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 .