src/HOL/Multivariate_Analysis/Integration.thy
author nipkow
Fri Apr 11 22:53:33 2014 +0200 (2014-04-11)
changeset 56541 0e3abadbef39
parent 56536 aefb4a8da31f
child 56544 b60d5d119489
permissions -rw-r--r--
made divide_pos_pos a simp rule
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(*  Author:     John Harrison
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    Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light)
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*)
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header {* Kurzweil-Henstock Gauge Integration in many dimensions. *}
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theory Integration
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imports
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  Derivative
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  "~~/src/HOL/Library/Indicator_Function"
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begin
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lemma cSup_abs_le: (* TODO: is this really needed? *)
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  fixes S :: "real set"
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  shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
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  by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2 bdd_aboveI)
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lemma cInf_abs_ge: (* TODO: is this really needed? *)
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  fixes S :: "real set"
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  shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Inf S\<bar> \<le> a"
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  by (simp add: Inf_real_def) (insert cSup_abs_le [of "uminus ` S"], auto)
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lemma cSup_asclose: (* TODO: is this really needed? *)
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  fixes S :: "real set"
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  assumes S: "S \<noteq> {}"
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    and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
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  shows "\<bar>Sup S - l\<bar> \<le> e"
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proof -
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  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e"
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    by arith
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  have "bdd_above S"
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    using b by (auto intro!: bdd_aboveI[of _ "l + e"])
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  with S b show ?thesis
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    unfolding th by (auto intro!: cSup_upper2 cSup_least)
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qed
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lemma cInf_asclose: (* TODO: is this really needed? *)
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  fixes S :: "real set"
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  assumes S: "S \<noteq> {}"
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    and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
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  shows "\<bar>Inf S - l\<bar> \<le> e"
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proof -
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  have "\<bar>- Sup (uminus ` S) - l\<bar> =  \<bar>Sup (uminus ` S) - (-l)\<bar>"
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    by auto
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  also have "\<dots> \<le> e"
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    apply (rule cSup_asclose)
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    using abs_minus_add_cancel b by (auto simp add: S)
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  finally have "\<bar>- Sup (uminus ` S) - l\<bar> \<le> e" .
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  then show ?thesis
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    by (simp add: Inf_real_def)
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qed
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lemma cSup_finite_ge_iff:
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  fixes S :: "real set"
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  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Sup S \<longleftrightarrow> (\<exists>x\<in>S. a \<le> x)"
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  by (metis cSup_eq_Max Max_ge_iff)
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lemma cSup_finite_le_iff:
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  fixes S :: "real set"
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  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Sup S \<longleftrightarrow> (\<forall>x\<in>S. a \<ge> x)"
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  by (metis cSup_eq_Max Max_le_iff)
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lemma cInf_finite_ge_iff:
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  fixes S :: "real set"
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  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
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  by (metis cInf_eq_Min Min_ge_iff)
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lemma cInf_finite_le_iff:
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  fixes S :: "real set"
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  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists>x\<in>S. a \<ge> x)"
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  by (metis cInf_eq_Min Min_le_iff)
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(*declare not_less[simp] not_le[simp]*)
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lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
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  scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
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  scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one
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lemma real_arch_invD:
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  "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
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  by (subst(asm) real_arch_inv)
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subsection {* Sundries *}
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lemma conjunctD2: assumes "a \<and> b" shows a b using assms by auto
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lemma conjunctD3: assumes "a \<and> b \<and> c" shows a b c using assms by auto
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lemma conjunctD4: assumes "a \<and> b \<and> c \<and> d" shows a b c d using assms by auto
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lemma conjunctD5: assumes "a \<and> b \<and> c \<and> d \<and> e" shows a b c d e using assms by auto
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declare norm_triangle_ineq4[intro]
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lemma simple_image: "{f x |x . x \<in> s} = f ` s"
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  by blast
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lemma linear_simps:
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  assumes "bounded_linear f"
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  shows
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    "f (a + b) = f a + f b"
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    "f (a - b) = f a - f b"
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    "f 0 = 0"
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    "f (- a) = - f a"
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    "f (s *\<^sub>R v) = s *\<^sub>R (f v)"
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proof -
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  interpret f: bounded_linear f by fact
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  show "f (a + b) = f a + f b" by (rule f.add)
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  show "f (a - b) = f a - f b" by (rule f.diff)
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  show "f 0 = 0" by (rule f.zero)
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  show "f (- a) = - f a" by (rule f.minus)
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  show "f (s *\<^sub>R v) = s *\<^sub>R (f v)" by (rule f.scaleR)
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qed
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lemma bounded_linearI:
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  assumes "\<And>x y. f (x + y) = f x + f y"
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    and "\<And>r x. f (r *\<^sub>R x) = r *\<^sub>R f x"
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    and "\<And>x. norm (f x) \<le> norm x * K"
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  shows "bounded_linear f"
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  using assms by (rule bounded_linear_intro) (* FIXME: duplicate *)
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lemma bounded_linear_component [intro]: "bounded_linear (\<lambda>x::'a::euclidean_space. x \<bullet> k)"
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  by (rule bounded_linear_inner_left)
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lemma transitive_stepwise_lt_eq:
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  assumes "(\<And>x y z::nat. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z)"
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  shows "((\<forall>m. \<forall>n>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n)))"
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  (is "?l = ?r")
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proof safe
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  assume ?r
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  fix n m :: nat
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  assume "m < n"
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  then show "R m n"
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  proof (induct n arbitrary: m)
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    case 0
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    then show ?case by auto
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  next
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    case (Suc n)
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    show ?case
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    proof (cases "m < n")
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      case True
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      show ?thesis
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        apply (rule assms[OF Suc(1)[OF True]])
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        using `?r`
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        apply auto
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        done
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    next
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      case False
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      then have "m = n"
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        using Suc(2) by auto
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      then show ?thesis
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        using `?r` by auto
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    qed
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  qed
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qed auto
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lemma transitive_stepwise_gt:
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  assumes "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" "\<And>n. R n (Suc n)"
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  shows "\<forall>n>m. R m n"
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proof -
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  have "\<forall>m. \<forall>n>m. R m n"
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    apply (subst transitive_stepwise_lt_eq)
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    apply (rule assms)
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    apply assumption
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    apply assumption
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    using assms(2) apply auto
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    done
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  then show ?thesis by auto
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qed
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lemma transitive_stepwise_le_eq:
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  assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
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  shows "(\<forall>m. \<forall>n\<ge>m. R m n) \<longleftrightarrow> (\<forall>n. R n (Suc n))"
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  (is "?l = ?r")
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proof safe
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  assume ?r
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  fix m n :: nat
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  assume "m \<le> n"
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  then show "R m n"
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  proof (induct n arbitrary: m)
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    case 0
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    with assms show ?case by auto
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  next
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    case (Suc n)
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    show ?case
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    proof (cases "m \<le> n")
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      case True
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      show ?thesis
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        apply (rule assms(2))
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        apply (rule Suc(1)[OF True])
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        using `?r` apply auto
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        done
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    next
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      case False
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      then have "m = Suc n"
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        using Suc(2) by auto
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      then show ?thesis
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        using assms(1) by auto
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    qed
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  qed
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qed auto
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lemma transitive_stepwise_le:
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  assumes "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z"
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    and "\<And>n. R n (Suc n)"
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  shows "\<forall>n\<ge>m. R m n"
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proof -
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  have "\<forall>m. \<forall>n\<ge>m. R m n"
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    apply (subst transitive_stepwise_le_eq)
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    apply (rule assms)
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    apply (rule assms,assumption,assumption)
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    using assms(3)
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    apply auto
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    done
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  then show ?thesis by auto
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qed
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subsection {* Some useful lemmas about intervals. *}
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abbreviation One :: "'a::euclidean_space"
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  where "One \<equiv> \<Sum>Basis"
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lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
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  using nonempty_Basis
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  by (fastforce simp add: set_eq_iff mem_box)
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lemma interior_subset_union_intervals:
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  assumes "i = cbox a b"
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    and "j = cbox c d"
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    and "interior j \<noteq> {}"
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    and "i \<subseteq> j \<union> s"
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    and "interior i \<inter> interior j = {}"
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  shows "interior i \<subseteq> interior s"
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proof -
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  have "box a b \<inter> cbox c d = {}"
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     using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
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     unfolding assms(1,2) interior_cbox by auto
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  moreover
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  have "box a b \<subseteq> cbox c d \<union> s"
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    apply (rule order_trans,rule box_subset_cbox)
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    using assms(4) unfolding assms(1,2)
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    apply auto
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    done
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  ultimately
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  show ?thesis
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    apply -
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    apply (rule interior_maximal)
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    defer
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    apply (rule open_interior)
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    unfolding assms(1,2) interior_cbox
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    apply auto
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    done
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qed
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lemma inter_interior_unions_intervals:
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  fixes f::"('a::euclidean_space) set set"
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  assumes "finite f"
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    and "open s"
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    and "\<forall>t\<in>f. \<exists>a b. t = cbox a b"
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    and "\<forall>t\<in>f. s \<inter> (interior t) = {}"
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  shows "s \<inter> interior (\<Union>f) = {}"
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proof (rule ccontr, unfold ex_in_conv[symmetric])
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  case goal1
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  have lem1: "\<And>x e s U. ball x e \<subseteq> s \<inter> interior U \<longleftrightarrow> ball x e \<subseteq> s \<inter> U"
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    apply rule
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    defer
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    apply (rule_tac Int_greatest)
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    unfolding open_subset_interior[OF open_ball]
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    using interior_subset
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    apply auto
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    done
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  have lem2: "\<And>x s P. \<exists>x\<in>s. P x \<Longrightarrow> \<exists>x\<in>insert x s. P x" by auto
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  have "\<And>f. finite f \<Longrightarrow> \<forall>t\<in>f. \<exists>a b. t = cbox a b \<Longrightarrow>
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    \<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"
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  proof -
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    case goal1
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    then show ?case
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    proof (induct rule: finite_induct)
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      case empty
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      obtain x where "x \<in> s \<inter> interior (\<Union>{})"
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        using empty(2) ..
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      then have False
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        unfolding Union_empty interior_empty by auto
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      then show ?case by auto
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    next
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      case (insert i f)
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      obtain x where x: "x \<in> s \<inter> interior (\<Union>insert i f)"
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        using insert(5) ..
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      then obtain e where e: "0 < e \<and> ball x e \<subseteq> s \<inter> interior (\<Union>insert i f)"
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        unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior], rule_format] ..
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      obtain a where "\<exists>b. i = cbox a b"
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        using insert(4)[rule_format,OF insertI1] ..
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      then obtain b where ab: "i = cbox a b" ..
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      show ?case
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      proof (cases "x \<in> i")
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        case False
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        then have "x \<in> UNIV - cbox a b"
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          unfolding ab by auto
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        then obtain d where "0 < d \<and> ball x d \<subseteq> UNIV - cbox a b"
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          unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_cbox],rule_format] ..
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        then have "0 < d" "ball x (min d e) \<subseteq> UNIV - i"
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          unfolding ab ball_min_Int by auto
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        then have "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)"
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          using e unfolding lem1 unfolding  ball_min_Int by auto
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        then have "x \<in> s \<inter> interior (\<Union>f)" using `d>0` e by auto
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        then have "\<exists>t\<in>f. \<exists>x e. 0 < e \<and> ball x e \<subseteq> s \<inter> t"
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          apply -
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          apply (rule insert(3))
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          using insert(4)
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          apply auto
wenzelm@49970
   310
          done
wenzelm@53399
   311
        then show ?thesis by auto
wenzelm@49970
   312
      next
wenzelm@49970
   313
        case True show ?thesis
immler@54775
   314
        proof (cases "x\<in>box a b")
wenzelm@49970
   315
          case True
immler@54775
   316
          then obtain d where "0 < d \<and> ball x d \<subseteq> box a b"
immler@56188
   317
            unfolding open_contains_ball_eq[OF open_box,rule_format] ..
wenzelm@53399
   318
          then show ?thesis
wenzelm@49970
   319
            apply (rule_tac x=i in bexI, rule_tac x=x in exI, rule_tac x="min d e" in exI)
wenzelm@49970
   320
            unfolding ab
immler@56188
   321
            using box_subset_cbox[of a b] and e
wenzelm@50945
   322
            apply fastforce+
wenzelm@49970
   323
            done
wenzelm@49970
   324
        next
wenzelm@49970
   325
          case False
wenzelm@53399
   326
          then obtain k where "x\<bullet>k \<le> a\<bullet>k \<or> x\<bullet>k \<ge> b\<bullet>k" and k: "k \<in> Basis"
immler@56188
   327
            unfolding mem_box by (auto simp add: not_less)
wenzelm@53399
   328
          then have "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
immler@56188
   329
            using True unfolding ab and mem_box
hoelzl@50526
   330
              apply (erule_tac x = k in ballE)
wenzelm@49970
   331
              apply auto
wenzelm@49970
   332
              done
wenzelm@53399
   333
          then have "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)"
wenzelm@53399
   334
          proof (rule disjE)
hoelzl@50526
   335
            let ?z = "x - (e/2) *\<^sub>R k"
hoelzl@50526
   336
            assume as: "x\<bullet>k = a\<bullet>k"
wenzelm@49970
   337
            have "ball ?z (e / 2) \<inter> i = {}"
wenzelm@49970
   338
              apply (rule ccontr)
wenzelm@53399
   339
              unfolding ex_in_conv[symmetric]
wenzelm@53399
   340
              apply (erule exE)
wenzelm@53399
   341
            proof -
wenzelm@49970
   342
              fix y
wenzelm@49970
   343
              assume "y \<in> ball ?z (e / 2) \<inter> i"
wenzelm@53399
   344
              then have "dist ?z y < e/2" and yi:"y\<in>i" by auto
wenzelm@53399
   345
              then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
hoelzl@50526
   346
                using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
wenzelm@53399
   347
              then have "y\<bullet>k < a\<bullet>k"
wenzelm@53399
   348
                using e[THEN conjunct1] k
haftmann@54230
   349
                by (auto simp add: field_simps abs_less_iff as inner_Basis inner_simps)
wenzelm@53399
   350
              then have "y \<notin> i"
immler@56188
   351
                unfolding ab mem_box by (auto intro!: bexI[OF _ k])
wenzelm@53399
   352
              then show False using yi by auto
wenzelm@49970
   353
            qed
wenzelm@49970
   354
            moreover
wenzelm@49970
   355
            have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
wenzelm@53399
   356
              apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
wenzelm@49970
   357
            proof
wenzelm@49970
   358
              fix y
wenzelm@53399
   359
              assume as: "y \<in> ball ?z (e/2)"
hoelzl@50526
   360
              have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R k)"
wenzelm@49970
   361
                apply -
hoelzl@50526
   362
                apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R k"])
hoelzl@50526
   363
                unfolding norm_scaleR norm_Basis[OF k]
wenzelm@49970
   364
                apply auto
wenzelm@49970
   365
                done
wenzelm@49970
   366
              also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2"
wenzelm@49970
   367
                apply (rule add_strict_left_mono)
wenzelm@50945
   368
                using as
wenzelm@50945
   369
                unfolding mem_ball dist_norm
wenzelm@50945
   370
                using e
wenzelm@50945
   371
                apply (auto simp add: field_simps)
wenzelm@49970
   372
                done
wenzelm@53399
   373
              finally show "y \<in> ball x e"
wenzelm@49970
   374
                unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
wenzelm@49970
   375
            qed
wenzelm@49970
   376
            ultimately show ?thesis
wenzelm@49970
   377
              apply (rule_tac x="?z" in exI)
wenzelm@49970
   378
              unfolding Union_insert
wenzelm@49970
   379
              apply auto
wenzelm@49970
   380
              done
wenzelm@49970
   381
          next
hoelzl@50526
   382
            let ?z = "x + (e/2) *\<^sub>R k"
hoelzl@50526
   383
            assume as: "x\<bullet>k = b\<bullet>k"
wenzelm@49970
   384
            have "ball ?z (e / 2) \<inter> i = {}"
wenzelm@49970
   385
              apply (rule ccontr)
wenzelm@53399
   386
              unfolding ex_in_conv[symmetric]
wenzelm@53408
   387
              apply (erule exE)
wenzelm@53408
   388
            proof -
wenzelm@49970
   389
              fix y
wenzelm@49970
   390
              assume "y \<in> ball ?z (e / 2) \<inter> i"
wenzelm@53408
   391
              then have "dist ?z y < e/2" and yi: "y \<in> i"
wenzelm@53408
   392
                by auto
wenzelm@53399
   393
              then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
wenzelm@53399
   394
                using Basis_le_norm[OF k, of "?z - y"]
wenzelm@53399
   395
                unfolding dist_norm by auto
wenzelm@53399
   396
              then have "y\<bullet>k > b\<bullet>k"
wenzelm@53399
   397
                using e[THEN conjunct1] k
wenzelm@53399
   398
                by (auto simp add:field_simps inner_simps inner_Basis as)
wenzelm@53399
   399
              then have "y \<notin> i"
immler@56188
   400
                unfolding ab mem_box by (auto intro!: bexI[OF _ k])
wenzelm@53399
   401
              then show False using yi by auto
wenzelm@49970
   402
            qed
wenzelm@49970
   403
            moreover
wenzelm@49970
   404
            have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
wenzelm@49970
   405
              apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
wenzelm@49970
   406
            proof
wenzelm@49970
   407
              fix y
wenzelm@49970
   408
              assume as: "y\<in> ball ?z (e/2)"
hoelzl@50526
   409
              have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R k)"
wenzelm@49970
   410
                apply -
wenzelm@53399
   411
                apply (rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R k"])
wenzelm@49970
   412
                unfolding norm_scaleR
hoelzl@50526
   413
                apply (auto simp: k)
wenzelm@49970
   414
                done
wenzelm@49970
   415
              also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2"
wenzelm@49970
   416
                apply (rule add_strict_left_mono)
wenzelm@49970
   417
                using as unfolding mem_ball dist_norm
wenzelm@49970
   418
                using e apply (auto simp add: field_simps)
wenzelm@49970
   419
                done
wenzelm@53399
   420
              finally show "y \<in> ball x e"
wenzelm@53399
   421
                unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
wenzelm@49970
   422
            qed
wenzelm@49970
   423
            ultimately show ?thesis
wenzelm@49970
   424
              apply (rule_tac x="?z" in exI)
wenzelm@49970
   425
              unfolding Union_insert
wenzelm@49970
   426
              apply auto
wenzelm@49970
   427
              done
wenzelm@53399
   428
          qed
wenzelm@53408
   429
          then obtain x where "ball x (e / 2) \<subseteq> s \<inter> \<Union>f" ..
wenzelm@53399
   430
          then have "x \<in> s \<inter> interior (\<Union>f)"
wenzelm@53408
   431
            unfolding lem1[where U="\<Union>f", symmetric]
wenzelm@49970
   432
            using centre_in_ball e[THEN conjunct1] by auto
wenzelm@53399
   433
          then show ?thesis
wenzelm@49970
   434
            apply -
wenzelm@49970
   435
            apply (rule lem2, rule insert(3))
wenzelm@53399
   436
            using insert(4)
wenzelm@53399
   437
            apply auto
wenzelm@49970
   438
            done
wenzelm@49970
   439
        qed
wenzelm@49970
   440
      qed
wenzelm@49970
   441
    qed
wenzelm@49970
   442
  qed
wenzelm@53408
   443
  from this[OF assms(1,3) goal1]
wenzelm@53408
   444
  obtain t x e where "t \<in> f" "0 < e" "ball x e \<subseteq> s \<inter> t"
wenzelm@53408
   445
    by blast
wenzelm@53408
   446
  then have "x \<in> s" "x \<in> interior t"
wenzelm@53399
   447
    using open_subset_interior[OF open_ball, of x e t]
wenzelm@53408
   448
    by auto
wenzelm@53399
   449
  then show False
wenzelm@53399
   450
    using `t \<in> f` assms(4) by auto
wenzelm@49970
   451
qed
wenzelm@49970
   452
immler@56188
   453
subsection {* Bounds on intervals where they exist. *}
immler@56188
   454
immler@56188
   455
definition interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
immler@56188
   456
  where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)"
immler@56188
   457
immler@56188
   458
definition interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
immler@56188
   459
   where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x:s. x\<bullet>i) *\<^sub>R i)"
immler@56188
   460
immler@56188
   461
lemma interval_upperbound[simp]:
immler@56188
   462
  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
immler@56188
   463
    interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
immler@56188
   464
  unfolding interval_upperbound_def euclidean_representation_setsum cbox_def SUP_def
immler@56188
   465
  by (safe intro!: cSup_eq) auto
immler@56188
   466
immler@56188
   467
lemma interval_lowerbound[simp]:
immler@56188
   468
  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
immler@56188
   469
    interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
immler@56188
   470
  unfolding interval_lowerbound_def euclidean_representation_setsum cbox_def INF_def
immler@56188
   471
  by (safe intro!: cInf_eq) auto
immler@56188
   472
immler@56188
   473
lemmas interval_bounds = interval_upperbound interval_lowerbound
immler@56188
   474
immler@56188
   475
lemma
immler@56188
   476
  fixes X::"real set"
immler@56188
   477
  shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
immler@56188
   478
    and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
immler@56188
   479
  by (auto simp: interval_upperbound_def interval_lowerbound_def SUP_def INF_def)
immler@56188
   480
immler@56188
   481
lemma interval_bounds'[simp]:
immler@56188
   482
  assumes "cbox a b \<noteq> {}"
immler@56188
   483
  shows "interval_upperbound (cbox a b) = b"
immler@56188
   484
    and "interval_lowerbound (cbox a b) = a"
immler@56188
   485
  using assms unfolding box_ne_empty by auto
wenzelm@53399
   486
himmelma@35172
   487
subsection {* Content (length, area, volume...) of an interval. *}
himmelma@35172
   488
immler@56188
   489
definition "content (s::('a::euclidean_space) set) =
immler@56188
   490
  (if s = {} then 0 else (\<Prod>i\<in>Basis. (interval_upperbound s)\<bullet>i - (interval_lowerbound s)\<bullet>i))"
immler@56188
   491
immler@56188
   492
lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> cbox a b \<noteq> {}"
immler@56188
   493
  unfolding box_eq_empty unfolding not_ex not_less by auto
immler@56188
   494
immler@56188
   495
lemma content_cbox:
immler@56188
   496
  fixes a :: "'a::euclidean_space"
hoelzl@50526
   497
  assumes "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
immler@56188
   498
  shows "content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
wenzelm@49970
   499
  using interval_not_empty[OF assms]
immler@54777
   500
  unfolding content_def
immler@56188
   501
  by (auto simp: )
immler@56188
   502
immler@56188
   503
lemma content_cbox':
immler@56188
   504
  fixes a :: "'a::euclidean_space"
immler@56188
   505
  assumes "cbox a b \<noteq> {}"
immler@56188
   506
  shows "content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
immler@56188
   507
  apply (rule content_cbox)
wenzelm@50945
   508
  using assms
immler@56188
   509
  unfolding box_ne_empty
wenzelm@49970
   510
  apply assumption
wenzelm@49970
   511
  done
wenzelm@49970
   512
wenzelm@53408
   513
lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
immler@56188
   514
  by (auto simp: interval_upperbound_def interval_lowerbound_def SUP_def INF_def content_def)
immler@56188
   515
hoelzl@50104
   516
lemma content_singleton[simp]: "content {a} = 0"
hoelzl@50104
   517
proof -
immler@56188
   518
  have "content (cbox a a) = 0"
immler@56188
   519
    by (subst content_cbox) (auto simp: ex_in_conv)
immler@56188
   520
  then show ?thesis by (simp add: cbox_sing)
immler@56188
   521
qed
immler@56188
   522
immler@56188
   523
lemma content_unit[intro]: "content(cbox 0 (One::'a::euclidean_space)) = 1"
immler@56188
   524
 proof -
immler@56188
   525
   have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i"
immler@56188
   526
    by auto
immler@56188
   527
  have "0 \<in> cbox 0 (One::'a)"
immler@56188
   528
    unfolding mem_box by auto
immler@56188
   529
  then show ?thesis
immler@56188
   530
     unfolding content_def interval_bounds[OF *] using setprod_1 by auto
immler@56188
   531
 qed
wenzelm@49970
   532
wenzelm@49970
   533
lemma content_pos_le[intro]:
immler@56188
   534
  fixes a::"'a::euclidean_space"
immler@56188
   535
  shows "0 \<le> content (cbox a b)"
immler@56188
   536
proof (cases "cbox a b = {}")
immler@56188
   537
  case False
immler@56188
   538
  then have *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
immler@56188
   539
    unfolding box_ne_empty .
immler@56188
   540
  have "0 \<le> (\<Prod>i\<in>Basis. interval_upperbound (cbox a b) \<bullet> i - interval_lowerbound (cbox a b) \<bullet> i)"
immler@56188
   541
    apply (rule setprod_nonneg)
immler@56188
   542
    unfolding interval_bounds[OF *]
immler@56188
   543
    using *
immler@56188
   544
    apply auto
immler@56188
   545
    done
immler@56188
   546
  also have "\<dots> = content (cbox a b)" using False by (simp add: content_def)
immler@56188
   547
  finally show ?thesis .
immler@56188
   548
qed (simp add: content_def)
wenzelm@49970
   549
wenzelm@49970
   550
lemma content_pos_lt:
immler@56188
   551
  fixes a :: "'a::euclidean_space"
hoelzl@50526
   552
  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
immler@56188
   553
  shows "0 < content (cbox a b)"
immler@54777
   554
  using assms
immler@56188
   555
  by (auto simp: content_def box_eq_empty intro!: setprod_pos)
wenzelm@49970
   556
wenzelm@53408
   557
lemma content_eq_0:
immler@56188
   558
  "content (cbox a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
immler@56188
   559
  by (auto simp: content_def box_eq_empty intro!: setprod_pos bexI)
himmelma@35172
   560
wenzelm@53408
   561
lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
wenzelm@53399
   562
  by auto
himmelma@35172
   563
immler@56188
   564
lemma content_cbox_cases:
immler@56188
   565
  "content (cbox a (b::'a::euclidean_space)) =
hoelzl@50526
   566
    (if \<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i then setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis else 0)"
immler@56188
   567
  by (auto simp: not_le content_eq_0 intro: less_imp_le content_cbox)
immler@56188
   568
immler@56188
   569
lemma content_eq_0_interior: "content (cbox a b) = 0 \<longleftrightarrow> interior(cbox a b) = {}"
immler@56188
   570
  unfolding content_eq_0 interior_cbox box_eq_empty
wenzelm@53408
   571
  by auto
himmelma@35172
   572
wenzelm@53399
   573
lemma content_pos_lt_eq:
immler@56188
   574
  "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
wenzelm@49970
   575
  apply rule
wenzelm@49970
   576
  defer
wenzelm@49970
   577
  apply (rule content_pos_lt, assumption)
wenzelm@49970
   578
proof -
immler@56188
   579
  assume "0 < content (cbox a b)"
immler@56188
   580
  then have "content (cbox a b) \<noteq> 0" by auto
wenzelm@53399
   581
  then show "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
wenzelm@49970
   582
    unfolding content_eq_0 not_ex not_le by fastforce
wenzelm@49970
   583
qed
wenzelm@49970
   584
wenzelm@53399
   585
lemma content_empty [simp]: "content {} = 0"
wenzelm@53399
   586
  unfolding content_def by auto
himmelma@35172
   587
wenzelm@49698
   588
lemma content_subset:
immler@56188
   589
  assumes "cbox a b \<subseteq> cbox c d"
immler@56188
   590
  shows "content (cbox a b) \<le> content (cbox c d)"
immler@56188
   591
proof (cases "cbox a b = {}")
immler@56188
   592
  case True
immler@56188
   593
  then show ?thesis
immler@56188
   594
    using content_pos_le[of c d] by auto
immler@56188
   595
next
immler@56188
   596
  case False
immler@56188
   597
  then have ab_ne: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
immler@56188
   598
    unfolding box_ne_empty by auto
immler@56188
   599
  then have ab_ab: "a\<in>cbox a b" "b\<in>cbox a b"
immler@56188
   600
    unfolding mem_box by auto
immler@56188
   601
  have "cbox c d \<noteq> {}" using assms False by auto
immler@56188
   602
  then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
immler@56188
   603
    using assms unfolding box_ne_empty by auto
immler@56188
   604
  show ?thesis
immler@56188
   605
    unfolding content_def
immler@56188
   606
    unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
immler@56188
   607
    unfolding if_not_P[OF False] if_not_P[OF `cbox c d \<noteq> {}`]
immler@56188
   608
    apply (rule setprod_mono)
immler@56188
   609
    apply rule
immler@56188
   610
  proof
immler@56188
   611
    fix i :: 'a
immler@56188
   612
    assume i: "i \<in> Basis"
immler@56188
   613
    show "0 \<le> b \<bullet> i - a \<bullet> i"
immler@56188
   614
      using ab_ne[THEN bspec, OF i] i by auto
immler@56188
   615
    show "b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
immler@56188
   616
      using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2),of i]
immler@56188
   617
      using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(1),of i]
immler@56188
   618
      using i by auto
immler@56188
   619
  qed
immler@56188
   620
qed
immler@56188
   621
immler@56188
   622
lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
nipkow@44890
   623
  unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
himmelma@35172
   624
wenzelm@49698
   625
himmelma@35172
   626
subsection {* The notion of a gauge --- simply an open set containing the point. *}
himmelma@35172
   627
wenzelm@53408
   628
definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
wenzelm@53399
   629
wenzelm@53399
   630
lemma gaugeI:
wenzelm@53399
   631
  assumes "\<And>x. x \<in> g x"
wenzelm@53399
   632
    and "\<And>x. open (g x)"
wenzelm@53399
   633
  shows "gauge g"
himmelma@35172
   634
  using assms unfolding gauge_def by auto
himmelma@35172
   635
wenzelm@53399
   636
lemma gaugeD[dest]:
wenzelm@53399
   637
  assumes "gauge d"
wenzelm@53399
   638
  shows "x \<in> d x"
wenzelm@53399
   639
    and "open (d x)"
wenzelm@49698
   640
  using assms unfolding gauge_def by auto
himmelma@35172
   641
himmelma@35172
   642
lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
wenzelm@53399
   643
  unfolding gauge_def by auto
wenzelm@53399
   644
wenzelm@53399
   645
lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
wenzelm@53399
   646
  unfolding gauge_def by auto
himmelma@35172
   647
wenzelm@49698
   648
lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)"
wenzelm@49698
   649
  by (rule gauge_ball) auto
himmelma@35172
   650
wenzelm@53408
   651
lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
wenzelm@53399
   652
  unfolding gauge_def by auto
himmelma@35172
   653
wenzelm@49698
   654
lemma gauge_inters:
wenzelm@53399
   655
  assumes "finite s"
wenzelm@53399
   656
    and "\<forall>d\<in>s. gauge (f d)"
wenzelm@53408
   657
  shows "gauge (\<lambda>x. \<Inter> {f d x | d. d \<in> s})"
wenzelm@49698
   658
proof -
wenzelm@53399
   659
  have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
wenzelm@53399
   660
    by auto
wenzelm@49698
   661
  show ?thesis
wenzelm@53399
   662
    unfolding gauge_def unfolding *
wenzelm@49698
   663
    using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
wenzelm@49698
   664
qed
wenzelm@49698
   665
wenzelm@53399
   666
lemma gauge_existence_lemma:
wenzelm@53408
   667
  "(\<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)"
wenzelm@53399
   668
  by (metis zero_less_one)
wenzelm@49698
   669
himmelma@35172
   670
himmelma@35172
   671
subsection {* Divisions. *}
himmelma@35172
   672
wenzelm@53408
   673
definition division_of (infixl "division'_of" 40)
wenzelm@53408
   674
where
wenzelm@53399
   675
  "s division_of i \<longleftrightarrow>
wenzelm@53399
   676
    finite s \<and>
immler@56188
   677
    (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = cbox a b)) \<and>
wenzelm@53399
   678
    (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
wenzelm@53399
   679
    (\<Union>s = i)"
himmelma@35172
   680
wenzelm@49698
   681
lemma division_ofD[dest]:
wenzelm@49698
   682
  assumes "s division_of i"
wenzelm@53408
   683
  shows "finite s"
wenzelm@53408
   684
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   685
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
immler@56188
   686
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
   687
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
wenzelm@53408
   688
    and "\<Union>s = i"
wenzelm@49698
   689
  using assms unfolding division_of_def by auto
himmelma@35172
   690
himmelma@35172
   691
lemma division_ofI:
wenzelm@53408
   692
  assumes "finite s"
wenzelm@53408
   693
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   694
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
immler@56188
   695
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
   696
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53399
   697
    and "\<Union>s = i"
wenzelm@53399
   698
  shows "s division_of i"
wenzelm@53399
   699
  using assms unfolding division_of_def by auto
himmelma@35172
   700
himmelma@35172
   701
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
himmelma@35172
   702
  unfolding division_of_def by auto
himmelma@35172
   703
immler@56188
   704
lemma division_of_self[intro]: "cbox a b \<noteq> {} \<Longrightarrow> {cbox a b} division_of (cbox a b)"
himmelma@35172
   705
  unfolding division_of_def by auto
himmelma@35172
   706
wenzelm@53399
   707
lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
wenzelm@53399
   708
  unfolding division_of_def by auto
himmelma@35172
   709
wenzelm@49698
   710
lemma division_of_sing[simp]:
immler@56188
   711
  "s division_of cbox a (a::'a::euclidean_space) \<longleftrightarrow> s = {cbox a a}"
wenzelm@53399
   712
  (is "?l = ?r")
wenzelm@49698
   713
proof
wenzelm@49698
   714
  assume ?r
wenzelm@53399
   715
  moreover
wenzelm@53399
   716
  {
wenzelm@49698
   717
    assume "s = {{a}}"
wenzelm@53399
   718
    moreover fix k assume "k\<in>s"
immler@56188
   719
    ultimately have"\<exists>x y. k = cbox x y"
wenzelm@50945
   720
      apply (rule_tac x=a in exI)+
immler@56188
   721
      unfolding cbox_sing
wenzelm@50945
   722
      apply auto
wenzelm@50945
   723
      done
wenzelm@49698
   724
  }
wenzelm@53399
   725
  ultimately show ?l
immler@56188
   726
    unfolding division_of_def cbox_sing by auto
wenzelm@49698
   727
next
wenzelm@49698
   728
  assume ?l
immler@56188
   729
  note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
wenzelm@53399
   730
  {
wenzelm@53399
   731
    fix x
wenzelm@53399
   732
    assume x: "x \<in> s" have "x = {a}"
wenzelm@53408
   733
      using *(2)[rule_format,OF x] by auto
wenzelm@53399
   734
  }
wenzelm@53408
   735
  moreover have "s \<noteq> {}"
wenzelm@53408
   736
    using *(4) by auto
wenzelm@53408
   737
  ultimately show ?r
immler@56188
   738
    unfolding cbox_sing by auto
wenzelm@49698
   739
qed
himmelma@35172
   740
himmelma@35172
   741
lemma elementary_empty: obtains p where "p division_of {}"
himmelma@35172
   742
  unfolding division_of_trivial by auto
himmelma@35172
   743
immler@56188
   744
lemma elementary_interval: obtains p where "p division_of (cbox a b)"
wenzelm@49698
   745
  by (metis division_of_trivial division_of_self)
himmelma@35172
   746
himmelma@35172
   747
lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
himmelma@35172
   748
  unfolding division_of_def by auto
himmelma@35172
   749
himmelma@35172
   750
lemma forall_in_division:
immler@56188
   751
  "d division_of i \<Longrightarrow> (\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. cbox a b \<in> d \<longrightarrow> P (cbox a b))"
nipkow@44890
   752
  unfolding division_of_def by fastforce
himmelma@35172
   753
wenzelm@53399
   754
lemma division_of_subset:
wenzelm@53399
   755
  assumes "p division_of (\<Union>p)"
wenzelm@53399
   756
    and "q \<subseteq> p"
wenzelm@53399
   757
  shows "q division_of (\<Union>q)"
wenzelm@53408
   758
proof (rule division_ofI)
wenzelm@53408
   759
  note * = division_ofD[OF assms(1)]
wenzelm@49698
   760
  show "finite q"
wenzelm@49698
   761
    apply (rule finite_subset)
wenzelm@53408
   762
    using *(1) assms(2)
wenzelm@53408
   763
    apply auto
wenzelm@49698
   764
    done
wenzelm@53399
   765
  {
wenzelm@53399
   766
    fix k
wenzelm@49698
   767
    assume "k \<in> q"
wenzelm@53408
   768
    then have kp: "k \<in> p"
wenzelm@53408
   769
      using assms(2) by auto
wenzelm@53408
   770
    show "k \<subseteq> \<Union>q"
wenzelm@53408
   771
      using `k \<in> q` by auto
immler@56188
   772
    show "\<exists>a b. k = cbox a b"
wenzelm@53408
   773
      using *(4)[OF kp] by auto
wenzelm@53408
   774
    show "k \<noteq> {}"
wenzelm@53408
   775
      using *(3)[OF kp] by auto
wenzelm@53399
   776
  }
wenzelm@49698
   777
  fix k1 k2
wenzelm@49698
   778
  assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
wenzelm@53408
   779
  then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
wenzelm@53399
   780
    using assms(2) by auto
wenzelm@53399
   781
  show "interior k1 \<inter> interior k2 = {}"
wenzelm@53408
   782
    using *(5)[OF **] by auto
wenzelm@49698
   783
qed auto
wenzelm@49698
   784
wenzelm@49698
   785
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@49698
   786
  unfolding division_of_def by auto
himmelma@35172
   787
wenzelm@49970
   788
lemma division_of_content_0:
immler@56188
   789
  assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
wenzelm@49970
   790
  shows "\<forall>k\<in>d. content k = 0"
wenzelm@49970
   791
  unfolding forall_in_division[OF assms(2)]
wenzelm@50945
   792
  apply (rule,rule,rule)
wenzelm@50945
   793
  apply (drule division_ofD(2)[OF assms(2)])
wenzelm@50945
   794
  apply (drule content_subset) unfolding assms(1)
wenzelm@49970
   795
proof -
wenzelm@49970
   796
  case goal1
wenzelm@53399
   797
  then show ?case using content_pos_le[of a b] by auto
wenzelm@49970
   798
qed
wenzelm@49970
   799
wenzelm@49970
   800
lemma division_inter:
immler@56188
   801
  fixes s1 s2 :: "'a::euclidean_space set"
wenzelm@53408
   802
  assumes "p1 division_of s1"
wenzelm@53408
   803
    and "p2 division_of s2"
wenzelm@49970
   804
  shows "{k1 \<inter> k2 | k1 k2 .k1 \<in> p1 \<and> k2 \<in> p2 \<and> k1 \<inter> k2 \<noteq> {}} division_of (s1 \<inter> s2)"
wenzelm@49970
   805
  (is "?A' division_of _")
wenzelm@49970
   806
proof -
wenzelm@49970
   807
  let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
wenzelm@53408
   808
  have *: "?A' = ?A" by auto
wenzelm@53399
   809
  show ?thesis
wenzelm@53399
   810
    unfolding *
wenzelm@49970
   811
  proof (rule division_ofI)
wenzelm@53399
   812
    have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
wenzelm@53399
   813
      by auto
wenzelm@53399
   814
    moreover have "finite (p1 \<times> p2)"
wenzelm@53399
   815
      using assms unfolding division_of_def by auto
wenzelm@49970
   816
    ultimately show "finite ?A" by auto
wenzelm@53399
   817
    have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
wenzelm@53399
   818
      by auto
wenzelm@49970
   819
    show "\<Union>?A = s1 \<inter> s2"
wenzelm@49970
   820
      apply (rule set_eqI)
wenzelm@49970
   821
      unfolding * and Union_image_eq UN_iff
wenzelm@49970
   822
      using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
wenzelm@49970
   823
      apply auto
wenzelm@49970
   824
      done
wenzelm@53399
   825
    {
wenzelm@53399
   826
      fix k
wenzelm@53399
   827
      assume "k \<in> ?A"
wenzelm@53408
   828
      then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
wenzelm@53399
   829
        by auto
wenzelm@53408
   830
      then show "k \<noteq> {}"
wenzelm@53408
   831
        by auto
wenzelm@49970
   832
      show "k \<subseteq> s1 \<inter> s2"
wenzelm@49970
   833
        using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
wenzelm@49970
   834
        unfolding k by auto
immler@56188
   835
      obtain a1 b1 where k1: "k1 = cbox a1 b1"
wenzelm@53408
   836
        using division_ofD(4)[OF assms(1) k(2)] by blast
immler@56188
   837
      obtain a2 b2 where k2: "k2 = cbox a2 b2"
wenzelm@53408
   838
        using division_ofD(4)[OF assms(2) k(3)] by blast
immler@56188
   839
      show "\<exists>a b. k = cbox a b"
wenzelm@53408
   840
        unfolding k k1 k2 unfolding inter_interval by auto
wenzelm@53408
   841
    }
wenzelm@49970
   842
    fix k1 k2
wenzelm@53408
   843
    assume "k1 \<in> ?A"
wenzelm@53408
   844
    then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
wenzelm@53408
   845
      by auto
wenzelm@53408
   846
    assume "k2 \<in> ?A"
wenzelm@53408
   847
    then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
wenzelm@53408
   848
      by auto
wenzelm@49970
   849
    assume "k1 \<noteq> k2"
wenzelm@53399
   850
    then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
wenzelm@53399
   851
      unfolding k1 k2 by auto
wenzelm@53408
   852
    have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
wenzelm@53408
   853
      interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
wenzelm@53408
   854
      interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
wenzelm@53408
   855
      interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
wenzelm@49970
   856
    show "interior k1 \<inter> interior k2 = {}"
wenzelm@49970
   857
      unfolding k1 k2
wenzelm@49970
   858
      apply (rule *)
wenzelm@49970
   859
      defer
wenzelm@49970
   860
      apply (rule_tac[1-4] interior_mono)
wenzelm@49970
   861
      using division_ofD(5)[OF assms(1) k1(2) k2(2)]
wenzelm@49970
   862
      using division_ofD(5)[OF assms(2) k1(3) k2(3)]
wenzelm@53408
   863
      using th
wenzelm@53408
   864
      apply auto
wenzelm@53399
   865
      done
wenzelm@49970
   866
  qed
wenzelm@49970
   867
qed
wenzelm@49970
   868
wenzelm@49970
   869
lemma division_inter_1:
wenzelm@53408
   870
  assumes "d division_of i"
immler@56188
   871
    and "cbox a (b::'a::euclidean_space) \<subseteq> i"
immler@56188
   872
  shows "{cbox a b \<inter> k | k. k \<in> d \<and> cbox a b \<inter> k \<noteq> {}} division_of (cbox a b)"
immler@56188
   873
proof (cases "cbox a b = {}")
wenzelm@49970
   874
  case True
wenzelm@53399
   875
  show ?thesis
wenzelm@53399
   876
    unfolding True and division_of_trivial by auto
wenzelm@49970
   877
next
wenzelm@49970
   878
  case False
immler@56188
   879
  have *: "cbox a b \<inter> i = cbox a b" using assms(2) by auto
wenzelm@53399
   880
  show ?thesis
wenzelm@53399
   881
    using division_inter[OF division_of_self[OF False] assms(1)]
wenzelm@53399
   882
    unfolding * by auto
wenzelm@49970
   883
qed
wenzelm@49970
   884
wenzelm@49970
   885
lemma elementary_inter:
immler@56188
   886
  fixes s t :: "'a::euclidean_space set"
wenzelm@53408
   887
  assumes "p1 division_of s"
wenzelm@53408
   888
    and "p2 division_of t"
himmelma@35172
   889
  shows "\<exists>p. p division_of (s \<inter> t)"
wenzelm@50945
   890
  apply rule
wenzelm@50945
   891
  apply (rule division_inter[OF assms])
wenzelm@50945
   892
  done
wenzelm@49970
   893
wenzelm@49970
   894
lemma elementary_inters:
wenzelm@53408
   895
  assumes "finite f"
wenzelm@53408
   896
    and "f \<noteq> {}"
immler@56188
   897
    and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)"
wenzelm@49970
   898
  shows "\<exists>p. p division_of (\<Inter> f)"
wenzelm@49970
   899
  using assms
wenzelm@49970
   900
proof (induct f rule: finite_induct)
wenzelm@49970
   901
  case (insert x f)
wenzelm@49970
   902
  show ?case
wenzelm@49970
   903
  proof (cases "f = {}")
wenzelm@49970
   904
    case True
wenzelm@53399
   905
    then show ?thesis
wenzelm@53399
   906
      unfolding True using insert by auto
wenzelm@49970
   907
  next
wenzelm@49970
   908
    case False
wenzelm@53408
   909
    obtain p where "p division_of \<Inter>f"
wenzelm@53408
   910
      using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
wenzelm@53408
   911
    moreover obtain px where "px division_of x"
wenzelm@53408
   912
      using insert(5)[rule_format,OF insertI1] ..
wenzelm@49970
   913
    ultimately show ?thesis
wenzelm@53408
   914
      apply -
wenzelm@49970
   915
      unfolding Inter_insert
wenzelm@53408
   916
      apply (rule elementary_inter)
wenzelm@49970
   917
      apply assumption
wenzelm@49970
   918
      apply assumption
wenzelm@49970
   919
      done
wenzelm@49970
   920
  qed
wenzelm@49970
   921
qed auto
himmelma@35172
   922
himmelma@35172
   923
lemma division_disjoint_union:
wenzelm@53408
   924
  assumes "p1 division_of s1"
wenzelm@53408
   925
    and "p2 division_of s2"
wenzelm@53408
   926
    and "interior s1 \<inter> interior s2 = {}"
wenzelm@50945
   927
  shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
wenzelm@50945
   928
proof (rule division_ofI)
wenzelm@53408
   929
  note d1 = division_ofD[OF assms(1)]
wenzelm@53408
   930
  note d2 = division_ofD[OF assms(2)]
wenzelm@53408
   931
  show "finite (p1 \<union> p2)"
wenzelm@53408
   932
    using d1(1) d2(1) by auto
wenzelm@53408
   933
  show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
wenzelm@53408
   934
    using d1(6) d2(6) by auto
wenzelm@50945
   935
  {
wenzelm@50945
   936
    fix k1 k2
wenzelm@50945
   937
    assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
wenzelm@50945
   938
    moreover
wenzelm@50945
   939
    let ?g="interior k1 \<inter> interior k2 = {}"
wenzelm@50945
   940
    {
wenzelm@50945
   941
      assume as: "k1\<in>p1" "k2\<in>p2"
wenzelm@50945
   942
      have ?g
wenzelm@50945
   943
        using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
wenzelm@50945
   944
        using assms(3) by blast
wenzelm@50945
   945
    }
wenzelm@50945
   946
    moreover
wenzelm@50945
   947
    {
wenzelm@50945
   948
      assume as: "k1\<in>p2" "k2\<in>p1"
wenzelm@50945
   949
      have ?g
wenzelm@50945
   950
        using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
wenzelm@50945
   951
        using assms(3) by blast
wenzelm@50945
   952
    }
wenzelm@53399
   953
    ultimately show ?g
wenzelm@53399
   954
      using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
wenzelm@50945
   955
  }
wenzelm@50945
   956
  fix k
wenzelm@50945
   957
  assume k: "k \<in> p1 \<union> p2"
wenzelm@53408
   958
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
   959
    using k d1(2) d2(2) by auto
wenzelm@53408
   960
  show "k \<noteq> {}"
wenzelm@53408
   961
    using k d1(3) d2(3) by auto
immler@56188
   962
  show "\<exists>a b. k = cbox a b"
wenzelm@53408
   963
    using k d1(4) d2(4) by auto
wenzelm@50945
   964
qed
himmelma@35172
   965
himmelma@35172
   966
lemma partial_division_extend_1:
immler@56188
   967
  fixes a b c d :: "'a::euclidean_space"
immler@56188
   968
  assumes incl: "cbox c d \<subseteq> cbox a b"
immler@56188
   969
    and nonempty: "cbox c d \<noteq> {}"
immler@56188
   970
  obtains p where "p division_of (cbox a b)" "cbox c d \<in> p"
hoelzl@50526
   971
proof
wenzelm@53408
   972
  let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
immler@56188
   973
    cbox (\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)"
wenzelm@53015
   974
  def p \<equiv> "?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
hoelzl@50526
   975
immler@56188
   976
  show "cbox c d \<in> p"
hoelzl@50526
   977
    unfolding p_def
immler@56188
   978
    by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
wenzelm@50945
   979
  {
wenzelm@50945
   980
    fix i :: 'a
wenzelm@50945
   981
    assume "i \<in> Basis"
hoelzl@50526
   982
    with incl nonempty have "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i"
immler@56188
   983
      unfolding box_eq_empty subset_box by (auto simp: not_le)
wenzelm@50945
   984
  }
hoelzl@50526
   985
  note ord = this
hoelzl@50526
   986
immler@56188
   987
  show "p division_of (cbox a b)"
hoelzl@50526
   988
  proof (rule division_ofI)
wenzelm@53399
   989
    show "finite p"
wenzelm@53399
   990
      unfolding p_def by (auto intro!: finite_PiE)
wenzelm@50945
   991
    {
wenzelm@50945
   992
      fix k
wenzelm@50945
   993
      assume "k \<in> p"
wenzelm@53015
   994
      then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
hoelzl@50526
   995
        by (auto simp: p_def)
immler@56188
   996
      then show "\<exists>a b. k = cbox a b"
wenzelm@53408
   997
        by auto
immler@56188
   998
      have "k \<subseteq> cbox a b \<and> k \<noteq> {}"
immler@56188
   999
      proof (simp add: k box_eq_empty subset_box not_less, safe)
wenzelm@53374
  1000
        fix i :: 'a
wenzelm@53374
  1001
        assume i: "i \<in> Basis"
wenzelm@50945
  1002
        with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
hoelzl@50526
  1003
          by (auto simp: PiE_iff)
wenzelm@53374
  1004
        with i ord[of i]
wenzelm@50945
  1005
        show "a \<bullet> i \<le> fst (f i) \<bullet> i" "snd (f i) \<bullet> i \<le> b \<bullet> i" "fst (f i) \<bullet> i \<le> snd (f i) \<bullet> i"
immler@54776
  1006
          by auto
hoelzl@50526
  1007
      qed
immler@56188
  1008
      then show "k \<noteq> {}" "k \<subseteq> cbox a b"
wenzelm@53408
  1009
        by auto
wenzelm@50945
  1010
      {
wenzelm@53408
  1011
        fix l
wenzelm@53408
  1012
        assume "l \<in> p"
wenzelm@53015
  1013
        then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
wenzelm@50945
  1014
          by (auto simp: p_def)
wenzelm@50945
  1015
        assume "l \<noteq> k"
wenzelm@50945
  1016
        have "\<exists>i\<in>Basis. f i \<noteq> g i"
wenzelm@50945
  1017
        proof (rule ccontr)
wenzelm@53408
  1018
          assume "\<not> ?thesis"
wenzelm@50945
  1019
          with f g have "f = g"
wenzelm@50945
  1020
            by (auto simp: PiE_iff extensional_def intro!: ext)
wenzelm@50945
  1021
          with `l \<noteq> k` show False
wenzelm@50945
  1022
            by (simp add: l k)
wenzelm@50945
  1023
        qed
wenzelm@53408
  1024
        then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
wenzelm@53408
  1025
        then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
wenzelm@50945
  1026
            "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
wenzelm@50945
  1027
          using f g by (auto simp: PiE_iff)
wenzelm@53408
  1028
        with * ord[of i] show "interior l \<inter> interior k = {}"
immler@56188
  1029
          by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i])
wenzelm@50945
  1030
      }
immler@56188
  1031
      note `k \<subseteq> cbox a b`
wenzelm@50945
  1032
    }
hoelzl@50526
  1033
    moreover
wenzelm@50945
  1034
    {
immler@56188
  1035
      fix x assume x: "x \<in> cbox a b"
hoelzl@50526
  1036
      have "\<forall>i\<in>Basis. \<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
hoelzl@50526
  1037
      proof
wenzelm@53408
  1038
        fix i :: 'a
wenzelm@53408
  1039
        assume "i \<in> Basis"
wenzelm@53399
  1040
        with x ord[of i]
hoelzl@50526
  1041
        have "(a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> c \<bullet> i) \<or> (c \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> d \<bullet> i) \<or>
hoelzl@50526
  1042
            (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
immler@56188
  1043
          by (auto simp: cbox_def)
hoelzl@50526
  1044
        then show "\<exists>l. x \<bullet> i \<in> {fst l \<bullet> i .. snd l \<bullet> i} \<and> l \<in> {(a, c), (c, d), (d, b)}"
hoelzl@50526
  1045
          by auto
hoelzl@50526
  1046
      qed
wenzelm@53408
  1047
      then obtain f where
wenzelm@53408
  1048
        f: "\<forall>i\<in>Basis. x \<bullet> i \<in> {fst (f i) \<bullet> i..snd (f i) \<bullet> i} \<and> f i \<in> {(a, c), (c, d), (d, b)}"
wenzelm@53408
  1049
        unfolding bchoice_iff ..
wenzelm@53374
  1050
      moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
hoelzl@50526
  1051
        by auto
hoelzl@50526
  1052
      moreover from f have "x \<in> ?B (restrict f Basis)"
immler@56188
  1053
        by (auto simp: mem_box)
hoelzl@50526
  1054
      ultimately have "\<exists>k\<in>p. x \<in> k"
wenzelm@53408
  1055
        unfolding p_def by blast
wenzelm@53408
  1056
    }
immler@56188
  1057
    ultimately show "\<Union>p = cbox a b"
hoelzl@50526
  1058
      by auto
hoelzl@50526
  1059
  qed
hoelzl@50526
  1060
qed
himmelma@35172
  1061
wenzelm@50945
  1062
lemma partial_division_extend_interval:
immler@56188
  1063
  assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
immler@56188
  1064
  obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
wenzelm@50945
  1065
proof (cases "p = {}")
wenzelm@50945
  1066
  case True
immler@56188
  1067
  obtain q where "q division_of (cbox a b)"
wenzelm@53408
  1068
    by (rule elementary_interval)
wenzelm@53399
  1069
  then show ?thesis
wenzelm@50945
  1070
    apply -
wenzelm@50945
  1071
    apply (rule that[of q])
wenzelm@50945
  1072
    unfolding True
wenzelm@50945
  1073
    apply auto
wenzelm@50945
  1074
    done
wenzelm@50945
  1075
next
wenzelm@50945
  1076
  case False
wenzelm@50945
  1077
  note p = division_ofD[OF assms(1)]
immler@56188
  1078
  have *: "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q"
wenzelm@50945
  1079
  proof
wenzelm@50945
  1080
    case goal1
immler@56188
  1081
    obtain c d where k: "k = cbox c d"
wenzelm@53408
  1082
      using p(4)[OF goal1] by blast
immler@56188
  1083
    have *: "cbox c d \<subseteq> cbox a b" "cbox c d \<noteq> {}"
immler@54775
  1084
      using p(2,3)[OF goal1, unfolded k] using assms(2)
immler@54776
  1085
      by (blast intro: order.trans)+
immler@56188
  1086
    obtain q where "q division_of cbox a b" "cbox c d \<in> q"
wenzelm@53408
  1087
      by (rule partial_division_extend_1[OF *])
wenzelm@53408
  1088
    then show ?case
wenzelm@53408
  1089
      unfolding k by auto
wenzelm@50945
  1090
  qed
immler@56188
  1091
  obtain q where q: "\<And>x. x \<in> p \<Longrightarrow> q x division_of cbox a b" "\<And>x. x \<in> p \<Longrightarrow> x \<in> q x"
wenzelm@53408
  1092
    using bchoice[OF *] by blast
wenzelm@53408
  1093
  have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
wenzelm@53408
  1094
    apply rule
wenzelm@53408
  1095
    apply (rule_tac p="q x" in division_of_subset)
wenzelm@50945
  1096
  proof -
wenzelm@50945
  1097
    fix x
wenzelm@53408
  1098
    assume x: "x \<in> p"
wenzelm@50945
  1099
    show "q x division_of \<Union>q x"
wenzelm@50945
  1100
      apply -
wenzelm@50945
  1101
      apply (rule division_ofI)
wenzelm@50945
  1102
      using division_ofD[OF q(1)[OF x]]
wenzelm@50945
  1103
      apply auto
wenzelm@50945
  1104
      done
wenzelm@53408
  1105
    show "q x - {x} \<subseteq> q x"
wenzelm@53408
  1106
      by auto
wenzelm@50945
  1107
  qed
wenzelm@53399
  1108
  then have "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)"
wenzelm@50945
  1109
    apply -
wenzelm@50945
  1110
    apply (rule elementary_inters)
wenzelm@50945
  1111
    apply (rule finite_imageI[OF p(1)])
wenzelm@50945
  1112
    unfolding image_is_empty
wenzelm@50945
  1113
    apply (rule False)
wenzelm@50945
  1114
    apply auto
wenzelm@50945
  1115
    done
wenzelm@53408
  1116
  then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
wenzelm@50945
  1117
  show ?thesis
wenzelm@50945
  1118
    apply (rule that[of "d \<union> p"])
wenzelm@50945
  1119
  proof -
wenzelm@53408
  1120
    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
immler@56188
  1121
    have *: "cbox a b = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
wenzelm@50945
  1122
      apply (rule *[OF False])
wenzelm@50945
  1123
    proof
wenzelm@50945
  1124
      fix i
wenzelm@53408
  1125
      assume i: "i \<in> p"
immler@56188
  1126
      show "\<Union>(q i - {i}) \<union> i = cbox a b"
wenzelm@50945
  1127
        using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
wenzelm@50945
  1128
    qed
immler@56188
  1129
    show "d \<union> p division_of (cbox a b)"
wenzelm@50945
  1130
      unfolding *
wenzelm@50945
  1131
      apply (rule division_disjoint_union[OF d assms(1)])
wenzelm@50945
  1132
      apply (rule inter_interior_unions_intervals)
wenzelm@50945
  1133
      apply (rule p open_interior ballI)+
wenzelm@53408
  1134
      apply assumption
wenzelm@53408
  1135
    proof
wenzelm@50945
  1136
      fix k
wenzelm@53408
  1137
      assume k: "k \<in> p"
wenzelm@53408
  1138
      have *: "\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}"
wenzelm@53408
  1139
        by auto
haftmann@52141
  1140
      show "interior (\<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)) \<inter> interior k = {}"
wenzelm@50945
  1141
        apply (rule *[of _ "interior (\<Union>(q k - {k}))"])
wenzelm@50945
  1142
        defer
wenzelm@50945
  1143
        apply (subst Int_commute)
wenzelm@50945
  1144
        apply (rule inter_interior_unions_intervals)
wenzelm@50945
  1145
      proof -
wenzelm@50945
  1146
        note qk=division_ofD[OF q(1)[OF k]]
immler@56188
  1147
        show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b"
wenzelm@53408
  1148
          using qk by auto
wenzelm@50945
  1149
        show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
wenzelm@50945
  1150
          using qk(5) using q(2)[OF k] by auto
wenzelm@53408
  1151
        have *: "\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x"
wenzelm@53408
  1152
          by auto
haftmann@52141
  1153
        show "interior (\<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)) \<subseteq> interior (\<Union>(q k - {k}))"
wenzelm@50945
  1154
          apply (rule interior_mono *)+
wenzelm@53408
  1155
          using k
wenzelm@53408
  1156
          apply auto
wenzelm@53408
  1157
          done
wenzelm@50945
  1158
      qed
wenzelm@50945
  1159
    qed
wenzelm@50945
  1160
  qed auto
wenzelm@50945
  1161
qed
himmelma@35172
  1162
wenzelm@53399
  1163
lemma elementary_bounded[dest]:
immler@56188
  1164
  fixes s :: "'a::euclidean_space set"
wenzelm@53408
  1165
  shows "p division_of s \<Longrightarrow> bounded s"
immler@56189
  1166
  unfolding division_of_def by (metis bounded_Union bounded_cbox)
wenzelm@53399
  1167
immler@56188
  1168
lemma elementary_subset_cbox:
immler@56188
  1169
  "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> cbox a (b::'a::euclidean_space)"
immler@56188
  1170
  by (meson elementary_bounded bounded_subset_cbox)
wenzelm@50945
  1171
wenzelm@50945
  1172
lemma division_union_intervals_exists:
immler@56188
  1173
  fixes a b :: "'a::euclidean_space"
immler@56188
  1174
  assumes "cbox a b \<noteq> {}"
immler@56188
  1175
  obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
immler@56188
  1176
proof (cases "cbox c d = {}")
wenzelm@50945
  1177
  case True
wenzelm@50945
  1178
  show ?thesis
wenzelm@50945
  1179
    apply (rule that[of "{}"])
wenzelm@50945
  1180
    unfolding True
wenzelm@50945
  1181
    using assms
wenzelm@50945
  1182
    apply auto
wenzelm@50945
  1183
    done
wenzelm@50945
  1184
next
wenzelm@50945
  1185
  case False
wenzelm@50945
  1186
  show ?thesis
immler@56188
  1187
  proof (cases "cbox a b \<inter> cbox c d = {}")
wenzelm@50945
  1188
    case True
wenzelm@53408
  1189
    have *: "\<And>a b. {a, b} = {a} \<union> {b}" by auto
wenzelm@50945
  1190
    show ?thesis
immler@56188
  1191
      apply (rule that[of "{cbox c d}"])
wenzelm@50945
  1192
      unfolding *
wenzelm@50945
  1193
      apply (rule division_disjoint_union)
immler@56188
  1194
      using `cbox c d \<noteq> {}` True assms
wenzelm@50945
  1195
      using interior_subset
wenzelm@50945
  1196
      apply auto
wenzelm@50945
  1197
      done
wenzelm@50945
  1198
  next
wenzelm@50945
  1199
    case False
immler@56188
  1200
    obtain u v where uv: "cbox a b \<inter> cbox c d = cbox u v"
wenzelm@50945
  1201
      unfolding inter_interval by auto
immler@56188
  1202
    have *: "cbox u v \<subseteq> cbox c d" using uv by auto
immler@56188
  1203
    obtain p where "p division_of cbox c d" "cbox u v \<in> p"
wenzelm@53408
  1204
      by (rule partial_division_extend_1[OF * False[unfolded uv]])
wenzelm@53408
  1205
    note p = this division_ofD[OF this(1)]
immler@56188
  1206
    have *: "cbox a b \<union> cbox c d = cbox a b \<union> \<Union>(p - {cbox u v})" "\<And>x s. insert x s = {x} \<union> s"
wenzelm@53399
  1207
      using p(8) unfolding uv[symmetric] by auto
wenzelm@50945
  1208
    show ?thesis
immler@56188
  1209
      apply (rule that[of "p - {cbox u v}"])
wenzelm@50945
  1210
      unfolding *(1)
wenzelm@50945
  1211
      apply (subst *(2))
wenzelm@50945
  1212
      apply (rule division_disjoint_union)
wenzelm@50945
  1213
      apply (rule, rule assms)
wenzelm@50945
  1214
      apply (rule division_of_subset[of p])
wenzelm@50945
  1215
      apply (rule division_of_union_self[OF p(1)])
wenzelm@50945
  1216
      defer
wenzelm@53399
  1217
      unfolding interior_inter[symmetric]
wenzelm@50945
  1218
    proof -
wenzelm@50945
  1219
      have *: "\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
immler@56188
  1220
      have "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = interior(cbox u v \<inter> \<Union>(p - {cbox u v}))"
wenzelm@50945
  1221
        apply (rule arg_cong[of _ _ interior])
wenzelm@50945
  1222
        apply (rule *[OF _ uv])
wenzelm@50945
  1223
        using p(8)
wenzelm@50945
  1224
        apply auto
wenzelm@50945
  1225
        done
wenzelm@50945
  1226
      also have "\<dots> = {}"
wenzelm@50945
  1227
        unfolding interior_inter
wenzelm@50945
  1228
        apply (rule inter_interior_unions_intervals)
wenzelm@50945
  1229
        using p(6) p(7)[OF p(2)] p(3)
wenzelm@50945
  1230
        apply auto
wenzelm@50945
  1231
        done
immler@56188
  1232
      finally show "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = {}" .
wenzelm@50945
  1233
    qed auto
wenzelm@50945
  1234
  qed
wenzelm@50945
  1235
qed
himmelma@35172
  1236
wenzelm@53399
  1237
lemma division_of_unions:
wenzelm@53399
  1238
  assumes "finite f"
wenzelm@53408
  1239
    and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@53399
  1240
    and "\<And>k1 k2. k1 \<in> \<Union>f \<Longrightarrow> k2 \<in> \<Union>f \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53399
  1241
  shows "\<Union>f division_of \<Union>\<Union>f"
wenzelm@53399
  1242
  apply (rule division_ofI)
wenzelm@53399
  1243
  prefer 5
wenzelm@53399
  1244
  apply (rule assms(3)|assumption)+
wenzelm@53399
  1245
  apply (rule finite_Union assms(1))+
wenzelm@53399
  1246
  prefer 3
wenzelm@53399
  1247
  apply (erule UnionE)
wenzelm@53399
  1248
  apply (rule_tac s=X in division_ofD(3)[OF assms(2)])
wenzelm@53399
  1249
  using division_ofD[OF assms(2)]
wenzelm@53399
  1250
  apply auto
wenzelm@53399
  1251
  done
wenzelm@53399
  1252
wenzelm@53399
  1253
lemma elementary_union_interval:
immler@56188
  1254
  fixes a b :: "'a::euclidean_space"
wenzelm@53399
  1255
  assumes "p division_of \<Union>p"
immler@56188
  1256
  obtains q where "q division_of (cbox a b \<union> \<Union>p)"
wenzelm@53399
  1257
proof -
wenzelm@53399
  1258
  note assm = division_ofD[OF assms]
wenzelm@53408
  1259
  have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
wenzelm@53399
  1260
    by auto
wenzelm@53399
  1261
  have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
wenzelm@53399
  1262
    by auto
wenzelm@53399
  1263
  {
wenzelm@53399
  1264
    presume "p = {} \<Longrightarrow> thesis"
immler@56188
  1265
      "cbox a b = {} \<Longrightarrow> thesis"
immler@56188
  1266
      "cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis"
immler@56188
  1267
      "p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis"
wenzelm@53399
  1268
    then show thesis by auto
wenzelm@53399
  1269
  next
wenzelm@53399
  1270
    assume as: "p = {}"
immler@56188
  1271
    obtain p where "p division_of (cbox a b)"
wenzelm@53408
  1272
      by (rule elementary_interval)
wenzelm@53399
  1273
    then show thesis
wenzelm@53408
  1274
      apply -
wenzelm@53408
  1275
      apply (rule that[of p])
wenzelm@53399
  1276
      unfolding as
wenzelm@53399
  1277
      apply auto
wenzelm@53399
  1278
      done
wenzelm@53399
  1279
  next
immler@56188
  1280
    assume as: "cbox a b = {}"
wenzelm@53399
  1281
    show thesis
wenzelm@53399
  1282
      apply (rule that)
wenzelm@53399
  1283
      unfolding as
wenzelm@53399
  1284
      using assms
wenzelm@53399
  1285
      apply auto
wenzelm@53399
  1286
      done
wenzelm@53399
  1287
  next
immler@56188
  1288
    assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}"
wenzelm@53399
  1289
    show thesis
immler@56188
  1290
      apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
wenzelm@53399
  1291
      unfolding finite_insert
wenzelm@53399
  1292
      apply (rule assm(1)) unfolding Union_insert
wenzelm@53399
  1293
      using assm(2-4) as
wenzelm@53399
  1294
      apply -
immler@54775
  1295
      apply (fast dest: assm(5))+
wenzelm@53399
  1296
      done
wenzelm@53399
  1297
  next
immler@56188
  1298
    assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}"
immler@56188
  1299
    have "\<forall>k\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
wenzelm@53399
  1300
    proof
wenzelm@53399
  1301
      case goal1
immler@56188
  1302
      from assm(4)[OF this] obtain c d where "k = cbox c d" by blast
wenzelm@53399
  1303
      then show ?case
wenzelm@53399
  1304
        apply -
wenzelm@53408
  1305
        apply (rule division_union_intervals_exists[OF as(3), of c d])
wenzelm@53399
  1306
        apply auto
wenzelm@53399
  1307
        done
wenzelm@53399
  1308
    qed
immler@56188
  1309
    from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert (cbox a b) (q x) division_of (cbox a b) \<union> x" ..
wenzelm@53408
  1310
    note q = division_ofD[OF this[rule_format]]
immler@56188
  1311
    let ?D = "\<Union>{insert (cbox a b) (q k) | k. k \<in> p}"
wenzelm@53399
  1312
    show thesis
wenzelm@53399
  1313
      apply (rule that[of "?D"])
wenzelm@53408
  1314
      apply (rule division_ofI)
wenzelm@53408
  1315
    proof -
immler@56188
  1316
      have *: "{insert (cbox a b) (q k) |k. k \<in> p} = (\<lambda>k. insert (cbox a b) (q k)) ` p"
wenzelm@53399
  1317
        by auto
wenzelm@53399
  1318
      show "finite ?D"
wenzelm@53399
  1319
        apply (rule finite_Union)
wenzelm@53399
  1320
        unfolding *
wenzelm@53399
  1321
        apply (rule finite_imageI)
wenzelm@53399
  1322
        using assm(1) q(1)
wenzelm@53399
  1323
        apply auto
wenzelm@53399
  1324
        done
immler@56188
  1325
      show "\<Union>?D = cbox a b \<union> \<Union>p"
wenzelm@53399
  1326
        unfolding * lem1
immler@56188
  1327
        unfolding lem2[OF as(1), of "cbox a b", symmetric]
wenzelm@53399
  1328
        using q(6)
wenzelm@53399
  1329
        by auto
wenzelm@53399
  1330
      fix k
wenzelm@53408
  1331
      assume k: "k \<in> ?D"
immler@56188
  1332
      then show "k \<subseteq> cbox a b \<union> \<Union>p"
wenzelm@53408
  1333
        using q(2) by auto
wenzelm@53399
  1334
      show "k \<noteq> {}"
wenzelm@53408
  1335
        using q(3) k by auto
immler@56188
  1336
      show "\<exists>a b. k = cbox a b"
wenzelm@53408
  1337
        using q(4) k by auto
wenzelm@53399
  1338
      fix k'
wenzelm@53408
  1339
      assume k': "k' \<in> ?D" "k \<noteq> k'"
immler@56188
  1340
      obtain x where x: "k \<in> insert (cbox a b) (q x)" "x\<in>p"
wenzelm@53408
  1341
        using k by auto
immler@56188
  1342
      obtain x' where x': "k'\<in>insert (cbox a b) (q x')" "x'\<in>p"
wenzelm@53399
  1343
        using k' by auto
wenzelm@53399
  1344
      show "interior k \<inter> interior k' = {}"
wenzelm@53399
  1345
      proof (cases "x = x'")
wenzelm@53399
  1346
        case True
wenzelm@53399
  1347
        show ?thesis
wenzelm@53399
  1348
          apply(rule q(5))
wenzelm@53399
  1349
          using x x' k'
wenzelm@53399
  1350
          unfolding True
wenzelm@53399
  1351
          apply auto
wenzelm@53399
  1352
          done
wenzelm@53399
  1353
      next
wenzelm@53399
  1354
        case False
wenzelm@53399
  1355
        {
immler@56188
  1356
          presume "k = cbox a b \<Longrightarrow> ?thesis"
immler@56188
  1357
            and "k' = cbox a b \<Longrightarrow> ?thesis"
immler@56188
  1358
            and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis"
wenzelm@53399
  1359
          then show ?thesis by auto
wenzelm@53399
  1360
        next
immler@56188
  1361
          assume as': "k  = cbox a b"
wenzelm@53399
  1362
          show ?thesis
wenzelm@53408
  1363
            apply (rule q(5))
wenzelm@53408
  1364
            using x' k'(2)
wenzelm@53408
  1365
            unfolding as'
wenzelm@53408
  1366
            apply auto
wenzelm@53408
  1367
            done
wenzelm@53399
  1368
        next
immler@56188
  1369
          assume as': "k' = cbox a b"
wenzelm@53399
  1370
          show ?thesis
wenzelm@53399
  1371
            apply (rule q(5))
wenzelm@53399
  1372
            using x  k'(2)
wenzelm@53399
  1373
            unfolding as'
wenzelm@53399
  1374
            apply auto
wenzelm@53399
  1375
            done
wenzelm@53399
  1376
        }
immler@56188
  1377
        assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b"
immler@56188
  1378
        obtain c d where k: "k = cbox c d"
wenzelm@53408
  1379
          using q(4)[OF x(2,1)] by blast
immler@56188
  1380
        have "interior k \<inter> interior (cbox a b) = {}"
wenzelm@53408
  1381
          apply (rule q(5))
wenzelm@53408
  1382
          using x k'(2)
wenzelm@53399
  1383
          using as'
wenzelm@53399
  1384
          apply auto
wenzelm@53399
  1385
          done
wenzelm@53399
  1386
        then have "interior k \<subseteq> interior x"
wenzelm@53399
  1387
          apply -
wenzelm@53408
  1388
          apply (rule interior_subset_union_intervals[OF k _ as(2) q(2)[OF x(2,1)]])
wenzelm@53399
  1389
          apply auto
wenzelm@53399
  1390
          done
wenzelm@53399
  1391
        moreover
immler@56188
  1392
        obtain c d where c_d: "k' = cbox c d"
wenzelm@53408
  1393
          using q(4)[OF x'(2,1)] by blast
immler@56188
  1394
        have "interior k' \<inter> interior (cbox a b) = {}"
wenzelm@53399
  1395
          apply (rule q(5))
wenzelm@53399
  1396
          using x' k'(2)
wenzelm@53399
  1397
          using as'
wenzelm@53399
  1398
          apply auto
wenzelm@53399
  1399
          done
wenzelm@53399
  1400
        then have "interior k' \<subseteq> interior x'"
wenzelm@53399
  1401
          apply -
wenzelm@53399
  1402
          apply (rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]])
wenzelm@53399
  1403
          apply auto
wenzelm@53399
  1404
          done
wenzelm@53399
  1405
        ultimately show ?thesis
wenzelm@53399
  1406
          using assm(5)[OF x(2) x'(2) False] by auto
wenzelm@53399
  1407
      qed
wenzelm@53399
  1408
    qed
wenzelm@53399
  1409
  }
wenzelm@53399
  1410
qed
himmelma@35172
  1411
himmelma@35172
  1412
lemma elementary_unions_intervals:
wenzelm@53399
  1413
  assumes fin: "finite f"
immler@56188
  1414
    and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a (b::'a::euclidean_space)"
wenzelm@53399
  1415
  obtains p where "p division_of (\<Union>f)"
wenzelm@53399
  1416
proof -
wenzelm@53399
  1417
  have "\<exists>p. p division_of (\<Union>f)"
wenzelm@53399
  1418
  proof (induct_tac f rule:finite_subset_induct)
himmelma@35172
  1419
    show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
wenzelm@53399
  1420
  next
wenzelm@53399
  1421
    fix x F
wenzelm@53399
  1422
    assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
wenzelm@53408
  1423
    from this(3) obtain p where p: "p division_of \<Union>F" ..
immler@56188
  1424
    from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
wenzelm@53399
  1425
    have *: "\<Union>F = \<Union>p"
wenzelm@53399
  1426
      using division_ofD[OF p] by auto
wenzelm@53399
  1427
    show "\<exists>p. p division_of \<Union>insert x F"
wenzelm@53399
  1428
      using elementary_union_interval[OF p[unfolded *], of a b]
wenzelm@53408
  1429
      unfolding Union_insert x * by auto
wenzelm@53408
  1430
  qed (insert assms, auto)
wenzelm@53399
  1431
  then show ?thesis
wenzelm@53399
  1432
    apply -
wenzelm@53399
  1433
    apply (erule exE)
wenzelm@53399
  1434
    apply (rule that)
wenzelm@53399
  1435
    apply auto
wenzelm@53399
  1436
    done
wenzelm@53399
  1437
qed
wenzelm@53399
  1438
wenzelm@53399
  1439
lemma elementary_union:
immler@56188
  1440
  fixes s t :: "'a::euclidean_space set"
wenzelm@53399
  1441
  assumes "ps division_of s"
wenzelm@53408
  1442
    and "pt division_of t"
himmelma@35172
  1443
  obtains p where "p division_of (s \<union> t)"
wenzelm@53399
  1444
proof -
wenzelm@53399
  1445
  have "s \<union> t = \<Union>ps \<union> \<Union>pt"
wenzelm@53399
  1446
    using assms unfolding division_of_def by auto
wenzelm@53399
  1447
  then have *: "\<Union>(ps \<union> pt) = s \<union> t" by auto
wenzelm@53399
  1448
  show ?thesis
wenzelm@53399
  1449
    apply -
wenzelm@53408
  1450
    apply (rule elementary_unions_intervals[of "ps \<union> pt"])
wenzelm@53399
  1451
    unfolding *
wenzelm@53399
  1452
    prefer 3
wenzelm@53399
  1453
    apply (rule_tac p=p in that)
wenzelm@53399
  1454
    using assms[unfolded division_of_def]
wenzelm@53399
  1455
    apply auto
wenzelm@53399
  1456
    done
wenzelm@53399
  1457
qed
wenzelm@53399
  1458
wenzelm@53399
  1459
lemma partial_division_extend:
immler@56188
  1460
  fixes t :: "'a::euclidean_space set"
wenzelm@53399
  1461
  assumes "p division_of s"
wenzelm@53399
  1462
    and "q division_of t"
wenzelm@53399
  1463
    and "s \<subseteq> t"
wenzelm@53399
  1464
  obtains r where "p \<subseteq> r" and "r division_of t"
wenzelm@53399
  1465
proof -
himmelma@35172
  1466
  note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
immler@56188
  1467
  obtain a b where ab: "t \<subseteq> cbox a b"
immler@56188
  1468
    using elementary_subset_cbox[OF assms(2)] by auto
immler@56188
  1469
  obtain r1 where "p \<subseteq> r1" "r1 division_of (cbox a b)"
wenzelm@53399
  1470
    apply (rule partial_division_extend_interval)
wenzelm@53399
  1471
    apply (rule assms(1)[unfolded divp(6)[symmetric]])
wenzelm@53399
  1472
    apply (rule subset_trans)
wenzelm@53399
  1473
    apply (rule ab assms[unfolded divp(6)[symmetric]])+
wenzelm@53408
  1474
    apply assumption
wenzelm@53399
  1475
    done
wenzelm@53399
  1476
  note r1 = this division_ofD[OF this(2)]
wenzelm@53408
  1477
  obtain p' where "p' division_of \<Union>(r1 - p)"
wenzelm@53399
  1478
    apply (rule elementary_unions_intervals[of "r1 - p"])
wenzelm@53399
  1479
    using r1(3,6)
wenzelm@53399
  1480
    apply auto
wenzelm@53399
  1481
    done
wenzelm@53399
  1482
  then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
wenzelm@53399
  1483
    apply -
wenzelm@53399
  1484
    apply (drule elementary_inter[OF _ assms(2)[unfolded divq(6)[symmetric]]])
wenzelm@53399
  1485
    apply auto
wenzelm@53399
  1486
    done
wenzelm@53399
  1487
  {
wenzelm@53399
  1488
    fix x
wenzelm@53399
  1489
    assume x: "x \<in> t" "x \<notin> s"
wenzelm@53399
  1490
    then have "x\<in>\<Union>r1"
wenzelm@53399
  1491
      unfolding r1 using ab by auto
wenzelm@53408
  1492
    then obtain r where r: "r \<in> r1" "x \<in> r"
wenzelm@53408
  1493
      unfolding Union_iff ..
wenzelm@53399
  1494
    moreover
wenzelm@53399
  1495
    have "r \<notin> p"
wenzelm@53399
  1496
    proof
wenzelm@53399
  1497
      assume "r \<in> p"
wenzelm@53399
  1498
      then have "x \<in> s" using divp(2) r by auto
wenzelm@53399
  1499
      then show False using x by auto
wenzelm@53399
  1500
    qed
wenzelm@53399
  1501
    ultimately have "x\<in>\<Union>(r1 - p)" by auto
wenzelm@53399
  1502
  }
wenzelm@53399
  1503
  then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
wenzelm@53399
  1504
    unfolding divp divq using assms(3) by auto
wenzelm@53399
  1505
  show ?thesis
wenzelm@53399
  1506
    apply (rule that[of "p \<union> r2"])
wenzelm@53399
  1507
    unfolding *
wenzelm@53399
  1508
    defer
wenzelm@53399
  1509
    apply (rule division_disjoint_union)
wenzelm@53399
  1510
    unfolding divp(6)
wenzelm@53399
  1511
    apply(rule assms r2)+
wenzelm@53399
  1512
  proof -
wenzelm@53399
  1513
    have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
wenzelm@53399
  1514
    proof (rule inter_interior_unions_intervals)
immler@56188
  1515
      show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b"
wenzelm@53399
  1516
        using r1 by auto
wenzelm@53399
  1517
      have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
wenzelm@53399
  1518
        by auto
wenzelm@53399
  1519
      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
wenzelm@53399
  1520
      proof
wenzelm@53399
  1521
        fix m x
wenzelm@53399
  1522
        assume as: "m \<in> r1 - p"
wenzelm@53399
  1523
        have "interior m \<inter> interior (\<Union>p) = {}"
wenzelm@53399
  1524
        proof (rule inter_interior_unions_intervals)
immler@56188
  1525
          show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b"
wenzelm@53399
  1526
            using divp by auto
wenzelm@53399
  1527
          show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
wenzelm@53399
  1528
            apply (rule, rule r1(7))
wenzelm@53399
  1529
            using as
wenzelm@53399
  1530
            using r1 
wenzelm@53399
  1531
            apply auto
wenzelm@53399
  1532
            done
wenzelm@53399
  1533
        qed
wenzelm@53399
  1534
        then show "interior s \<inter> interior m = {}"
wenzelm@53399
  1535
          unfolding divp by auto
wenzelm@53399
  1536
      qed
wenzelm@53399
  1537
    qed
wenzelm@53399
  1538
    then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
wenzelm@53399
  1539
      using interior_subset by auto
wenzelm@53399
  1540
  qed auto
wenzelm@53399
  1541
qed
wenzelm@53399
  1542
himmelma@35172
  1543
himmelma@35172
  1544
subsection {* Tagged (partial) divisions. *}
himmelma@35172
  1545
wenzelm@53408
  1546
definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
wenzelm@53408
  1547
  where "s tagged_partial_division_of i \<longleftrightarrow>
wenzelm@53408
  1548
    finite s \<and>
immler@56188
  1549
    (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and>
wenzelm@53408
  1550
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1551
      interior k1 \<inter> interior k2 = {})"
wenzelm@53408
  1552
wenzelm@53408
  1553
lemma tagged_partial_division_ofD[dest]:
wenzelm@53408
  1554
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1555
  shows "finite s"
wenzelm@53408
  1556
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1557
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1558
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1559
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
wenzelm@53408
  1560
      (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1561
  using assms unfolding tagged_partial_division_of_def by blast+
wenzelm@53408
  1562
wenzelm@53408
  1563
definition tagged_division_of (infixr "tagged'_division'_of" 40)
wenzelm@53408
  1564
  where "s tagged_division_of i \<longleftrightarrow> s tagged_partial_division_of i \<and> (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1565
huffman@44167
  1566
lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
himmelma@35172
  1567
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1568
himmelma@35172
  1569
lemma tagged_division_of:
wenzelm@53408
  1570
  "s tagged_division_of i \<longleftrightarrow>
wenzelm@53408
  1571
    finite s \<and>
immler@56188
  1572
    (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and>
wenzelm@53408
  1573
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1574
      interior k1 \<inter> interior k2 = {}) \<and>
wenzelm@53408
  1575
    (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1576
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1577
wenzelm@53408
  1578
lemma tagged_division_ofI:
wenzelm@53408
  1579
  assumes "finite s"
wenzelm@53408
  1580
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1581
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1582
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1583
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1584
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1585
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1586
  shows "s tagged_division_of i"
wenzelm@53408
  1587
  unfolding tagged_division_of
wenzelm@53408
  1588
  apply rule
wenzelm@53408
  1589
  defer
wenzelm@53408
  1590
  apply rule
wenzelm@53408
  1591
  apply (rule allI impI conjI assms)+
wenzelm@53408
  1592
  apply assumption
wenzelm@53408
  1593
  apply rule
wenzelm@53408
  1594
  apply (rule assms)
wenzelm@53408
  1595
  apply assumption
wenzelm@53408
  1596
  apply (rule assms)
wenzelm@53408
  1597
  apply assumption
wenzelm@53408
  1598
  using assms(1,5-)
wenzelm@53408
  1599
  apply blast+
wenzelm@53408
  1600
  done
wenzelm@53408
  1601
wenzelm@53408
  1602
lemma tagged_division_ofD[dest]:
wenzelm@53408
  1603
  assumes "s tagged_division_of i"
wenzelm@53408
  1604
  shows "finite s"
wenzelm@53408
  1605
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1606
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1607
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1608
    and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1609
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1610
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
wenzelm@53408
  1611
  using assms unfolding tagged_division_of by blast+
wenzelm@53408
  1612
wenzelm@53408
  1613
lemma division_of_tagged_division:
wenzelm@53408
  1614
  assumes "s tagged_division_of i"
wenzelm@53408
  1615
  shows "(snd ` s) division_of i"
wenzelm@53408
  1616
proof (rule division_ofI)
wenzelm@53408
  1617
  note assm = tagged_division_ofD[OF assms]
wenzelm@53408
  1618
  show "\<Union>(snd ` s) = i" "finite (snd ` s)"
wenzelm@53408
  1619
    using assm by auto
wenzelm@53408
  1620
  fix k
wenzelm@53408
  1621
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1622
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1623
    by auto
immler@56188
  1624
  then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = cbox a b"
wenzelm@53408
  1625
    using assm by fastforce+
wenzelm@53408
  1626
  fix k'
wenzelm@53408
  1627
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1628
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1629
    by auto
wenzelm@53408
  1630
  then show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1631
    apply -
wenzelm@53408
  1632
    apply (rule assm(5))
wenzelm@53408
  1633
    apply (rule xk xk')+
wenzelm@53408
  1634
    using k'
wenzelm@53408
  1635
    apply auto
wenzelm@53408
  1636
    done
himmelma@35172
  1637
qed
himmelma@35172
  1638
wenzelm@53408
  1639
lemma partial_division_of_tagged_division:
wenzelm@53408
  1640
  assumes "s tagged_partial_division_of i"
himmelma@35172
  1641
  shows "(snd ` s) division_of \<Union>(snd ` s)"
wenzelm@53408
  1642
proof (rule division_ofI)
wenzelm@53408
  1643
  note assm = tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1644
  show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
wenzelm@53408
  1645
    using assm by auto
wenzelm@53408
  1646
  fix k
wenzelm@53408
  1647
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1648
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1649
    by auto
immler@56188
  1650
  then show "k \<noteq> {}" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>(snd ` s)"
wenzelm@53408
  1651
    using assm by auto
wenzelm@53408
  1652
  fix k'
wenzelm@53408
  1653
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1654
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1655
    by auto
wenzelm@53408
  1656
  then show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1657
    apply -
wenzelm@53408
  1658
    apply (rule assm(5))
wenzelm@53408
  1659
    apply(rule xk xk')+
wenzelm@53408
  1660
    using k'
wenzelm@53408
  1661
    apply auto
wenzelm@53408
  1662
    done
himmelma@35172
  1663
qed
himmelma@35172
  1664
wenzelm@53408
  1665
lemma tagged_partial_division_subset:
wenzelm@53408
  1666
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1667
    and "t \<subseteq> s"
himmelma@35172
  1668
  shows "t tagged_partial_division_of i"
wenzelm@53408
  1669
  using assms
wenzelm@53408
  1670
  unfolding tagged_partial_division_of_def
wenzelm@53408
  1671
  using finite_subset[OF assms(2)]
wenzelm@53408
  1672
  by blast
wenzelm@53408
  1673
wenzelm@53408
  1674
lemma setsum_over_tagged_division_lemma:
immler@56188
  1675
  fixes d :: "'m::euclidean_space set \<Rightarrow> 'a::real_normed_vector"
wenzelm@53408
  1676
  assumes "p tagged_division_of i"
immler@56188
  1677
    and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> content (cbox u v) = 0 \<Longrightarrow> d (cbox u v) = 0"
himmelma@35172
  1678
  shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
wenzelm@53408
  1679
proof -
wenzelm@53408
  1680
  note assm = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1681
  have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
wenzelm@53408
  1682
    unfolding o_def by (rule ext) auto
wenzelm@53408
  1683
  show ?thesis
wenzelm@53408
  1684
    unfolding *
wenzelm@53408
  1685
    apply (subst eq_commute)
wenzelm@53408
  1686
  proof (rule setsum_reindex_nonzero)
wenzelm@53408
  1687
    show "finite p"
wenzelm@53408
  1688
      using assm by auto
wenzelm@53408
  1689
    fix x y
wenzelm@53408
  1690
    assume as: "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
immler@56188
  1691
    obtain a b where ab: "snd x = cbox a b"
wenzelm@53408
  1692
      using assm(4)[of "fst x" "snd x"] as(1) by auto
wenzelm@53408
  1693
    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
wenzelm@53408
  1694
      unfolding as(4)[symmetric] using as(1-3) by auto
wenzelm@53408
  1695
    then have "interior (snd x) \<inter> interior (snd y) = {}"
wenzelm@53408
  1696
      apply -
wenzelm@53408
  1697
      apply (rule assm(5)[of "fst x" _ "fst y"])
wenzelm@53408
  1698
      using as
wenzelm@53408
  1699
      apply auto
wenzelm@53408
  1700
      done
immler@56188
  1701
    then have "content (cbox a b) = 0"
wenzelm@53408
  1702
      unfolding as(4)[symmetric] ab content_eq_0_interior by auto
immler@56188
  1703
    then have "d (cbox a b) = 0"
wenzelm@53408
  1704
      apply -
wenzelm@53408
  1705
      apply (rule assms(2))
wenzelm@53408
  1706
      using assm(2)[of "fst x" "snd x"] as(1)
wenzelm@53408
  1707
      unfolding ab[symmetric]
wenzelm@53408
  1708
      apply auto
wenzelm@53408
  1709
      done
wenzelm@53408
  1710
    then show "d (snd x) = 0"
wenzelm@53408
  1711
      unfolding ab by auto
wenzelm@53408
  1712
  qed
wenzelm@53408
  1713
qed
wenzelm@53408
  1714
wenzelm@53408
  1715
lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
wenzelm@53408
  1716
  by auto
himmelma@35172
  1717
himmelma@35172
  1718
lemma tagged_division_of_empty: "{} tagged_division_of {}"
himmelma@35172
  1719
  unfolding tagged_division_of by auto
himmelma@35172
  1720
wenzelm@53408
  1721
lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1722
  unfolding tagged_partial_division_of_def by auto
himmelma@35172
  1723
wenzelm@53408
  1724
lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1725
  unfolding tagged_division_of by auto
himmelma@35172
  1726
immler@56188
  1727
lemma tagged_division_of_self: "x \<in> cbox a b \<Longrightarrow> {(x,cbox a b)} tagged_division_of (cbox a b)"
wenzelm@53408
  1728
  by (rule tagged_division_ofI) auto
himmelma@35172
  1729
immler@56188
  1730
lemma tagged_division_of_self_real: "x \<in> {a .. b::real} \<Longrightarrow> {(x,{a .. b})} tagged_division_of {a .. b}"
immler@56188
  1731
  unfolding box_real[symmetric]
immler@56188
  1732
  by (rule tagged_division_of_self)
immler@56188
  1733
himmelma@35172
  1734
lemma tagged_division_union:
wenzelm@53408
  1735
  assumes "p1 tagged_division_of s1"
wenzelm@53408
  1736
    and "p2 tagged_division_of s2"
wenzelm@53408
  1737
    and "interior s1 \<inter> interior s2 = {}"
himmelma@35172
  1738
  shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
wenzelm@53408
  1739
proof (rule tagged_division_ofI)
wenzelm@53408
  1740
  note p1 = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1741
  note p2 = tagged_division_ofD[OF assms(2)]
wenzelm@53408
  1742
  show "finite (p1 \<union> p2)"
wenzelm@53408
  1743
    using p1(1) p2(1) by auto
wenzelm@53408
  1744
  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
wenzelm@53408
  1745
    using p1(6) p2(6) by blast
wenzelm@53408
  1746
  fix x k
wenzelm@53408
  1747
  assume xk: "(x, k) \<in> p1 \<union> p2"
immler@56188
  1748
  show "x \<in> k" "\<exists>a b. k = cbox a b"
wenzelm@53408
  1749
    using xk p1(2,4) p2(2,4) by auto
wenzelm@53408
  1750
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
  1751
    using xk p1(3) p2(3) by blast
wenzelm@53408
  1752
  fix x' k'
wenzelm@53408
  1753
  assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1754
  have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1755
    using assms(3) interior_mono by blast
wenzelm@53408
  1756
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1757
    apply (cases "(x, k) \<in> p1")
wenzelm@53408
  1758
    apply (case_tac[!] "(x',k') \<in> p1")
wenzelm@53408
  1759
    apply (rule p1(5))
wenzelm@53408
  1760
    prefer 4
wenzelm@53408
  1761
    apply (rule *)
wenzelm@53408
  1762
    prefer 6
wenzelm@53408
  1763
    apply (subst Int_commute)
wenzelm@53408
  1764
    apply (rule *)
wenzelm@53408
  1765
    prefer 8
wenzelm@53408
  1766
    apply (rule p2(5))
wenzelm@53408
  1767
    using p1(3) p2(3)
wenzelm@53408
  1768
    using xk xk'
wenzelm@53408
  1769
    apply auto
wenzelm@53408
  1770
    done
wenzelm@53408
  1771
qed
himmelma@35172
  1772
himmelma@35172
  1773
lemma tagged_division_unions:
wenzelm@53408
  1774
  assumes "finite iset"
wenzelm@53408
  1775
    and "\<forall>i\<in>iset. pfn i tagged_division_of i"
wenzelm@53408
  1776
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
himmelma@35172
  1777
  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
wenzelm@53408
  1778
proof (rule tagged_division_ofI)
himmelma@35172
  1779
  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
wenzelm@53408
  1780
  show "finite (\<Union>(pfn ` iset))"
wenzelm@53408
  1781
    apply (rule finite_Union)
wenzelm@53408
  1782
    using assms
wenzelm@53408
  1783
    apply auto
wenzelm@53408
  1784
    done
wenzelm@53408
  1785
  have "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>((\<lambda>i. \<Union>{k. \<exists>x. (x, k) \<in> pfn i}) ` iset)"
wenzelm@53408
  1786
    by blast
wenzelm@53408
  1787
  also have "\<dots> = \<Union>iset"
wenzelm@53408
  1788
    using assm(6) by auto
wenzelm@53399
  1789
  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
wenzelm@53408
  1790
  fix x k
wenzelm@53408
  1791
  assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
wenzelm@53408
  1792
  then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
wenzelm@53408
  1793
    by auto
immler@56188
  1794
  show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
wenzelm@53408
  1795
    using assm(2-4)[OF i] using i(1) by auto
wenzelm@53408
  1796
  fix x' k'
wenzelm@53408
  1797
  assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1798
  then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
wenzelm@53408
  1799
    by auto
wenzelm@53408
  1800
  have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1801
    using i(1) i'(1)
wenzelm@53408
  1802
    using assms(3)[rule_format] interior_mono
wenzelm@53408
  1803
    by blast
wenzelm@53408
  1804
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1805
    apply (cases "i = i'")
wenzelm@53408
  1806
    using assm(5)[OF i _ xk'(2)] i'(2)
wenzelm@53408
  1807
    using assm(3)[OF i] assm(3)[OF i']
wenzelm@53408
  1808
    defer
wenzelm@53408
  1809
    apply -
wenzelm@53408
  1810
    apply (rule *)
wenzelm@53408
  1811
    apply auto
wenzelm@53408
  1812
    done
himmelma@35172
  1813
qed
himmelma@35172
  1814
himmelma@35172
  1815
lemma tagged_partial_division_of_union_self:
wenzelm@53408
  1816
  assumes "p tagged_partial_division_of s"
himmelma@35172
  1817
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1818
  apply (rule tagged_division_ofI)
wenzelm@53408
  1819
  using tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1820
  apply auto
wenzelm@53408
  1821
  done
wenzelm@53408
  1822
wenzelm@53408
  1823
lemma tagged_division_of_union_self:
wenzelm@53408
  1824
  assumes "p tagged_division_of s"
wenzelm@53408
  1825
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1826
  apply (rule tagged_division_ofI)
wenzelm@53408
  1827
  using tagged_division_ofD[OF assms]
wenzelm@53408
  1828
  apply auto
wenzelm@53408
  1829
  done
wenzelm@53408
  1830
himmelma@35172
  1831
himmelma@35172
  1832
subsection {* Fine-ness of a partition w.r.t. a gauge. *}
himmelma@35172
  1833
wenzelm@53408
  1834
definition fine  (infixr "fine" 46)
wenzelm@53408
  1835
  where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
wenzelm@53408
  1836
wenzelm@53408
  1837
lemma fineI:
wenzelm@53408
  1838
  assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1839
  shows "d fine s"
wenzelm@53408
  1840
  using assms unfolding fine_def by auto
wenzelm@53408
  1841
wenzelm@53408
  1842
lemma fineD[dest]:
wenzelm@53408
  1843
  assumes "d fine s"
wenzelm@53408
  1844
  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1845
  using assms unfolding fine_def by auto
himmelma@35172
  1846
himmelma@35172
  1847
lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
himmelma@35172
  1848
  unfolding fine_def by auto
himmelma@35172
  1849
himmelma@35172
  1850
lemma fine_inters:
himmelma@35172
  1851
 "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
himmelma@35172
  1852
  unfolding fine_def by blast
himmelma@35172
  1853
wenzelm@53408
  1854
lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
himmelma@35172
  1855
  unfolding fine_def by blast
himmelma@35172
  1856
wenzelm@53408
  1857
lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
himmelma@35172
  1858
  unfolding fine_def by auto
himmelma@35172
  1859
wenzelm@53408
  1860
lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
himmelma@35172
  1861
  unfolding fine_def by blast
himmelma@35172
  1862
wenzelm@53408
  1863
himmelma@35172
  1864
subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
himmelma@35172
  1865
wenzelm@53408
  1866
definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
wenzelm@53408
  1867
  where "(f has_integral_compact_interval y) i \<longleftrightarrow>
wenzelm@53408
  1868
    (\<forall>e>0. \<exists>d. gauge d \<and>
wenzelm@53408
  1869
      (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
wenzelm@53408
  1870
        norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1871
wenzelm@53408
  1872
definition has_integral ::
immler@56188
  1873
    "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
wenzelm@53408
  1874
  (infixr "has'_integral" 46)
wenzelm@53408
  1875
  where "(f has_integral y) i \<longleftrightarrow>
immler@56188
  1876
    (if \<exists>a b. i = cbox a b
wenzelm@53408
  1877
     then (f has_integral_compact_interval y) i
immler@56188
  1878
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1879
      (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) (cbox a b) \<and>
wenzelm@53408
  1880
        norm (z - y) < e)))"
himmelma@35172
  1881
himmelma@35172
  1882
lemma has_integral:
immler@56188
  1883
  "(f has_integral y) (cbox a b) \<longleftrightarrow>
wenzelm@53408
  1884
    (\<forall>e>0. \<exists>d. gauge d \<and>
immler@56188
  1885
      (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
wenzelm@53408
  1886
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1887
  unfolding has_integral_def has_integral_compact_interval_def
wenzelm@53408
  1888
  by auto
wenzelm@53408
  1889
immler@56188
  1890
lemma has_integral_real:
immler@56188
  1891
  "(f has_integral y) {a .. b::real} \<longleftrightarrow>
immler@56188
  1892
    (\<forall>e>0. \<exists>d. gauge d \<and>
immler@56188
  1893
      (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
immler@56188
  1894
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
immler@56188
  1895
  unfolding box_real[symmetric]
immler@56188
  1896
  by (rule has_integral)
immler@56188
  1897
wenzelm@53408
  1898
lemma has_integralD[dest]:
immler@56188
  1899
  assumes "(f has_integral y) (cbox a b)"
wenzelm@53408
  1900
    and "e > 0"
wenzelm@53408
  1901
  obtains d where "gauge d"
immler@56188
  1902
    and "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d fine p \<Longrightarrow>
wenzelm@53408
  1903
      norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
himmelma@35172
  1904
  using assms unfolding has_integral by auto
himmelma@35172
  1905
himmelma@35172
  1906
lemma has_integral_alt:
wenzelm@53408
  1907
  "(f has_integral y) i \<longleftrightarrow>
immler@56188
  1908
    (if \<exists>a b. i = cbox a b
wenzelm@53408
  1909
     then (f has_integral y) i
immler@56188
  1910
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1911
      (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)))"
wenzelm@53408
  1912
  unfolding has_integral
wenzelm@53408
  1913
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1914
  by auto
himmelma@35172
  1915
himmelma@35172
  1916
lemma has_integral_altD:
wenzelm@53408
  1917
  assumes "(f has_integral y) i"
immler@56188
  1918
    and "\<not> (\<exists>a b. i = cbox a b)"
wenzelm@53408
  1919
    and "e>0"
wenzelm@53408
  1920
  obtains B where "B > 0"
immler@56188
  1921
    and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1922
      (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - y) < e)"
wenzelm@53408
  1923
  using assms
wenzelm@53408
  1924
  unfolding has_integral
wenzelm@53408
  1925
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1926
  by auto
wenzelm@53408
  1927
wenzelm@53408
  1928
definition integrable_on (infixr "integrable'_on" 46)
wenzelm@53408
  1929
  where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
wenzelm@53408
  1930
wenzelm@53408
  1931
definition "integral i f = (SOME y. (f has_integral y) i)"
himmelma@35172
  1932
wenzelm@53409
  1933
lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
wenzelm@53409
  1934
  unfolding integrable_on_def integral_def by (rule someI_ex)
himmelma@35172
  1935
himmelma@35172
  1936
lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
himmelma@35172
  1937
  unfolding integrable_on_def by auto
himmelma@35172
  1938
wenzelm@53409
  1939
lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
himmelma@35172
  1940
  by auto
himmelma@35172
  1941
himmelma@35172
  1942
lemma setsum_content_null:
immler@56188
  1943
  assumes "content (cbox a b) = 0"
immler@56188
  1944
    and "p tagged_division_of (cbox a b)"
himmelma@35172
  1945
  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
wenzelm@53409
  1946
proof (rule setsum_0', rule)
wenzelm@53409
  1947
  fix y
wenzelm@53409
  1948
  assume y: "y \<in> p"
wenzelm@53409
  1949
  obtain x k where xk: "y = (x, k)"
wenzelm@53409
  1950
    using surj_pair[of y] by blast
himmelma@35172
  1951
  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
immler@56188
  1952
  from this(2) obtain c d where k: "k = cbox c d" by blast
wenzelm@53409
  1953
  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
wenzelm@53409
  1954
    unfolding xk by auto
wenzelm@53409
  1955
  also have "\<dots> = 0"
wenzelm@53409
  1956
    using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
wenzelm@53409
  1957
    unfolding assms(1) k
wenzelm@53409
  1958
    by auto
himmelma@35172
  1959
  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
himmelma@35172
  1960
qed
himmelma@35172
  1961
wenzelm@53409
  1962
himmelma@35172
  1963
subsection {* Some basic combining lemmas. *}
himmelma@35172
  1964
himmelma@35172
  1965
lemma tagged_division_unions_exists:
wenzelm@53409
  1966
  assumes "finite iset"
wenzelm@53409
  1967
    and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
wenzelm@53409
  1968
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
wenzelm@53409
  1969
    and "\<Union>iset = i"
wenzelm@53409
  1970
   obtains p where "p tagged_division_of i" and "d fine p"
wenzelm@53409
  1971
proof -
wenzelm@53409
  1972
  obtain pfn where pfn:
wenzelm@53409
  1973
    "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
wenzelm@53409
  1974
    "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
wenzelm@53409
  1975
    using bchoice[OF assms(2)] by auto
wenzelm@53409
  1976
  show thesis
wenzelm@53409
  1977
    apply (rule_tac p="\<Union>(pfn ` iset)" in that)
wenzelm@53409
  1978
    unfolding assms(4)[symmetric]
wenzelm@53409
  1979
    apply (rule tagged_division_unions[OF assms(1) _ assms(3)])
wenzelm@53409
  1980
    defer
wenzelm@53409
  1981
    apply (rule fine_unions)
wenzelm@53409
  1982
    using pfn
wenzelm@53409
  1983
    apply auto
wenzelm@53409
  1984
    done
himmelma@35172
  1985
qed
himmelma@35172
  1986
wenzelm@53409
  1987
himmelma@35172
  1988
subsection {* The set we're concerned with must be closed. *}
himmelma@35172
  1989
wenzelm@53409
  1990
lemma division_of_closed:
immler@56189
  1991
  fixes i :: "'n::euclidean_space set"
wenzelm@53409
  1992
  shows "s division_of i \<Longrightarrow> closed i"
nipkow@44890
  1993
  unfolding division_of_def by fastforce
himmelma@35172
  1994
himmelma@35172
  1995
subsection {* General bisection principle for intervals; might be useful elsewhere. *}
himmelma@35172
  1996
wenzelm@53409
  1997
lemma interval_bisection_step:
immler@56188
  1998
  fixes type :: "'a::euclidean_space"
wenzelm@53409
  1999
  assumes "P {}"
wenzelm@53409
  2000
    and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
immler@56188
  2001
    and "\<not> P (cbox a (b::'a))"
immler@56188
  2002
  obtains c d where "\<not> P (cbox c d)"
wenzelm@53409
  2003
    and "\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
wenzelm@53409
  2004
proof -
immler@56188
  2005
  have "cbox a b \<noteq> {}"
immler@54776
  2006
    using assms(1,3) by metis
wenzelm@53409
  2007
  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
immler@56188
  2008
    by (force simp: mem_box)
wenzelm@53409
  2009
  {
wenzelm@53409
  2010
    fix f
wenzelm@53409
  2011
    have "finite f \<Longrightarrow>
wenzelm@53409
  2012
      \<forall>s\<in>f. P s \<Longrightarrow>
immler@56188
  2013
      \<forall>s\<in>f. \<exists>a b. s = cbox a b \<Longrightarrow>
wenzelm@53409
  2014
      \<forall>s\<in>f.\<forall>t\<in>f. s \<noteq> t \<longrightarrow> interior s \<inter> interior t = {} \<Longrightarrow> P (\<Union>f)"
wenzelm@53409
  2015
    proof (induct f rule: finite_induct)
wenzelm@53409
  2016
      case empty
wenzelm@53409
  2017
      show ?case
wenzelm@53409
  2018
        using assms(1) by auto
wenzelm@53409
  2019
    next
wenzelm@53409
  2020
      case (insert x f)
wenzelm@53409
  2021
      show ?case
wenzelm@53409
  2022
        unfolding Union_insert
wenzelm@53409
  2023
        apply (rule assms(2)[rule_format])
wenzelm@53409
  2024
        apply rule
wenzelm@53409
  2025
        defer
wenzelm@53409
  2026
        apply rule
wenzelm@53409
  2027
        defer
wenzelm@53409
  2028
        apply (rule inter_interior_unions_intervals)
wenzelm@53409
  2029
        using insert
wenzelm@53409
  2030
        apply auto
wenzelm@53409
  2031
        done
wenzelm@53409
  2032
    qed
wenzelm@53409
  2033
  } note * = this
immler@56188
  2034
  let ?A = "{cbox c d | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or>
wenzelm@53409
  2035
    (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
hoelzl@50526
  2036
  let ?PP = "\<lambda>c d. \<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> c\<bullet>i \<le> d\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i \<and> 2 * (d\<bullet>i - c\<bullet>i) \<le> b\<bullet>i - a\<bullet>i"
wenzelm@53409
  2037
  {
immler@56188
  2038
    presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False"
wenzelm@53409
  2039
    then show thesis
wenzelm@53409
  2040
      unfolding atomize_not not_all
wenzelm@53409
  2041
      apply -
wenzelm@53409
  2042
      apply (erule exE)+
wenzelm@53409
  2043
      apply (rule_tac c=x and d=xa in that)
wenzelm@53409
  2044
      apply auto
wenzelm@53409
  2045
      done
wenzelm@53409
  2046
  }
immler@56188
  2047
  assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)"
wenzelm@53409
  2048
  have "P (\<Union> ?A)"
wenzelm@53409
  2049
    apply (rule *)
wenzelm@53409
  2050
    apply (rule_tac[2-] ballI)
wenzelm@53409
  2051
    apply (rule_tac[4] ballI)
wenzelm@53409
  2052
    apply (rule_tac[4] impI)
wenzelm@53409
  2053
  proof -
immler@56188
  2054
    let ?B = "(\<lambda>s. cbox (\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i::'a)
immler@56188
  2055
      (\<Sum>i\<in>Basis. (if i \<in> s then (a\<bullet>i + b\<bullet>i) / 2 else b\<bullet>i) *\<^sub>R i)) ` {s. s \<subseteq> Basis}"
wenzelm@53409
  2056
    have "?A \<subseteq> ?B"
wenzelm@53409
  2057
    proof
wenzelm@53409
  2058
      case goal1
immler@56188
  2059
      then obtain c d where x: "x = cbox c d"
wenzelm@53409
  2060
        "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  2061
          c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  2062
          c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
immler@56188
  2063
      have *: "\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> cbox a b = cbox c d"
wenzelm@53409
  2064
        by auto
wenzelm@53409
  2065
      show "x \<in> ?B"
wenzelm@53409
  2066
        unfolding image_iff
wenzelm@53409
  2067
        apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
wenzelm@53409
  2068
        unfolding x
wenzelm@53409
  2069
        apply (rule *)
hoelzl@50526
  2070
        apply (simp_all only: euclidean_eq_iff[where 'a='a] inner_setsum_left_Basis mem_Collect_eq simp_thms
wenzelm@53409
  2071
          cong: ball_cong)
hoelzl@50526
  2072
        apply safe
wenzelm@53409
  2073
      proof -
wenzelm@53409
  2074
        fix i :: 'a
wenzelm@53409
  2075
        assume i: "i \<in> Basis"
wenzelm@53409
  2076
        then show "c \<bullet> i = (if c \<bullet> i = a \<bullet> i then a \<bullet> i else (a \<bullet> i + b \<bullet> i) / 2)"
wenzelm@53409
  2077
          and "d \<bullet> i = (if c \<bullet> i = a \<bullet> i then (a \<bullet> i + b \<bullet> i) / 2 else b \<bullet> i)"
wenzelm@53409
  2078
          using x(2)[of i] ab[OF i] by (auto simp add:field_simps)
wenzelm@53409
  2079
      qed
wenzelm@53409
  2080
    qed
wenzelm@53409
  2081
    then show "finite ?A"
wenzelm@53409
  2082
      by (rule finite_subset) auto
wenzelm@53409
  2083
    fix s
wenzelm@53409
  2084
    assume "s \<in> ?A"
wenzelm@53409
  2085
    then obtain c d where s:
immler@56188
  2086
      "s = cbox c d"
wenzelm@53409
  2087
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  2088
         c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  2089
         c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
wenzelm@53409
  2090
      by blast
wenzelm@53409
  2091
    show "P s"
wenzelm@53409
  2092
      unfolding s
wenzelm@53409
  2093
      apply (rule as[rule_format])
wenzelm@53409
  2094
    proof -
wenzelm@53409
  2095
      case goal1
wenzelm@53409
  2096
      then show ?case
wenzelm@53409
  2097
        using s(2)[of i] using ab[OF `i \<in> Basis`] by auto
wenzelm@53409
  2098
    qed
immler@56188
  2099
    show "\<exists>a b. s = cbox a b"
wenzelm@53409
  2100
      unfolding s by auto
wenzelm@53409
  2101
    fix t
wenzelm@53409
  2102
    assume "t \<in> ?A"
wenzelm@53409
  2103
    then obtain e f where t:
immler@56188
  2104
      "t = cbox e f"
wenzelm@53409
  2105
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  2106
        e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  2107
        e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
wenzelm@53409
  2108
      by blast
wenzelm@53409
  2109
    assume "s \<noteq> t"
wenzelm@53409
  2110
    then have "\<not> (c = e \<and> d = f)"
wenzelm@53409
  2111
      unfolding s t by auto
wenzelm@53409
  2112
    then obtain i where "c\<bullet>i \<noteq> e\<bullet>i \<or> d\<bullet>i \<noteq> f\<bullet>i" and i': "i \<in> Basis"
hoelzl@50526
  2113
      unfolding euclidean_eq_iff[where 'a='a] by auto
wenzelm@53409
  2114
    then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
wenzelm@53409
  2115
      apply -
wenzelm@53409
  2116
      apply(erule_tac[!] disjE)
wenzelm@53409
  2117
    proof -
wenzelm@53409
  2118
      assume "c\<bullet>i \<noteq> e\<bullet>i"
wenzelm@53409
  2119
      then show "d\<bullet>i \<noteq> f\<bullet>i"
wenzelm@53409
  2120
        using s(2)[OF i'] t(2)[OF i'] by fastforce
wenzelm@53409
  2121
    next
wenzelm@53409
  2122
      assume "d\<bullet>i \<noteq> f\<bullet>i"
wenzelm@53409
  2123
      then show "c\<bullet>i \<noteq> e\<bullet>i"
wenzelm@53409
  2124
        using s(2)[OF i'] t(2)[OF i'] by fastforce
wenzelm@53409
  2125
    qed
wenzelm@53409
  2126
    have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
wenzelm@53409
  2127
      by auto
wenzelm@53409
  2128
    show "interior s \<inter> interior t = {}"
immler@56188
  2129
      unfolding s t interior_cbox
wenzelm@53409
  2130
    proof (rule *)
wenzelm@53409
  2131
      fix x
immler@54775
  2132
      assume "x \<in> box c d" "x \<in> box e f"
wenzelm@53409
  2133
      then have x: "c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i"
immler@56188
  2134
        unfolding mem_box using i'
wenzelm@53409
  2135
        apply -
wenzelm@53409
  2136
        apply (erule_tac[!] x=i in ballE)+
wenzelm@53409
  2137
        apply auto
wenzelm@53409
  2138
        done
wenzelm@53409
  2139
      show False
wenzelm@53409
  2140
        using s(2)[OF i']
wenzelm@53409
  2141
        apply -
wenzelm@53409
  2142
        apply (erule_tac disjE)
wenzelm@53409
  2143
        apply (erule_tac[!] conjE)
wenzelm@53409
  2144
      proof -
wenzelm@53409
  2145
        assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
wenzelm@53409
  2146
        show False
wenzelm@53409
  2147
          using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
wenzelm@53409
  2148
      next
wenzelm@53409
  2149
        assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
wenzelm@53409
  2150
        show False
wenzelm@53409
  2151
          using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
wenzelm@53409
  2152
      qed
wenzelm@53409
  2153
    qed
wenzelm@53409
  2154
  qed
immler@56188
  2155
  also have "\<Union> ?A = cbox a b"
wenzelm@53409
  2156
  proof (rule set_eqI,rule)
wenzelm@53409
  2157
    fix x
wenzelm@53409
  2158
    assume "x \<in> \<Union>?A"
wenzelm@53409
  2159
    then obtain c d where x:
immler@56188
  2160
      "x \<in> cbox c d"
wenzelm@53409
  2161
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  2162
        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  2163
        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
immler@56188
  2164
    show "x\<in>cbox a b"
immler@56188
  2165
      unfolding mem_box
wenzelm@53409
  2166
    proof safe
wenzelm@53409
  2167
      fix i :: 'a
wenzelm@53409
  2168
      assume i: "i \<in> Basis"
wenzelm@53409
  2169
      then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
immler@56188
  2170
        using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
wenzelm@53409
  2171
    qed
wenzelm@53409
  2172
  next
wenzelm@53409
  2173
    fix x
immler@56188
  2174
    assume x: "x \<in> cbox a b"
wenzelm@53409
  2175
    have "\<forall>i\<in>Basis.
wenzelm@53409
  2176
      \<exists>c d. (c = a\<bullet>i \<and> d = (a\<bullet>i + b\<bullet>i) / 2 \<or> c = (a\<bullet>i + b\<bullet>i) / 2 \<and> d = b\<bullet>i) \<and> c\<le>x\<bullet>i \<and> x\<bullet>i \<le> d"
wenzelm@53409
  2177
      (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
immler@56188
  2178
      unfolding mem_box
hoelzl@50526
  2179
    proof
wenzelm@53409
  2180
      fix i :: 'a
wenzelm@53409
  2181
      assume i: "i \<in> Basis"
hoelzl@50526
  2182
      have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)"
immler@56188
  2183
        using x[unfolded mem_box,THEN bspec, OF i] by auto
wenzelm@53409
  2184
      then show "\<exists>c d. ?P i c d"
wenzelm@53409
  2185
        by blast
hoelzl@50526
  2186
    qed
wenzelm@53409
  2187
    then show "x\<in>\<Union>?A"
hoelzl@50526
  2188
      unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
wenzelm@53409
  2189
      apply -
wenzelm@53409
  2190
      apply (erule exE)+
immler@56188
  2191
      apply (rule_tac x="cbox xa xaa" in exI)
immler@56188
  2192
      unfolding mem_box
wenzelm@53409
  2193
      apply auto
wenzelm@53409
  2194
      done
wenzelm@53409
  2195
  qed
wenzelm@53409
  2196
  finally show False
wenzelm@53409
  2197
    using assms by auto
wenzelm@53409
  2198
qed
wenzelm@53409
  2199
wenzelm@53409
  2200
lemma interval_bisection:
immler@56188
  2201
  fixes type :: "'a::euclidean_space"
wenzelm@53409
  2202
  assumes "P {}"
wenzelm@53409
  2203
    and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
immler@56188
  2204
    and "\<not> P (cbox a (b::'a))"
immler@56188
  2205
  obtains x where "x \<in> cbox a b"
immler@56188
  2206
    and "\<forall>e>0. \<exists>c d. x \<in> cbox c d \<and> cbox c d \<subseteq> ball x e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
immler@56188
  2207
proof -
immler@56188
  2208
  have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and>
hoelzl@50526
  2209
    (\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and>
wenzelm@53409
  2210
       2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))"
wenzelm@53409
  2211
  proof
wenzelm@53409
  2212
    case goal1
wenzelm@53409
  2213
    then show ?case
wenzelm@53409
  2214
    proof -
immler@56188
  2215
      presume "\<not> P (cbox (fst x) (snd x)) \<Longrightarrow> ?thesis"
immler@56188
  2216
      then show ?thesis by (cases "P (cbox (fst x) (snd x))") auto
wenzelm@53409
  2217
    next
immler@56188
  2218
      assume as: "\<not> P (cbox (fst x) (snd x))"
immler@56188
  2219
      obtain c d where "\<not> P (cbox c d)"
wenzelm@53409
  2220
        "\<forall>i\<in>Basis.
wenzelm@53409
  2221
           fst x \<bullet> i \<le> c \<bullet> i \<and>
wenzelm@53409
  2222
           c \<bullet> i \<le> d \<bullet> i \<and>
wenzelm@53409
  2223
           d \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@53409
  2224
           2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
wenzelm@53409
  2225
        by (rule interval_bisection_step[of P, OF assms(1-2) as])
wenzelm@53409
  2226
      then show ?thesis
wenzelm@53409
  2227
        apply -
wenzelm@53409
  2228
        apply (rule_tac x="(c,d)" in exI)
wenzelm@53409
  2229
        apply auto
wenzelm@53409
  2230
        done
wenzelm@53409
  2231
    qed
wenzelm@53409
  2232
  qed
wenzelm@55751
  2233
  then obtain f where f:
wenzelm@55751
  2234
    "\<forall>x.
immler@56188
  2235
      \<not> P (cbox (fst x) (snd x)) \<longrightarrow>
immler@56188
  2236
      \<not> P (cbox (fst (f x)) (snd (f x))) \<and>
wenzelm@55751
  2237
        (\<forall>i\<in>Basis.
wenzelm@55751
  2238
            fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and>
wenzelm@55751
  2239
            fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and>
wenzelm@55751
  2240
            snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@55751
  2241
            2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)"
wenzelm@53409
  2242
    apply -
wenzelm@53409
  2243
    apply (drule choice)
wenzelm@55751
  2244
    apply blast
wenzelm@55751
  2245
    done
wenzelm@53409
  2246
  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)"
wenzelm@53409
  2247
  def A \<equiv> "\<lambda>n. fst(AB n)"
wenzelm@53409
  2248
  def B \<equiv> "\<lambda>n. snd(AB n)"
wenzelm@53409
  2249
  note ab_def = A_def B_def AB_def
immler@56188
  2250
  have "A 0 = a" "B 0 = b" "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and>
wenzelm@53399
  2251
    (\<forall>i\<in>Basis. A(n)\<bullet>i \<le> A(Suc n)\<bullet>i \<and> A(Suc n)\<bullet>i \<le> B(Suc n)\<bullet>i \<and> B(Suc n)\<bullet>i \<le> B(n)\<bullet>i \<and>
hoelzl@50526
  2252
    2 * (B(Suc n)\<bullet>i - A(Suc n)\<bullet>i) \<le> B(n)\<bullet>i - A(n)\<bullet>i)" (is "\<And>n. ?P n")
wenzelm@53409
  2253
  proof -
wenzelm@53409
  2254
    show "A 0 = a" "B 0 = b"
wenzelm@53409
  2255
      unfolding ab_def by auto
wenzelm@53409
  2256
    case goal3
wenzelm@53409
  2257
    note S = ab_def funpow.simps o_def id_apply
wenzelm@53409
  2258
    show ?case
wenzelm@53409
  2259
    proof (induct n)
wenzelm@53409
  2260
      case 0
wenzelm@53409
  2261
      then show ?case
wenzelm@53409
  2262
        unfolding S
wenzelm@53409
  2263
        apply (rule f[rule_format]) using assms(3)
wenzelm@53409
  2264
        apply auto
wenzelm@53409
  2265
        done
wenzelm@53409
  2266
    next
wenzelm@53409
  2267
      case (Suc n)
wenzelm@53409
  2268
      show ?case
wenzelm@53409
  2269
        unfolding S
wenzelm@53409
  2270
        apply (rule f[rule_format])
wenzelm@53409
  2271
        using Suc
wenzelm@53409
  2272
        unfolding S
wenzelm@53409
  2273
        apply auto
wenzelm@53409
  2274
        done
wenzelm@53409
  2275
    qed
wenzelm@53409
  2276
  qed
wenzelm@53409
  2277
  note AB = this(1-2) conjunctD2[OF this(3),rule_format]
wenzelm@53409
  2278
immler@56188
  2279
  have interv: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e"
wenzelm@53409
  2280
  proof -
wenzelm@53409
  2281
    case goal1
wenzelm@53409
  2282
    obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
wenzelm@53409
  2283
      using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] ..
wenzelm@53409
  2284
    show ?case
wenzelm@53409
  2285
      apply (rule_tac x=n in exI)
wenzelm@53409
  2286
      apply rule
wenzelm@53409
  2287
      apply rule
wenzelm@53409
  2288
    proof -
wenzelm@53409
  2289
      fix x y
immler@56188
  2290
      assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
wenzelm@53409
  2291
      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis"
wenzelm@53409
  2292
        unfolding dist_norm by(rule norm_le_l1)
hoelzl@50526
  2293
      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
wenzelm@53409
  2294
      proof (rule setsum_mono)
wenzelm@53409
  2295
        fix i :: 'a
wenzelm@53409
  2296
        assume i: "i \<in> Basis"
wenzelm@53409
  2297
        show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
immler@56188
  2298
          using xy[unfolded mem_box,THEN bspec, OF i]
wenzelm@53409
  2299
          by (auto simp: inner_diff_left)
wenzelm@53409
  2300
      qed
wenzelm@53409
  2301
      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
wenzelm@53409
  2302
        unfolding setsum_divide_distrib
wenzelm@53409
  2303
      proof (rule setsum_mono)
wenzelm@53409
  2304
        case goal1
wenzelm@53409
  2305
        then show ?case
wenzelm@53409
  2306
        proof (induct n)
wenzelm@53409
  2307
          case 0
wenzelm@53409
  2308
          then show ?case
wenzelm@53409
  2309
            unfolding AB by auto
wenzelm@53409
  2310
        next
wenzelm@53409
  2311
          case (Suc n)
wenzelm@53409
  2312
          have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
hoelzl@37489
  2313
            using AB(4)[of i n] using goal1 by auto
wenzelm@53409
  2314
          also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
wenzelm@53409
  2315
            using Suc by (auto simp add:field_simps)
wenzelm@53409
  2316
          finally show ?case .
wenzelm@53409
  2317
        qed
wenzelm@53409
  2318
      qed
wenzelm@53409
  2319
      also have "\<dots> < e"
wenzelm@53409
  2320
        using n using goal1 by (auto simp add:field_simps)
wenzelm@53409
  2321
      finally show "dist x y < e" .
wenzelm@53409
  2322
    qed
wenzelm@53409
  2323
  qed
wenzelm@53409
  2324
  {
wenzelm@53409
  2325
    fix n m :: nat
immler@56188
  2326
    assume "m \<le> n" then have "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)"
hoelzl@54411
  2327
    proof (induction rule: inc_induct)
wenzelm@53409
  2328
      case (step i)
wenzelm@53409
  2329
      show ?case
immler@56188
  2330
        using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
wenzelm@53409
  2331
    qed simp
wenzelm@53409
  2332
  } note ABsubset = this
immler@56188
  2333
  have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)"
immler@56188
  2334
    by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
immler@54776
  2335
      (metis nat.exhaust AB(1-3) assms(1,3))
immler@56188
  2336
  then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
wenzelm@53409
  2337
    by blast
wenzelm@53409
  2338
  show thesis
wenzelm@53409
  2339
  proof (rule that[rule_format, of x0])
immler@56188
  2340
    show "x0\<in>cbox a b"
wenzelm@53409
  2341
      using x0[of 0] unfolding AB .
wenzelm@53409
  2342
    fix e :: real
wenzelm@53409
  2343
    assume "e > 0"
wenzelm@53409
  2344
    from interv[OF this] obtain n
immler@56188
  2345
      where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" ..
immler@56188
  2346
    show "\<exists>c d. x0 \<in> cbox c d \<and> cbox c d \<subseteq> ball x0 e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
wenzelm@53409
  2347
      apply (rule_tac x="A n" in exI)
wenzelm@53409
  2348
      apply (rule_tac x="B n" in exI)
wenzelm@53409
  2349
      apply rule
wenzelm@53409
  2350
      apply (rule x0)
wenzelm@53409
  2351
      apply rule
wenzelm@53409
  2352
      defer
wenzelm@53409
  2353
      apply rule
wenzelm@53409
  2354
    proof -
immler@56188
  2355
      show "\<not> P (cbox (A n) (B n))"
wenzelm@53409
  2356
        apply (cases "0 < n")
wenzelm@53409
  2357
        using AB(3)[of "n - 1"] assms(3) AB(1-2)
wenzelm@53409
  2358
        apply auto
wenzelm@53409
  2359
        done
immler@56188
  2360
      show "cbox (A n) (B n) \<subseteq> ball x0 e"
wenzelm@53409
  2361
        using n using x0[of n] by auto
immler@56188
  2362
      show "cbox (A n) (B n) \<subseteq> cbox a b"
wenzelm@53409
  2363
        unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
wenzelm@53409
  2364
    qed
wenzelm@53409
  2365
  qed
wenzelm@53409
  2366
qed
wenzelm@53409
  2367
himmelma@35172
  2368
himmelma@35172
  2369
subsection {* Cousin's lemma. *}
himmelma@35172
  2370
wenzelm@53409
  2371
lemma fine_division_exists:
immler@56188
  2372
  fixes a b :: "'a::euclidean_space"
wenzelm@53409
  2373
  assumes "gauge g"
immler@56188
  2374
  obtains p where "p tagged_division_of (cbox a b)" "g fine p"
immler@56188
  2375
proof -
immler@56188
  2376
  presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
immler@56188
  2377
  then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
wenzelm@53410
  2378
    by blast
wenzelm@53409
  2379
  then show thesis ..
wenzelm@53409
  2380
next
immler@56188
  2381
  assume as: "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
wenzelm@55751
  2382
  obtain x where x:
immler@56188
  2383
    "x \<in> (cbox a b)"
wenzelm@55751
  2384
    "\<And>e. 0 < e \<Longrightarrow>
wenzelm@55751
  2385
      \<exists>c d.
immler@56188
  2386
        x \<in> cbox c d \<and>
immler@56188
  2387
        cbox c d \<subseteq> ball x e \<and>
immler@56188
  2388
        cbox c d \<subseteq> (cbox a b) \<and>
immler@56188
  2389
        \<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
wenzelm@53410
  2390
    apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
wenzelm@53410
  2391
    apply (rule_tac x="{}" in exI)
wenzelm@53410
  2392
    defer
wenzelm@53410
  2393
    apply (erule conjE exE)+
wenzelm@53409
  2394
  proof -
wenzelm@53410
  2395
    show "{} tagged_division_of {} \<and> g fine {}"
wenzelm@53410
  2396
      unfolding fine_def by auto
wenzelm@53410
  2397
    fix s t p p'
wenzelm@53410
  2398
    assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'"
wenzelm@53410
  2399
      "interior s \<inter> interior t = {}"
wenzelm@53409
  2400
    then show "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p"
wenzelm@53409
  2401
      apply -
wenzelm@53409
  2402
      apply (rule_tac x="p \<union> p'" in exI)
wenzelm@53409
  2403
      apply rule
wenzelm@53409
  2404
      apply (rule tagged_division_union)
wenzelm@53409
  2405
      prefer 4
wenzelm@53409
  2406
      apply (rule fine_union)
wenzelm@53409
  2407
      apply auto
wenzelm@53409
  2408
      done
wenzelm@55751
  2409
  qed blast
wenzelm@53410
  2410
  obtain e where e: "e > 0" "ball x e \<subseteq> g x"
wenzelm@53409
  2411
    using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
wenzelm@53410
  2412
  from x(2)[OF e(1)] obtain c d where c_d:
immler@56188
  2413
    "x \<in> cbox c d"
immler@56188
  2414
    "cbox c d \<subseteq> ball x e"
immler@56188
  2415
    "cbox c d \<subseteq> cbox a b"
immler@56188
  2416
    "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
wenzelm@53410
  2417
    by blast
immler@56188
  2418
  have "g fine {(x, cbox c d)}"
wenzelm@53409
  2419
    unfolding fine_def using e using c_d(2) by auto
wenzelm@53410
  2420
  then show False
wenzelm@53410
  2421
    using tagged_division_of_self[OF c_d(1)] using c_d by auto
wenzelm@53409
  2422
qed
wenzelm@53409
  2423
immler@56188
  2424
lemma fine_division_exists_real:
immler@56188
  2425
  fixes a b :: real
immler@56188
  2426
  assumes "gauge g"
immler@56188
  2427
  obtains p where "p tagged_division_of {a .. b}" "g fine p"
immler@56188
  2428
  by (metis assms box_real(2) fine_division_exists)
himmelma@35172
  2429
himmelma@35172
  2430
subsection {* Basic theorems about integrals. *}
himmelma@35172
  2431
wenzelm@53409
  2432
lemma has_integral_unique:
immler@56188
  2433
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2434
  assumes "(f has_integral k1) i"
wenzelm@53410
  2435
    and "(f has_integral k2) i"
wenzelm@53409
  2436
  shows "k1 = k2"
wenzelm@53410
  2437
proof (rule ccontr)
wenzelm@53842
  2438
  let ?e = "norm (k1 - k2) / 2"
wenzelm@53410
  2439
  assume as:"k1 \<noteq> k2"
wenzelm@53410
  2440
  then have e: "?e > 0"
wenzelm@53410
  2441
    by auto
wenzelm@53410
  2442
  have lem: "\<And>f::'n \<Rightarrow> 'a.  \<And>a b k1 k2.
immler@56188
  2443
    (f has_integral k1) (cbox a b) \<Longrightarrow> (f has_integral k2) (cbox a b) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
wenzelm@53410
  2444
  proof -
wenzelm@53410
  2445
    case goal1
wenzelm@53410
  2446
    let ?e = "norm (k1 - k2) / 2"
wenzelm@53410
  2447
    from goal1(3) have e: "?e > 0" by auto
wenzelm@55751
  2448
    obtain d1 where d1:
wenzelm@55751
  2449
        "gauge d1"
immler@56188
  2450
        "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
wenzelm@55751
  2451
          d1 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k1) < norm (k1 - k2) / 2"
wenzelm@55751
  2452
      by (rule has_integralD[OF goal1(1) e]) blast
wenzelm@55751
  2453
    obtain d2 where d2:
wenzelm@55751
  2454
        "gauge d2"
immler@56188
  2455
        "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
wenzelm@55751
  2456
          d2 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k2) < norm (k1 - k2) / 2"
wenzelm@55751
  2457
      by (rule has_integralD[OF goal1(2) e]) blast
wenzelm@55751
  2458
    obtain p where p:
immler@56188
  2459
        "p tagged_division_of cbox a b"
wenzelm@55751
  2460
        "(\<lambda>x. d1 x \<inter> d2 x) fine p"
wenzelm@55751
  2461
      by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)]])
wenzelm@53410
  2462
    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
wenzelm@53410
  2463
    have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
wenzelm@53410
  2464
      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
wenzelm@53410
  2465
      by (auto simp add:algebra_simps norm_minus_commute)
himmelma@35172
  2466
    also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2467
      apply (rule add_strict_mono)
wenzelm@53410
  2468
      apply (rule_tac[!] d2(2) d1(2))
wenzelm@53410
  2469
      using p unfolding fine_def
wenzelm@53410
  2470
      apply auto
wenzelm@53410
  2471
      done
himmelma@35172
  2472
    finally show False by auto
wenzelm@53410
  2473
  qed
wenzelm@53410
  2474
  {
immler@56188
  2475
    presume "\<not> (\<exists>a b. i = cbox a b) \<Longrightarrow> False"
wenzelm@53410
  2476
    then show False
wenzelm@53410
  2477
      apply -
immler@56188
  2478
      apply (cases "\<exists>a b. i = cbox a b")
wenzelm@53410
  2479
      using assms
wenzelm@53410
  2480
      apply (auto simp add:has_integral intro:lem[OF _ _ as])
wenzelm@53410
  2481
      done
wenzelm@53410
  2482
  }
immler@56188
  2483
  assume as: "\<not> (\<exists>a b. i = cbox a b)"
wenzelm@55751
  2484
  obtain B1 where B1:
wenzelm@55751
  2485
      "0 < B1"
immler@56188
  2486
      "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2487
        \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
wenzelm@55751
  2488
          norm (z - k1) < norm (k1 - k2) / 2"
wenzelm@55751
  2489
    by (rule has_integral_altD[OF assms(1) as,OF e]) blast
wenzelm@55751
  2490
  obtain B2 where B2:
wenzelm@55751
  2491
      "0 < B2"
immler@56188
  2492
      "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2493
        \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
wenzelm@55751
  2494
          norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@55751
  2495
    by (rule has_integral_altD[OF assms(2) as,OF e]) blast
immler@56188
  2496
  have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> cbox a b"
immler@56188
  2497
    apply (rule bounded_subset_cbox)
wenzelm@53410
  2498
    using bounded_Un bounded_ball
wenzelm@53410
  2499
    apply auto
wenzelm@53410
  2500
    done
immler@56188
  2501
  then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
wenzelm@53410
  2502
    by blast
wenzelm@53410
  2503
  obtain w where w:
immler@56188
  2504
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) (cbox a b)"
wenzelm@53410
  2505
    "norm (w - k1) < norm (k1 - k2) / 2"
wenzelm@53410
  2506
    using B1(2)[OF ab(1)] by blast
wenzelm@53410
  2507
  obtain z where z:
immler@56188
  2508
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b)"
wenzelm@53410
  2509
    "norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@53410
  2510
    using B2(2)[OF ab(2)] by blast
wenzelm@53410
  2511
  have "z = w"
wenzelm@53410
  2512
    using lem[OF w(1) z(1)] by auto
wenzelm@53410
  2513
  then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
wenzelm@53410
  2514
    using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
wenzelm@53410
  2515
    by (auto simp add: norm_minus_commute)
wenzelm@53410
  2516
  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2517
    apply (rule add_strict_mono)
wenzelm@53410
  2518
    apply (rule_tac[!] z(2) w(2))
wenzelm@53410
  2519
    done
wenzelm@53410
  2520
  finally show False by auto
wenzelm@53410
  2521
qed
wenzelm@53410
  2522
wenzelm@53410
  2523
lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
wenzelm@53410
  2524
  unfolding integral_def