<Rightarrow> 'a::real_normed_vector"

   assumes "p tagged_division_of i" "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"

   shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"

 proof- note assm=tagged_division_ofD[OF assms(1)]

   have *:"(\<lambda>(x,k). d k) = d \<circ> snd" unfolding o_def apply(rule ext) by auto

   show ?thesis unfolding * apply(subst eq_commute) proof(rule setsum_reindex_nonzero)

     show "finite p" using assm by auto

     fix x y assume as:"x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y" 

     obtain a b where ab:"snd x = {a..b}" using assm(4)[of "fst x" "snd x"] as(1) by auto

     have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y" unfolding as(4)[THEN sym] using as(1-3) by auto

     hence "interior (snd x) \<inter> interior (snd y) = {}" apply-apply(rule assm(5)[of "fst x" _ "fst y"]) using as by auto 

     hence "content {a..b} = 0" unfolding as(4)[THEN sym] ab content_eq_0_interior by auto

     hence "d {a..b} = 0" apply-apply(rule assms(2)) using assm(2)[of "fst x" "snd x"] as(1) unfolding ab[THEN sym] by auto

     thus "d (snd x) = 0" 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> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"

   apply(rule tagged_division_ofI) by auto

 lemma tagged_division_union:

   assumes "p1 tagged_division_of s1"  "p2 tagged_division_of s2" "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)] and 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 = {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) subset_interior by blast

   show "interior k \<inter> interior k' = {}" apply(cases "(x,k)\<in>p1", case_tac[!] "(x',k')\<in>p1")

     apply(rule p1(5)) prefer 4 apply(rule *) prefer 6 apply(subst Int_commute,rule *) prefer 8 apply(rule p2(5))

     using p1(3) p2(3) using xk xk' by auto qed 

 lemma tagged_division_unions:

   assumes "finite iset" "\<forall>i\<in>iset. (pfn(i) tagged_division_of i)"

   "\<forall>i1 \<in> iset. \<forall>i2 \<in> iset. ~(i1 = 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)" apply(rule finite_Union) 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 = {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] subset_interior by blast

   show "interior k \<inter> interior k' = {}" apply(cases "i=i'")

     using assm(5)[OF i _ xk'(2)]  i'(2) using assm(3)[OF i] assm(3)[OF i'] defer apply-apply(rule *) 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] by auto

 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] by auto

 subsection {* Fine-ness of a partition w.r.t. a gauge. *}

 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 {* Gauge integral. Define on compact intervals first, then use a limit. *}

 definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46) where

   "(f has_integral_compact_interval y) i \<equiv>

         (\<forall>e>0. \<exists>d. gauge d \<and>

           (\<forall>p. p tagged_division_of i \<and> d fine p

                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"

 definition has_integral (infixr "has'_integral" 46) where 

 "((f::(real^'n \<Rightarrow> 'b::real_normed_vector)) has_integral y) i \<equiv>

         if (\<exists>a b. i = {a..b}) then (f has_integral_compact_interval y) i

         else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}

               \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>

                                        norm(z - y) < e))"

 lemma has_integral:

  "(f has_integral y) ({a..b}) \<longleftrightarrow>

         (\<forall>e>0. \<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p

                         \<longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"

   unfolding has_integral_def has_integral_compact_interval_def by auto

 lemma has_integralD[dest]: assumes

  "(f has_integral y) ({a..b})" "e>0"

   obtains d where "gauge d" "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p

                         \<Longrightarrow> norm(setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"

   using assms unfolding has_integral by auto

 lemma has_integral_alt:

  "(f has_integral y) i \<longleftrightarrow>

       (if (\<exists>a b. i = {a..b}) then (f has_integral y) i

        else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b}

                                \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0)

                                         has_integral z) ({a..b}) \<and>

                                        norm(z - y) < e)))"

   unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto

 lemma has_integral_altD:

   assumes "(f has_integral y) i" "\<not> (\<exists>a b. i = {a..b})" "e>0"

   obtains B where "B>0" "\<forall>a b. ball 0 B \<subseteq> {a..b}\<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"

   using assms unfolding has_integral unfolding has_integral_compact_interval_def has_integral_def by auto

 definition integrable_on (infixr "integrable'_on" 46) where

   "(f integrable_on i) \<equiv> \<exists>y. (f has_integral y) i"

 definition "integral i f \<equiv> SOME y. (f has_integral y) i"

 lemma integrable_integral[dest]:

  "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"

   unfolding integrable_on_def integral_def by(rule someI_ex)

 lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"

   unfolding integrable_on_def by auto

 lemma has_integral_integral:"f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"

   by auto

 lemma setsum_content_null:

   assumes "content({a..b}) = 0" "p tagged_division_of {a..b}"

   shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"

 proof(rule setsum_0',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) guess c .. then guess d .. note c_d=this

   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 c_d]] content_pos_le[of c d]

     unfolding assms(1) c_d by auto

   finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .

 qed

 subsection {* Some basic combining lemmas. *}

 lemma tagged_division_unions_exists:

   assumes "finite iset" "\<forall>i \<in> iset. \<exists>p. p tagged_division_of i \<and> d fine p"

   "\<forall>i1\<in>iset. \<forall>i2\<in>iset. ~(i1 = i2) \<longrightarrow> (interior(i1) \<inter> interior(i2) = {})" "(\<Union>iset = i)"

    obtains p where "p tagged_division_of i" "d fine p"

 proof- guess pfn using bchoice[OF assms(2)] .. note pfn = conjunctD2[OF this[rule_format]]

   show thesis apply(rule_tac p="\<Union>(pfn ` iset)" in that) unfolding assms(4)[THEN sym]

     apply(rule tagged_division_unions[OF assms(1) _ assms(3)]) defer 

     apply(rule fine_unions) using pfn by auto

 qed

 subsection {* The set we're concerned with must be closed. *}

 lemma division_of_closed: "s division_of i \<Longrightarrow> closed (i::(real^'n) set)"

   unfolding division_of_def by(fastsimp intro!: closed_Union closed_interval)

 subsection {* General bisection principle for intervals; might be useful elsewhere. *}

 lemma interval_bisection_step:

   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "~(P {a..b::real^'n})"

   obtains c d where "~(P{c..d})"

   "\<forall>i. a$i \<le> c$i \<and> c$i \<le> d$i \<and> d$i \<le> b$i \<and> 2 * (d$i - c$i) \<le> b$i - a$i"

 proof- have "{a..b} \<noteq> {}" using assms(1,3) by auto

   note ab=this[unfolded interval_eq_empty not_ex not_less]

   { fix f have "finite f \<Longrightarrow>

         (\<forall>s\<in>f. P s) \<Longrightarrow>

         (\<forall>s\<in>f. \<exists>a b. s = {a..b}) \<Longrightarrow>

         (\<forall>s\<in>f.\<forall>t\<in>f. ~(s = t) \<longrightarrow> interior(s) \<inter> interior(t) = {}) \<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])

         apply rule defer apply rule defer apply(rule inter_interior_unions_intervals)

         using insert by auto

     qed } note * = this

   let ?A = "{{c..d} | c d. \<forall>i. (c$i = a$i) \<and> (d$i = (a$i + b$i) / 2) \<or> (c$i = (a$i + b$i) / 2) \<and> (d$i = b$i)}"

   let ?PP = "\<lambda>c d. \<forall>i. a$i \<le> c$i \<and> c$i \<le> d$i \<and> d$i \<le> b$i \<and> 2 * (d$i - c$i) \<le> b$i - a$i"

   { presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"

     thus thesis unfolding atomize_not not_all apply-apply(erule exE)+ apply(rule_tac c=x and d=xa in that) by auto }

   assume as:"\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"

   have "P (\<Union> ?A)" proof(rule *, rule_tac[2-] ballI, rule_tac[4] ballI, rule_tac[4] impI) 

     let ?B = "(\<lambda>s.{(\<chi> i. if i \<in> s then a$i else (a$i + b$i) / 2) ..

       (\<chi> i. if i \<in> s then (a$i + b$i) / 2 else b$i)}) ` {s. s \<subseteq> UNIV}"

     have "?A \<subseteq> ?B" proof case goal1

       then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+) note c_d=this[rule_format]

       have *:"\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}" by auto

       show "x\<in>?B" unfolding image_iff apply(rule_tac x="{i. c$i = a$i}" in bexI)

         unfolding c_d apply(rule * ) unfolding Cart_eq cond_component Cart_lambda_beta

       proof(rule_tac[1-2] allI) fix i show "c $ i = (if i \<in> {i. c $ i = a $ i} then a $ i else (a $ i + b $ i) / 2)"

           "d $ i = (if i \<in> {i. c $ i = a $ i} then (a $ i + b $ i) / 2 else b $ i)"

           using c_d(2)[of i] ab[THEN spec[where x=i]] by(auto simp add:field_simps)

       qed auto qed

     thus "finite ?A" apply(rule finite_subset[of _ ?B]) by auto

     fix s assume "s\<in>?A" then guess c unfolding mem_Collect_eq .. then guess d apply- by(erule exE,(erule conjE)+)

     note c_d=this[rule_format]

     show "P s" unfolding c_d apply(rule as[rule_format]) proof- case goal1 show ?case 

         using c_d(2)[of i] using ab[THEN spec[where x=i]] by auto qed

     show "\<exists>a b. s = {a..b}" unfolding c_d by auto

     fix t assume "t\<in>?A" then guess e unfolding mem_Collect_eq .. then guess f apply- by(erule exE,(erule conjE)+)

     note e_f=this[rule_format]

     assume "s \<noteq> t" hence "\<not> (c = e \<and> d = f)" unfolding c_d e_f by auto

     then obtain i where "c$i \<noteq> e$i \<or> d$i \<noteq> f$i" unfolding de_Morgan_conj Cart_eq by auto

     hence i:"c$i \<noteq> e$i" "d$i \<noteq> f$i" apply- apply(erule_tac[!] disjE)

     proof- assume "c$i \<noteq> e$i" thus "d$i \<noteq> f$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp

     next   assume "d$i \<noteq> f$i" thus "c$i \<noteq> e$i" using c_d(2)[of i] e_f(2)[of i] by fastsimp

     qed 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 e_f c_d interior_closed_interval proof(rule *)

       fix x assume "x\<in>{c<..<d}" "x\<in>{e<..<f}"

       hence x:"c$i < d$i" "e$i < f$i" "c$i < f$i" "e$i < d$i" unfolding mem_interval apply-apply(erule_tac[!] x=i in allE)+ by auto

       show False using c_d(2)[of i] apply- apply(erule_tac disjE)

       proof(erule_tac[!] conjE) assume as:"c $ i = a $ i" "d $ i = (a $ i + b $ i) / 2"

         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)

       next assume as:"c $ i = (a $ i + b $ i) / 2" "d $ i = b $ i"

         show False using e_f(2)[of i] and i x unfolding as by(fastsimp simp add:field_simps)

       qed qed qed

   also have "\<Union> ?A = {a..b}" proof(rule set_ext,rule)

     fix x assume "x\<in>\<Union>?A" then guess Y unfolding Union_iff ..

     from this(1) guess c unfolding mem_Collect_eq .. then guess d ..

     note c_d = this[THEN conjunct2,rule_format] `x\<in>Y`[unfolded this[THEN conjunct1]]

     show "x\<in>{a..b}" unfolding mem_interval proof 

       fix i show "a $ i \<le> x $ i \<and> x $ i \<le> b $ i"

         using c_d(1)[of i] c_d(2)[unfolded mem_interval,THEN spec[where x=i]] by auto qed

   next fix x assume x:"x\<in>{a..b}"

     have "\<forall>i. \<exists>c d. (c = a$i \<and> d = (a$i + b$i) / 2 \<or> c = (a$i + b$i) / 2 \<and> d = b$i) \<and> c\<le>x$i \<and> x$i \<le> d"

       (is "\<forall>i. \<exists>c d. ?P i c d") unfolding mem_interval proof fix i

       have "?P i (a$i) ((a $ i + b $ i) / 2) \<or> ?P i ((a $ i + b $ i) / 2) (b$i)"

         using x[unfolded mem_interval,THEN spec[where x=i]] by auto thus "\<exists>c d. ?P i c d" by blast

     qed thus "x\<in>\<Union>?A" unfolding Union_iff lambda_skolem unfolding Bex_def mem_Collect_eq

       apply-apply(erule exE)+ apply(rule_tac x="{xa..xaa}" in exI) unfolding mem_interval by auto

   qed finally show False using assms by auto qed

 lemma interval_bisection:

   assumes "P {}" "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))" "\<not> P {a..b::real^'n}"

   obtains x where "x \<in> {a..b}" "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> ~P({c..d})"

 proof-

   have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and> (\<forall>i. fst x$i \<le> fst y$i \<and> fst y$i \<le> snd y$i \<and> snd y$i \<le> snd x$i \<and>

                            2 * (snd y$i - fst y$i) \<le> snd x$i - fst x$i))" proof case goal1 thus ?case proof-

       presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"

       thus ?thesis apply(cases "P {fst x..snd x}") by auto

     next assume as:"\<not> P {fst x..snd x}" from interval_bisection_step[of P, OF assms(1-2) as] guess c d . 

       thus ?thesis apply- apply(rule_tac x="(c,d)" in exI) by auto

     qed qed then guess f apply-apply(drule choice) by(erule exE) note f=this

   def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)" def A \<equiv> "\<lambda>n. fst(AB n)" and B \<equiv> "\<lambda>n. snd(AB n)" note ab_def = this AB_def

   have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>

     (\<forall>i. A(n)$i \<le> A(Suc n)$i \<and> A(Suc n)$i \<le> B(Suc n)$i \<and> B(Suc n)$i \<le> B(n)$i \<and> 

     2 * (B(Suc n)$i - A(Suc n)$i) \<le> B(n)$i - A(n)$i)" (is "\<And>n. ?P n")

   proof- show "A 0 = a" "B 0 = b" unfolding ab_def by auto

     case goal3 note S = ab_def funpow.simps o_def id_apply show ?case

     proof(induct n) case 0 thus ?case unfolding S apply(rule f[rule_format]) using assms(3) by auto

     next case (Suc n) show ?case unfolding S apply(rule f[rule_format]) using Suc unfolding S by auto

     qed qed note AB = this(1-2) conjunctD2[OF this(3),rule_format]

   have interv:"\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"

   proof- case goal1 guess n using real_arch_pow2[of "(setsum (\<lambda>i. b$i - a$i) UNIV) / e"] .. note n=this

     show ?case apply(rule_tac x=n in exI) proof(rule,rule)

       fix x y assume xy:"x\<in>{A n..B n}" "y\<in>{A n..B n}"

       have "dist x y \<le> setsum (\<lambda>i. abs((x - y)$i)) UNIV" unfolding vector_dist_norm by(rule norm_le_l1)

       also have "\<dots> \<le> setsum (\<lambda>i. B n$i - A n$i) UNIV"

       proof(rule setsum_mono) fix i show "\<bar>(x - y) $ i\<bar> \<le> B n $ i - A n $ i"

           using xy[unfolded mem_interval,THEN spec[where x=i]]

           unfolding vector_minus_component by auto qed

       also have "\<dots> \<le> setsum (\<lambda>i. b$i - a$i) UNIV / 2^n" unfolding setsum_divide_distrib

       proof(rule setsum_mono) case goal1 thus ?case

         proof(induct n) case 0 thus ?case unfolding AB by auto

         next case (Suc n) have "B (Suc n) $ i - A (Suc n) $ i \<le> (B n $ i - A n $ i) / 2" using AB(4)[of n i] by auto

           also have "\<dots> \<le> (b $ i - a $ i) / 2 ^ Suc n" using Suc by(auto simp add:field_simps) finally show ?case .

         qed qed

       also have "\<dots> < e" using n using goal1 by(auto simp add:field_simps) finally show "dist x y < e" .

     qed qed

   { fix n m ::nat assume "m \<le> n" then guess d unfolding le_Suc_ex_iff .. note d=this

     have "{A n..B n} \<subseteq> {A m..B m}" unfolding d 

     proof(induct d) case 0 thus ?case by auto

     next case (Suc d) show ?case apply(rule subset_trans[OF _ Suc])

         apply(rule) unfolding mem_interval apply(rule,erule_tac x=i in allE)

       proof- case goal1 thus ?case using AB(4)[of "m + d" i] by(auto simp add:field_simps)

       qed qed } note ABsubset = this 

   have "\<exists>a. \<forall>n. a\<in>{A n..B n}" apply(rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])

   proof- fix n show "{A n..B n} \<noteq> {}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(1,3) AB(1-2) by auto qed auto

   then guess x0 .. note x0=this[rule_format]

   show thesis proof(rule that[rule_format,of x0])

     show "x0\<in>{a..b}" using x0[of 0] unfolding AB .

     fix e assume "0 < (e::real)" from interv[OF this] guess n .. note n=this

     show "\<exists>c d. x0 \<in> {c..d} \<and> {c..d} \<subseteq> ball x0 e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"

       apply(rule_tac x="A n" in exI,rule_tac x="B n" in exI) apply(rule,rule x0) apply rule defer 

     proof show "\<not> P {A n..B n}" apply(cases "0<n") using AB(3)[of "n - 1"] assms(3) AB(1-2) by auto

       show "{A n..B n} \<subseteq> ball x0 e" using n using x0[of n] by auto

       show "{A n..B n} \<subseteq> {a..b}" unfolding AB(1-2)[symmetric] apply(rule ABsubset) by auto

     qed qed qed 

 subsection {* Cousin's lemma. *}

 lemma fine_division_exists: assumes "gauge g" 

   obtains p where "p tagged_division_of {a..b::real^'n}" "g fine p"

 proof- presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"

   then guess p unfolding atomize_not not_not .. thus thesis apply-apply(rule that[of p]) by auto

 next assume as:"\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"

   guess x apply(rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])

     apply(rule_tac x="{}" in exI) defer apply(erule conjE exE)+

   proof- show "{} tagged_division_of {} \<and> g fine {}" unfolding fine_def by auto

     fix s t p p' assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'" "interior s \<inter> interior t = {}"

     thus "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p" apply-apply(rule_tac x="p \<union> p'" in exI) apply rule

       apply(rule tagged_division_union) prefer 4 apply(rule fine_union) by auto

   qed note x=this

   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)] guess c d apply-apply(erule exE conjE)+ . note c_d = this

   have "g fine {(x, {c..d})}" unfolding fine_def using e using c_d(2) by auto

   thus False using tagged_division_of_self[OF c_d(1)] using c_d by auto qed

 subsection {* Basic theorems about integrals. *}

 lemma has_integral_unique: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"

   assumes "(f has_integral k1) i" "(f has_integral k2) i" shows "k1 = k2"

 proof(rule ccontr) let ?e = "norm(k1 - k2) / 2" assume as:"k1 \<noteq> k2" hence e:"?e > 0" by auto

   have lem:"\<And>f::real^'n \<Rightarrow> 'a.  \<And> a b k1 k2.

     (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"

   proof- case goal1 let ?e = "norm(k1 - k2) / 2" from goal1(3) have e:"?e > 0" by auto

     guess d1 by(rule has_integralD[OF goal1(1) e]) note d1=this

     guess d2 by(rule has_integralD[OF goal1(2) e]) note d2=this

     guess p by(rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this

     let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"

       using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"] by(auto simp add:group_simps norm_minus_commute)

     also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"

       apply(rule add_strict_mono) apply(rule_tac[!] d2(2) d1(2)) using p unfolding fine_def by auto

     finally show False by auto

   qed { presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"

     thus False apply-apply(cases "\<exists>a b. i = {a..b}")

       using assms by(auto simp add:has_integral intro:lem[OF _ _ as]) }

   assume as:"\<not> (\<exists>a b. i = {a..b})"

   guess B1 by(rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]

   guess B2 by(rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]

   have "\<exists>a b::real^'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}" apply(rule bounded_subset_closed_interval)

     using bounded_Un bounded_ball by auto then guess a b apply-by(erule exE)+

   note ab=conjunctD2[OF this[unfolded Un_subset_iff]]

   guess w using B1(2)[OF ab(1)] .. note w=conjunctD2[OF this]

   guess z using B2(2)[OF ab(2)] .. note z=conjunctD2[OF this]

   have "z = w" using lem[OF w(1) z(1)] by auto

   hence "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"

     using norm_triangle_ineq4[of "k1 - w" "k2 - z"] by(auto simp add: norm_minus_commute) 

   also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2" apply(rule add_strict_mono) by(rule_tac[!] z(2) w(2))

   finally show False by auto qed

 lemma integral_unique[intro]:

   "(f has_integral y) k \<Longrightarrow> integral k f = y"

   unfolding integral_def apply(rule some_equality) by(auto intro: has_integral_unique) 

 lemma has_integral_is_0: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 

   assumes "\<forall>x\<in>s. f x = 0" shows "(f has_integral 0) s"

 proof- have lem:"\<And>a b. \<And>f::real^'n \<Rightarrow> 'a.

     (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})" unfolding has_integral

   proof(rule,rule) fix a b e and f::"real^'n \<Rightarrow> 'a"

     assume as:"\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"

     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"

       apply(rule_tac x="\<lambda>x. ball x 1" in exI)  apply(rule,rule gaugeI) unfolding centre_in_ball defer apply(rule open_ball)

     proof(rule,rule,erule conjE) case goal1

       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0" proof(rule setsum_0',rule)

         fix x assume x:"x\<in>p" have "f (fst x) = 0" using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto

         thus "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0" apply(subst surjective_pairing[of x]) unfolding split_conv by auto

       qed thus ?case using as by auto

     qed auto qed  { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"

     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}")

       using assms by(auto simp add:has_integral intro:lem) }

   have *:"(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)" apply(rule ext) using assms by auto

   assume "\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P *

   apply(rule,rule,rule_tac x=1 in exI,rule) defer apply(rule,rule,rule)

   proof- fix e::real and a b assume "e>0"

     thus "\<exists>z. ((\<lambda>x::real^'n. 0::'a) has_integral z) {a..b} \<and> norm (z - 0) < e"

       apply(rule_tac x=0 in exI) apply(rule,rule lem) by auto

   qed auto qed

 lemma has_integral_0[simp]: "((\<lambda>x::real^'n. 0) has_integral 0) s"

   apply(rule has_integral_is_0) by auto 

 lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"

   using has_integral_unique[OF has_integral_0] by auto

 lemma has_integral_linear: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"

   assumes "(f has_integral y) s" "bounded_linear h" shows "((h o f) has_integral ((h y))) s"

 proof- interpret bounded_linear h using assms(2) . from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]

   have lem:"\<And>f::real^'n \<Rightarrow> 'a. \<And> y a b.

     (f has_integral y) ({a..b}) \<Longrightarrow> ((h o f) has_integral h(y)) ({a..b})"

   proof(subst has_integral,rule,rule) case goal1

     from pos_bounded guess B .. note B=conjunctD2[OF this,rule_format]

     have *:"e / B > 0" apply(rule divide_pos_pos) using goal1(2) B by auto

     guess g using has_integralD[OF goal1(1) *] . note g=this

     show ?case apply(rule_tac x=g in exI) apply(rule,rule g(1))

     proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "g fine p" 

       have *:"\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x" by auto

       have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = setsum (h \<circ> (\<lambda>(x, k). content k *\<^sub>R f x)) p"

         unfolding o_def unfolding scaleR[THEN sym] * by simp

       also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto

       finally have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .

       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e" unfolding * diff[THEN sym]

         apply(rule le_less_trans[OF B(2)]) using g(2)[OF as] B(1) by(auto simp add:field_simps)

     qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"

     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }

   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P

   proof(rule,rule) fix e::real  assume e:"0<e"

     have *:"0 < e/B" by(rule divide_pos_pos,rule e,rule B(1))

     guess M using has_integral_altD[OF assms(1) as *,rule_format] . note M=this

     show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) {a..b} \<and> norm (z - h y) < e)"

       apply(rule_tac x=M in exI) apply(rule,rule M(1))

     proof(rule,rule,rule) case goal1 guess z using M(2)[OF goal1(1)] .. note z=conjunctD2[OF this]

       have *:"(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"

         unfolding o_def apply(rule ext) using zero by auto

       show ?case apply(rule_tac x="h z" in exI,rule) unfolding * apply(rule lem[OF z(1)]) unfolding diff[THEN sym]

         apply(rule le_less_trans[OF B(2)]) using B(1) z(2) by(auto simp add:field_simps)

     qed qed qed

 lemma has_integral_cmul:

   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"

   unfolding o_def[THEN sym] apply(rule has_integral_linear,assumption)

   by(rule scaleR.bounded_linear_right)

 lemma has_integral_neg:

   shows "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. -(f x)) has_integral (-k)) s"

   apply(drule_tac c="-1" in has_integral_cmul) by auto

 lemma has_integral_add: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector" 

   assumes "(f has_integral k) s" "(g has_integral l) s"

   shows "((\<lambda>x. f x + g x) has_integral (k + l)) s"

 proof- have lem:"\<And>f g::real^'n \<Rightarrow> 'a. \<And>a b k l.

     (f has_integral k) ({a..b}) \<Longrightarrow> (g has_integral l) ({a..b}) \<Longrightarrow>

      ((\<lambda>x. f(x) + g(x)) has_integral (k + l)) ({a..b})" proof- case goal1

     show ?case unfolding has_integral proof(rule,rule) fix e::real assume e:"e>0" hence *:"e/2>0" by auto

       guess d1 using has_integralD[OF goal1(1) *] . note d1=this

       guess d2 using has_integralD[OF goal1(2) *] . note d2=this

       show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e)"

         apply(rule_tac x="\<lambda>x. (d1 x) \<inter> (d2 x)" in exI) apply(rule,rule gauge_inter[OF d1(1) d2(1)])

       proof(rule,rule,erule conjE) fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d1 x \<inter> d2 x) fine p"

         have *:"(\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) = (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p. content k *\<^sub>R g x)"

           unfolding scaleR_right_distrib setsum_addf[of "\<lambda>(x,k). content k *\<^sub>R f x" "\<lambda>(x,k). content k *\<^sub>R g x" p,THEN sym]

           by(rule setsum_cong2,auto)

         have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) = norm (((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k) + ((\<Sum>(x, k)\<in>p. content k *\<^sub>R g x) - l))"

           unfolding * by(auto simp add:group_simps) also let ?res = "\<dots>"

         from as have *:"d1 fine p" "d2 fine p" unfolding fine_inter by auto

         have "?res < e/2 + e/2" apply(rule le_less_trans[OF norm_triangle_ineq])

           apply(rule add_strict_mono) using d1(2)[OF as(1) *(1)] and d2(2)[OF as(1) *(2)] by auto

         finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (f x + g x)) - (k + l)) < e" by auto

       qed qed qed { presume "\<not> (\<exists>a b. s = {a..b}) \<Longrightarrow> ?thesis"

     thus ?thesis apply-apply(cases "\<exists>a b. s = {a..b}") using assms by(auto simp add:has_integral intro!:lem) }

   assume as:"\<not> (\<exists>a b. s = {a..b})" thus ?thesis apply(subst has_integral_alt) unfolding if_not_P

   proof(rule,rule) case goal1 hence *:"e/2 > 0" by auto

     from has_integral_altD[OF assms(1) as *] guess B1 . note B1=this[rule_format]

     from has_integral_altD[OF assms(2) as *] guess B2 . note B2=this[rule_format]

     show ?case apply(rule_tac x="max B1 B2" in exI) apply(rule,rule min_max.less_supI1,rule B1)

     proof(rule,rule,rule) fix a b assume "ball 0 (max B1 B2) \<subseteq> {a..b::real^'n}"

       hence *:"ball 0 B1 \<subseteq> {a..b::real^'n}" "ball 0 B2 \<subseteq> {a..b::real^'n}" by auto

       guess w using B1(2)[OF *(1)] .. note w=conjunctD2[OF this]

       guess z using B2(2)[OF *(2)] .. note z=conjunctD2[OF this]

       have *:"\<And>x. (if x \<in> s then f x + g x else 0) = (if x \<in> s then f x else 0) + (if x \<in> s then g x else 0)" by auto

       show "\<exists>z. ((\<lambda>x. if x \<in> s then f x + g x else 0) has_integral z) {a..b} \<and> norm (z - (k + l)) < e"

         apply(rule_tac x="w + z" in exI) apply(rule,rule lem[OF w(1) z(1), unfolded *[THEN sym]])

         using norm_triangle_ineq[of "w - k" "z - l"] w(2) z(2) by(auto simp add:field_simps)

     qed qed qed

 lemma has_integral_sub:

   shows "(f has_integral k) s \<Longrightarrow> (g has_integral l) s \<Longrightarrow> ((\<lambda>x. f(x) - g(x)) has_integral (k - l)) s"

   using has_integral_add[OF _ has_integral_neg,of f k s g l] unfolding group_simps by auto

 lemma integral_0: "integral s (\<lambda>x::real^'n. 0::real^'m) = 0"

   by(rule integral_unique has_integral_0)+

 lemma integral_add:

   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow>

    integral s (\<lambda>x. f x + g x) = integral s f + integral s g"

   apply(rule integral_unique) apply(drule integrable_integral)+

   apply(rule has_integral_add) by assumption+

 lemma integral_cmul:

   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. c *\<^sub>R f x) = c *\<^sub>R integral s f"

   apply(rule integral_unique) apply(drule integrable_integral)+

   apply(rule has_integral_cmul) by assumption+

 lemma integral_neg:

   shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. - f x) = - integral s f"

   apply(rule integral_unique) apply(drule integrable_integral)+

   apply(rule has_integral_neg) by assumption+

 lemma integral_sub:

   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x - g x) = integral s f - integral s g"

   apply(rule integral_unique) apply(drule integrable_integral)+

   apply(rule has_integral_sub) by assumption+

 lemma integrable_0: "(\<lambda>x. 0) integrable_on s"

   unfolding integrable_on_def using has_integral_0 by auto

 lemma integrable_add:

   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x + g x) integrable_on s"

   unfolding integrable_on_def by(auto intro: has_integral_add)

 lemma integrable_cmul:

   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) integrable_on s"

   unfolding integrable_on_def by(auto intro: has_integral_cmul)

 lemma integrable_neg:

   shows "f integrable_on s \<Longrightarrow> (\<lambda>x. -f(x)) integrable_on s"

   unfolding integrable_on_def by(auto intro: has_integral_neg)

 lemma integrable_sub:

   shows "f integrable_on s \<Longrightarrow> g integrable_on s \<Longrightarrow> (\<lambda>x. f x - g x) integrable_on s"

   unfolding integrable_on_def by(auto intro: has_integral_sub)

 lemma integrable_linear:

   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> (h o f) integrable_on s"

   unfolding integrable_on_def by(auto intro: has_integral_linear)

 lemma integral_linear:

   shows "f integrable_on s \<Longrightarrow> bounded_linear h \<Longrightarrow> integral s (h o f) = h(integral s f)"

   apply(rule has_integral_unique) defer unfolding has_integral_integral 

   apply(drule has_integral_linear,assumption,assumption) unfolding has_integral_integral[THEN sym]

   apply(rule integrable_linear) by assumption+

 lemma has_integral_setsum:

   assumes "finite t" "\<forall>a\<in>t. ((f a) has_integral (i a)) s"

   shows "((\<lambda>x. setsum (\<lambda>a. f a x) t) has_integral (setsum i t)) s"

 proof(insert assms(1) subset_refl[of t],induct rule:finite_subset_induct)

   case (insert x F) show ?case unfolding setsum_insert[OF insert(1,3)]

     apply(rule has_integral_add) using insert assms by auto

 qed auto

 lemma integral_setsum:

   shows "finite t \<Longrightarrow> \<forall>a\<in>t. (f a) integrable_on s \<Longrightarrow>

   integral s (\<lambda>x. setsum (\<lambda>a. f a x) t) = setsum (\<lambda>a. integral s (f a)) t"

   apply(rule integral_unique) apply(rule has_integral_setsum)

   using integrable_integral by auto

 lemma integrable_setsum:

   shows "finite t \<Longrightarrow> \<forall>a \<in> t.(f a) integrable_on s \<Longrightarrow> (\<lambda>x. setsum (\<lambda>a. f a x) t) integrable_on s"

   unfolding integrable_on_def apply(drule bchoice) using has_integral_setsum[of t] by auto

 lemma has_integral_eq:

   assumes "\<forall>x\<in>s. f x = g x" "(f has_integral k) s" shows "(g has_integral k) s"

   using has_integral_sub[OF assms(2), of "\<lambda>x. f x - g x" 0]

   using has_integral_is_0[of s "\<lambda>x. f x - g x"] using assms(1) by auto

 lemma integrable_eq:

   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> f integrable_on s \<Longrightarrow> g integrable_on s"

   unfolding integrable_on_def using has_integral_eq[of s f g] by auto

 lemma has_integral_eq_eq:

   shows "\<forall>x\<in>s. f x = g x \<Longrightarrow> ((f has_integral k) s \<longleftrightarrow> (g has_integral k) s)"

   using has_integral_eq[of s f g] has_integral_eq[of s g f] by auto 

 lemma has_integral_null[dest]:

   assumes "content({a..b}) = 0" shows  "(f has_integral 0) ({a..b})"

   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI,rule) defer

 proof(rule,rule,erule conjE) fix e::real assume e:"e>0" thus "gauge (\<lambda>x. ball x 1)" by auto

   fix p assume p:"p tagged_division_of {a..b}" (*"(\<lambda>x. ball x 1) fine p"*)

   have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) = 0" unfolding norm_eq_zero diff_0_right

     using setsum_content_null[OF assms(1) p, of f] . 

   thus "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e" using e by auto qed

 lemma has_integral_null_eq[simp]:

   shows "content({a..b}) = 0 \<Longrightarrow> ((f has_integral i) ({a..b}) \<longleftrightarrow> i = 0)"

   apply rule apply(rule has_integral_unique,assumption) 

   apply(drule has_integral_null,assumption)

   apply(drule has_integral_null) by auto

 lemma integral_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> integral({a..b}) f = 0"

   by(rule integral_unique,drule has_integral_null)

 lemma integrable_on_null[dest]: shows "content({a..b}) = 0 \<Longrightarrow> f integrable_on {a..b}"

   unfolding integrable_on_def apply(drule has_integral_null) by auto

 lemma has_integral_empty[intro]: shows "(f has_integral 0) {}"

   unfolding empty_as_interval apply(rule has_integral_null) 

   using content_empty unfolding empty_as_interval .

 lemma has_integral_empty_eq[simp]: shows "(f has_integral i) {} \<longleftrightarrow> i = 0"

   apply(rule,rule has_integral_unique,assumption) by auto

 lemma integrable_on_empty[intro]: shows "f integrable_on {}" unfolding integrable_on_def by auto

 lemma integral_empty[simp]: shows "integral {} f = 0"

   apply(rule integral_unique) using has_integral_empty .

 lemma has_integral_refl[intro]: shows "(f has_integral 0) {a..a}"

   apply(rule has_integral_null) unfolding content_eq_0_interior

   unfolding interior_closed_interval using interval_sing by auto

 lemma integrable_on_refl[intro]: shows "f integrable_on {a..a}" unfolding integrable_on_def by auto

 lemma integral_refl: shows "integral {a..a} f = 0" apply(rule integral_unique) by auto

 subsection {* Cauchy-type criterion for integrability. *}

 lemma integrable_cauchy: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" 

   shows "f integrable_on {a..b} \<longleftrightarrow>

   (\<forall>e>0.\<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<and> d fine p1 \<and>

                             p2 tagged_division_of {a..b} \<and> d fine p2

                             \<longrightarrow> norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 -

                                      setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) < e))" (is "?l = (\<forall>e>0. \<exists>d. ?P e d)")

 proof assume ?l

   then guess y unfolding integrable_on_def has_integral .. note y=this

   show "\<forall>e>0. \<exists>d. ?P e d" proof(rule,rule) case goal1 hence "e/2 > 0" by auto

     then guess d apply- apply(drule y[rule_format]) by(erule exE,erule conjE) note d=this[rule_format]

     show ?case apply(rule_tac x=d in exI,rule,rule d) apply(rule,rule,rule,(erule conjE)+)

     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b}" "d fine p1" "p2 tagged_division_of {a..b}" "d fine p2"

       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"

         apply(rule dist_triangle_half_l[where y=y,unfolded vector_dist_norm])

         using d(2)[OF conjI[OF as(1-2)]] d(2)[OF conjI[OF as(3-4)]] .

     qed qed

 next assume "\<forall>e>0. \<exists>d. ?P e d" hence "\<forall>n::nat. \<exists>d. ?P (inverse(real (n + 1))) d" by auto

   from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format],rule_format]

   have "\<And>n. gauge (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}})" apply(rule gauge_inters) using d(1) by auto

   hence "\<forall>n. \<exists>p. p tagged_division_of {a..b} \<and> (\<lambda>x. \<Inter>{d i x |i. i \<in> {0..n}}) fine p" apply-

   proof case goal1 from this[of n] show ?case apply(drule_tac fine_division_exists) by auto qed

   from choice[OF this] guess p .. note p = conjunctD2[OF this[rule_format]]

   have dp:"\<And>i n. i\<le>n \<Longrightarrow> d i fine p n" using p(2) unfolding fine_inters by auto

   have "Cauchy (\<lambda>n. setsum (\<lambda>(x,k). content k *\<^sub>R (f x)) (p n))"

   proof(rule CauchyI) case goal1 then guess N unfolding real_arch_inv[of e] .. note N=this

     show ?case apply(rule_tac x=N in exI)

     proof(rule,rule,rule,rule) fix m n assume mn:"N \<le> m" "N \<le> n" have *:"N = (N - 1) + 1" using N by auto

       show "norm ((\<Sum>(x, k)\<in>p m. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p n. content k *\<^sub>R f x)) < e"

         apply(rule less_trans[OF _ N[THEN conjunct2,THEN conjunct2]]) apply(subst *) apply(rule d(2))

         using dp p(1) using mn by auto 

     qed qed

   then guess y unfolding convergent_eq_cauchy[THEN sym] .. note y=this[unfolded Lim_sequentially,rule_format]

   show ?l unfolding integrable_on_def has_integral apply(rule_tac x=y in exI)

   proof(rule,rule) fix e::real assume "e>0" hence *:"e/2 > 0" by auto

     then guess N1 unfolding real_arch_inv[of "e/2"] .. note N1=this hence N1':"N1 = N1 - 1 + 1" by auto

     guess N2 using y[OF *] .. note N2=this

     show "\<exists>d. gauge d \<and> (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e)"

       apply(rule_tac x="d (N1 + N2)" in exI) apply rule defer 

     proof(rule,rule,erule conjE) show "gauge (d (N1 + N2))" using d by auto

       fix q assume as:"q tagged_division_of {a..b}" "d (N1 + N2) fine q"

       have *:"inverse (real (N1 + N2 + 1)) < e / 2" apply(rule less_trans) using N1 by auto

       show "norm ((\<Sum>(x, k)\<in>q. content k *\<^sub>R f x) - y) < e" apply(rule norm_triangle_half_r)

         apply(rule less_trans[OF _ *]) apply(subst N1', rule d(2)[of "p (N1+N2)"]) defer

         using N2[rule_format,unfolded vector_dist_norm,of "N1+N2"]

         using as dp[of "N1 - 1 + 1 + N2" "N1 + N2"] using p(1)[of "N1 + N2"] using N1 by auto qed qed qed

 subsection {* Additivity of integral on abutting intervals. *}

 lemma interval_split:

   "{a..b::real^'n} \<inter> {x. x$k \<le> c} = {a .. (\<chi> i. if i = k then min (b$k) c else b$i)}"

   "{a..b} \<inter> {x. x$k \<ge> c} = {(\<chi> i. if i = k then max (a$k) c else a$i) .. b}"

   apply(rule_tac[!] set_ext) unfolding Int_iff mem_interval mem_Collect_eq

   unfolding Cart_lambda_beta by auto

 lemma content_split:

   "content {a..b::real^'n} = content({a..b} \<inter> {x. x$k \<le> c}) + content({a..b} \<inter> {x. x$k >= c})"

 proof- note simps = interval_split content_closed_interval_cases Cart_lambda_beta vector_le_def

   { presume "a\<le>b \<Longrightarrow> ?thesis" thus ?thesis apply(cases "a\<le>b") unfolding simps by auto }

   have *:"UNIV = insert k (UNIV - {k})" "\<And>x. finite (UNIV-{x::'n})" "\<And>x. x\<notin>UNIV-{x}" by auto

   have *:"\<And>X Y Z. (\<Prod>i\<in>UNIV. Z i (if i = k then X else Y i)) = Z k X * (\<Prod>i\<in>UNIV-{k}. Z i (Y i))"

     "(\<Prod>i\<in>UNIV. b$i - a$i) = (\<Prod>i\<in>UNIV-{k}. b$i - a$i) * (b$k - a$k)" 

     apply(subst *(1)) defer apply(subst *(1)) unfolding setprod_insert[OF *(2-)] by auto

   assume as:"a\<le>b" moreover have "\<And>x. min (b $ k) c = max (a $ k) c

     \<Longrightarrow> x* (b$k - a$k) = x*(max (a $ k) c - a $ k) + x*(b $ k - max (a $ k) c)"

     by  (auto simp add:field_simps)

   moreover have "\<not> a $ k \<le> c \<Longrightarrow> \<not> c \<le> b $ k \<Longrightarrow> False"

     unfolding not_le using as[unfolded vector_le_def,rule_format,of k] by auto

   ultimately show ?thesis 

     unfolding simps unfolding *(1)[of "\<lambda>i x. b$i - x"] *(1)[of "\<lambda>i x. x - a$i"] *(2) by(auto)

 qed

 lemma division_split_left_inj:

   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"

   "k1 \<inter> {x::real^'n. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}"

   shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"

 proof- note d=division_ofD[OF assms(1)]

   have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k \<le> c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k \<le> c}) = {})"

     unfolding interval_split content_eq_0_interior by auto

   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this

   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule 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)]])

     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed

 lemma division_split_right_inj:

   assumes "d division_of i" "k1 \<in> d" "k2 \<in> d"  "k1 \<noteq> k2"

   "k1 \<inter> {x::real^'n. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}"

   shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"

 proof- note d=division_ofD[OF assms(1)]

   have *:"\<And>a b::real^'n. \<And> c k. (content({a..b} \<inter> {x. x$k >= c}) = 0 \<longleftrightarrow> interior({a..b} \<inter> {x. x$k >= c}) = {})"

     unfolding interval_split content_eq_0_interior by auto

   guess u1 v1 using d(4)[OF assms(2)] apply-by(erule exE)+ note uv1=this

   guess u2 v2 using d(4)[OF assms(3)] apply-by(erule 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)]])

     defer apply(subst assms(5)[unfolded uv1 uv2]) unfolding uv1 uv2 by auto qed

 lemma tagged_division_split_left_inj:

   assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$k \<le> c} = k2 \<inter> {x. x$k \<le> c}" 

   shows "content(k1 \<inter> {x. x$k \<le> c}) = 0"

 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto

   show ?thesis apply(rule division_split_left_inj[OF division_of_tagged_division[OF assms(1)]])

     apply(rule_tac[1-2] *) using assms(2-) by auto qed

 lemma tagged_division_split_right_inj:

   assumes "d tagged_division_of i" "(x1,k1) \<in> d" "(x2,k2) \<in> d" "k1 \<noteq> k2"  "k1 \<inter> {x. x$k \<ge> c} = k2 \<inter> {x. x$k \<ge> c}" 

   shows "content(k1 \<inter> {x. x$k \<ge> c}) = 0"

 proof- have *:"\<And>a b c. (a,b) \<in> c \<Longrightarrow> b \<in> snd ` c" unfolding image_iff apply(rule_tac x="(a,b)" in bexI) by auto

   show ?thesis apply(rule division_split_right_inj[OF division_of_tagged_division[OF assms(1)]])

     apply(rule_tac[1-2] *) using assms(2-) by auto qed

 lemma division_split:

   assumes "p division_of {a..b::real^'n}"

   shows "{l \<inter> {x. x$k \<le> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$k \<le> c} = {})} division_of ({a..b} \<inter> {x. x$k \<le> c})" (is "?p1 division_of ?I1") and 

         "{l \<inter> {x. x$k \<ge> c} | l. l \<in> p \<and> ~(l \<inter> {x. x$k \<ge> c} = {})} division_of ({a..b} \<inter> {x. x$k \<ge> c})" (is "?p2 division_of ?I2")

 proof(rule_tac[!] division_ofI) note p=division_ofD[OF assms]

   show "finite ?p1" "finite ?p2" using p(1) by auto show "\<Union>?p1 = ?I1" "\<Union>?p2 = ?I2" unfolding p(6)[THEN sym] by auto

   { fix k assume "k\<in>?p1" then guess l unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l=this

     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this

     show "k\<subseteq>?I1" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l

       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto

     fix k' assume "k'\<in>?p1" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this

     assume "k\<noteq>k'" thus "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 apply-by(erule exE,(erule conjE)+) note l=this

     guess u v using p(4)[OF l(2)] apply-by(erule exE)+ note uv=this

     show "k\<subseteq>?I2" "k \<noteq> {}" "\<exists>a b. k = {a..b}" unfolding l

       using p(2-3)[OF l(2)] l(3) unfolding uv apply- prefer 3 apply(subst interval_split) by auto

     fix k' assume "k'\<in>?p2" then guess l' unfolding mem_Collect_eq apply-by(erule exE,(erule conjE)+) note l'=this

     assume "k\<noteq>k'" thus "interior k \<inter> interior k' = {}" unfolding l l' using p(5)[OF l(2) l'(2)] by auto }

 qed

 lemma has_integral_split: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"

   assumes "(f has_integral i) ({a..b} \<inter> {x. x$k \<le> c})"  "(f has_integral j) ({a..b} \<inter> {x. x$k \<ge> c})"

   shows "(f has_integral (i + j)) ({a..b})"

 proof(unfold has_integral,rule,rule) case goal1 hence e:"e/2>0" by auto

   guess d1 using has_integralD[OF assms(1)[unfolded interval_split] e] . note d1=this[unfolded interval_split[THEN sym]]

   guess d2 using has_integralD[OF assms(2)[unfolded interval_split] e] . note d2=this[unfolded interval_split[THEN sym]]

   let ?d = "\<lambda>x. if x$k = c then (d1 x \<inter> d2 x) else ball x (abs(x$k - c)) \<inter> d1 x \<inter> d2 x"

   show ?case apply(rule_tac x="?d" in exI,rule) defer apply(rule,rule,(erule conjE)+)

   proof- show "gauge ?d" using d1(1) d2(1) unfolding gauge_def by auto

     fix p assume "p tagged_division_of {a..b}" "?d fine p" note p = this tagged_division_ofD[OF this(1)]

     have lem0:"\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"

          "\<And>x kk. (x,kk) \<in> p \<Longrightarrow> ~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"

     proof- fix x kk assume as:"(x,kk)\<in>p"

       show "~(kk \<inter> {x. x$k \<le> c} = {}) \<Longrightarrow> x$k \<le> c"

       proof(rule ccontr) case goal1

         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"

           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto

         hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<le> c}" using goal1(1) by blast 

         then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<le> c" apply-apply(rule le_less_trans)

           using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)

         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)

       qed

       show "~(kk \<inter> {x. x$k \<ge> c} = {}) \<Longrightarrow> x$k \<ge> c"

       proof(rule ccontr) case goal1

         from this(2)[unfolded not_le] have "kk \<subseteq> ball x \<bar>x $ k - c\<bar>"

           using p(2)[unfolded fine_def,rule_format,OF as,unfolded split_conv] by auto

         hence "\<exists>y. y \<in> ball x \<bar>x $ k - c\<bar> \<inter> {x. x $ k \<ge> c}" using goal1(1) by blast 

         then guess y .. hence "\<bar>x $ k - y $ k\<bar> < \<bar>x $ k - c\<bar>" "y$k \<ge> c" apply-apply(rule le_less_trans)

           using component_le_norm[of "x - y" k,unfolded vector_minus_component] by(auto simp add:vector_dist_norm)

         thus False using goal1(2)[unfolded not_le] by(auto simp add:field_simps)

       qed

     qed

     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 lem2: "\<And>f s P f. finite s \<Longrightarrow> finite {(x,f k) | x k. (x,k) \<in> s \<and> P x k}"

     proof- case goal1 thus ?case apply-apply(rule finite_subset[of _ "(\<lambda>(x,k). (x,f k)) ` s"]) by auto qed

     have lem3: "\<And>g::(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool. finite p \<Longrightarrow>

       setsum (\<lambda>(x,k). content k *\<^sub>R f x) {(x,g k) |x k. (x,k) \<in> p \<and> ~(g k = {})}

                = setsum (\<lambda>(x,k). content k *\<^sub>R f x) ((\<lambda>(x,k). (x,g k)) ` p)"

       apply(rule setsum_mono_zero_left) prefer 3

     proof fix g::"(real ^ 'n \<Rightarrow> bool) \<Rightarrow> real ^ 'n \<Rightarrow> bool" and i::"(real^'n) \<times> ((real^'n) set)"

       assume "i \<in> (\<lambda>(x, k). (x, g k)) ` p - {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}"

       then obtain x k where xk:"i=(x,g k)" "(x,k)\<in>p" "(x,g k) \<notin> {(x, g k) |x k. (x, k) \<in> p \<and> g k \<noteq> {}}" by auto

       have "content (g k) = 0" using xk using content_empty by auto

       thus "(\<lambda>(x, k). content k *\<^sub>R f x) i = 0" unfolding xk split_conv by auto

     qed auto

     have lem4:"\<And>g. (\<lambda>(x,l). content (g l) *\<^sub>R f x) = (\<lambda>(x,l). content l *\<^sub>R f x) o (\<lambda>(x,l). (x,g l))" apply(rule ext) by auto

     let ?M1 = "{(x,kk \<inter> {x. x$k \<le> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<le> c} \<noteq> {}}"

     have "norm ((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) < e/2" apply(rule d1(2),rule tagged_division_ofI)

       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)

     proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M1} = {a..b} \<inter> {x. x$k \<le> c}" unfolding p(8)[THEN sym] by auto

       fix x l assume xl:"(x,l)\<in>?M1"

       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this

       have "l' \<subseteq> d1 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto

       thus "l \<subseteq> d1 x" unfolding xl' by auto

       show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $ k \<le> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)

         using lem0(1)[OF xl'(3-4)] by auto

       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[where c=c and k=k])

       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M1"

       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this

       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"

       proof(cases "l' = r' \<longrightarrow> x' = y'")

         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto

       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto

         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto

       qed qed moreover

     let ?M2 = "{(x,kk \<inter> {x. x$k \<ge> c}) |x kk. (x,kk) \<in> p \<and> kk \<inter> {x. x$k \<ge> c} \<noteq> {}}" 

     have "norm ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j) < e/2" apply(rule d2(2),rule tagged_division_ofI)

       apply(rule lem2 p(3))+ prefer 6 apply(rule fineI)

     proof- show "\<Union>{k. \<exists>x. (x, k) \<in> ?M2} = {a..b} \<inter> {x. x$k \<ge> c}" unfolding p(8)[THEN sym] by auto

       fix x l assume xl:"(x,l)\<in>?M2"

       then guess x' l' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note xl'=this

       have "l' \<subseteq> d2 x'" apply(rule order_trans[OF fineD[OF p(2) xl'(3)]]) by auto

       thus "l \<subseteq> d2 x" unfolding xl' by auto

       show "x\<in>l" "l \<subseteq> {a..b} \<inter> {x. x $ k \<ge> c}" unfolding xl' using p(4-6)[OF xl'(3)] using xl'(4)

         using lem0(2)[OF xl'(3-4)] by auto

       show  "\<exists>a b. l = {a..b}" unfolding xl' using p(6)[OF xl'(3)] by(fastsimp simp add: interval_split[where c=c and k=k])

       fix y r let ?goal = "interior l \<inter> interior r = {}" assume yr:"(y,r)\<in>?M2"

       then guess y' r' unfolding mem_Collect_eq apply- unfolding Pair_eq apply((erule exE)+,(erule conjE)+) .  note yr'=this

       assume as:"(x,l) \<noteq> (y,r)" show "interior l \<inter> interior r = {}"

       proof(cases "l' = r' \<longrightarrow> x' = y'")

         case False thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto

       next case True hence "l' \<noteq> r'" using as unfolding xl' yr' by auto

         thus ?thesis using p(7)[OF xl'(3) yr'(3)] using as unfolding xl' yr' by auto

       qed qed ultimately

     have "norm (((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)) < e/2 + e/2"

       apply- apply(rule norm_triangle_lt) by auto

     also { have *:"\<And>x y. x = (0::real) \<Longrightarrow> x *\<^sub>R (y::'a) = 0" using scaleR_zero_left by auto

       have "((\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) - i) + ((\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - j)

        = (\<Sum>(x, k)\<in>?M1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>?M2. content k *\<^sub>R f x) - (i + j)" by auto

       also have "\<dots> = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. x $ k \<le> c}) *\<^sub>R f x) + (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. c \<le> x $ k}) *\<^sub>R f x) - (i + j)"

         unfolding lem3[OF p(3)] apply(subst setsum_reindex_nonzero[OF p(3)]) defer apply(subst setsum_reindex_nonzero[OF p(3)])

         defer unfolding lem4[THEN sym] apply(rule refl) unfolding split_paired_all split_conv apply(rule_tac[!] *)

       proof- case goal1 thus ?case apply- apply(rule tagged_division_split_left_inj [OF p(1), of a b aa ba]) by auto

       next case   goal2 thus ?case apply- apply(rule tagged_division_split_right_inj[OF p(1), of a b aa ba]) by auto

       qed also note setsum_addf[THEN sym]

       also have *:"\<And>x. x\<in>p \<Longrightarrow> (\<lambda>(x, ka). content (ka \<inter> {x. x $ k \<le> c}) *\<^sub>R f x) x + (\<lambda>(x, ka). content (ka \<inter> {x. c \<le> x $ k}) *\<^sub>R f x) x

         = (\<lambda>(x,ka). content ka *\<^sub>R f x) x" unfolding split_paired_all split_conv

       proof- fix a b assume "(a,b) \<in> p" from p(6)[OF this] guess u v apply-by(erule exE)+ note uv=this

         thus "content (b \<inter> {x. x $ k \<le> c}) *\<^sub>R f a + content (b \<inter> {x. c \<le> x $ k}) *\<^sub>R f a = content b *\<^sub>R f a"

           unfolding scaleR_left_distrib[THEN sym] unfolding uv content_split[of u v k c] by auto

       qed note setsum_cong2[OF this]

       finally have "(\<Sum>(x, k)\<in>{(x, kk \<inter> {x. x $ k \<le> c}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. x $ k \<le> c} \<noteq> {}}. content k *\<^sub>R f x) - i +

         ((\<Sum>(x, k)\<in>{(x, kk \<inter> {x. c \<le> x $ k}) |x kk. (x, kk) \<in> p \<and> kk \<inter> {x. c \<le> x $ k} \<noteq> {}}. content k *\<^sub>R f x) - j) =

         (\<Sum>(x, ka)\<in>p. content ka *\<^sub>R f x) - (i + j)" by auto }

     finally show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - (i + j)) < e" by auto qed qed

 subsection {* A sort of converse, integrability on subintervals. *}

 lemma tagged_division_union_interval:

   assumes "p1 tagged_division_of ({a..b} \<inter> {x::real^'n. x$k \<le> (c::real)})"  "p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c})"

   shows "(p1 \<union> p2) tagged_division_of ({a..b})"

 proof- have *:"{a..b} = ({a..b} \<inter> {x. x$k \<le> c}) \<union> ({a..b} \<inter> {x. x$k \<ge> c})" by auto

   show ?thesis apply(subst *) apply(rule tagged_division_union[OF assms])

     unfolding interval_split interior_closed_interval

     by(auto simp add: vector_less_def Cart_lambda_beta elim!:allE[where x=k]) qed

 lemma has_integral_separate_sides: fixes f::"real^'m \<Rightarrow> 'a::real_normed_vector"

   assumes "(f has_integral i) ({a..b})" "e>0"

   obtains d where "gauge d" "(\<forall>p1 p2. p1 tagged_division_of ({a..b} \<inter> {x. x$k \<le> c}) \<and> d fine p1 \<and>

                                 p2 tagged_division_of ({a..b} \<inter> {x. x$k \<ge> c}) \<and> d fine p2

                                 \<longrightarrow> norm((setsum (\<lambda>(x,k). content k *\<^sub>R f x) p1 +

                                           setsum (\<lambda>(x,k). content k *\<^sub>R f x) p2) - i) < e)"

 proof- guess d using has_integralD[OF assms] . note d=this

   show ?thesis apply(rule that[of d]) apply(rule d) apply(rule,rule,rule,(erule conjE)+)

   proof- fix p1 p2 assume "p1 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c}" "d fine p1" note p1=tagged_division_ofD[OF this(1)] this

                    assume "p2 tagged_division_of {a..b} \<inter> {x. c \<le> x $ k}" "d fine p2" note p2=tagged_division_ofD[OF this(1)] this

     note tagged_division_union_interval[OF p1(7) p2(7)] note p12 = tagged_division_ofD[OF this] this

     have "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) = norm ((\<Sum>(x, k)\<in>p1 \<union> p2. content k *\<^sub>R f x) - i)"

       apply(subst setsum_Un_zero) apply(rule p1 p2)+ apply(rule) unfolding split_paired_all split_conv

     proof- fix a b assume ab:"(a,b) \<in> p1 \<inter> p2"

       have "(a,b) \<in> p1" using ab by auto from p1(4)[OF this] guess u v apply-by(erule exE)+ note uv =this

       have "b \<subseteq> {x. x$k = c}" using ab p1(3)[of a b] p2(3)[of a b] by fastsimp

       moreover have "interior {x. x $ k = c} = {}" 

       proof(rule ccontr) case goal1 then obtain x where x:"x\<in>interior {x. x$k = c}" by auto

         then guess e unfolding mem_interior .. note e=this

         have x:"x$k = c" using x interior_subset by fastsimp

         have *:"\<And>i. \<bar>(x - (x + (\<chi> i. if i = k then e / 2 else 0))) $ i\<bar> = (if i = k then e/2 else 0)" using e by auto

         have "x + (\<chi> i. if i = k then e/2 else 0) \<in> ball x e" unfolding mem_ball vector_dist_norm 

           apply(rule le_less_trans[OF norm_le_l1]) unfolding * 

           unfolding setsum_delta[OF finite_UNIV] using e by auto 

         hence "x + (\<chi> i. if i = k then e/2 else 0) \<in> {x. x$k = c}" using e by auto

         thus False unfolding mem_Collect_eq using e x by auto

       qed ultimately have "content b = 0" unfolding uv content_eq_0_interior apply-apply(drule subset_interior) by auto

       thus "content b *\<^sub>R f a = 0" by auto

     qed auto

     also have "\<dots> < e" by(rule d(2) p12 fine_union p1 p2)+

     finally show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) + (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x) - i) < e" . qed qed

 lemma integrable_split[intro]: fixes f::"real^'n \<Rightarrow> 'a::{real_normed_vector,complete_space}" assumes "f integrable_on {a..b}"

   shows "f integrable_on ({a..b} \<inter> {x. x$k \<le> c})" (is ?t1) and "f integrable_on ({a..b} \<inter> {x. x$k \<ge> c})" (is ?t2) 

 proof- guess y using assms unfolding integrable_on_def .. note y=this

   def b' \<equiv> "(\<chi> i. if i = k then min (b$k) c else b$i)::real^'n"

   and a' \<equiv> "(\<chi> i. if i = k then max (a$k) c else a$i)::real^'n"

   show ?t1 ?t2 unfolding interval_split integrable_cauchy unfolding interval_split[THEN sym]

   proof(rule_tac[!] allI impI)+ fix e::real assume "e>0" hence "e/2>0" by auto

     from has_integral_separate_sides[OF y this,of k c] guess d . note d=this[rule_format]

     let ?P = "\<lambda>A. \<exists>d. gauge d \<and> (\<forall>p1 p2. p1 tagged_division_of {a..b} \<inter> A \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> A \<and> d fine p2 \<longrightarrow>

                               norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e)"

     show "?P {x. x $ k \<le> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)

     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c} \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> {x. x $ k \<le> c} \<and> d fine p2"

       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"

       proof- guess p using fine_division_exists[OF d(1), of a' b] . note p=this

         show ?thesis using norm_triangle_half_l[OF d(2)[of p1 p] d(2)[of p2 p]]

           using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]

           using p using assms by(auto simp add:group_simps)

       qed qed  

     show "?P {x. x $ k \<ge> c}" apply(rule_tac x=d in exI) apply(rule,rule d) apply(rule,rule,rule)

     proof- fix p1 p2 assume as:"p1 tagged_division_of {a..b} \<inter> {x. x $ k \<ge> c} \<and> d fine p1 \<and> p2 tagged_division_of {a..b} \<inter> {x. x $ k \<ge> c} \<and> d fine p2"

       show "norm ((\<Sum>(x, k)\<in>p1. content k *\<^sub>R f x) - (\<Sum>(x, k)\<in>p2. content k *\<^sub>R f x)) < e"

       proof- guess p using fine_division_exists[OF d(1), of a b'] . note p=this

         show ?thesis using norm_triangle_half_l[OF d(2)[of p p1] d(2)[of p p2]]

           using as unfolding interval_split b'_def[symmetric] a'_def[symmetric]

           using p using assms by(auto simp add:group_simps) qed qed qed qed

 subsection {* Generalized notion of additivity. *}

 definition "neutral opp = (SOME x. \<forall>y. opp x y = y \<and> opp y x = y)"

 definition operative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ((real^'n) set \<Rightarrow> 'a) \<Rightarrow> bool" where

   "operative opp f \<equiv> 

     (\<forall>a b. content {a..b} = 0 \<longrightarrow> f {a..b} = neutral(opp)) \<and>

     (\<forall>a b c k. f({a..b}) =

                    opp (f({a..b} \<inter> {x. x$k \<le> c}))

                        (f({a..b} \<inter> {x. x$k \<ge> c})))"

 lemma operativeD[dest]: assumes "operative opp f"

   shows "\<And>a b. content {a..b} = 0 \<Longrightarrow> f {a..b} = neutral(opp)"

   "\<And>a b c k. f({a..b}) = opp (f({a..b} \<inter> {x. x$k \<le> c})) (f({a..b} \<inter> {x. x$k \<ge> c}))"

   using assms unfolding operative_def by auto

 lemma operative_trivial:

  "operative opp f \<Longrightarrow> content({a..b}) = 0 \<Longrightarrow> f({a..b}) = neutral opp"

   unfolding operative_def by auto

 lemma property_empty_interval:

  "(\<forall>a b. content({a..b}) = 0 \<longrightarrow> P({a..b})) \<Longrightarrow> P {}" 

   using content_empty unfolding empty_as_interval by auto

 lemma operative_empty: "operative opp f \<Longrightarrow> f {} = neutral opp"

   unfolding operative_def apply(rule property_empty_interval) by auto

 subsection {* Using additivity of lifted function to encode definedness. *}

 lemma forall_option: "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>x. P(Some x))"

   by (metis map_of.simps option.nchotomy)

 lemma exists_option:

  "(\<exists>x. P x) \<longleftrightarrow> P None \<or> (\<exists>x. P(Some x))" 

   by (metis map_of.simps option.nchotomy)

 fun lifted where 

   "lifted (opp::'a\<Rightarrow>'a\<Rightarrow>'b) (Some x) (Some y) = Some(opp x y)" |

   "lifted opp None _ = (None::'b option)" |

   "lifted opp _ None = None"

 lemma lifted_simp_1[simp]: "lifted opp v None = None"

   apply(induct v) by auto

 definition "monoidal opp \<equiv>  (\<forall>x y. opp x y = opp y x) \<and>

                    (\<forall>x y z. opp x (opp y z) = opp (opp x y) z) \<and>

                    (\<forall>x. opp (neutral opp) x = x)"

 lemma monoidalI: assumes "\<And>x y. opp x y = opp y x"

   "\<And>x y z. opp x (opp y z) = opp (opp x y) z"

   "\<And>x. opp (neutral opp) x = x" shows "monoidal opp"

   unfolding monoidal_def using assms by fastsimp

 lemma monoidal_ac: assumes "monoidal opp"

   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a" "opp a b = opp b a"

   "opp (opp a b) c = opp a (opp b c)"  "opp a (opp b c) = opp b (opp a c)"

   using assms unfolding monoidal_def apply- by metis+

 lemma monoidal_simps[simp]: assumes "monoidal opp"

   shows "opp (neutral opp) a = a" "opp a (neutral opp) = a"

   using monoidal_ac[OF assms] by auto

 lemma neutral_lifted[cong]: assumes "monoidal opp"

   shows "neutral (lifted opp) = Some(neutral opp)"

   apply(subst neutral_def) apply(rule some_equality) apply(rule,induct_tac y) prefer 3

 proof- fix x assume "\<forall>y. lifted opp x y = y \<and> lifted opp y x = y"

   thus "x = Some (neutral opp)" apply(induct x) defer

     apply rule apply(subst neutral_def) apply(subst eq_commute,rule some_equality)

     apply(rule,erule_tac x="Some y" in allE) defer apply(erule_tac x="Some x" in allE) by auto

 qed(auto simp add:monoidal_ac[OF assms])

 lemma monoidal_lifted[intro]: assumes "monoidal opp" shows "monoidal(lifted opp)"

   unfolding monoidal_def forall_option neutral_lifted[OF assms] using monoidal_ac[OF assms] by auto

 definition "support opp f s = {x. x\<in>s \<and> f x \<noteq> neutral opp}"

 definition "fold' opp e s \<equiv> (if finite s then fold opp e s else e)"

 definition "iterate opp s f \<equiv> fold' (\<lambda>x a. opp (f x) a) (neutral opp) (support opp f s)"

 lemma support_subset[intro]:"support opp f s \<subseteq> s" unfolding support_def by auto

 lemma support_empty[simp]:"support opp f {} = {}" using support_subset[of opp f "{}"] by auto

 lemma fun_left_comm_monoidal[intro]: assumes "monoidal opp" shows "fun_left_comm opp"

   unfolding fun_left_comm_def using monoidal_ac[OF assms] by auto

 lemma support_clauses:

   "\<And>f g s. support opp f {} = {}"

   "\<And>f g s. support opp f (insert x s) = (if f(x) = neutral opp then support opp f s else insert x (support opp f s))"

   "\<And>f g s. support opp f (s - {x}) = (support opp f s) - {x}"

   "\<And>f g s. support opp f (s \<union> t) = (support opp f s) \<union> (support opp f t)"

   "\<And>f g s. support opp f (s \<inter> t) = (support opp f s) \<inter> (support opp f t)"

   "\<And>f g s. support opp f (s - t) = (support opp f s) - (support opp f t)"

   "\<And>f g s. support opp g (f ` s) = f ` (support opp (g o f) s)"

 unfolding support_def by auto

 lemma finite_support[intro]:"finite s \<Longrightarrow> finite (support opp f s)"

   unfolding support_def by auto

 lemma iterate_empty[simp]:"iterate opp {} f = neutral opp"

   unfolding iterate_def fold'_def by auto 

 lemma iterate_insert[simp]: assumes "monoidal opp" "finite s"

   shows "iterate opp (insert x s) f = (if x \<in> s then iterate opp s f else opp (f x) (iterate opp s f))" 

 proof(cases "x\<in>s") case True hence *:"insert x s = s" by auto

   show ?thesis unfolding iterate_def if_P[OF True] * by auto

 next case False note x=this

   note * = fun_left_comm.fun_left_comm_apply[OF fun_left_comm_monoidal[OF assms(1)]]

   show ?thesis proof(cases "f x = neutral opp")

     case True show ?thesis unfolding iterate_def if_not_P[OF x] support_clauses if_P[OF True]

       unfolding True monoidal_simps[OF assms(1)] by auto

   next case False show ?thesis unfolding iterate_def fold'_def  if_not_P[OF x] support_clauses if_not_P[OF False]

       apply(subst fun_left_comm.fold_insert[OF * finite_support])

       using `finite s` unfolding support_def using False x by auto qed qed 

 lemma iterate_some:

   assumes "monoidal opp"  "finite s"

   shows "iterate (lifted opp) s (\<lambda>x. Some(f x)) = Some (iterate opp s f)" using assms(2)

 proof(induct s) case empty thus ?case using assms by auto

 next case (insert x F) show ?case apply(subst iterate_insert) prefer 3 apply(subst if_not_P)

     defer unfolding insert(3) lifted.simps apply rule using assms insert by auto qed

 subsection {* Two key instances of additivity. *}

 lemma neutral_add[simp]:

   "neutral op + = (0::_::comm_monoid_add)" unfolding neutral_def 

   apply(rule some_equality) defer apply(erule_tac x=0 in allE) by auto

 lemma operative_content[intro]: "operative (op +) content"

   unfolding operative_def content_split[THEN sym] neutral_add by auto

 lemma neutral_monoid[simp]: "neutral ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a) = 0"

   unfolding neutral_def apply(rule some_equality) defer

   apply(erule_tac x=0 in allE) by auto

 lemma monoidal_monoid[intro]:

   shows "monoidal ((op +)::('a::comm_monoid_add) \<Rightarrow> 'a \<Rightarrow> 'a)"

   unfolding monoidal_def neutral_monoid by(auto simp add: group_simps) 

 lemma operative_integral: fixes f::"real^'n \<Rightarrow> 'a::banach"

   shows "operative (lifted(op +)) (\<lambda>i. if f integrable_on i then Some(integral i f) else None)"

   unfolding operative_def unfolding neutral_lifted[OF monoidal_monoid] neutral_add

   apply(rule,rule,rule,rule) defer apply(rule allI)+

 proof- fix a b c k show "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) =

               lifted op + (if f integrable_on {a..b} \<inter> {x. x $ k \<le> c} then Some (integral ({a..b} \<inter> {x. x $ k \<le> c}) f) else None)

                (if f integrable_on {a..b} \<inter> {x. c \<le> x $ k} then Some (integral ({a..b} \<inter> {x. c \<le> x $ k}) f) else None)"

   proof(cases "f integrable_on {a..b}") 

     case True show ?thesis unfolding if_P[OF True]

       unfolding if_P[OF integrable_split(1)[OF True]] if_P[OF integrable_split(2)[OF True]]

       unfolding lifted.simps option.inject apply(rule integral_unique) apply(rule has_integral_split) 

       apply(rule_tac[!] integrable_integral integrable_split)+ using True by assumption+

   next case False have "(\<not> (f integrable_on {a..b} \<inter> {x. x $ k \<le> c})) \<or> (\<not> ( f integrable_on {a..b} \<inter> {x. c \<le> x $ k}))"

     proof(rule ccontr) case goal1 hence "f integrable_on {a..b}" apply- unfolding integrable_on_def

         apply(rule_tac x="integral ({a..b} \<inter> {x. x $ k \<le> c}) f + integral ({a..b} \<inter> {x. x $ k \<ge> c}) f" in exI)

         apply(rule has_integral_split) apply(rule_tac[!] integrable_integral) by auto

       thus False using False by auto

     qed thus ?thesis using False by auto 

   qed next 

   fix a b assume as:"content {a..b::real^'n} = 0"

   thus "(if f integrable_on {a..b} then Some (integral {a..b} f) else None) = Some 0"

     unfolding if_P[OF integrable_on_null[OF as]] using has_integral_null_eq[OF as] by auto qed

 subsection {* Points of division of a partition. *}

 definition "division_points (k::(real^'n) set) d = 

     {(j,x). (interval_lowerbound k)$j < x \<and> x < (interval_upperbound k)$j \<and>

            (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"

 lemma division_points_finite: 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)$j < x \<and> x < (interval_upperbound i)$j \<and>

            (\<exists>i\<in>d. (interval_lowerbound i)$j = x \<or> (interval_upperbound i)$j = x)}"

   have *:"division_points i d = \<Union>(?M ` UNIV)"

     unfolding division_points_def by auto

   show ?thesis unfolding * using assm by auto qed

 lemma division_points_subset:

   assumes "d division_of {a..b}" "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"

   shows "division_points ({a..b} \<inter> {x. x$k \<le> c}) {l \<inter> {x. x$k \<le> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<le> c} = {})}

                   \<subseteq> division_points ({a..b}) d" (is ?t1) and

         "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l . l \<in> d \<and> ~(l \<inter> {x. x$k \<ge> c} = {})}

                   \<subseteq> division_points ({a..b}) d" (is ?t2)

 proof- note assm = division_ofD[OF assms(1)]

   have *:"\<forall>i. a$i \<le> b$i"   "\<forall>i. a$i \<le> (\<chi> i. if i = k then min (b $ k) c else b $ i) $ i"

     "\<forall>i. (\<chi> i. if i = k then max (a $ k) c else a $ i) $ i \<le> b$i"  "min (b $ k) c = c" "max (a $ k) c = c"

     using assms using less_imp_le by auto

   show ?t1 unfolding division_points_def interval_split[of a b]

     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *

     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+

   proof- fix i l x assume as:"a $ fst x < snd x" "snd x < (if fst x = k then c else b $ fst x)"

       "interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x"  "i = l \<inter> {x. x $ k \<le> c}" "l \<in> d" "l \<inter> {x. x $ k \<le> c} \<noteq> {}"

     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this

     have *:"\<forall>i. u $ i \<le> (\<chi> i. if i = k then min (v $ k) c else v $ i) $ i" using as(6) unfolding l interval_split interval_ne_empty as .

     have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto

     show "a $ fst x < snd x \<and> snd x < b $ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x)"

       using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-

       apply(rule,assumption,rule) defer apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]

       apply(case_tac[!] "fst x = k") using assms by auto

   qed

   show ?t2 unfolding division_points_def interval_split[of a b]

     unfolding interval_bounds[OF *(1)] interval_bounds[OF *(2)] interval_bounds[OF *(3)] Cart_lambda_beta unfolding *

     unfolding subset_eq apply(rule) unfolding mem_Collect_eq split_beta apply(erule bexE conjE)+ unfolding mem_Collect_eq apply(erule exE conjE)+

   proof- fix i l x assume as:"(if fst x = k then c else a $ fst x) < snd x" "snd x < b $ fst x" "interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x"

       "i = l \<inter> {x. c \<le> x $ k}" "l \<in> d" "l \<inter> {x. c \<le> x $ k} \<noteq> {}"

     from assm(4)[OF this(5)] guess u v apply-by(erule exE)+ note l=this

     have *:"\<forall>i. (\<chi> i. if i = k then max (u $ k) c else u $ i) $ i \<le> v $ i" using as(6) unfolding l interval_split interval_ne_empty as .

     have **:"\<forall>i. u$i \<le> v$i" using l using as(6) unfolding interval_ne_empty[THEN sym] by auto

     show "a $ fst x < snd x \<and> snd x < b $ fst x \<and> (\<exists>i\<in>d. interval_lowerbound i $ fst x = snd x \<or> interval_upperbound i $ fst x = snd x)"

       using as(1-3,5) unfolding l interval_split interval_ne_empty as interval_bounds[OF *] Cart_lambda_beta apply-

       apply rule defer apply(rule,assumption) apply(rule_tac x="{u..v}" in bexI) unfolding interval_bounds[OF **]

       apply(case_tac[!] "fst x = k") using assms by auto qed qed

 lemma division_points_psubset:

   assumes "d division_of {a..b}"  "\<forall>i. a$i < b$i"  "a$k < c" "c < b$k"

   "l \<in> d" "interval_lowerbound l$k = c \<or> interval_upperbound l$k = c"

   shows "division_points ({a..b} \<inter> {x. x$k \<le> c}) {l \<inter> {x. x$k \<le> c} | l. l\<in>d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D1 \<subset> ?D") 

         "division_points ({a..b} \<inter> {x. x$k \<ge> c}) {l \<inter> {x. x$k \<ge> c} | l. l\<in>d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}} \<subset> division_points ({a..b}) d" (is "?D2 \<subset> ?D") 

 proof- have ab:"\<forall>i. a$i \<le> b$i" using assms(2) by(auto intro!:less_imp_le)

   guess u v using division_ofD(4)[OF assms(1,5)] apply-by(erule exE)+ note l=this

   have uv:"\<forall>i. u$i \<le> v$i" "\<forall>i. a$i \<le> u$i \<and> v$i \<le> b$i" using division_ofD(2,2,3)[OF assms(1,5)] unfolding l interval_ne_empty

     unfolding subset_eq apply- defer apply(erule_tac x=u in ballE, erule_tac x=v in ballE) unfolding mem_interval by auto

   have *:"interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"

          "interval_upperbound ({a..b} \<inter> {x. x $ k \<le> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"

     unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)

     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto

   have "\<exists>x. x \<in> ?D - ?D1" using assms(2-) apply-apply(erule disjE)

     apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer

     apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)

     unfolding division_points_def unfolding interval_bounds[OF ab]

     apply (auto simp add:interval_bounds) unfolding * by auto

   thus "?D1 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto

   have *:"interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_lowerbound l $ k}) $ k = interval_lowerbound l $ k"

          "interval_lowerbound ({a..b} \<inter> {x. x $ k \<ge> interval_upperbound l $ k}) $ k = interval_upperbound l $ k"

     unfolding interval_split apply(subst interval_bounds) prefer 3 apply(subst interval_bounds)

     unfolding l interval_bounds[OF uv(1)] using uv[rule_format,of k] ab by auto

   have "\<exists>x. x \<in> ?D - ?D2" using assms(2-) apply-apply(erule disjE)

     apply(rule_tac x="(k,(interval_lowerbound l)$k)" in exI) defer

     apply(rule_tac x="(k,(interval_upperbound l)$k)" in exI)

     unfolding division_points_def unfolding interval_bounds[OF ab]

     apply (auto simp add:interval_bounds) unfolding * by auto

   thus "?D2 \<subset> ?D" apply-apply(rule,rule division_points_subset[OF assms(1-4)]) by auto qed

 subsection {* Preservation by divisions and tagged divisions. *}

 lemma support_support[simp]:"support opp f (support opp f s) = support opp f s"

   unfolding support_def by auto

 lemma iterate_support[simp]: "iterate opp (support opp f s) f = iterate opp s f"

   unfolding iterate_def support_support by auto

 lemma iterate_expand_cases:

   "iterate opp s f = (if finite(support opp f s) then iterate opp (support opp f s) f else neutral opp)"

   apply(cases) apply(subst if_P,assumption) unfolding iterate_def support_support fold'_def by auto 

 lemma iterate_image: assumes "monoidal opp"  "inj_on f s"

   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)"

 proof- have *:"\<And>s. finite s \<Longrightarrow>  \<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<longrightarrow> x = y \<Longrightarrow>

      iterate opp (f ` s) g = iterate opp s (g \<circ> f)"

   proof- case goal1 show ?case using goal1

     proof(induct s) case empty thus ?case using assms(1) by auto

     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)]

         unfolding if_not_P[OF insert(2)] apply(subst insert(3)[THEN sym])

         unfolding image_insert defer apply(subst iterate_insert[OF assms(1)])

         apply(rule finite_imageI insert)+ apply(subst if_not_P)

         unfolding image_iff o_def using insert(2,4) by auto

     qed qed

   show ?thesis 

     apply(cases "finite (support opp g (f ` s))")

     apply(subst (1) iterate_support[THEN sym],subst (2) iterate_support[THEN sym])

     unfolding support_clauses apply(rule *)apply(rule finite_imageD,assumption) unfolding inj_on_def[symmetric]

     apply(rule subset_inj_on[OF assms(2) support_subset])+

     apply(subst iterate_expand_cases) unfolding support_clauses apply(simp only: if_False)

     apply(subst iterate_expand_cases) apply(subst if_not_P) by auto qed

 (* This lemma about iterations comes up in a few places.                     *)

 lemma iterate_nonzero_image_lemma:

   assumes "monoidal opp" "finite s" "g(a) = neutral opp"

   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = neutral opp"

   shows "iterate opp {f x | x. x \<in> s \<and> f x \<noteq> a} g = iterate opp s (g \<circ> f)"

 proof- have *:"{f x |x. x \<in> s \<and> ~(f x = a)} = f ` {x. x \<in> s \<and> ~(f x = a)}" by auto

   have **:"support opp (g \<circ> f) {x \<in> s. f x \<noteq> a} = support opp (g \<circ> f) s"

     unfolding support_def using assms(3) by auto

   show ?thesis unfolding *

     apply(subst iterate_support[THEN sym]) unfolding support_clauses

     apply(subst iterate_image[OF assms(1)]) defer

     apply(subst(2) iterate_support[THEN sym]) apply(subst **)

     unfolding inj_on_def using assms(3,4) unfolding support_def by auto qed

 lemma iterate_eq_neutral:

   assumes "monoidal opp"  "\<forall>x \<in> s. (f(x) = neutral opp)"

   shows "(iterate opp s f = neutral opp)"

 proof- have *:"support opp f s = {}" unfolding support_def using assms(2) by auto

   show ?thesis apply(subst iterate_support[THEN sym]) 

     unfolding * using assms(1) by auto qed

 lemma iterate_op: assumes "monoidal opp" "finite s"

   shows "iterate opp s (\<lambda>x. opp (f x) (g x)) = opp (iterate opp s f) (iterate opp s g)" using assms(2)

 proof(induct s) case empty thus ?case unfolding iterate_insert[OF assms(1)] using assms(1) by auto

 next case (insert x F) show ?case unfolding iterate_insert[OF assms(1) insert(1)] if_not_P[OF insert(2)] insert(3)

     unfolding monoidal_ac[OF assms(1)] by(rule refl) qed

 lemma iterate_eq: assumes "monoidal opp" "\<And>x. x \<in> s \<Longrightarrow> f x = g x"

   shows "iterate opp s f = iterate opp s g"

 proof- have *:"support opp g s = support opp f s"

     unfolding support_def using assms(2) by auto

   show ?thesis

   proof(cases "finite (support opp f s)")

     case False thus ?thesis apply(subst iterate_expand_cases,subst(2) iterate_expand_cases)

       unfolding * by auto

   next def su \<equiv> "support opp f s"

     case True note support_subset[of opp f s] 

     thus ?thesis apply- apply(subst iterate_support[THEN sym],subst(2) iterate_support[THEN sym]) unfolding * using True

       unfolding su_def[symmetric]

     proof(induct su) case empty show ?case by auto

     next case (insert x s) show ?case unfolding iterate_insert[OF assms(1) insert(1)] 

         unfolding if_not_P[OF insert(2)] apply(subst insert(3))

         defer apply(subst assms(2)[of x]) using insert by auto qed qed qed

 lemma nonempty_witness: assumes "s \<noteq> {}" obtains x where "x \<in> s" using assms by auto

 lemma operative_division: fixes f::"(real^'n) set \<Rightarrow> 'a"

   assumes "monoidal opp" "operative opp f" "d division_of {a..b}"

   shows "iterate opp d f = f {a..b}"

 proof- def C \<equiv> "card (division_points {a..b} d)" thus ?thesis using assms

   proof(induct C arbitrary:a b d rule:full_nat_induct)

     case goal1

     { presume *:"content {a..b} \<noteq> 0 \<Longrightarrow> ?case"

       thus ?case apply-apply(cases) defer apply assumption

       proof- assume as:"content {a..b} = 0"

         show ?case unfolding operativeD(1)[OF assms(2) as] apply(rule iterate_eq_neutral[OF goal1(2)])

         proof fix x assume x:"x\<in>d"

           then guess u v apply(drule_tac division_ofD(4)[OF goal1(4)]) by(erule exE)+

           thus "f x = neutral opp" using division_of_content_0[OF as goal1(4)] 

             using operativeD(1)[OF assms(2)] x by auto

         qed qed }

     assume "content {a..b} \<noteq> 0" note ab = this[unfolded content_lt_nz[THEN sym] content_pos_lt_eq]

     hence ab':"\<forall>i. a$i \<le> b$i" by (auto intro!: less_imp_le) show ?case 

     proof(cases "division_points {a..b} d = {}")

       case True have d':"\<forall>i\<in>d. \<exists>u v. i = {u..v} \<and>

         (\<forall>j. u$j = a$j \<and> v$j = a$j \<or> u$j = b$j \<and> v$j = b$j \<or> u$j = a$j \<and> v$j = b$j)"

         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule)

         apply(rule_tac x=a in exI,rule_tac x=b in exI) apply(rule,rule refl) apply(rule)

       proof- fix u v j assume as:"{u..v} \<in> d" note division_ofD(3)[OF goal1(4) this]

         hence uv:"\<forall>i. u$i \<le> v$i" "u$j \<le> v$j" unfolding interval_ne_empty by auto

         have *:"\<And>p r Q. p \<or> r \<or> (\<forall>x\<in>d. Q x) \<Longrightarrow> p \<or> r \<or> (Q {u..v})" using as by auto

         have "(j, u$j) \<notin> division_points {a..b} d"

           "(j, v$j) \<notin> division_points {a..b} d" using True by auto

         note this[unfolded de_Morgan_conj division_points_def mem_Collect_eq split_conv interval_bounds[OF ab'] bex_simps]

         note *[OF this(1)] *[OF this(2)] note this[unfolded interval_bounds[OF uv(1)]]

         moreover have "a$j \<le> u$j" "v$j \<le> b$j" using division_ofD(2,2,3)[OF goal1(4) as] 

           unfolding subset_eq apply- apply(erule_tac x=u in ballE,erule_tac[3] x=v in ballE)

           unfolding interval_ne_empty mem_interval by auto

         ultimately show "u$j = a$j \<and> v$j = a$j \<or> u$j = b$j \<and> v$j = b$j \<or> u$j = a$j \<and> v$j = b$j"

           unfolding not_less de_Morgan_disj using ab[rule_format,of j] uv(2) by auto

       qed have "(1/2) *\<^sub>R (a+b) \<in> {a..b}" unfolding mem_interval using ab by(auto intro!:less_imp_le)

       note this[unfolded division_ofD(6)[OF goal1(4),THEN sym] Union_iff]

       then guess i .. note i=this guess u v using d'[rule_format,OF i(1)] apply-by(erule exE conjE)+ note uv=this

       have "{a..b} \<in> d"

       proof- { presume "i = {a..b}" thus ?thesis using i by auto }

         { presume "u = a" "v = b" thus "i = {a..b}" using uv by auto }

         show "u = a" "v = b" unfolding Cart_eq

         proof(rule_tac[!] allI) fix j note i(2)[unfolded uv mem_interval,rule_format,of j]

           thus "u $ j = a $ j" "v $ j = b $ j" using uv(2)[rule_format,of j] by auto

         qed qed

       hence *:"d = insert {a..b} (d - {{a..b}})" by auto

       have "iterate opp (d - {{a..b}}) f = neutral opp" apply(rule iterate_eq_neutral[OF goal1(2)])

       proof fix x assume x:"x \<in> d - {{a..b}}" hence "x\<in>d" by auto note d'[rule_format,OF this]

         then guess u v apply-by(erule exE conjE)+ note uv=this

         have "u\<noteq>a \<or> v\<noteq>b" using x[unfolded uv] by auto  

         then obtain j where "u$j \<noteq> a$j \<or> v$j \<noteq> b$j" unfolding Cart_eq by auto

         hence "u$j = v$j" using uv(2)[rule_format,of j] by auto

         hence "content {u..v} = 0"  unfolding content_eq_0 apply(rule_tac x=j in exI) by auto

         thus "f x = neutral opp" unfolding uv(1) by(rule operativeD(1)[OF goal1(3)])

       qed thus "iterate opp d f = f {a..b}" apply-apply(subst *) 

         apply(subst iterate_insert[OF goal1(2)]) using goal1(2,4) by auto

     next case False hence "\<exists>x. x\<in>division_points {a..b} d" by auto

       then guess k c unfolding split_paired_Ex apply- unfolding division_points_def mem_Collect_eq split_conv

         by(erule exE conjE)+ note kc=this[unfolded interval_bounds[OF ab']]

       from this(3) guess j .. note j=this

       def d1 \<equiv> "{l \<inter> {x. x$k \<le> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<le> c} \<noteq> {}}"

       def d2 \<equiv> "{l \<inter> {x. x$k \<ge> c} | l. l \<in> d \<and> l \<inter> {x. x$k \<ge> c} \<noteq> {}}"

       def cb \<equiv> "(\<chi> i. if i = k then c else b$i)" and ca \<equiv> "(\<chi> i. if i = k then c else a$i)"

       note division_points_psubset[OF goal1(4) ab kc(1-2) j]

       note psubset_card_mono[OF _ this(1)] psubset_card_mono[OF _ this(2)]

       hence *:"(iterate opp d1 f) = f ({a..b} \<inter> {x. x$k \<le> c})" "(iterate opp d2 f) = f ({a..b} \<inter> {x. x$k \<ge> c})"

         apply- unfolding interval_split apply(rule_tac[!] goal1(1)[rule_format])

         using division_split[OF goal1(4), where k=k and c=c]

         unfolding interval_split d1_def[symmetric] d2_def[symmetric] unfolding goal1(2) Suc_le_mono

         using goal1(2-3) using division_points_finite[OF goal1(4)] by auto

       have "f {a..b} = opp (iterate opp d1 f) (iterate opp d2 f)" (is "_ = ?prev")

         unfolding * apply(rule operativeD(2)) using goal1(3) .

       also have "iterate opp d1 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<le> c}))"

         unfolding d1_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])

         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+

         unfolding empty_as_interval[THEN sym] apply(rule content_empty)

       proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. x $ k \<le> c} = y \<inter> {x. x $ k \<le> c}" "l \<noteq> y" 

         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this

         show "f (l \<inter> {x. x $ k \<le> c}) = neutral opp" unfolding l interval_split

           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_left_inj)

           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+

       qed also have "iterate opp d2 f = iterate opp d (\<lambda>l. f(l \<inter> {x. x$k \<ge> c}))"

         unfolding d2_def apply(rule iterate_nonzero_image_lemma[unfolded o_def])

         unfolding empty_as_interval apply(rule goal1 division_of_finite operativeD[OF goal1(3)])+

         unfolding empty_as_interval[THEN sym] apply(rule content_empty)

       proof(rule,rule,rule,erule conjE) fix l y assume as:"l \<in> d" "y \<in> d" "l \<inter> {x. c \<le> x $ k} = y \<inter> {x. c \<le> x $ k}" "l \<noteq> y" 

         from division_ofD(4)[OF goal1(4) this(1)] guess u v apply-by(erule exE)+ note l=this

         show "f (l \<inter> {x. x $ k \<ge> c}) = neutral opp" unfolding l interval_split

           apply(rule operativeD(1) goal1)+ unfolding interval_split[THEN sym] apply(rule division_split_right_inj)

           apply(rule goal1) unfolding l[THEN sym] apply(rule as(1),rule as(2)) by(rule as)+

       qed also have *:"\<forall>x\<in>d. f x = opp (f (x \<inter> {x. x $ k \<le> c})) (f (x \<inter> {x. c \<le> x $ k}))"

         unfolding forall_in_division[OF goal1(4)] apply(rule,rule,rule,rule operativeD(2)) using goal1(3) .

       have "opp (iterate opp d (\<lambda>l. f (l \<inter> {x. x $ k \<le> c}))) (iterate opp d (\<lambda>l. f (l \<inter> {x. c \<le> x $ k})))

         = iterate opp d f" apply(subst(3) iterate_eq[OF _ *[rule_format]]) prefer 3

         apply(rule iterate_op[THEN sym]) using goal1 by auto

       finally show ?thesis by auto

     qed qed qed 

 lemma iterate_image_nonzero: assumes "monoidal opp"

   "finite s" "\<forall>x\<in>s. \<forall>y\<in>s. ~(x = y) \<and> f x = f y \<longrightarrow> g(f x) = neutral opp"

   shows "iterate opp (f ` s) g = iterate opp s (g \<circ> f)" using assms

 proof(induct rule:finite_subset_induct[OF assms(2) subset_refl])

   case goal1 show ?case using assms(1) by auto

 next case goal2 have *:"\<And>x y. y = neutral opp \<Longrightarrow> x = opp y x" using assms(1) by auto

   show ?case unfolding image_insert apply(subst iterate_insert[OF assms(1)])

     apply(rule finite_imageI goal2)+

     apply(cases "f a \<in> f ` F") unfolding if_P if_not_P apply(subst goal2(4)[OF assms(1) goal2(1)]) defer

     apply(subst iterate_insert[OF assms(1) goal2(1)]) defer

     apply(subst iterate_insert[OF assms(1) goal2(1)])

     unfolding if_not_P[OF goal2(3)] defer unfolding image_iff defer apply(erule bexE)

     apply(rule *) unfolding o_def apply(rule_tac y=x in goal2(7)[rule_format])

     using goal2 unfolding o_def by auto qed 

 lemma operative_tagged_division: assumes "monoidal opp" "operative opp f" "d tagged_division_of {a..b}"

   shows "iterate(opp) d (\<lambda>(x,l). f l) = f {a..b}"

 proof- have *:"(\<lambda>(x,l). f l) = (f o snd)" unfolding o_def by(rule,auto) note assm = tagged_division_ofD[OF assms(3)]

   have "iterate(opp) d (\<lambda>(x,l). f l) = iterate opp (snd ` d) f" unfolding *

     apply(rule iterate_image_nonzero[THEN sym,OF assms(1)]) apply(rule tagged_division_of_finite assms)+ 

     unfolding Ball_def split_paired_All snd_conv apply(rule,rule,rule,rule,rule,rule,rule,erule conjE)

   proof- fix a b aa ba assume as:"(a, b) \<in> d" "(aa, ba) \<in> d" "(a, b) \<noteq> (aa, ba)" "b = ba"

     guess u v using assm(4)[OF as(1)] apply-by(erule exE)+ note uv=this

     show "f b = neutral opp" unfolding uv apply(rule operativeD(1)[OF assms(2)])

       unfolding content_eq_0_interior using tagged_division_ofD(5)[OF assms(3) as(1-3)]

       unfolding as(4)[THEN sym] uv by auto

   qed also have "\<dots> = f {a..b}" 

     using operative_division[OF assms(1-2) division_of_tagged_division[OF assms(3)]] .

   finally show ?thesis . qed

 subsection {* Additivity of content. *}

 lemma setsum_iterate:assumes "finite s" shows "setsum f s = iterate op + s f"

 proof- have *:"setsum f s = setsum f (support op + f s)"

     apply(rule setsum_mono_zero_right)

     unfolding support_def neutral_monoid using assms by auto

   thus ?thesis unfolding * setsum_def iterate_def fold_image_def fold'_def

     unfolding neutral_monoid . qed

 lemma additive_content_division: assumes "d division_of {a..b}"

   shows "setsum content d = content({a..b})"

   unfolding operative_division[OF monoidal_monoid operative_content assms,THEN sym]

   apply(subst setsum_iterate) using assms by auto

 lemma additive_content_tagged_division:

   assumes "d tagged_division_of {a..b}"

   shows "setsum (\<lambda>(x,l). content l) d = content({a..b})"

   unfolding operative_tagged_division[OF monoidal_monoid operative_content assms,THEN sym]

   apply(subst setsum_iterate) using assms by auto

 subsection {* Finally, the integral of a constant\<forall> *}

 lemma has_integral_const[intro]:

   "((\<lambda>x. c) has_integral (content({a..b::real^'n}) *\<^sub>R c)) ({a..b})"

   unfolding has_integral apply(rule,rule,rule_tac x="\<lambda>x. ball x 1" in exI)

   apply(rule,rule gauge_trivial)apply(rule,rule,erule conjE)

   unfolding split_def apply(subst scaleR_left.setsum[THEN sym, unfolded o_def])

   defer apply(subst additive_content_tagged_division[unfolded split_def]) apply assumption by auto

 subsection {* Bounds on the norm of Riemann sums and the integral itself. *}

 lemma dsum_bound: assumes "p division_of {a..b}" "norm(c) \<le> e"

   shows "norm(setsum (\<lambda>l. content l *\<^sub>R c) p) \<le> e * content({a..b})" (is "?l \<le> ?r")

   apply(rule order_trans,rule setsum_norm) defer unfolding norm_scaleR setsum_left_distrib[THEN sym]

   apply(rule order_trans[OF mult_left_mono],rule assms,rule setsum_abs_ge_zero)

   apply(subst real_mult_commute) apply(rule mult_left_mono)

   apply(rule order_trans[of _ "setsum content p"]) apply(rule eq_refl,rule setsum_cong2)

   apply(subst abs_of_nonneg) unfolding additive_content_division[OF assms(1)]

 proof- from order_trans[OF norm_ge_zero[of c] assms(2)] show "0 \<le> e" .

   fix x assume "x\<in>p" from division_ofD(4)[OF assms(1) this] guess u v apply-by(erule exE)+

   thus "0 \<le> content x" using content_pos_le by auto

 qed(insert assms,auto)

 lemma rsum_bound: assumes "p tagged_division_of {a..b}" "\<forall>x\<in>{a..b}. norm(f x) \<le> e"

   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p) \<le> e * content({a..b})"

 proof(cases "{a..b} = {}") case True

   show ?thesis using assms(1) unfolding True tagged_division_of_trivial by auto

 next case False show ?thesis

     apply(rule order_trans,rule setsum_norm) defer unfolding split_def norm_scaleR

     apply(rule order_trans[OF setsum_mono]) apply(rule mult_left_mono[OF _ abs_ge_zero, of _ e]) defer

     unfolding setsum_left_distrib[THEN sym] apply(subst real_mult_commute) apply(rule mult_left_mono)

     apply(rule order_trans[of _ "setsum (content \<circ> snd) p"]) apply(rule eq_refl,rule setsum_cong2)

     apply(subst o_def, rule abs_of_nonneg)

   proof- show "setsum (content \<circ> snd) p \<le> content {a..b}" apply(rule eq_refl)

       unfolding additive_content_tagged_division[OF assms(1),THEN sym] split_def by auto

     guess w using nonempty_witness[OF False] .

     thus "e\<ge>0" apply-apply(rule order_trans) defer apply(rule assms(2)[rule_format],assumption) by auto

     fix xk assume *:"xk\<in>p" guess x k  using surj_pair[of xk] apply-by(erule exE)+ note xk = this *[unfolded this]

     from tagged_division_ofD(4)[OF assms(1) xk(2)] guess u v apply-by(erule exE)+ note uv=this

     show "0\<le> content (snd xk)" unfolding xk snd_conv uv by(rule content_pos_le)

     show "norm (f (fst xk)) \<le> e" unfolding xk fst_conv using tagged_division_ofD(2,3)[OF assms(1) xk(2)] assms(2) by auto

   qed(insert assms,auto) qed

 lemma rsum_diff_bound:

   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e"

   shows "norm(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - setsum (\<lambda>(x,k). content k *\<^sub>R g x) p) \<le> e * content({a..b})"

   apply(rule order_trans[OF _ rsum_bound[OF assms]]) apply(rule eq_refl) apply(rule arg_cong[where f=norm])

   unfolding setsum_subtractf[THEN sym] apply(rule setsum_cong2) unfolding scaleR.diff_right by auto

 lemma has_integral_bound: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"

   assumes "0 \<le> B" "(f has_integral i) ({a..b})" "\<forall>x\<in>{a..b}. norm(f x) \<le> B"

   shows "norm i \<le> B * content {a..b}"

 proof- let ?P = "content {a..b} > 0" { presume "?P \<Longrightarrow> ?thesis"

     thus ?thesis proof(cases ?P) case False

       hence *:"content {a..b} = 0" using content_lt_nz by auto

       hence **:"i = 0" using assms(2) apply(subst has_integral_null_eq[THEN sym]) by auto

       show ?thesis unfolding * ** using assms(1) by auto

     qed auto } assume ab:?P

   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }

   assume "\<not> ?thesis" hence *:"norm i - B * content {a..b} > 0" by auto

   from assms(2)[unfolded has_integral,rule_format,OF *] guess d apply-by(erule exE conjE)+ note d=this[rule_format]

   from fine_division_exists[OF this(1), of a b] guess p . note p=this

   have *:"\<And>s B. norm s \<le> B \<Longrightarrow> \<not> (norm (s - i) < norm i - B)"

   proof- case goal1 thus ?case unfolding not_less

     using norm_triangle_sub[of i s] unfolding norm_minus_commute by auto

   qed show False using d(2)[OF conjI[OF p]] *[OF rsum_bound[OF p(1) assms(3)]] by auto qed

 subsection {* Similar theorems about relationship among components. *}

 lemma rsum_component_le: fixes f::"real^'n \<Rightarrow> real^'m"

   assumes "p tagged_division_of {a..b}"  "\<forall>x\<in>{a..b}. (f x)$i \<le> (g x)$i"

   shows "(setsum (\<lambda>(x,k). content k *\<^sub>R f x) p)$i \<le> (setsum (\<lambda>(x,k). content k *\<^sub>R g x) p)$i"

   unfolding setsum_component apply(rule setsum_mono)

   apply(rule mp) defer apply assumption apply(induct_tac x,rule) unfolding split_conv

 proof- fix a b assume ab:"(a,b) \<in> p" note assm = tagged_division_ofD(2-4)[OF assms(1) ab]

   from this(3) guess u v apply-by(erule exE)+ note b=this

   show "(content b *\<^sub>R f a) $ i \<le> (content b *\<^sub>R g a) $ i" unfolding b

     unfolding Cart_nth.scaleR real_scaleR_def apply(rule mult_left_mono)

     defer apply(rule content_pos_le,rule assms(2)[rule_format]) using assm by auto qed

 lemma has_integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"

   assumes "(f has_integral i) s" "(g has_integral j) s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"

   shows "i$k \<le> j$k"

 proof- have lem:"\<And>a b g i j. \<And>f::real^'n \<Rightarrow> real^'m. (f has_integral i) ({a..b}) \<Longrightarrow> 

     (g has_integral j) ({a..b}) \<Longrightarrow> \<forall>x\<in>{a..b}. (f x)$k \<le> (g x)$k \<Longrightarrow> i$k \<le> j$k"

   proof(rule ccontr) case goal1 hence *:"0 < (i$k - j$k) / 3" by auto

     guess d1 using goal1(1)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d1=this[rule_format]

     guess d2 using goal1(2)[unfolded has_integral,rule_format,OF *] apply-by(erule exE conjE)+ note d2=this[rule_format]

     guess p using fine_division_exists[OF gauge_inter[OF d1(1) d2(1)], of a b] unfolding fine_inter .

     note p = this(1) conjunctD2[OF this(2)]  note le_less_trans[OF component_le_norm, of _ _ k]

     note this[OF d1(2)[OF conjI[OF p(1,2)]]] this[OF d2(2)[OF conjI[OF p(1,3)]]]

     thus False unfolding Cart_nth.diff using rsum_component_le[OF p(1) goal1(3)] by smt

   qed let ?P = "\<exists>a b. s = {a..b}"

   { presume "\<not> ?P \<Longrightarrow> ?thesis" thus ?thesis proof(cases ?P)

       case True then guess a b apply-by(erule exE)+ note s=this

       show ?thesis apply(rule lem) using assms[unfolded s] by auto

     qed auto } assume as:"\<not> ?P"

   { presume "\<not> ?thesis \<Longrightarrow> False" thus ?thesis by auto }

   assume "\<not> i$k \<le> j$k" hence ij:"(i$k - j$k) / 3 > 0" by auto

   note has_integral_altD[OF _ as this] from this[OF assms(1)] this[OF assms(2)] guess B1 B2 . note B=this[rule_format]

   have "bounded (ball 0 B1 \<union> ball (0::real^'n) B2)" unfolding bounded_Un by(rule conjI bounded_ball)+

   from bounded_subset_closed_interval[OF this] guess a b apply- by(erule exE)+

   note ab = conjunctD2[OF this[unfolded Un_subset_iff]]

   guess w1 using B(2)[OF ab(1)] .. note w1=conjunctD2[OF this]

   guess w2 using B(4)[OF ab(2)] .. note w2=conjunctD2[OF this]

   have *:"\<And>w1 w2 j i::real .\<bar>w1 - i\<bar> < (i - j) / 3 \<Longrightarrow> \<bar>w2 - j\<bar> < (i - j) / 3 \<Longrightarrow> w1 \<le> w2 \<Longrightarrow> False" by smt(*SMTSMT*)

   note le_less_trans[OF component_le_norm[of _ k]] note this[OF w1(2)] this[OF w2(2)] moreover

   have "w1$k \<le> w2$k" apply(rule lem[OF w1(1) w2(1)]) using assms by auto ultimately

   show False unfolding Cart_nth.diff by(rule *) qed

 lemma integral_component_le: fixes f::"real^'n \<Rightarrow> real^'m"

   assumes "f integrable_on s" "g integrable_on s"  "\<forall>x\<in>s. (f x)$k \<le> (g x)$k"

   shows "(integral s f)$k \<le> (integral s g)$k"

   apply(rule has_integral_component_le) using integrable_integral assms by auto

 lemma has_integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"

   assumes "(f has_integral i) s"  "(g has_integral j) s" "\<forall>x\<in>s. f x \<le> g x"

   shows "dest_vec1 i \<le> dest_vec1 j" apply(rule has_integral_component_le[OF assms(1-2)])

   using assms(3) unfolding vector_le_def by auto

 lemma integral_dest_vec1_le: fixes f::"real^'n \<Rightarrow> real^1"

   assumes "f integrable_on s" "g integrable_on s" "\<forall>x\<in>s. f x \<le> g x"

   shows "dest_vec1(integral s f) \<le> dest_vec1(integral s g)"

   apply(rule has_integral_dest_vec1_le) apply(rule_tac[1-2] integrable_integral) using assms by auto

 lemma has_integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"

   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> i$k"

   using has_integral_component_le[OF has_integral_0 assms(1)] using assms(2) by auto

 lemma integral_component_pos: fixes f::"real^'n \<Rightarrow> real^'m"

   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> (f x)$k" shows "0 \<le> (integral s f)$k"

   apply(rule has_integral_component_pos) using assms by auto

 lemma has_integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"

   assumes "(f has_integral i) s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> i"

   using has_integral_component_pos[OF assms(1), of 1]

   using assms(2) unfolding vector_le_def by auto

 lemma integral_dest_vec1_pos: fixes f::"real^'n \<Rightarrow> real^1"

   assumes "f integrable_on s" "\<forall>x\<in>s. 0 \<le> f x" shows "0 \<le> integral s f"

   apply(rule has_integral_dest_vec1_pos) using assms by auto

 lemma has_integral_component_neg: fixes f::"real^'n \<Rightarrow> real^'m"

   assumes "(f has_integral i) s" "\<forall>x\<in>s. (f x)$k \<le> 0" shows "i$k \<le> 0"

   using has_integral_component_le[OF assms(1) has_integral_0] assms(2) by auto

 lemma has_integral_dest_vec1_neg: fixes f::"real^'n \<Rightarrow> real^1"

   assumes "(f has_integral i) s" "\<forall>x\<in>s. f x \<le> 0" shows "i \<le> 0"

   using has_integral_component_neg[OF assms(1),of 1] using assms(2) by auto

 lemma has_integral_component_lbound:

   assumes "(f has_integral i) {a..b}"  "\<forall>x\<in>{a..b}. B \<le> f(x)$k" shows "B * content {a..b} \<le> i$k"

   using has_integral_component_le[OF has_integral_const assms(1),of "(\<chi> i. B)" k] assms(2)

   unfolding Cart_lambda_beta vector_scaleR_component by(auto simp add:field_simps)

 lemma has_integral_component_ubound: 

   assumes "(f has_integral i) {a..b}" "\<forall>x\<in>{a..b}. f x$k \<le> B"

   shows "i$k \<le> B * content({a..b})"

   using has_integral_component_le[OF assms(1) has_integral_const, of k "vec B"]

   unfolding vec_component Cart_nth.scaleR using assms(2) by(auto simp add:field_simps)

 lemma integral_component_lbound:

   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. B \<le> f(x)$k"

   shows "B * content({a..b}) \<le> (integral({a..b}) f)$k"

   apply(rule has_integral_component_lbound) using assms unfolding has_integral_integral by auto

 lemma integral_component_ubound:

   assumes "f integrable_on {a..b}" "\<forall>x\<in>{a..b}. f(x)$k \<le> B"

   shows "(integral({a..b}) f)$k \<le> B * content({a..b})"

   apply(rule has_integral_component_ubound) using assms unfolding has_integral_integral by auto

 subsection {* Uniform limit of integrable functions is integrable. *}

 lemma real_arch_invD:

   "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"

   by(subst(asm) real_arch_inv)

 lemma integrable_uniform_limit: fixes f::"real^'n \<Rightarrow> 'a::banach"

   assumes "\<forall>e>0. \<exists>g. (\<forall>x\<in>{a..b}. norm(f x - g x) \<le> e) \<and> g integrable_on {a..b}"

   shows "f integrable_on {a..b}"

 proof- { presume *:"content {a..b} > 0 \<Longrightarrow> ?thesis"

     show ?thesis apply cases apply(rule *,assumption)

       unfolding content_lt_nz integrable_on_def using has_integral_null by auto }

   assume as:"content {a..b} > 0"

   have *:"\<And>P. \<forall>e>(0::real). P e \<Longrightarrow> \<forall>n::nat. P (inverse (real n+1))" by auto

   from choice[OF *[OF assms]] guess g .. note g=conjunctD2[OF this[rule_format],rule_format]

   from choice[OF allI[OF g(2)[unfolded integrable_on_def], of "\<lambda>x. x"]] guess i .. note i=this[rule_format]

   have "Cauchy i" unfolding Cauchy_def

   proof(rule,rule) fix e::real assume "e>0"

     hence "e / 4 / content {a..b} > 0" using as by(auto simp add:field_simps)

     then guess M apply-apply(subst(asm) real_arch_inv) by(erule exE conjE)+ note M=this

     show "\<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (i m) (i n) < e" apply(rule_tac x=M in exI,rule,rule,rule,rule)

     proof- case goal1 have "e/4>0" using `e>0` by auto note * = i[unfolded has_integral,rule_format,OF this]

       from *[of m] guess gm apply-by(erule conjE exE)+ note gm=this[rule_format]

       from *[of n] guess gn apply-by(erule conjE exE)+ note gn=this[rule_format]

       from fine_division_exists[OF gauge_inter[OF gm(1) gn(1)], of a b] guess p . note p=this

       have lem2:"\<And>s1 s2 i1 i2. norm(s2 - s1) \<le> e/2 \<Longrightarrow> norm(s1 - i1) < e / 4 \<Longrightarrow> norm(s2 - i2) < e / 4 \<Longrightarrow>norm(i1 - i2) < e"

       proof- case goal1 have "norm (i1 - i2) \<le> norm (i1 - s1) + norm (s1 - s2) + norm (s2 - i2)"

           using norm_triangle_ineq[of "i1 - s1" "s1 - i2"]

           using norm_triangle_ineq[of "s1 - s2" "s2 - i2"] by(auto simp add:group_simps)

         also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)

         finally show ?case .

       qed

       show ?case unfolding vector_dist_norm apply(rule lem2) defer

         apply(rule gm(2)[OF conjI[OF p(1)]],rule_tac[2] gn(2)[OF conjI[OF p(1)]])

         using conjunctD2[OF p(2)[unfolded fine_inter]] apply- apply assumption+ apply(rule order_trans)

         apply(rule rsum_diff_bound[OF p(1), where e="2 / real M"])

       proof show "2 / real M * content {a..b} \<le> e / 2" unfolding divide_inverse 

           using M as by(auto simp add:field_simps)

         fix x assume x:"x \<in> {a..b}"

         have "norm (f x - g n x) + norm (f x - g m x) \<le> inverse (real n + 1) + inverse (real m + 1)" 

             using g(1)[OF x, of n] g(1)[OF x, of m] by auto

         also have "\<dots> \<le> inverse (real M) + inverse (real M)" apply(rule add_mono)

           apply(rule_tac[!] le_imp_inverse_le) using goal1 M by auto

         also have "\<dots> = 2 / real M" unfolding real_divide_def by auto

         finally show "norm (g n x - g m x) \<le> 2 / real M"

           using norm_triangle_le[of "g n x - f x" "f x - g m x" "2 / real M"]

           by(auto simp add:group_simps simp add:norm_minus_commute)

       qed qed qed

   from this[unfolded convergent_eq_cauchy[THEN sym]] guess s .. note s=this

   show ?thesis unfolding integrable_on_def apply(rule_tac x=s in exI) unfolding has_integral

   proof(rule,rule)  

     case goal1 hence *:"e/3 > 0" by auto

     from s[unfolded Lim_sequentially,rule_format,OF this] guess N1 .. note N1=this

     from goal1 as have "e / 3 / content {a..b} > 0" by(auto simp add:field_simps)

     from real_arch_invD[OF this] guess N2 apply-by(erule exE conjE)+ note N2=this

     from i[of "N1 + N2",unfolded has_integral,rule_format,OF *] guess g' .. note g'=conjunctD2[OF this,rule_format]

     have lem:"\<And>sf sg i. norm(sf - sg) \<le> e / 3 \<Longrightarrow> norm(i - s) < e / 3 \<Longrightarrow> norm(sg - i) < e / 3 \<Longrightarrow> norm(sf - s) < e"

     proof- case goal1 have "norm (sf - s) \<le> norm (sf - sg) + norm (sg - i) + norm (i - s)"

         using norm_triangle_ineq[of "sf - sg" "sg - s"]

         using norm_triangle_ineq[of "sg -  i" " i - s"] by(auto simp add:group_simps)

       also have "\<dots> < e" using goal1 unfolding norm_minus_commute by(auto simp add:group_simps)

       finally show ?case .

     qed

     show ?case apply(rule_tac x=g' in exI) apply(rule,rule g')

     proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> g' fine p" note * = g'(2)[OF this]

       show "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - s) < e" apply-apply(rule lem[OF _ _ *])

         apply(rule order_trans,rule rsum_diff_bound[OF p[THEN conjunct1]]) apply(rule,rule g,assumption)

       proof- have "content {a..b} < e / 3 * (real N2)"

           using N2 unfolding inverse_eq_divide using as by(auto simp add:field_simps)

         hence "content {a..b} < e / 3 * (real (N1 + N2) + 1)"

           apply-apply(rule less_le_trans,assumption) using `e>0` by auto 

         thus "inverse (real (N1 + N2) + 1) * content {a..b} \<le> e / 3"

           unfolding inverse_eq_divide by(auto simp add:field_simps)

         show "norm (i (N1 + N2) - s) < e / 3" by(rule N1[rule_format,unfolded vector_dist_norm],auto)

       qed qed qed qed

 subsection {* Negligible sets. *}

 definition "indicator s \<equiv> (\<lambda>x. if x \<in> s then 1 else (0::real))"

 lemma dest_vec1_indicator:

  "indicator s x = (if x \<in> s then 1 else 0)" unfolding indicator_def by auto

 lemma indicator_pos_le[intro]: "0 \<le> (indicator s x)" unfolding indicator_def by auto

 lemma indicator_le_1[intro]: "(indicator s x) \<le> 1" unfolding indicator_def by auto

 lemma dest_vec1_indicator_abs_le_1: "abs(indicator s x) \<le> 1"

   unfolding indicator_def by auto

 definition "negligible (s::(real^'n) set) \<equiv> (\<forall>a b. ((indicator s) has_integral 0) {a..b})"

 lemma indicator_simps[simp]:"x\<in>s \<Longrightarrow> indicator s x = 1" "x\<notin>s \<Longrightarrow> indicator s x = 0"

   unfolding indicator_def by auto

 subsection {* Negligibility of hyperplane. *}

 lemma vsum_nonzero_image_lemma: 

   assumes "finite s" "g(a) = 0"

   "\<forall>x\<in>s. \<forall>y\<in>s. f x = f y \<and> x \<noteq> y \<longrightarrow> g(f x) = 0"

   shows "setsum g {f x |x. x \<in> s \<and> f x \<noteq> a} = setsum (g o f) s"

   unfolding setsum_iterate[OF assms(1)] apply(subst setsum_iterate) defer

   apply(rule iterate_nonzero_image_lemma) apply(rule assms monoidal_monoid)+

   unfolding assms using neutral_add unfolding neutral_add using assms by auto 

 lemma interval_doublesplit: shows "{a..b} \<inter> {x . abs(x$k - c) \<le> (e::real)} =

   {(\<chi> i. if i = k then max (a$k) (c - e) else a$i) .. (\<chi> i. if i = k then min (b$k) (c + e) else b$i)}"

 proof- have *:"\<And>x c e::real. abs(x - c) \<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 by(rule refl) qed

 lemma division_doublesplit: assumes "p division_of {a..b::real^'n}" 

   shows "{l \<inter> {x. abs(x$k - c) \<le> e} |l. l \<in> p \<and> l \<inter> {x. abs(x$k - c) \<le> e} \<noteq> {}} division_of ({a..b} \<inter> {x. abs(x$k - c) \<le> e})"

 proof- have *:"\<And>x c. abs(x - c) \<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" and k=k,unfolded interval_split]

   note division_split(2)[OF this, where c="c-e" and k=k] 

   thus ?thesis apply(rule **) unfolding interval_doublesplit unfolding * unfolding interval_split interval_doublesplit

     apply(rule set_ext) unfolding mem_Collect_eq apply rule apply(erule conjE exE)+ apply(rule_tac x=la in exI) defer

     apply(erule conjE exE)+ apply(rule_tac x="l \<inter> {x. c + e \<ge> x $ k}" in exI) apply rule defer apply rule

     apply(rule_tac x=l in exI) by blast+ qed

 lemma content_doublesplit: assumes "0 < e"

   obtains d where "0 < d" "content({a..b} \<inter> {x. abs(x$k - c) \<le> d}) < e"

 proof(cases "content {a..b} = 0")

   case True show ?thesis apply(rule that[of 1]) defer unfolding interval_doublesplit

     apply(rule le_less_trans[OF content_subset]) defer apply(subst True)

     unfolding interval_doublesplit[THEN sym] using assms by auto 

 next case False def d \<equiv> "e / 3 / setprod (\<lambda>i. b$i - a$i) (UNIV - {k})"

   note False[unfolded content_eq_0 not_ex not_le, rule_format]

   hence prod0:"0 < setprod (\<lambda>i. b$i - a$i) (UNIV - {k})" apply-apply(rule setprod_pos) by smt

   hence "d > 0" unfolding d_def using assms by(auto simp add:field_simps) thus ?thesis

   proof(rule that[of d]) have *:"UNIV = insert k (UNIV - {k})" by auto

     have **:"{a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {} \<Longrightarrow> 

       (\<Prod>i\<in>UNIV - {k}. interval_upperbound ({a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) $ i - interval_lowerbound ({a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) $ i)

       = (\<Prod>i\<in>UNIV - {k}. b$i - a$i)" apply(rule setprod_cong,rule refl)

       unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds by auto

     show "content ({a..b} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) < e" apply(cases) unfolding content_def apply(subst if_P,assumption,rule assms)

       unfolding if_not_P apply(subst *) apply(subst setprod_insert) unfolding **

       unfolding interval_doublesplit interval_eq_empty not_ex not_less unfolding interval_bounds unfolding Cart_lambda_beta if_P[OF refl]

     proof- have "(min (b $ k) (c + d) - max (a $ k) (c - d)) \<le> 2 * d" by auto

       also have "... < e / (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i)" unfolding d_def using assms prod0 by(auto simp add:field_simps)

       finally show "(min (b $ k) (c + d) - max (a $ k) (c - d)) * (\<Prod>i\<in>UNIV - {k}. b $ i - a $ i) < e"

         unfolding pos_less_divide_eq[OF prod0] . qed auto qed qed

 lemma negligible_standard_hyperplane[intro]: "negligible {x. x$k = (c::real)}" 

   unfolding negligible_def has_integral apply(rule,rule,rule,rule)

 proof- case goal1 from content_doublesplit[OF this,of a b k c] guess d . note d=this let ?i = "indicator {x. x$k = c}"

   show ?case apply(rule_tac x="\<lambda>x. ball x d" in exI) apply(rule,rule gauge_ball,rule d)

   proof(rule,rule) fix p assume p:"p tagged_division_of {a..b} \<and> (\<lambda>x. ball x d) fine p"

     have *:"(\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) = (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. abs(x$k - c) \<le> d}) *\<^sub>R ?i x)"

       apply(rule setsum_cong2) unfolding split_paired_all real_scaleR_def mult_cancel_right split_conv

       apply(cases,rule disjI1,assumption,rule disjI2)

     proof- fix x l assume as:"(x,l)\<in>p" "?i x \<noteq> 0" hence xk:"x$k = c" unfolding indicator_def apply-by(rule ccontr,auto)

       show "content l = content (l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})" apply(rule arg_cong[where f=content])

         apply(rule set_ext,rule,rule) unfolding mem_Collect_eq

       proof- fix y assume y:"y\<in>l" note p[THEN conjunct2,unfolded fine_def,rule_format,OF as(1),unfolded split_conv]

         note this[unfolded subset_eq mem_ball vector_dist_norm,rule_format,OF y] note le_less_trans[OF component_le_norm[of _ k] this]

         thus "\<bar>y $ k - c\<bar> \<le> d" unfolding Cart_nth.diff xk by auto

       qed auto qed

     note p'= tagged_division_ofD[OF p[THEN conjunct1]] and p''=division_of_tagged_division[OF p[THEN conjunct1]]

     show "norm ((\<Sum>(x, ka)\<in>p. content ka *\<^sub>R ?i x) - 0) < e" unfolding diff_0_right * unfolding real_scaleR_def real_norm_def

       apply(subst abs_of_nonneg) apply(rule setsum_nonneg,rule) unfolding split_paired_all split_conv

       apply(rule mult_nonneg_nonneg) apply(drule p'(4)) apply(erule exE)+ apply(rule_tac b=b in back_subst)

       prefer 2 apply(subst(asm) eq_commute) apply assumption

       apply(subst interval_doublesplit) apply(rule content_pos_le) apply(rule indicator_pos_le)

     proof- have "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) \<le> (\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}))"

         apply(rule setsum_mono) unfolding split_paired_all split_conv 

         apply(rule mult_right_le_one_le) apply(drule p'(4)) by(auto simp add:interval_doublesplit intro!:content_pos_le)

       also have "... < e" apply(subst setsum_over_tagged_division_lemma[OF p[THEN conjunct1]])

       proof- case goal1 have "content ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<le> content {u..v}"

           unfolding interval_doublesplit apply(rule content_subset) unfolding interval_doublesplit[THEN sym] by auto

         thus ?case unfolding goal1 unfolding interval_doublesplit using content_pos_le by smt

       next have *:"setsum content {l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} |l. l \<in> snd ` p \<and> l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} \<noteq> {}} \<ge> 0"

           apply(rule setsum_nonneg,rule) unfolding mem_Collect_eq image_iff apply(erule exE bexE conjE)+ unfolding split_paired_all 

         proof- fix x l a b assume as:"x = l \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}" "(a, b) \<in> p" "l = snd (a, b)"

           guess u v using p'(4)[OF as(2)] apply-by(erule exE)+ note * = this

           show "content x \<ge> 0" unfolding as snd_conv * interval_doublesplit by(rule content_pos_le)

         qed have **:"norm (1::real) \<le> 1" by auto note division_doublesplit[OF p'',unfolded interval_doublesplit]

         note dsum_bound[OF this **,unfolded interval_doublesplit[THEN sym]]

         note this[unfolded real_scaleR_def real_norm_def class_semiring.semiring_rules, of k c d] note le_less_trans[OF this d(2)]

         from this[unfolded abs_of_nonneg[OF *]] show "(\<Sum>ka\<in>snd ` p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d})) < e"

           apply(subst vsum_nonzero_image_lemma[of "snd ` p" content "{}", unfolded o_def,THEN sym])

           apply(rule finite_imageI p' content_empty)+ unfolding forall_in_division[OF p'']

         proof(rule,rule,rule,rule,rule,rule,rule,erule conjE) fix m n u v

           assume as:"{m..n} \<in> snd ` p" "{u..v} \<in> snd ` p" "{m..n} \<noteq> {u..v}"  "{m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d} = {u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}"

           have "({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<inter> ({u..v} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) \<subseteq> {m..n} \<inter> {u..v}" by blast

           note subset_interior[OF this, unfolded division_ofD(5)[OF p'' as(1-3)] interior_inter[of "{m..n}"]]

           hence "interior ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = {}" unfolding as Int_absorb by auto

           thus "content ({m..n} \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) = 0" unfolding interval_doublesplit content_eq_0_interior[THEN sym] .

         qed qed

       finally show "(\<Sum>(x, ka)\<in>p. content (ka \<inter> {x. \<bar>x $ k - c\<bar> \<le> d}) * ?i x) < e" .

     qed qed qed

 subsection {* A technical lemma about "refinement" of division. *}

 lemma tagged_division_finer: fixes p::"((real^'n) \<times> ((real^'n) set)) set"

   assumes "p tagged_division_of {a..b}" "gauge d"

   obtains q where "q tagged_division_of {a..b}" "d fine q" "\<forall>(x,k) \<in> p. k \<subseteq> d(x) \<longrightarrow> (x,k) \<in> q"

 proof-

   let ?P = "\<lambda>p. p tagged_partial_division_of {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 {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

     thus ?thesis apply-apply(rule that[of q]) unfolding tagged_division_ofD[OF assms(1)] by auto

   } fix p::"((real^'n) \<times> ((real^'n) set)) set" assume as:"finite p"

   show "?P p" apply(rule,rule) using as proof(induct p) 

     case empty show ?case apply(rule_tac x="{}" in exI) unfolding fine_def by auto

   next case (insert xk p) guess x k using surj_pair[of xk] apply- by(erule 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 apply-by(erule exE)+ note uv=this

     have "finite {k. \<exists>x. (x, k) \<in> p}" 

       apply(rule finite_subset[of _ "snd ` p"],rule) unfolding subset_eq image_iff mem_Collect_eq

       apply(erule exE,rule_tac x="(xa,x)" in bexI) using p by auto

     hence int:"interior {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],assumption)      

       apply(rule p(5))  unfolding uv xk apply(rule insertI1,rule insertI2) apply assumption

       using insert(2) unfolding uv xk by auto

     show ?case proof(cases "{u..v} \<subseteq> d x")

       case True thus ?thesis apply(rule_tac x="{(x,{u..v})} \<union> q1" in exI) apply rule

         unfolding * uv apply(rule tagged_division_union,rule tagged_division_of_self)

         apply(rule p[unfolded xk uv] insertI1)+  apply(rule q1,rule int) 

         apply(rule,rule fine_union,subst fine_def) defer apply(rule q1)

         unfolding Ball_def split_paired_All split_conv apply(rule,rule,rule,rule)

         apply(erule insertE) defer apply(rule UnI2) apply(drule q1(3)[rule_format]) unfolding xk uv by auto

     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,rule,rule,rule) apply(erule insertE)

         apply(rule UnI2) defer apply(drule q1(3)[rule_format])using False unfolding xk uv by auto

     qed qed qed

 subsection {* Hence the main theorem about negligible sets. *}

 lemma finite_product_dependent: assumes "finite s" "\<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 by auto qed auto

 lemma sum_sum_product: assumes "finite s" "\<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_Un_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 by auto

     show "finite {(i, j) |i j. i \<in> s \<and> j \<in> t i}" apply(rule finite_product_dependent) using insert by auto

   qed(insert insert, auto) qed auto

 lemma has_integral_negligible: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"

   assumes "negligible s" "\<forall>x\<in>(t - s). f x = 0"

   shows "(f has_integral 0) t"

 proof- presume P:"\<And>f::real^'n \<Rightarrow> 'a. \<And>a b. (\<forall>x. ~(x \<in> s) \<longrightarrow> f x = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"

   let ?f = "(\<lambda>x. if x \<in> t then f x else 0)"

   show ?thesis apply(rule_tac f="?f" in has_integral_eq) apply(rule) unfolding if_P apply(rule refl)

     apply(subst has_integral_alt) apply(cases,subst if_P,assumption) unfolding if_not_P

   proof- assume "\<exists>a b. t = {a..b}" then guess a b apply-by(erule exE)+ note t = this

     show "(?f has_integral 0) t" unfolding t apply(rule P) using assms(2) unfolding t by auto

   next show "\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> t then ?f x else 0) has_integral z) {a..b} \<and> norm (z - 0) < e)"

       apply(safe,rule_tac x=1 in exI,rule) apply(rule zero_less_one,safe) apply(rule_tac x=0 in exI)

       apply(rule,rule P) using assms(2) by auto

   qed

 next fix f::"real^'n \<Rightarrow> 'a" and a b::"real^'n" assume assm:"\<forall>x. x \<notin> s \<longrightarrow> f x = 0" 

   show "(f has_integral 0) {a..b}" unfolding has_integral

   proof(safe) case goal1

     hence "\<And>n. e / 2 / ((real n+1) * (2 ^ n)) > 0" 

       apply-apply(rule divide_pos_pos) defer apply(rule mult_pos_pos) by(auto simp add:field_simps)

     note assms(1)[unfolded negligible_def has_integral,rule_format,OF this,of a b] note allI[OF this,of "\<lambda>x. x"] 

     from choice[OF this] guess d .. note d=conjunctD2[OF this[rule_format]]

     show ?case apply(rule_tac x="\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x" in exI) 

     proof safe show "gauge (\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x)" using d(1) unfolding gauge_def by auto

       fix p assume as:"p tagged_division_of {a..b}" "(\<lambda>x. d (nat \<lfloor>norm (f x)\<rfloor>) x) fine p" 

       let ?goal = "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"

       { presume "p\<noteq>{} \<Longrightarrow> ?goal" thus ?goal apply(cases "p={}") using goal1 by auto  }

       assume as':"p \<noteq> {}" from real_arch_simple[of "Sup((\<lambda>(x,k). norm(f x)) ` p)"] guess N ..

       hence N:"\<forall>x\<in>(\<lambda>(x, k). norm (f x)) ` p. x \<le> real N" apply(subst(asm) Sup_finite_le_iff) using as as' by auto

       have "\<forall>i. \<exists>q. q tagged_division_of {a..b} \<and> (d i) fine q \<and> (\<forall>(x, k)\<in>p. k \<subseteq> (d i) x \<longrightarrow> (x, k) \<in> q)"

         apply(rule,rule tagged_division_finer[OF as(1) d(1)]) by auto

       from choice[OF this] guess q .. note q=conjunctD3[OF this[rule_format]]

       have *:"\<And>i. (\<Sum>(x, k)\<in>q i. content k *\<^sub>R indicator s x) \<ge> 0" apply(rule setsum_nonneg,safe) 

         unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) apply(drule tagged_division_ofD(4)[OF q(1)]) by auto

       have **:"\<And>f g s t. finite s \<Longrightarrow> finite t \<Longrightarrow> (\<forall>(x,y) \<in> t. (0::real) \<le> g(x,y)) \<Longrightarrow> (\<forall>y\<in>s. \<exists>x. (x,y) \<in> t \<and> f(y) \<le> g(x,y)) \<Longrightarrow> setsum f s \<le> setsum g t"

       proof- case goal1 thus ?case apply-apply(rule setsum_le_included[of s t g snd f]) prefer 4

           apply safe apply(erule_tac x=x in ballE) apply(erule exE) apply(rule_tac x="(xa,x)" in bexI) by auto qed

       have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) \<le> setsum (\<lambda>i. (real i + 1) *

                      norm(setsum (\<lambda>(x,k). content k *\<^sub>R indicator s x) (q i))) {0..N+1}"

         unfolding real_norm_def setsum_right_distrib abs_of_nonneg[OF *] diff_0_right

         apply(rule order_trans,rule setsum_norm) defer apply(subst sum_sum_product) prefer 3 

       proof(rule **,safe) show "finite {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i}" apply(rule finite_product_dependent) using q by auto

         fix i a b assume as'':"(a,b) \<in> q i" show "0 \<le> (real i + 1) * (content b *\<^sub>R indicator s a)"

           unfolding real_scaleR_def apply(rule mult_nonneg_nonneg) defer apply(rule mult_nonneg_nonneg)

           using tagged_division_ofD(4)[OF q(1) as''] by auto

       next fix i::nat show "finite (q i)" using q by auto

       next fix x k assume xk:"(x,k) \<in> p" def n \<equiv> "nat \<lfloor>norm (f x)\<rfloor>"

         have *:"norm (f x) \<in> (\<lambda>(x, k). norm (f x)) ` p" using xk by auto

         have nfx:"real n \<le> norm(f x)" "norm(f x) \<le> real n + 1" unfolding n_def by auto

         hence "n \<in> {0..N + 1}" using N[rule_format,OF *] by auto

         moreover  note as(2)[unfolded fine_def,rule_format,OF xk,unfolded split_conv]

         note q(3)[rule_format,OF xk,unfolded split_conv,rule_format,OF this] note this[unfolded n_def[symmetric]]

         moreover have "norm (content k *\<^sub>R f x) \<le> (real n + 1) * (content k * indicator s x)"

         proof(cases "x\<in>s") case False thus ?thesis using assm by auto

         next case True have *:"content k \<ge> 0" using tagged_division_ofD(4)[OF as(1) xk] by auto

           moreover have "content k * norm (f x) \<le> content k * (real n + 1)" apply(rule mult_mono) using nfx * by auto

           ultimately show ?thesis unfolding abs_mult using nfx True by(auto simp add:field_simps)

         qed ultimately show "\<exists>y. (y, x, k) \<in> {(i, j) |i j. i \<in> {0..N + 1} \<and> j \<in> q i} \<and> norm (content k *\<^sub>R f x) \<le> (real y + 1) * (content k *\<^sub>R indicator s x)"

           apply(rule_tac x=n in exI,safe) apply(rule_tac x=n in exI,rule_tac x="(x,k)" in exI,safe) by auto

       qed(insert as, auto)

       also have "... \<le> setsum (\<lambda>i. e / 2 / 2 ^ i) {0..N+1}" apply(rule setsum_mono) 

       proof- case goal1 thus ?case apply(subst mult_commute, subst pos_le_divide_eq[THEN sym])

           using d(2)[rule_format,of "q i" i] using q[rule_format] by(auto simp add:field_simps)

       qed also have "... < e * inverse 2 * 2" unfolding real_divide_def setsum_right_distrib[THEN sym]

         apply(rule mult_strict_left_mono) unfolding power_inverse atLeastLessThanSuc_atLeastAtMost[THEN sym]

         apply(subst sumr_geometric) using goal1 by auto

       finally show "?goal" by auto qed qed qed

 lemma has_integral_spike: fixes f::"real^'n \<Rightarrow> 'a::real_normed_vector"

   assumes "negligible s" "(\<forall>x\<in>(t - s). g x = f x)" "(f has_integral y) t"

   shows "(g has_integral y) t"

 proof- { fix a b::"real^'n" and f g ::"real^'n \<Rightarrow> 'a" and y::'a

     assume as:"\<forall>x \<in> {a..b} - s. g x = f x" "(f has_integral y) {a..b}"

     have "((\<lambda>x. f x + (g x - f x)) has_integral (y + 0)) {a..b}" apply(rule has_integral_add[OF as(2)])

       apply(rule has_integral_negligible[OF assms(1)]) using as by auto

     hence "(g has_integral y) {a..b}" by auto } note * = this

   show ?thesis apply(subst has_integral_alt) using assms(2-) apply-apply(rule cond_cases,safe)

     apply(rule *, assumption+) apply(subst(asm) has_integral_alt) unfolding if_not_P

     apply(erule_tac x=e in allE,safe,rule_tac x=B in exI,safe) apply(erule_tac x=a in allE,erule_tac x=b in allE,safe)

     apply(rule_tac x=z in exI,safe) apply(rule *[where fa2="\<lambda>x. if x\<in>t then f x else 0"]) by auto qed

 lemma has_integral_spike_eq:

   assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"

   shows "((f has_integral y) t \<longleftrightarrow> (g has_integral y) t)"

   apply rule apply(rule_tac[!] has_integral_spike[OF assms(1)]) using assms(2) by auto

 lemma integrable_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x" "f integrable_on t"

   shows "g integrable_on  t"

   using assms unfolding integrable_on_def apply-apply(erule exE)

   apply(rule,rule has_integral_spike) by fastsimp+

 lemma integral_spike: assumes "negligible s" "\<forall>x\<in>(t - s). g x = f x"

   shows "integral t f = integral t g"

   unfolding integral_def using has_integral_spike_eq[OF assms] by auto

 subsection {* Some other trivialities about negligible sets. *}

 lemma negligible_subset[intro]: assumes "negligible s" "t \<subseteq> s" shows "negligible t" unfolding negligible_def 

 proof(safe) case goal1 show ?case using assms(1)[unfolded negligible_def,rule_format,of a b]

     apply-apply(rule has_integral_spike[OF assms(1)]) defer apply assumption

     using assms(2) unfolding indicator_def by auto qed

 lemma negligible_diff[intro?]: assumes "negligible s" shows "negligible(s - t)" using assms by auto

 lemma negligible_inter: assumes "negligible s \<or> negligible t" shows "negligible(s \<inter> t)" using assms by auto

 lemma negligible_union: assumes "negligible s" "negligible t" shows "negligible (s \<union> t)" unfolding negligible_def 

 proof safe case goal1 note assm = assms[unfolded negligible_def,rule_format,of a b]

   thus ?case apply(subst has_integral_spike_eq[OF assms(2)])

     defer apply assumption unfolding indicator_def by auto qed

 lemma negligible_union_eq[simp]: "negligible (s \<union> t) \<longleftrightarrow> (negligible s \<and> negligible t)"

   using negligible_union by auto

 lemma negligible_sing[intro]: "negligible {a::real^'n}" 

 proof- guess x using UNIV_witness[where 'a='n] ..

   show ?thesis using negligible_standard_hyperplane[of x "a$x"] by auto qed

 lemma negligible_insert[simp]: "negligible(insert a s) \<longleftrightarrow> negligible s"

   apply(subst insert_is_Un) unfolding negligible_union_eq by auto

 lemma negligible_empty[intro]: "negligible {}" by auto

 lemma negligible_finite[intro]: assumes "finite s" shows "negligible s"

   using assms apply(induct s) by auto

 lemma negligible_unions[intro]: assumes "finite s" "\<forall>t\<in>s. negligible t" shows "negligible(\<Union>s)"

   using assms by(induct,auto) 

 lemma negligible:  "negligible s \<longleftrightarrow> (\<forall>t::(real^'n) set. (indicator s has_integral 0) t)"

   apply safe defer apply(subst negligible_def)

 proof- fix t::"(real^'n) set" assume as:"negligible s"