src/HOL/Multivariate_Analysis/Integration.thy
author immler
Mon Dec 16 17:08:22 2013 +0100 (2013-12-16)
changeset 54775 2d3df8633dad
parent 54411 f72e58a5a75f
child 54776 db890d9fc5c2
permissions -rw-r--r--
prefer box over greaterThanLessThan on euclidean_space
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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) (rule cSup_abs_le, 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: "{} = {One..(0::'a::ordered_euclidean_space)}"
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  apply (auto simp: eucl_le[where 'a='a])
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  by (metis (mono_tags) eucl_less eucl_less_not_refl order.strict_trans2 zero_less_one)
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lemma interior_subset_union_intervals:
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  assumes "i = {a..b::'a::ordered_euclidean_space}"
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    and "j = {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> {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_closed_interval by auto
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  moreover
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  have "box a b \<subseteq> {c..d} \<union> s"
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    apply (rule order_trans,rule interval_open_subset_closed)
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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_closed_interval
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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::ordered_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 = {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 = {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 = {a..b}"
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        using insert(4)[rule_format,OF insertI1] ..
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      then obtain b where ab: "i = {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 - {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 - {a..b}"
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          unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_interval],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 -
wenzelm@49970
   307
          apply (rule insert(3))
wenzelm@49970
   308
          using insert(4)
wenzelm@49970
   309
          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"
wenzelm@53408
   317
            unfolding open_contains_ball_eq[OF open_interval,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
wenzelm@50945
   321
            using interval_open_subset_closed[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"
wenzelm@49970
   327
            unfolding mem_interval by (auto simp add: not_less)
wenzelm@53399
   328
          then have "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
wenzelm@49970
   329
            using True unfolding ab and mem_interval
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"
hoelzl@50526
   351
                unfolding ab mem_interval 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"
hoelzl@50526
   400
                unfolding ab mem_interval 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
hoelzl@37489
   453
himmelma@35172
   454
subsection {* Bounds on intervals where they exist. *}
himmelma@35172
   455
wenzelm@53399
   456
definition interval_upperbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a"
hoelzl@54263
   457
  where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)"
wenzelm@53399
   458
wenzelm@53399
   459
definition interval_lowerbound :: "('a::ordered_euclidean_space) set \<Rightarrow> 'a"
hoelzl@54263
   460
  where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x:s. x\<bullet>i) *\<^sub>R i)"
wenzelm@49970
   461
wenzelm@49970
   462
lemma interval_upperbound[simp]:
hoelzl@50526
   463
  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
hoelzl@50526
   464
    interval_upperbound {a..b} = (b::'a::ordered_euclidean_space)"
hoelzl@50526
   465
  unfolding interval_upperbound_def euclidean_representation_setsum
hoelzl@54263
   466
  by (auto simp del: ex_simps simp add: Bex_def ex_simps[symmetric] eucl_le[where 'a='a] SUP_def
hoelzl@54263
   467
           intro!: cSup_eq)
wenzelm@49970
   468
wenzelm@49970
   469
lemma interval_lowerbound[simp]:
hoelzl@50526
   470
  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
hoelzl@50526
   471
    interval_lowerbound {a..b} = (a::'a::ordered_euclidean_space)"
hoelzl@50526
   472
  unfolding interval_lowerbound_def euclidean_representation_setsum
hoelzl@54263
   473
  by (auto simp del: ex_simps simp add: Bex_def ex_simps[symmetric] eucl_le[where 'a='a] INF_def
hoelzl@54263
   474
           intro!: cInf_eq)
himmelma@35172
   475
himmelma@35172
   476
lemmas interval_bounds = interval_upperbound interval_lowerbound
himmelma@35172
   477
wenzelm@49970
   478
lemma interval_bounds'[simp]:
wenzelm@53399
   479
  assumes "{a..b} \<noteq> {}"
wenzelm@53399
   480
  shows "interval_upperbound {a..b} = b"
wenzelm@53399
   481
    and "interval_lowerbound {a..b} = a"
himmelma@35172
   482
  using assms unfolding interval_ne_empty by auto
himmelma@35172
   483
wenzelm@53399
   484
himmelma@35172
   485
subsection {* Content (length, area, volume...) of an interval. *}
himmelma@35172
   486
hoelzl@37489
   487
definition "content (s::('a::ordered_euclidean_space) set) =
hoelzl@50526
   488
  (if s = {} then 0 else (\<Prod>i\<in>Basis. (interval_upperbound s)\<bullet>i - (interval_lowerbound s)\<bullet>i))"
hoelzl@50526
   489
wenzelm@53399
   490
lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> {a..b::'a::ordered_euclidean_space} \<noteq> {}"
hoelzl@37489
   491
  unfolding interval_eq_empty unfolding not_ex not_less by auto
hoelzl@37489
   492
wenzelm@49970
   493
lemma content_closed_interval:
wenzelm@53399
   494
  fixes a :: "'a::ordered_euclidean_space"
hoelzl@50526
   495
  assumes "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
hoelzl@50526
   496
  shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
wenzelm@49970
   497
  using interval_not_empty[OF assms]
wenzelm@49970
   498
  unfolding content_def interval_upperbound[OF assms] interval_lowerbound[OF assms]
wenzelm@49970
   499
  by auto
wenzelm@49970
   500
wenzelm@49970
   501
lemma content_closed_interval':
wenzelm@53399
   502
  fixes a :: "'a::ordered_euclidean_space"
wenzelm@53399
   503
  assumes "{a..b} \<noteq> {}"
hoelzl@50526
   504
  shows "content {a..b} = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
wenzelm@49970
   505
  apply (rule content_closed_interval)
wenzelm@50945
   506
  using assms
wenzelm@50945
   507
  unfolding interval_ne_empty
wenzelm@49970
   508
  apply assumption
wenzelm@49970
   509
  done
wenzelm@49970
   510
wenzelm@53408
   511
lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
wenzelm@53408
   512
  unfolding content_def by auto
hoelzl@37489
   513
hoelzl@50104
   514
lemma content_singleton[simp]: "content {a} = 0"
hoelzl@50104
   515
proof -
hoelzl@50104
   516
  have "content {a .. a} = 0"
hoelzl@50526
   517
    by (subst content_closed_interval) (auto simp: ex_in_conv)
hoelzl@50104
   518
  then show ?thesis by simp
hoelzl@50104
   519
qed
hoelzl@50104
   520
wenzelm@49970
   521
lemma content_unit[intro]: "content{0..One::'a::ordered_euclidean_space} = 1"
wenzelm@49970
   522
proof -
wenzelm@53408
   523
  have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i"
wenzelm@53408
   524
    by auto
wenzelm@53408
   525
  have "0 \<in> {0..One::'a}"
wenzelm@53408
   526
    unfolding mem_interval by auto
wenzelm@53399
   527
  then show ?thesis
wenzelm@53399
   528
    unfolding content_def interval_bounds[OF *] using setprod_1 by auto
wenzelm@49970
   529
qed
wenzelm@49970
   530
wenzelm@49970
   531
lemma content_pos_le[intro]:
wenzelm@49970
   532
  fixes a::"'a::ordered_euclidean_space"
wenzelm@49970
   533
  shows "0 \<le> content {a..b}"
wenzelm@49970
   534
proof (cases "{a..b} = {}")
wenzelm@49970
   535
  case False
wenzelm@53399
   536
  then have *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
wenzelm@53399
   537
    unfolding interval_ne_empty .
hoelzl@50526
   538
  have "(\<Prod>i\<in>Basis. interval_upperbound {a..b} \<bullet> i - interval_lowerbound {a..b} \<bullet> i) \<ge> 0"
wenzelm@49970
   539
    apply (rule setprod_nonneg)
wenzelm@49970
   540
    unfolding interval_bounds[OF *]
wenzelm@49970
   541
    using *
hoelzl@50526
   542
    apply (erule_tac x=x in ballE)
wenzelm@49970
   543
    apply auto
wenzelm@49970
   544
    done
wenzelm@53408
   545
  then show ?thesis
wenzelm@53408
   546
    unfolding content_def by (auto simp del:interval_bounds')
wenzelm@53399
   547
next
wenzelm@53399
   548
  case True
wenzelm@53408
   549
  then show ?thesis
wenzelm@53408
   550
    unfolding content_def by auto
wenzelm@53399
   551
qed
wenzelm@49970
   552
wenzelm@49970
   553
lemma content_pos_lt:
wenzelm@53399
   554
  fixes a :: "'a::ordered_euclidean_space"
hoelzl@50526
   555
  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
wenzelm@49970
   556
  shows "0 < content {a..b}"
wenzelm@49970
   557
proof -
hoelzl@50526
   558
  have help_lemma1: "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i \<Longrightarrow> \<forall>i\<in>Basis. a\<bullet>i \<le> ((b\<bullet>i)::real)"
wenzelm@53408
   559
    apply rule
wenzelm@53408
   560
    apply (erule_tac x=i in ballE)
wenzelm@49970
   561
    apply auto
wenzelm@49970
   562
    done
wenzelm@53408
   563
  show ?thesis
wenzelm@53408
   564
    unfolding content_closed_interval[OF help_lemma1[OF assms]]
wenzelm@53399
   565
    apply (rule setprod_pos)
wenzelm@53399
   566
    using assms
wenzelm@53399
   567
    apply (erule_tac x=x in ballE)
wenzelm@49970
   568
    apply auto
wenzelm@49970
   569
    done
wenzelm@49970
   570
qed
wenzelm@49970
   571
wenzelm@53408
   572
lemma content_eq_0:
wenzelm@53408
   573
  "content{a..b::'a::ordered_euclidean_space} = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
wenzelm@49970
   574
proof (cases "{a..b} = {}")
wenzelm@49970
   575
  case True
wenzelm@53399
   576
  then show ?thesis
wenzelm@49970
   577
    unfolding content_def if_P[OF True]
wenzelm@49970
   578
    unfolding interval_eq_empty
wenzelm@49970
   579
    apply -
hoelzl@50526
   580
    apply (rule, erule bexE)
hoelzl@50526
   581
    apply (rule_tac x = i in bexI)
wenzelm@49970
   582
    apply auto
wenzelm@49970
   583
    done
wenzelm@49970
   584
next
wenzelm@49970
   585
  case False
wenzelm@53408
   586
  then have "\<forall>i\<in>Basis. b \<bullet> i \<ge> a \<bullet> i"
wenzelm@53408
   587
    unfolding interval_eq_empty not_ex not_less
wenzelm@53408
   588
    by fastforce
wenzelm@53408
   589
  then show ?thesis
hoelzl@50526
   590
    unfolding content_def if_not_P[OF False] setprod_zero_iff[OF finite_Basis]
hoelzl@50526
   591
    by (auto intro!: bexI)
wenzelm@49970
   592
qed
himmelma@35172
   593
wenzelm@53408
   594
lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
wenzelm@53399
   595
  by auto
himmelma@35172
   596
himmelma@35172
   597
lemma content_closed_interval_cases:
wenzelm@49970
   598
  "content {a..b::'a::ordered_euclidean_space} =
hoelzl@50526
   599
    (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)"
hoelzl@50526
   600
  by (auto simp: not_le content_eq_0 intro: less_imp_le content_closed_interval)
himmelma@35172
   601
himmelma@35172
   602
lemma content_eq_0_interior: "content {a..b} = 0 \<longleftrightarrow> interior({a..b}) = {}"
wenzelm@53408
   603
  unfolding content_eq_0 interior_closed_interval interval_eq_empty
wenzelm@53408
   604
  by auto
himmelma@35172
   605
wenzelm@53399
   606
lemma content_pos_lt_eq:
wenzelm@53399
   607
  "0 < content {a..b::'a::ordered_euclidean_space} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
wenzelm@49970
   608
  apply rule
wenzelm@49970
   609
  defer
wenzelm@49970
   610
  apply (rule content_pos_lt, assumption)
wenzelm@49970
   611
proof -
wenzelm@49970
   612
  assume "0 < content {a..b}"
wenzelm@53399
   613
  then have "content {a..b} \<noteq> 0" by auto
wenzelm@53399
   614
  then show "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
wenzelm@49970
   615
    unfolding content_eq_0 not_ex not_le by fastforce
wenzelm@49970
   616
qed
wenzelm@49970
   617
wenzelm@53399
   618
lemma content_empty [simp]: "content {} = 0"
wenzelm@53399
   619
  unfolding content_def by auto
himmelma@35172
   620
wenzelm@49698
   621
lemma content_subset:
wenzelm@49698
   622
  assumes "{a..b} \<subseteq> {c..d}"
wenzelm@49698
   623
  shows "content {a..b::'a::ordered_euclidean_space} \<le> content {c..d}"
wenzelm@49698
   624
proof (cases "{a..b} = {}")
wenzelm@49698
   625
  case True
wenzelm@53399
   626
  then show ?thesis
wenzelm@53399
   627
    using content_pos_le[of c d] by auto
wenzelm@49698
   628
next
wenzelm@49698
   629
  case False
wenzelm@53399
   630
  then have ab_ne: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
wenzelm@53399
   631
    unfolding interval_ne_empty by auto
wenzelm@53399
   632
  then have ab_ab: "a\<in>{a..b}" "b\<in>{a..b}"
wenzelm@53399
   633
    unfolding mem_interval by auto
himmelma@35172
   634
  have "{c..d} \<noteq> {}" using assms False by auto
wenzelm@53399
   635
  then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
wenzelm@53399
   636
    using assms unfolding interval_ne_empty by auto
wenzelm@49698
   637
  show ?thesis
wenzelm@49698
   638
    unfolding content_def
wenzelm@49698
   639
    unfolding interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
wenzelm@49698
   640
    unfolding if_not_P[OF False] if_not_P[OF `{c..d} \<noteq> {}`]
wenzelm@53408
   641
    apply (rule setprod_mono)
wenzelm@53408
   642
    apply rule
wenzelm@49698
   643
  proof
hoelzl@50526
   644
    fix i :: 'a
wenzelm@53399
   645
    assume i: "i \<in> Basis"
wenzelm@53399
   646
    show "0 \<le> b \<bullet> i - a \<bullet> i"
wenzelm@53399
   647
      using ab_ne[THEN bspec, OF i] i by auto
hoelzl@50526
   648
    show "b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
himmelma@35172
   649
      using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(2),of i]
wenzelm@49698
   650
      using assms[unfolded subset_eq mem_interval,rule_format,OF ab_ab(1),of i]
wenzelm@49698
   651
      using i by auto
wenzelm@49698
   652
  qed
wenzelm@49698
   653
qed
himmelma@35172
   654
himmelma@35172
   655
lemma content_lt_nz: "0 < content {a..b} \<longleftrightarrow> content {a..b} \<noteq> 0"
nipkow@44890
   656
  unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
himmelma@35172
   657
wenzelm@49698
   658
himmelma@35172
   659
subsection {* The notion of a gauge --- simply an open set containing the point. *}
himmelma@35172
   660
wenzelm@53408
   661
definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
wenzelm@53399
   662
wenzelm@53399
   663
lemma gaugeI:
wenzelm@53399
   664
  assumes "\<And>x. x \<in> g x"
wenzelm@53399
   665
    and "\<And>x. open (g x)"
wenzelm@53399
   666
  shows "gauge g"
himmelma@35172
   667
  using assms unfolding gauge_def by auto
himmelma@35172
   668
wenzelm@53399
   669
lemma gaugeD[dest]:
wenzelm@53399
   670
  assumes "gauge d"
wenzelm@53399
   671
  shows "x \<in> d x"
wenzelm@53399
   672
    and "open (d x)"
wenzelm@49698
   673
  using assms unfolding gauge_def by auto
himmelma@35172
   674
himmelma@35172
   675
lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
wenzelm@53399
   676
  unfolding gauge_def by auto
wenzelm@53399
   677
wenzelm@53399
   678
lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
wenzelm@53399
   679
  unfolding gauge_def by auto
himmelma@35172
   680
wenzelm@49698
   681
lemma gauge_trivial[intro]: "gauge (\<lambda>x. ball x 1)"
wenzelm@49698
   682
  by (rule gauge_ball) auto
himmelma@35172
   683
wenzelm@53408
   684
lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
wenzelm@53399
   685
  unfolding gauge_def by auto
himmelma@35172
   686
wenzelm@49698
   687
lemma gauge_inters:
wenzelm@53399
   688
  assumes "finite s"
wenzelm@53399
   689
    and "\<forall>d\<in>s. gauge (f d)"
wenzelm@53408
   690
  shows "gauge (\<lambda>x. \<Inter> {f d x | d. d \<in> s})"
wenzelm@49698
   691
proof -
wenzelm@53399
   692
  have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
wenzelm@53399
   693
    by auto
wenzelm@49698
   694
  show ?thesis
wenzelm@53399
   695
    unfolding gauge_def unfolding *
wenzelm@49698
   696
    using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
wenzelm@49698
   697
qed
wenzelm@49698
   698
wenzelm@53399
   699
lemma gauge_existence_lemma:
wenzelm@53408
   700
  "(\<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
   701
  by (metis zero_less_one)
wenzelm@49698
   702
himmelma@35172
   703
himmelma@35172
   704
subsection {* Divisions. *}
himmelma@35172
   705
wenzelm@53408
   706
definition division_of (infixl "division'_of" 40)
wenzelm@53408
   707
where
wenzelm@53399
   708
  "s division_of i \<longleftrightarrow>
wenzelm@53399
   709
    finite s \<and>
wenzelm@53399
   710
    (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = {a..b})) \<and>
wenzelm@53399
   711
    (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
wenzelm@53399
   712
    (\<Union>s = i)"
himmelma@35172
   713
wenzelm@49698
   714
lemma division_ofD[dest]:
wenzelm@49698
   715
  assumes "s division_of i"
wenzelm@53408
   716
  shows "finite s"
wenzelm@53408
   717
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   718
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
wenzelm@53408
   719
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
wenzelm@53408
   720
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
wenzelm@53408
   721
    and "\<Union>s = i"
wenzelm@49698
   722
  using assms unfolding division_of_def by auto
himmelma@35172
   723
himmelma@35172
   724
lemma division_ofI:
wenzelm@53408
   725
  assumes "finite s"
wenzelm@53408
   726
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   727
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
wenzelm@53408
   728
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
wenzelm@53408
   729
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53399
   730
    and "\<Union>s = i"
wenzelm@53399
   731
  shows "s division_of i"
wenzelm@53399
   732
  using assms unfolding division_of_def by auto
himmelma@35172
   733
himmelma@35172
   734
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
himmelma@35172
   735
  unfolding division_of_def by auto
himmelma@35172
   736
himmelma@35172
   737
lemma division_of_self[intro]: "{a..b} \<noteq> {} \<Longrightarrow> {{a..b}} division_of {a..b}"
himmelma@35172
   738
  unfolding division_of_def by auto
himmelma@35172
   739
wenzelm@53399
   740
lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
wenzelm@53399
   741
  unfolding division_of_def by auto
himmelma@35172
   742
wenzelm@49698
   743
lemma division_of_sing[simp]:
wenzelm@53399
   744
  "s division_of {a..a::'a::ordered_euclidean_space} \<longleftrightarrow> s = {{a..a}}"
wenzelm@53399
   745
  (is "?l = ?r")
wenzelm@49698
   746
proof
wenzelm@49698
   747
  assume ?r
wenzelm@53399
   748
  moreover
wenzelm@53399
   749
  {
wenzelm@49698
   750
    assume "s = {{a}}"
wenzelm@53399
   751
    moreover fix k assume "k\<in>s"
wenzelm@49698
   752
    ultimately have"\<exists>x y. k = {x..y}"
wenzelm@50945
   753
      apply (rule_tac x=a in exI)+
wenzelm@50945
   754
      unfolding interval_sing
wenzelm@50945
   755
      apply auto
wenzelm@50945
   756
      done
wenzelm@49698
   757
  }
wenzelm@53399
   758
  ultimately show ?l
wenzelm@53399
   759
    unfolding division_of_def interval_sing by auto
wenzelm@49698
   760
next
wenzelm@49698
   761
  assume ?l
wenzelm@53408
   762
  note * = conjunctD4[OF this[unfolded division_of_def interval_sing]]
wenzelm@53399
   763
  {
wenzelm@53399
   764
    fix x
wenzelm@53399
   765
    assume x: "x \<in> s" have "x = {a}"
wenzelm@53408
   766
      using *(2)[rule_format,OF x] by auto
wenzelm@53399
   767
  }
wenzelm@53408
   768
  moreover have "s \<noteq> {}"
wenzelm@53408
   769
    using *(4) by auto
wenzelm@53408
   770
  ultimately show ?r
wenzelm@53408
   771
    unfolding interval_sing by auto
wenzelm@49698
   772
qed
himmelma@35172
   773
himmelma@35172
   774
lemma elementary_empty: obtains p where "p division_of {}"
himmelma@35172
   775
  unfolding division_of_trivial by auto
himmelma@35172
   776
wenzelm@49698
   777
lemma elementary_interval: obtains p where "p division_of {a..b}"
wenzelm@49698
   778
  by (metis division_of_trivial division_of_self)
himmelma@35172
   779
himmelma@35172
   780
lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
himmelma@35172
   781
  unfolding division_of_def by auto
himmelma@35172
   782
himmelma@35172
   783
lemma forall_in_division:
wenzelm@53408
   784
  "d division_of i \<Longrightarrow> (\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. {a..b} \<in> d \<longrightarrow> P {a..b})"
nipkow@44890
   785
  unfolding division_of_def by fastforce
himmelma@35172
   786
wenzelm@53399
   787
lemma division_of_subset:
wenzelm@53399
   788
  assumes "p division_of (\<Union>p)"
wenzelm@53399
   789
    and "q \<subseteq> p"
wenzelm@53399
   790
  shows "q division_of (\<Union>q)"
wenzelm@53408
   791
proof (rule division_ofI)
wenzelm@53408
   792
  note * = division_ofD[OF assms(1)]
wenzelm@49698
   793
  show "finite q"
wenzelm@49698
   794
    apply (rule finite_subset)
wenzelm@53408
   795
    using *(1) assms(2)
wenzelm@53408
   796
    apply auto
wenzelm@49698
   797
    done
wenzelm@53399
   798
  {
wenzelm@53399
   799
    fix k
wenzelm@49698
   800
    assume "k \<in> q"
wenzelm@53408
   801
    then have kp: "k \<in> p"
wenzelm@53408
   802
      using assms(2) by auto
wenzelm@53408
   803
    show "k \<subseteq> \<Union>q"
wenzelm@53408
   804
      using `k \<in> q` by auto
wenzelm@53408
   805
    show "\<exists>a b. k = {a..b}"
wenzelm@53408
   806
      using *(4)[OF kp] by auto
wenzelm@53408
   807
    show "k \<noteq> {}"
wenzelm@53408
   808
      using *(3)[OF kp] by auto
wenzelm@53399
   809
  }
wenzelm@49698
   810
  fix k1 k2
wenzelm@49698
   811
  assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
wenzelm@53408
   812
  then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
wenzelm@53399
   813
    using assms(2) by auto
wenzelm@53399
   814
  show "interior k1 \<inter> interior k2 = {}"
wenzelm@53408
   815
    using *(5)[OF **] by auto
wenzelm@49698
   816
qed auto
wenzelm@49698
   817
wenzelm@49698
   818
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@49698
   819
  unfolding division_of_def by auto
himmelma@35172
   820
wenzelm@49970
   821
lemma division_of_content_0:
wenzelm@49970
   822
  assumes "content {a..b} = 0" "d division_of {a..b}"
wenzelm@49970
   823
  shows "\<forall>k\<in>d. content k = 0"
wenzelm@49970
   824
  unfolding forall_in_division[OF assms(2)]
wenzelm@50945
   825
  apply (rule,rule,rule)
wenzelm@50945
   826
  apply (drule division_ofD(2)[OF assms(2)])
wenzelm@50945
   827
  apply (drule content_subset) unfolding assms(1)
wenzelm@49970
   828
proof -
wenzelm@49970
   829
  case goal1
wenzelm@53399
   830
  then show ?case using content_pos_le[of a b] by auto
wenzelm@49970
   831
qed
wenzelm@49970
   832
wenzelm@49970
   833
lemma division_inter:
wenzelm@53408
   834
  fixes s1 s2 :: "'a::ordered_euclidean_space set"
wenzelm@53408
   835
  assumes "p1 division_of s1"
wenzelm@53408
   836
    and "p2 division_of s2"
wenzelm@49970
   837
  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
   838
  (is "?A' division_of _")
wenzelm@49970
   839
proof -
wenzelm@49970
   840
  let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
wenzelm@53408
   841
  have *: "?A' = ?A" by auto
wenzelm@53399
   842
  show ?thesis
wenzelm@53399
   843
    unfolding *
wenzelm@49970
   844
  proof (rule division_ofI)
wenzelm@53399
   845
    have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
wenzelm@53399
   846
      by auto
wenzelm@53399
   847
    moreover have "finite (p1 \<times> p2)"
wenzelm@53399
   848
      using assms unfolding division_of_def by auto
wenzelm@49970
   849
    ultimately show "finite ?A" by auto
wenzelm@53399
   850
    have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
wenzelm@53399
   851
      by auto
wenzelm@49970
   852
    show "\<Union>?A = s1 \<inter> s2"
wenzelm@49970
   853
      apply (rule set_eqI)
wenzelm@49970
   854
      unfolding * and Union_image_eq UN_iff
wenzelm@49970
   855
      using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
wenzelm@49970
   856
      apply auto
wenzelm@49970
   857
      done
wenzelm@53399
   858
    {
wenzelm@53399
   859
      fix k
wenzelm@53399
   860
      assume "k \<in> ?A"
wenzelm@53408
   861
      then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
wenzelm@53399
   862
        by auto
wenzelm@53408
   863
      then show "k \<noteq> {}"
wenzelm@53408
   864
        by auto
wenzelm@49970
   865
      show "k \<subseteq> s1 \<inter> s2"
wenzelm@49970
   866
        using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
wenzelm@49970
   867
        unfolding k by auto
wenzelm@53408
   868
      obtain a1 b1 where k1: "k1 = {a1..b1}"
wenzelm@53408
   869
        using division_ofD(4)[OF assms(1) k(2)] by blast
wenzelm@53408
   870
      obtain a2 b2 where k2: "k2 = {a2..b2}"
wenzelm@53408
   871
        using division_ofD(4)[OF assms(2) k(3)] by blast
wenzelm@49970
   872
      show "\<exists>a b. k = {a..b}"
wenzelm@53408
   873
        unfolding k k1 k2 unfolding inter_interval by auto
wenzelm@53408
   874
    }
wenzelm@49970
   875
    fix k1 k2
wenzelm@53408
   876
    assume "k1 \<in> ?A"
wenzelm@53408
   877
    then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
wenzelm@53408
   878
      by auto
wenzelm@53408
   879
    assume "k2 \<in> ?A"
wenzelm@53408
   880
    then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
wenzelm@53408
   881
      by auto
wenzelm@49970
   882
    assume "k1 \<noteq> k2"
wenzelm@53399
   883
    then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
wenzelm@53399
   884
      unfolding k1 k2 by auto
wenzelm@53408
   885
    have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
wenzelm@53408
   886
      interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
wenzelm@53408
   887
      interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
wenzelm@53408
   888
      interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
wenzelm@49970
   889
    show "interior k1 \<inter> interior k2 = {}"
wenzelm@49970
   890
      unfolding k1 k2
wenzelm@49970
   891
      apply (rule *)
wenzelm@49970
   892
      defer
wenzelm@49970
   893
      apply (rule_tac[1-4] interior_mono)
wenzelm@49970
   894
      using division_ofD(5)[OF assms(1) k1(2) k2(2)]
wenzelm@49970
   895
      using division_ofD(5)[OF assms(2) k1(3) k2(3)]
wenzelm@53408
   896
      using th
wenzelm@53408
   897
      apply auto
wenzelm@53399
   898
      done
wenzelm@49970
   899
  qed
wenzelm@49970
   900
qed
wenzelm@49970
   901
wenzelm@49970
   902
lemma division_inter_1:
wenzelm@53408
   903
  assumes "d division_of i"
wenzelm@53408
   904
    and "{a..b::'a::ordered_euclidean_space} \<subseteq> i"
wenzelm@53408
   905
  shows "{{a..b} \<inter> k | k. k \<in> d \<and> {a..b} \<inter> k \<noteq> {}} division_of {a..b}"
wenzelm@49970
   906
proof (cases "{a..b} = {}")
wenzelm@49970
   907
  case True
wenzelm@53399
   908
  show ?thesis
wenzelm@53399
   909
    unfolding True and division_of_trivial by auto
wenzelm@49970
   910
next
wenzelm@49970
   911
  case False
wenzelm@49970
   912
  have *: "{a..b} \<inter> i = {a..b}" using assms(2) by auto
wenzelm@53399
   913
  show ?thesis
wenzelm@53399
   914
    using division_inter[OF division_of_self[OF False] assms(1)]
wenzelm@53399
   915
    unfolding * by auto
wenzelm@49970
   916
qed
wenzelm@49970
   917
wenzelm@49970
   918
lemma elementary_inter:
wenzelm@53408
   919
  fixes s t :: "'a::ordered_euclidean_space set"
wenzelm@53408
   920
  assumes "p1 division_of s"
wenzelm@53408
   921
    and "p2 division_of t"
himmelma@35172
   922
  shows "\<exists>p. p division_of (s \<inter> t)"
wenzelm@50945
   923
  apply rule
wenzelm@50945
   924
  apply (rule division_inter[OF assms])
wenzelm@50945
   925
  done
wenzelm@49970
   926
wenzelm@49970
   927
lemma elementary_inters:
wenzelm@53408
   928
  assumes "finite f"
wenzelm@53408
   929
    and "f \<noteq> {}"
wenzelm@53408
   930
    and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::ordered_euclidean_space) set)"
wenzelm@49970
   931
  shows "\<exists>p. p division_of (\<Inter> f)"
wenzelm@49970
   932
  using assms
wenzelm@49970
   933
proof (induct f rule: finite_induct)
wenzelm@49970
   934
  case (insert x f)
wenzelm@49970
   935
  show ?case
wenzelm@49970
   936
  proof (cases "f = {}")
wenzelm@49970
   937
    case True
wenzelm@53399
   938
    then show ?thesis
wenzelm@53399
   939
      unfolding True using insert by auto
wenzelm@49970
   940
  next
wenzelm@49970
   941
    case False
wenzelm@53408
   942
    obtain p where "p division_of \<Inter>f"
wenzelm@53408
   943
      using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
wenzelm@53408
   944
    moreover obtain px where "px division_of x"
wenzelm@53408
   945
      using insert(5)[rule_format,OF insertI1] ..
wenzelm@49970
   946
    ultimately show ?thesis
wenzelm@53408
   947
      apply -
wenzelm@49970
   948
      unfolding Inter_insert
wenzelm@53408
   949
      apply (rule elementary_inter)
wenzelm@49970
   950
      apply assumption
wenzelm@49970
   951
      apply assumption
wenzelm@49970
   952
      done
wenzelm@49970
   953
  qed
wenzelm@49970
   954
qed auto
himmelma@35172
   955
himmelma@35172
   956
lemma division_disjoint_union:
wenzelm@53408
   957
  assumes "p1 division_of s1"
wenzelm@53408
   958
    and "p2 division_of s2"
wenzelm@53408
   959
    and "interior s1 \<inter> interior s2 = {}"
wenzelm@50945
   960
  shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
wenzelm@50945
   961
proof (rule division_ofI)
wenzelm@53408
   962
  note d1 = division_ofD[OF assms(1)]
wenzelm@53408
   963
  note d2 = division_ofD[OF assms(2)]
wenzelm@53408
   964
  show "finite (p1 \<union> p2)"
wenzelm@53408
   965
    using d1(1) d2(1) by auto
wenzelm@53408
   966
  show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
wenzelm@53408
   967
    using d1(6) d2(6) by auto
wenzelm@50945
   968
  {
wenzelm@50945
   969
    fix k1 k2
wenzelm@50945
   970
    assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
wenzelm@50945
   971
    moreover
wenzelm@50945
   972
    let ?g="interior k1 \<inter> interior k2 = {}"
wenzelm@50945
   973
    {
wenzelm@50945
   974
      assume as: "k1\<in>p1" "k2\<in>p2"
wenzelm@50945
   975
      have ?g
wenzelm@50945
   976
        using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
wenzelm@50945
   977
        using assms(3) by blast
wenzelm@50945
   978
    }
wenzelm@50945
   979
    moreover
wenzelm@50945
   980
    {
wenzelm@50945
   981
      assume as: "k1\<in>p2" "k2\<in>p1"
wenzelm@50945
   982
      have ?g
wenzelm@50945
   983
        using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
wenzelm@50945
   984
        using assms(3) by blast
wenzelm@50945
   985
    }
wenzelm@53399
   986
    ultimately show ?g
wenzelm@53399
   987
      using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
wenzelm@50945
   988
  }
wenzelm@50945
   989
  fix k
wenzelm@50945
   990
  assume k: "k \<in> p1 \<union> p2"
wenzelm@53408
   991
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
   992
    using k d1(2) d2(2) by auto
wenzelm@53408
   993
  show "k \<noteq> {}"
wenzelm@53408
   994
    using k d1(3) d2(3) by auto
wenzelm@53408
   995
  show "\<exists>a b. k = {a..b}"
wenzelm@53408
   996
    using k d1(4) d2(4) by auto
wenzelm@50945
   997
qed
himmelma@35172
   998
himmelma@35172
   999
lemma partial_division_extend_1:
wenzelm@53408
  1000
  fixes a b c d :: "'a::ordered_euclidean_space"
wenzelm@53408
  1001
  assumes incl: "{c..d} \<subseteq> {a..b}"
wenzelm@50945
  1002
    and nonempty: "{c..d} \<noteq> {}"
himmelma@35172
  1003
  obtains p where "p division_of {a..b}" "{c..d} \<in> p"
hoelzl@50526
  1004
proof
wenzelm@53408
  1005
  let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
wenzelm@53408
  1006
    {(\<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
  1007
  def p \<equiv> "?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
hoelzl@50526
  1008
hoelzl@50526
  1009
  show "{c .. d} \<in> p"
hoelzl@50526
  1010
    unfolding p_def
hoelzl@50526
  1011
    by (auto simp add: interval_eq_empty eucl_le[where 'a='a]
wenzelm@53399
  1012
        intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
wenzelm@50945
  1013
  {
wenzelm@50945
  1014
    fix i :: 'a
wenzelm@50945
  1015
    assume "i \<in> Basis"
hoelzl@50526
  1016
    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"
wenzelm@50945
  1017
      unfolding interval_eq_empty subset_interval by (auto simp: not_le)
wenzelm@50945
  1018
  }
hoelzl@50526
  1019
  note ord = this
hoelzl@50526
  1020
hoelzl@50526
  1021
  show "p division_of {a..b}"
hoelzl@50526
  1022
  proof (rule division_ofI)
wenzelm@53399
  1023
    show "finite p"
wenzelm@53399
  1024
      unfolding p_def by (auto intro!: finite_PiE)
wenzelm@50945
  1025
    {
wenzelm@50945
  1026
      fix k
wenzelm@50945
  1027
      assume "k \<in> p"
wenzelm@53015
  1028
      then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
hoelzl@50526
  1029
        by (auto simp: p_def)
wenzelm@53408
  1030
      then show "\<exists>a b. k = {a..b}"
wenzelm@53408
  1031
        by auto
hoelzl@50526
  1032
      have "k \<subseteq> {a..b} \<and> k \<noteq> {}"
hoelzl@50526
  1033
      proof (simp add: k interval_eq_empty subset_interval not_less, safe)
wenzelm@53374
  1034
        fix i :: 'a
wenzelm@53374
  1035
        assume i: "i \<in> Basis"
wenzelm@50945
  1036
        with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
hoelzl@50526
  1037
          by (auto simp: PiE_iff)
wenzelm@53374
  1038
        with i ord[of i]
wenzelm@50945
  1039
        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"
hoelzl@50526
  1040
          by (auto simp: subset_iff eucl_le[where 'a='a])
hoelzl@50526
  1041
      qed
wenzelm@53408
  1042
      then show "k \<noteq> {}" "k \<subseteq> {a .. b}"
wenzelm@53408
  1043
        by auto
wenzelm@50945
  1044
      {
wenzelm@53408
  1045
        fix l
wenzelm@53408
  1046
        assume "l \<in> p"
wenzelm@53015
  1047
        then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
wenzelm@50945
  1048
          by (auto simp: p_def)
wenzelm@50945
  1049
        assume "l \<noteq> k"
wenzelm@50945
  1050
        have "\<exists>i\<in>Basis. f i \<noteq> g i"
wenzelm@50945
  1051
        proof (rule ccontr)
wenzelm@53408
  1052
          assume "\<not> ?thesis"
wenzelm@50945
  1053
          with f g have "f = g"
wenzelm@50945
  1054
            by (auto simp: PiE_iff extensional_def intro!: ext)
wenzelm@50945
  1055
          with `l \<noteq> k` show False
wenzelm@50945
  1056
            by (simp add: l k)
wenzelm@50945
  1057
        qed
wenzelm@53408
  1058
        then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
wenzelm@53408
  1059
        then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
wenzelm@50945
  1060
            "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
wenzelm@50945
  1061
          using f g by (auto simp: PiE_iff)
wenzelm@53408
  1062
        with * ord[of i] show "interior l \<inter> interior k = {}"
wenzelm@50945
  1063
          by (auto simp add: l k interior_closed_interval disjoint_interval intro!: bexI[of _ i])
wenzelm@50945
  1064
      }
wenzelm@53408
  1065
      note `k \<subseteq> {a.. b}`
wenzelm@50945
  1066
    }
hoelzl@50526
  1067
    moreover
wenzelm@50945
  1068
    {
wenzelm@50945
  1069
      fix x assume x: "x \<in> {a .. b}"
hoelzl@50526
  1070
      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
  1071
      proof
wenzelm@53408
  1072
        fix i :: 'a
wenzelm@53408
  1073
        assume "i \<in> Basis"
wenzelm@53399
  1074
        with x ord[of i]
hoelzl@50526
  1075
        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
  1076
            (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
hoelzl@50526
  1077
          by (auto simp: eucl_le[where 'a='a])
hoelzl@50526
  1078
        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
  1079
          by auto
hoelzl@50526
  1080
      qed
wenzelm@53408
  1081
      then obtain f where
wenzelm@53408
  1082
        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
  1083
        unfolding bchoice_iff ..
wenzelm@53374
  1084
      moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
hoelzl@50526
  1085
        by auto
hoelzl@50526
  1086
      moreover from f have "x \<in> ?B (restrict f Basis)"
hoelzl@50526
  1087
        by (auto simp: mem_interval eucl_le[where 'a='a])
hoelzl@50526
  1088
      ultimately have "\<exists>k\<in>p. x \<in> k"
wenzelm@53408
  1089
        unfolding p_def by blast
wenzelm@53408
  1090
    }
hoelzl@50526
  1091
    ultimately show "\<Union>p = {a..b}"
hoelzl@50526
  1092
      by auto
hoelzl@50526
  1093
  qed
hoelzl@50526
  1094
qed
himmelma@35172
  1095
wenzelm@50945
  1096
lemma partial_division_extend_interval:
wenzelm@50945
  1097
  assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> {a..b}"
wenzelm@50945
  1098
  obtains q where "p \<subseteq> q" "q division_of {a..b::'a::ordered_euclidean_space}"
wenzelm@50945
  1099
proof (cases "p = {}")
wenzelm@50945
  1100
  case True
wenzelm@53408
  1101
  obtain q where "q division_of {a..b}"
wenzelm@53408
  1102
    by (rule elementary_interval)
wenzelm@53399
  1103
  then show ?thesis
wenzelm@50945
  1104
    apply -
wenzelm@50945
  1105
    apply (rule that[of q])
wenzelm@50945
  1106
    unfolding True
wenzelm@50945
  1107
    apply auto
wenzelm@50945
  1108
    done
wenzelm@50945
  1109
next
wenzelm@50945
  1110
  case False
wenzelm@50945
  1111
  note p = division_ofD[OF assms(1)]
wenzelm@53408
  1112
  have *: "\<forall>k\<in>p. \<exists>q. q division_of {a..b} \<and> k \<in> q"
wenzelm@50945
  1113
  proof
wenzelm@50945
  1114
    case goal1
wenzelm@53408
  1115
    obtain c d where k: "k = {c..d}"
wenzelm@53408
  1116
      using p(4)[OF goal1] by blast
wenzelm@50945
  1117
    have *: "{c..d} \<subseteq> {a..b}" "{c..d} \<noteq> {}"
immler@54775
  1118
      using p(2,3)[OF goal1, unfolded k] using assms(2)
immler@54775
  1119
        by auto
wenzelm@53408
  1120
    obtain q where "q division_of {a..b}" "{c..d} \<in> q"
wenzelm@53408
  1121
      by (rule partial_division_extend_1[OF *])
wenzelm@53408
  1122
    then show ?case
wenzelm@53408
  1123
      unfolding k by auto
wenzelm@50945
  1124
  qed
wenzelm@53408
  1125
  obtain q where q: "\<And>x. x \<in> p \<Longrightarrow> q x division_of {a..b}" "\<And>x. x \<in> p \<Longrightarrow> x \<in> q x"
wenzelm@53408
  1126
    using bchoice[OF *] by blast
wenzelm@53408
  1127
  have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
wenzelm@53408
  1128
    apply rule
wenzelm@53408
  1129
    apply (rule_tac p="q x" in division_of_subset)
wenzelm@50945
  1130
  proof -
wenzelm@50945
  1131
    fix x
wenzelm@53408
  1132
    assume x: "x \<in> p"
wenzelm@50945
  1133
    show "q x division_of \<Union>q x"
wenzelm@50945
  1134
      apply -
wenzelm@50945
  1135
      apply (rule division_ofI)
wenzelm@50945
  1136
      using division_ofD[OF q(1)[OF x]]
wenzelm@50945
  1137
      apply auto
wenzelm@50945
  1138
      done
wenzelm@53408
  1139
    show "q x - {x} \<subseteq> q x"
wenzelm@53408
  1140
      by auto
wenzelm@50945
  1141
  qed
wenzelm@53399
  1142
  then have "\<exists>d. d division_of \<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)"
wenzelm@50945
  1143
    apply -
wenzelm@50945
  1144
    apply (rule elementary_inters)
wenzelm@50945
  1145
    apply (rule finite_imageI[OF p(1)])
wenzelm@50945
  1146
    unfolding image_is_empty
wenzelm@50945
  1147
    apply (rule False)
wenzelm@50945
  1148
    apply auto
wenzelm@50945
  1149
    done
wenzelm@53408
  1150
  then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
wenzelm@50945
  1151
  show ?thesis
wenzelm@50945
  1152
    apply (rule that[of "d \<union> p"])
wenzelm@50945
  1153
  proof -
wenzelm@53408
  1154
    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
wenzelm@53408
  1155
    have *: "{a..b} = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
wenzelm@50945
  1156
      apply (rule *[OF False])
wenzelm@50945
  1157
    proof
wenzelm@50945
  1158
      fix i
wenzelm@53408
  1159
      assume i: "i \<in> p"
wenzelm@50945
  1160
      show "\<Union>(q i - {i}) \<union> i = {a..b}"
wenzelm@50945
  1161
        using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
wenzelm@50945
  1162
    qed
wenzelm@50945
  1163
    show "d \<union> p division_of {a..b}"
wenzelm@50945
  1164
      unfolding *
wenzelm@50945
  1165
      apply (rule division_disjoint_union[OF d assms(1)])
wenzelm@50945
  1166
      apply (rule inter_interior_unions_intervals)
wenzelm@50945
  1167
      apply (rule p open_interior ballI)+
wenzelm@53408
  1168
      apply assumption
wenzelm@53408
  1169
    proof
wenzelm@50945
  1170
      fix k
wenzelm@53408
  1171
      assume k: "k \<in> p"
wenzelm@53408
  1172
      have *: "\<And>u t s. u \<subseteq> s \<Longrightarrow> s \<inter> t = {} \<Longrightarrow> u \<inter> t = {}"
wenzelm@53408
  1173
        by auto
haftmann@52141
  1174
      show "interior (\<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)) \<inter> interior k = {}"
wenzelm@50945
  1175
        apply (rule *[of _ "interior (\<Union>(q k - {k}))"])
wenzelm@50945
  1176
        defer
wenzelm@50945
  1177
        apply (subst Int_commute)
wenzelm@50945
  1178
        apply (rule inter_interior_unions_intervals)
wenzelm@50945
  1179
      proof -
wenzelm@50945
  1180
        note qk=division_ofD[OF q(1)[OF k]]
wenzelm@53408
  1181
        show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = {a..b}"
wenzelm@53408
  1182
          using qk by auto
wenzelm@50945
  1183
        show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
wenzelm@50945
  1184
          using qk(5) using q(2)[OF k] by auto
wenzelm@53408
  1185
        have *: "\<And>x s. x \<in> s \<Longrightarrow> \<Inter>s \<subseteq> x"
wenzelm@53408
  1186
          by auto
haftmann@52141
  1187
        show "interior (\<Inter> ((\<lambda>i. \<Union>(q i - {i})) ` p)) \<subseteq> interior (\<Union>(q k - {k}))"
wenzelm@50945
  1188
          apply (rule interior_mono *)+
wenzelm@53408
  1189
          using k
wenzelm@53408
  1190
          apply auto
wenzelm@53408
  1191
          done
wenzelm@50945
  1192
      qed
wenzelm@50945
  1193
    qed
wenzelm@50945
  1194
  qed auto
wenzelm@50945
  1195
qed
himmelma@35172
  1196
wenzelm@53399
  1197
lemma elementary_bounded[dest]:
wenzelm@53408
  1198
  fixes s :: "'a::ordered_euclidean_space set"
wenzelm@53408
  1199
  shows "p division_of s \<Longrightarrow> bounded s"
wenzelm@53408
  1200
  unfolding division_of_def by (metis bounded_Union bounded_interval)
wenzelm@53399
  1201
wenzelm@53399
  1202
lemma elementary_subset_interval:
wenzelm@53399
  1203
  "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> {a..b::'a::ordered_euclidean_space}"
wenzelm@50945
  1204
  by (meson elementary_bounded bounded_subset_closed_interval)
wenzelm@50945
  1205
wenzelm@50945
  1206
lemma division_union_intervals_exists:
wenzelm@53408
  1207
  fixes a b :: "'a::ordered_euclidean_space"
wenzelm@53408
  1208
  assumes "{a..b} \<noteq> {}"
wenzelm@50945
  1209
  obtains p where "(insert {a..b} p) division_of ({a..b} \<union> {c..d})"
wenzelm@50945
  1210
proof (cases "{c..d} = {}")
wenzelm@50945
  1211
  case True
wenzelm@50945
  1212
  show ?thesis
wenzelm@50945
  1213
    apply (rule that[of "{}"])
wenzelm@50945
  1214
    unfolding True
wenzelm@50945
  1215
    using assms
wenzelm@50945
  1216
    apply auto
wenzelm@50945
  1217
    done
wenzelm@50945
  1218
next
wenzelm@50945
  1219
  case False
wenzelm@50945
  1220
  show ?thesis
wenzelm@50945
  1221
  proof (cases "{a..b} \<inter> {c..d} = {}")
wenzelm@50945
  1222
    case True
wenzelm@53408
  1223
    have *: "\<And>a b. {a, b} = {a} \<union> {b}" by auto
wenzelm@50945
  1224
    show ?thesis
wenzelm@50945
  1225
      apply (rule that[of "{{c..d}}"])
wenzelm@50945
  1226
      unfolding *
wenzelm@50945
  1227
      apply (rule division_disjoint_union)
wenzelm@53408
  1228
      using `{c..d} \<noteq> {}` True assms
wenzelm@50945
  1229
      using interior_subset
wenzelm@50945
  1230
      apply auto
wenzelm@50945
  1231
      done
wenzelm@50945
  1232
  next
wenzelm@50945
  1233
    case False
wenzelm@50945
  1234
    obtain u v where uv: "{a..b} \<inter> {c..d} = {u..v}"
wenzelm@50945
  1235
      unfolding inter_interval by auto
wenzelm@50945
  1236
    have *: "{u..v} \<subseteq> {c..d}" using uv by auto
wenzelm@53408
  1237
    obtain p where "p division_of {c..d}" "{u..v} \<in> p"
wenzelm@53408
  1238
      by (rule partial_division_extend_1[OF * False[unfolded uv]])
wenzelm@53408
  1239
    note p = this division_ofD[OF this(1)]
wenzelm@50945
  1240
    have *: "{a..b} \<union> {c..d} = {a..b} \<union> \<Union>(p - {{u..v}})" "\<And>x s. insert x s = {x} \<union> s"
wenzelm@53399
  1241
      using p(8) unfolding uv[symmetric] by auto
wenzelm@50945
  1242
    show ?thesis
wenzelm@50945
  1243
      apply (rule that[of "p - {{u..v}}"])
wenzelm@50945
  1244
      unfolding *(1)
wenzelm@50945
  1245
      apply (subst *(2))
wenzelm@50945
  1246
      apply (rule division_disjoint_union)
wenzelm@50945
  1247
      apply (rule, rule assms)
wenzelm@50945
  1248
      apply (rule division_of_subset[of p])
wenzelm@50945
  1249
      apply (rule division_of_union_self[OF p(1)])
wenzelm@50945
  1250
      defer
wenzelm@53399
  1251
      unfolding interior_inter[symmetric]
wenzelm@50945
  1252
    proof -
wenzelm@50945
  1253
      have *: "\<And>cd p uv ab. p \<subseteq> cd \<Longrightarrow> ab \<inter> cd = uv \<Longrightarrow> ab \<inter> p = uv \<inter> p" by auto
wenzelm@53399
  1254
      have "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = interior({u..v} \<inter> \<Union>(p - {{u..v}}))"
wenzelm@50945
  1255
        apply (rule arg_cong[of _ _ interior])
wenzelm@50945
  1256
        apply (rule *[OF _ uv])
wenzelm@50945
  1257
        using p(8)
wenzelm@50945
  1258
        apply auto
wenzelm@50945
  1259
        done
wenzelm@50945
  1260
      also have "\<dots> = {}"
wenzelm@50945
  1261
        unfolding interior_inter
wenzelm@50945
  1262
        apply (rule inter_interior_unions_intervals)
wenzelm@50945
  1263
        using p(6) p(7)[OF p(2)] p(3)
wenzelm@50945
  1264
        apply auto
wenzelm@50945
  1265
        done
wenzelm@50945
  1266
      finally show "interior ({a..b} \<inter> \<Union>(p - {{u..v}})) = {}" .
wenzelm@50945
  1267
    qed auto
wenzelm@50945
  1268
  qed
wenzelm@50945
  1269
qed
himmelma@35172
  1270
wenzelm@53399
  1271
lemma division_of_unions:
wenzelm@53399
  1272
  assumes "finite f"
wenzelm@53408
  1273
    and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@53399
  1274
    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
  1275
  shows "\<Union>f division_of \<Union>\<Union>f"
wenzelm@53399
  1276
  apply (rule division_ofI)
wenzelm@53399
  1277
  prefer 5
wenzelm@53399
  1278
  apply (rule assms(3)|assumption)+
wenzelm@53399
  1279
  apply (rule finite_Union assms(1))+
wenzelm@53399
  1280
  prefer 3
wenzelm@53399
  1281
  apply (erule UnionE)
wenzelm@53399
  1282
  apply (rule_tac s=X in division_ofD(3)[OF assms(2)])
wenzelm@53399
  1283
  using division_ofD[OF assms(2)]
wenzelm@53399
  1284
  apply auto
wenzelm@53399
  1285
  done
wenzelm@53399
  1286
wenzelm@53399
  1287
lemma elementary_union_interval:
wenzelm@53408
  1288
  fixes a b :: "'a::ordered_euclidean_space"
wenzelm@53399
  1289
  assumes "p division_of \<Union>p"
wenzelm@53408
  1290
  obtains q where "q division_of ({a..b} \<union> \<Union>p)"
wenzelm@53399
  1291
proof -
wenzelm@53399
  1292
  note assm = division_ofD[OF assms]
wenzelm@53408
  1293
  have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
wenzelm@53399
  1294
    by auto
wenzelm@53399
  1295
  have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
wenzelm@53399
  1296
    by auto
wenzelm@53399
  1297
  {
wenzelm@53399
  1298
    presume "p = {} \<Longrightarrow> thesis"
wenzelm@53399
  1299
      "{a..b} = {} \<Longrightarrow> thesis"
wenzelm@53399
  1300
      "{a..b} \<noteq> {} \<Longrightarrow> interior {a..b} = {} \<Longrightarrow> thesis"
wenzelm@53399
  1301
      "p \<noteq> {} \<Longrightarrow> interior {a..b}\<noteq>{} \<Longrightarrow> {a..b} \<noteq> {} \<Longrightarrow> thesis"
wenzelm@53399
  1302
    then show thesis by auto
wenzelm@53399
  1303
  next
wenzelm@53399
  1304
    assume as: "p = {}"
wenzelm@53408
  1305
    obtain p where "p division_of {a..b}"
wenzelm@53408
  1306
      by (rule elementary_interval)
wenzelm@53399
  1307
    then show thesis
wenzelm@53408
  1308
      apply -
wenzelm@53408
  1309
      apply (rule that[of p])
wenzelm@53399
  1310
      unfolding as
wenzelm@53399
  1311
      apply auto
wenzelm@53399
  1312
      done
wenzelm@53399
  1313
  next
wenzelm@53399
  1314
    assume as: "{a..b} = {}"
wenzelm@53399
  1315
    show thesis
wenzelm@53399
  1316
      apply (rule that)
wenzelm@53399
  1317
      unfolding as
wenzelm@53399
  1318
      using assms
wenzelm@53399
  1319
      apply auto
wenzelm@53399
  1320
      done
wenzelm@53399
  1321
  next
wenzelm@53399
  1322
    assume as: "interior {a..b} = {}" "{a..b} \<noteq> {}"
wenzelm@53399
  1323
    show thesis
wenzelm@53399
  1324
      apply (rule that[of "insert {a..b} p"],rule division_ofI)
wenzelm@53399
  1325
      unfolding finite_insert
wenzelm@53399
  1326
      apply (rule assm(1)) unfolding Union_insert
wenzelm@53399
  1327
      using assm(2-4) as
wenzelm@53399
  1328
      apply -
immler@54775
  1329
      apply (fast dest: assm(5))+
wenzelm@53399
  1330
      done
wenzelm@53399
  1331
  next
wenzelm@53399
  1332
    assume as: "p \<noteq> {}" "interior {a..b} \<noteq> {}" "{a..b} \<noteq> {}"
wenzelm@53399
  1333
    have "\<forall>k\<in>p. \<exists>q. (insert {a..b} q) division_of ({a..b} \<union> k)"
wenzelm@53399
  1334
    proof
wenzelm@53399
  1335
      case goal1
wenzelm@53408
  1336
      from assm(4)[OF this] obtain c d where "k = {c..d}" by blast
wenzelm@53399
  1337
      then show ?case
wenzelm@53399
  1338
        apply -
wenzelm@53408
  1339
        apply (rule division_union_intervals_exists[OF as(3), of c d])
wenzelm@53399
  1340
        apply auto
wenzelm@53399
  1341
        done
wenzelm@53399
  1342
    qed
wenzelm@53408
  1343
    from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert {a..b} (q x) division_of {a..b} \<union> x" ..
wenzelm@53408
  1344
    note q = division_ofD[OF this[rule_format]]
wenzelm@53399
  1345
    let ?D = "\<Union>{insert {a..b} (q k) | k. k \<in> p}"
wenzelm@53399
  1346
    show thesis
wenzelm@53399
  1347
      apply (rule that[of "?D"])
wenzelm@53408
  1348
      apply (rule division_ofI)
wenzelm@53408
  1349
    proof -
wenzelm@53399
  1350
      have *: "{insert {a..b} (q k) |k. k \<in> p} = (\<lambda>k. insert {a..b} (q k)) ` p"
wenzelm@53399
  1351
        by auto
wenzelm@53399
  1352
      show "finite ?D"
wenzelm@53399
  1353
        apply (rule finite_Union)
wenzelm@53399
  1354
        unfolding *
wenzelm@53399
  1355
        apply (rule finite_imageI)
wenzelm@53399
  1356
        using assm(1) q(1)
wenzelm@53399
  1357
        apply auto
wenzelm@53399
  1358
        done
wenzelm@53399
  1359
      show "\<Union>?D = {a..b} \<union> \<Union>p"
wenzelm@53399
  1360
        unfolding * lem1
wenzelm@53408
  1361
        unfolding lem2[OF as(1), of "{a..b}", symmetric]
wenzelm@53399
  1362
        using q(6)
wenzelm@53399
  1363
        by auto
wenzelm@53399
  1364
      fix k
wenzelm@53408
  1365
      assume k: "k \<in> ?D"
wenzelm@53408
  1366
      then show "k \<subseteq> {a..b} \<union> \<Union>p"
wenzelm@53408
  1367
        using q(2) by auto
wenzelm@53399
  1368
      show "k \<noteq> {}"
wenzelm@53408
  1369
        using q(3) k by auto
wenzelm@53408
  1370
      show "\<exists>a b. k = {a..b}"
wenzelm@53408
  1371
        using q(4) k by auto
wenzelm@53399
  1372
      fix k'
wenzelm@53408
  1373
      assume k': "k' \<in> ?D" "k \<noteq> k'"
wenzelm@53408
  1374
      obtain x where x: "k \<in> insert {a..b} (q x)" "x\<in>p"
wenzelm@53408
  1375
        using k by auto
wenzelm@53399
  1376
      obtain x' where x': "k'\<in>insert {a..b} (q x')" "x'\<in>p"
wenzelm@53399
  1377
        using k' by auto
wenzelm@53399
  1378
      show "interior k \<inter> interior k' = {}"
wenzelm@53399
  1379
      proof (cases "x = x'")
wenzelm@53399
  1380
        case True
wenzelm@53399
  1381
        show ?thesis
wenzelm@53399
  1382
          apply(rule q(5))
wenzelm@53399
  1383
          using x x' k'
wenzelm@53399
  1384
          unfolding True
wenzelm@53399
  1385
          apply auto
wenzelm@53399
  1386
          done
wenzelm@53399
  1387
      next
wenzelm@53399
  1388
        case False
wenzelm@53399
  1389
        {
wenzelm@53399
  1390
          presume "k = {a..b} \<Longrightarrow> ?thesis"
wenzelm@53399
  1391
            and "k' = {a..b} \<Longrightarrow> ?thesis"
wenzelm@53399
  1392
            and "k \<noteq> {a..b} \<Longrightarrow> k' \<noteq> {a..b} \<Longrightarrow> ?thesis"
wenzelm@53399
  1393
          then show ?thesis by auto
wenzelm@53399
  1394
        next
wenzelm@53399
  1395
          assume as': "k  = {a..b}"
wenzelm@53399
  1396
          show ?thesis
wenzelm@53408
  1397
            apply (rule q(5))
wenzelm@53408
  1398
            using x' k'(2)
wenzelm@53408
  1399
            unfolding as'
wenzelm@53408
  1400
            apply auto
wenzelm@53408
  1401
            done
wenzelm@53399
  1402
        next
wenzelm@53399
  1403
          assume as': "k' = {a..b}"
wenzelm@53399
  1404
          show ?thesis
wenzelm@53399
  1405
            apply (rule q(5))
wenzelm@53399
  1406
            using x  k'(2)
wenzelm@53399
  1407
            unfolding as'
wenzelm@53399
  1408
            apply auto
wenzelm@53399
  1409
            done
wenzelm@53399
  1410
        }
wenzelm@53399
  1411
        assume as': "k \<noteq> {a..b}" "k' \<noteq> {a..b}"
wenzelm@53408
  1412
        obtain c d where k: "k = {c..d}"
wenzelm@53408
  1413
          using q(4)[OF x(2,1)] by blast
wenzelm@53408
  1414
        have "interior k \<inter> interior {a..b} = {}"
wenzelm@53408
  1415
          apply (rule q(5))
wenzelm@53408
  1416
          using x k'(2)
wenzelm@53399
  1417
          using as'
wenzelm@53399
  1418
          apply auto
wenzelm@53399
  1419
          done
wenzelm@53399
  1420
        then have "interior k \<subseteq> interior x"
wenzelm@53399
  1421
          apply -
wenzelm@53408
  1422
          apply (rule interior_subset_union_intervals[OF k _ as(2) q(2)[OF x(2,1)]])
wenzelm@53399
  1423
          apply auto
wenzelm@53399
  1424
          done
wenzelm@53399
  1425
        moreover
wenzelm@53408
  1426
        obtain c d where c_d: "k' = {c..d}"
wenzelm@53408
  1427
          using q(4)[OF x'(2,1)] by blast
wenzelm@53399
  1428
        have "interior k' \<inter> interior {a..b} = {}"
wenzelm@53399
  1429
          apply (rule q(5))
wenzelm@53399
  1430
          using x' k'(2)
wenzelm@53399
  1431
          using as'
wenzelm@53399
  1432
          apply auto
wenzelm@53399
  1433
          done
wenzelm@53399
  1434
        then have "interior k' \<subseteq> interior x'"
wenzelm@53399
  1435
          apply -
wenzelm@53399
  1436
          apply (rule interior_subset_union_intervals[OF c_d _ as(2) q(2)[OF x'(2,1)]])
wenzelm@53399
  1437
          apply auto
wenzelm@53399
  1438
          done
wenzelm@53399
  1439
        ultimately show ?thesis
wenzelm@53399
  1440
          using assm(5)[OF x(2) x'(2) False] by auto
wenzelm@53399
  1441
      qed
wenzelm@53399
  1442
    qed
wenzelm@53399
  1443
  }
wenzelm@53399
  1444
qed
himmelma@35172
  1445
himmelma@35172
  1446
lemma elementary_unions_intervals:
wenzelm@53399
  1447
  assumes fin: "finite f"
wenzelm@53399
  1448
    and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = {a..b::'a::ordered_euclidean_space}"
wenzelm@53399
  1449
  obtains p where "p division_of (\<Union>f)"
wenzelm@53399
  1450
proof -
wenzelm@53399
  1451
  have "\<exists>p. p division_of (\<Union>f)"
wenzelm@53399
  1452
  proof (induct_tac f rule:finite_subset_induct)
himmelma@35172
  1453
    show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
wenzelm@53399
  1454
  next
wenzelm@53399
  1455
    fix x F
wenzelm@53399
  1456
    assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
wenzelm@53408
  1457
    from this(3) obtain p where p: "p division_of \<Union>F" ..
wenzelm@53408
  1458
    from assms(2)[OF as(4)] obtain a b where x: "x = {a..b}" by blast
wenzelm@53399
  1459
    have *: "\<Union>F = \<Union>p"
wenzelm@53399
  1460
      using division_ofD[OF p] by auto
wenzelm@53399
  1461
    show "\<exists>p. p division_of \<Union>insert x F"
wenzelm@53399
  1462
      using elementary_union_interval[OF p[unfolded *], of a b]
wenzelm@53408
  1463
      unfolding Union_insert x * by auto
wenzelm@53408
  1464
  qed (insert assms, auto)
wenzelm@53399
  1465
  then show ?thesis
wenzelm@53399
  1466
    apply -
wenzelm@53399
  1467
    apply (erule exE)
wenzelm@53399
  1468
    apply (rule that)
wenzelm@53399
  1469
    apply auto
wenzelm@53399
  1470
    done
wenzelm@53399
  1471
qed
wenzelm@53399
  1472
wenzelm@53399
  1473
lemma elementary_union:
wenzelm@53408
  1474
  fixes s t :: "'a::ordered_euclidean_space set"
wenzelm@53399
  1475
  assumes "ps division_of s"
wenzelm@53408
  1476
    and "pt division_of t"
himmelma@35172
  1477
  obtains p where "p division_of (s \<union> t)"
wenzelm@53399
  1478
proof -
wenzelm@53399
  1479
  have "s \<union> t = \<Union>ps \<union> \<Union>pt"
wenzelm@53399
  1480
    using assms unfolding division_of_def by auto
wenzelm@53399
  1481
  then have *: "\<Union>(ps \<union> pt) = s \<union> t" by auto
wenzelm@53399
  1482
  show ?thesis
wenzelm@53399
  1483
    apply -
wenzelm@53408
  1484
    apply (rule elementary_unions_intervals[of "ps \<union> pt"])
wenzelm@53399
  1485
    unfolding *
wenzelm@53399
  1486
    prefer 3
wenzelm@53399
  1487
    apply (rule_tac p=p in that)
wenzelm@53399
  1488
    using assms[unfolded division_of_def]
wenzelm@53399
  1489
    apply auto
wenzelm@53399
  1490
    done
wenzelm@53399
  1491
qed
wenzelm@53399
  1492
wenzelm@53399
  1493
lemma partial_division_extend:
wenzelm@53399
  1494
  fixes t :: "'a::ordered_euclidean_space set"
wenzelm@53399
  1495
  assumes "p division_of s"
wenzelm@53399
  1496
    and "q division_of t"
wenzelm@53399
  1497
    and "s \<subseteq> t"
wenzelm@53399
  1498
  obtains r where "p \<subseteq> r" and "r division_of t"
wenzelm@53399
  1499
proof -
himmelma@35172
  1500
  note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
wenzelm@53399
  1501
  obtain a b where ab: "t \<subseteq> {a..b}"
wenzelm@53399
  1502
    using elementary_subset_interval[OF assms(2)] by auto
wenzelm@53408
  1503
  obtain r1 where "p \<subseteq> r1" "r1 division_of {a..b}"
wenzelm@53399
  1504
    apply (rule partial_division_extend_interval)
wenzelm@53399
  1505
    apply (rule assms(1)[unfolded divp(6)[symmetric]])
wenzelm@53399
  1506
    apply (rule subset_trans)
wenzelm@53399
  1507
    apply (rule ab assms[unfolded divp(6)[symmetric]])+
wenzelm@53408
  1508
    apply assumption
wenzelm@53399
  1509
    done
wenzelm@53399
  1510
  note r1 = this division_ofD[OF this(2)]
wenzelm@53408
  1511
  obtain p' where "p' division_of \<Union>(r1 - p)"
wenzelm@53399
  1512
    apply (rule elementary_unions_intervals[of "r1 - p"])
wenzelm@53399
  1513
    using r1(3,6)
wenzelm@53399
  1514
    apply auto
wenzelm@53399
  1515
    done
wenzelm@53399
  1516
  then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
wenzelm@53399
  1517
    apply -
wenzelm@53399
  1518
    apply (drule elementary_inter[OF _ assms(2)[unfolded divq(6)[symmetric]]])
wenzelm@53399
  1519
    apply auto
wenzelm@53399
  1520
    done
wenzelm@53399
  1521
  {
wenzelm@53399
  1522
    fix x
wenzelm@53399
  1523
    assume x: "x \<in> t" "x \<notin> s"
wenzelm@53399
  1524
    then have "x\<in>\<Union>r1"
wenzelm@53399
  1525
      unfolding r1 using ab by auto
wenzelm@53408
  1526
    then obtain r where r: "r \<in> r1" "x \<in> r"
wenzelm@53408
  1527
      unfolding Union_iff ..
wenzelm@53399
  1528
    moreover
wenzelm@53399
  1529
    have "r \<notin> p"
wenzelm@53399
  1530
    proof
wenzelm@53399
  1531
      assume "r \<in> p"
wenzelm@53399
  1532
      then have "x \<in> s" using divp(2) r by auto
wenzelm@53399
  1533
      then show False using x by auto
wenzelm@53399
  1534
    qed
wenzelm@53399
  1535
    ultimately have "x\<in>\<Union>(r1 - p)" by auto
wenzelm@53399
  1536
  }
wenzelm@53399
  1537
  then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
wenzelm@53399
  1538
    unfolding divp divq using assms(3) by auto
wenzelm@53399
  1539
  show ?thesis
wenzelm@53399
  1540
    apply (rule that[of "p \<union> r2"])
wenzelm@53399
  1541
    unfolding *
wenzelm@53399
  1542
    defer
wenzelm@53399
  1543
    apply (rule division_disjoint_union)
wenzelm@53399
  1544
    unfolding divp(6)
wenzelm@53399
  1545
    apply(rule assms r2)+
wenzelm@53399
  1546
  proof -
wenzelm@53399
  1547
    have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
wenzelm@53399
  1548
    proof (rule inter_interior_unions_intervals)
wenzelm@53408
  1549
      show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = {a..b}"
wenzelm@53399
  1550
        using r1 by auto
wenzelm@53399
  1551
      have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
wenzelm@53399
  1552
        by auto
wenzelm@53399
  1553
      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
wenzelm@53399
  1554
      proof
wenzelm@53399
  1555
        fix m x
wenzelm@53399
  1556
        assume as: "m \<in> r1 - p"
wenzelm@53399
  1557
        have "interior m \<inter> interior (\<Union>p) = {}"
wenzelm@53399
  1558
        proof (rule inter_interior_unions_intervals)
wenzelm@53399
  1559
          show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = {a..b}"
wenzelm@53399
  1560
            using divp by auto
wenzelm@53399
  1561
          show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
wenzelm@53399
  1562
            apply (rule, rule r1(7))
wenzelm@53399
  1563
            using as
wenzelm@53399
  1564
            using r1 
wenzelm@53399
  1565
            apply auto
wenzelm@53399
  1566
            done
wenzelm@53399
  1567
        qed
wenzelm@53399
  1568
        then show "interior s \<inter> interior m = {}"
wenzelm@53399
  1569
          unfolding divp by auto
wenzelm@53399
  1570
      qed
wenzelm@53399
  1571
    qed
wenzelm@53399
  1572
    then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
wenzelm@53399
  1573
      using interior_subset by auto
wenzelm@53399
  1574
  qed auto
wenzelm@53399
  1575
qed
wenzelm@53399
  1576
himmelma@35172
  1577
himmelma@35172
  1578
subsection {* Tagged (partial) divisions. *}
himmelma@35172
  1579
wenzelm@53408
  1580
definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
wenzelm@53408
  1581
  where "s tagged_partial_division_of i \<longleftrightarrow>
wenzelm@53408
  1582
    finite s \<and>
wenzelm@53408
  1583
    (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
wenzelm@53408
  1584
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1585
      interior k1 \<inter> interior k2 = {})"
wenzelm@53408
  1586
wenzelm@53408
  1587
lemma tagged_partial_division_ofD[dest]:
wenzelm@53408
  1588
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1589
  shows "finite s"
wenzelm@53408
  1590
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1591
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
  1592
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
wenzelm@53408
  1593
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
wenzelm@53408
  1594
      (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1595
  using assms unfolding tagged_partial_division_of_def by blast+
wenzelm@53408
  1596
wenzelm@53408
  1597
definition tagged_division_of (infixr "tagged'_division'_of" 40)
wenzelm@53408
  1598
  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
  1599
huffman@44167
  1600
lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
himmelma@35172
  1601
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1602
himmelma@35172
  1603
lemma tagged_division_of:
wenzelm@53408
  1604
  "s tagged_division_of i \<longleftrightarrow>
wenzelm@53408
  1605
    finite s \<and>
wenzelm@53408
  1606
    (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = {a..b})) \<and>
wenzelm@53408
  1607
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1608
      interior k1 \<inter> interior k2 = {}) \<and>
wenzelm@53408
  1609
    (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1610
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1611
wenzelm@53408
  1612
lemma tagged_division_ofI:
wenzelm@53408
  1613
  assumes "finite s"
wenzelm@53408
  1614
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1615
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
  1616
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
wenzelm@53408
  1617
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1618
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1619
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1620
  shows "s tagged_division_of i"
wenzelm@53408
  1621
  unfolding tagged_division_of
wenzelm@53408
  1622
  apply rule
wenzelm@53408
  1623
  defer
wenzelm@53408
  1624
  apply rule
wenzelm@53408
  1625
  apply (rule allI impI conjI assms)+
wenzelm@53408
  1626
  apply assumption
wenzelm@53408
  1627
  apply rule
wenzelm@53408
  1628
  apply (rule assms)
wenzelm@53408
  1629
  apply assumption
wenzelm@53408
  1630
  apply (rule assms)
wenzelm@53408
  1631
  apply assumption
wenzelm@53408
  1632
  using assms(1,5-)
wenzelm@53408
  1633
  apply blast+
wenzelm@53408
  1634
  done
wenzelm@53408
  1635
wenzelm@53408
  1636
lemma tagged_division_ofD[dest]:
wenzelm@53408
  1637
  assumes "s tagged_division_of i"
wenzelm@53408
  1638
  shows "finite s"
wenzelm@53408
  1639
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1640
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
  1641
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = {a..b}"
wenzelm@53408
  1642
    and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1643
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1644
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
wenzelm@53408
  1645
  using assms unfolding tagged_division_of by blast+
wenzelm@53408
  1646
wenzelm@53408
  1647
lemma division_of_tagged_division:
wenzelm@53408
  1648
  assumes "s tagged_division_of i"
wenzelm@53408
  1649
  shows "(snd ` s) division_of i"
wenzelm@53408
  1650
proof (rule division_ofI)
wenzelm@53408
  1651
  note assm = tagged_division_ofD[OF assms]
wenzelm@53408
  1652
  show "\<Union>(snd ` s) = i" "finite (snd ` s)"
wenzelm@53408
  1653
    using assm by auto
wenzelm@53408
  1654
  fix k
wenzelm@53408
  1655
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1656
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1657
    by auto
wenzelm@53408
  1658
  then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = {a..b}"
wenzelm@53408
  1659
    using assm by fastforce+
wenzelm@53408
  1660
  fix k'
wenzelm@53408
  1661
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1662
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1663
    by auto
wenzelm@53408
  1664
  then show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1665
    apply -
wenzelm@53408
  1666
    apply (rule assm(5))
wenzelm@53408
  1667
    apply (rule xk xk')+
wenzelm@53408
  1668
    using k'
wenzelm@53408
  1669
    apply auto
wenzelm@53408
  1670
    done
himmelma@35172
  1671
qed
himmelma@35172
  1672
wenzelm@53408
  1673
lemma partial_division_of_tagged_division:
wenzelm@53408
  1674
  assumes "s tagged_partial_division_of i"
himmelma@35172
  1675
  shows "(snd ` s) division_of \<Union>(snd ` s)"
wenzelm@53408
  1676
proof (rule division_ofI)
wenzelm@53408
  1677
  note assm = tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1678
  show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
wenzelm@53408
  1679
    using assm by auto
wenzelm@53408
  1680
  fix k
wenzelm@53408
  1681
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1682
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1683
    by auto
wenzelm@53408
  1684
  then show "k \<noteq> {}" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>(snd ` s)"
wenzelm@53408
  1685
    using assm by auto
wenzelm@53408
  1686
  fix k'
wenzelm@53408
  1687
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1688
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1689
    by auto
wenzelm@53408
  1690
  then show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1691
    apply -
wenzelm@53408
  1692
    apply (rule assm(5))
wenzelm@53408
  1693
    apply(rule xk xk')+
wenzelm@53408
  1694
    using k'
wenzelm@53408
  1695
    apply auto
wenzelm@53408
  1696
    done
himmelma@35172
  1697
qed
himmelma@35172
  1698
wenzelm@53408
  1699
lemma tagged_partial_division_subset:
wenzelm@53408
  1700
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1701
    and "t \<subseteq> s"
himmelma@35172
  1702
  shows "t tagged_partial_division_of i"
wenzelm@53408
  1703
  using assms
wenzelm@53408
  1704
  unfolding tagged_partial_division_of_def
wenzelm@53408
  1705
  using finite_subset[OF assms(2)]
wenzelm@53408
  1706
  by blast
wenzelm@53408
  1707
wenzelm@53408
  1708
lemma setsum_over_tagged_division_lemma:
wenzelm@53408
  1709
  fixes d :: "'m::ordered_euclidean_space set \<Rightarrow> 'a::real_normed_vector"
wenzelm@53408
  1710
  assumes "p tagged_division_of i"
wenzelm@53408
  1711
    and "\<And>u v. {u..v} \<noteq> {} \<Longrightarrow> content {u..v} = 0 \<Longrightarrow> d {u..v} = 0"
himmelma@35172
  1712
  shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
wenzelm@53408
  1713
proof -
wenzelm@53408
  1714
  note assm = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1715
  have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
wenzelm@53408
  1716
    unfolding o_def by (rule ext) auto
wenzelm@53408
  1717
  show ?thesis
wenzelm@53408
  1718
    unfolding *
wenzelm@53408
  1719
    apply (subst eq_commute)
wenzelm@53408
  1720
  proof (rule setsum_reindex_nonzero)
wenzelm@53408
  1721
    show "finite p"
wenzelm@53408
  1722
      using assm by auto
wenzelm@53408
  1723
    fix x y
wenzelm@53408
  1724
    assume as: "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
wenzelm@53408
  1725
    obtain a b where ab: "snd x = {a..b}"
wenzelm@53408
  1726
      using assm(4)[of "fst x" "snd x"] as(1) by auto
wenzelm@53408
  1727
    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
wenzelm@53408
  1728
      unfolding as(4)[symmetric] using as(1-3) by auto
wenzelm@53408
  1729
    then have "interior (snd x) \<inter> interior (snd y) = {}"
wenzelm@53408
  1730
      apply -
wenzelm@53408
  1731
      apply (rule assm(5)[of "fst x" _ "fst y"])
wenzelm@53408
  1732
      using as
wenzelm@53408
  1733
      apply auto
wenzelm@53408
  1734
      done
wenzelm@53408
  1735
    then have "content {a..b} = 0"
wenzelm@53408
  1736
      unfolding as(4)[symmetric] ab content_eq_0_interior by auto
wenzelm@53408
  1737
    then have "d {a..b} = 0"
wenzelm@53408
  1738
      apply -
wenzelm@53408
  1739
      apply (rule assms(2))
wenzelm@53408
  1740
      using assm(2)[of "fst x" "snd x"] as(1)
wenzelm@53408
  1741
      unfolding ab[symmetric]
wenzelm@53408
  1742
      apply auto
wenzelm@53408
  1743
      done
wenzelm@53408
  1744
    then show "d (snd x) = 0"
wenzelm@53408
  1745
      unfolding ab by auto
wenzelm@53408
  1746
  qed
wenzelm@53408
  1747
qed
wenzelm@53408
  1748
wenzelm@53408
  1749
lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
wenzelm@53408
  1750
  by auto
himmelma@35172
  1751
himmelma@35172
  1752
lemma tagged_division_of_empty: "{} tagged_division_of {}"
himmelma@35172
  1753
  unfolding tagged_division_of by auto
himmelma@35172
  1754
wenzelm@53408
  1755
lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1756
  unfolding tagged_partial_division_of_def by auto
himmelma@35172
  1757
wenzelm@53408
  1758
lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1759
  unfolding tagged_division_of by auto
himmelma@35172
  1760
wenzelm@53408
  1761
lemma tagged_division_of_self: "x \<in> {a..b} \<Longrightarrow> {(x,{a..b})} tagged_division_of {a..b}"
wenzelm@53408
  1762
  by (rule tagged_division_ofI) auto
himmelma@35172
  1763
himmelma@35172
  1764
lemma tagged_division_union:
wenzelm@53408
  1765
  assumes "p1 tagged_division_of s1"
wenzelm@53408
  1766
    and "p2 tagged_division_of s2"
wenzelm@53408
  1767
    and "interior s1 \<inter> interior s2 = {}"
himmelma@35172
  1768
  shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
wenzelm@53408
  1769
proof (rule tagged_division_ofI)
wenzelm@53408
  1770
  note p1 = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1771
  note p2 = tagged_division_ofD[OF assms(2)]
wenzelm@53408
  1772
  show "finite (p1 \<union> p2)"
wenzelm@53408
  1773
    using p1(1) p2(1) by auto
wenzelm@53408
  1774
  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
wenzelm@53408
  1775
    using p1(6) p2(6) by blast
wenzelm@53408
  1776
  fix x k
wenzelm@53408
  1777
  assume xk: "(x, k) \<in> p1 \<union> p2"
wenzelm@53408
  1778
  show "x \<in> k" "\<exists>a b. k = {a..b}"
wenzelm@53408
  1779
    using xk p1(2,4) p2(2,4) by auto
wenzelm@53408
  1780
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
  1781
    using xk p1(3) p2(3) by blast
wenzelm@53408
  1782
  fix x' k'
wenzelm@53408
  1783
  assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1784
  have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1785
    using assms(3) interior_mono by blast
wenzelm@53408
  1786
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1787
    apply (cases "(x, k) \<in> p1")
wenzelm@53408
  1788
    apply (case_tac[!] "(x',k') \<in> p1")
wenzelm@53408
  1789
    apply (rule p1(5))
wenzelm@53408
  1790
    prefer 4
wenzelm@53408
  1791
    apply (rule *)
wenzelm@53408
  1792
    prefer 6
wenzelm@53408
  1793
    apply (subst Int_commute)
wenzelm@53408
  1794
    apply (rule *)
wenzelm@53408
  1795
    prefer 8
wenzelm@53408
  1796
    apply (rule p2(5))
wenzelm@53408
  1797
    using p1(3) p2(3)
wenzelm@53408
  1798
    using xk xk'
wenzelm@53408
  1799
    apply auto
wenzelm@53408
  1800
    done
wenzelm@53408
  1801
qed
himmelma@35172
  1802
himmelma@35172
  1803
lemma tagged_division_unions:
wenzelm@53408
  1804
  assumes "finite iset"
wenzelm@53408
  1805
    and "\<forall>i\<in>iset. pfn i tagged_division_of i"
wenzelm@53408
  1806
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
himmelma@35172
  1807
  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
wenzelm@53408
  1808
proof (rule tagged_division_ofI)
himmelma@35172
  1809
  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
wenzelm@53408
  1810
  show "finite (\<Union>(pfn ` iset))"
wenzelm@53408
  1811
    apply (rule finite_Union)
wenzelm@53408
  1812
    using assms
wenzelm@53408
  1813
    apply auto
wenzelm@53408
  1814
    done
wenzelm@53408
  1815
  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
  1816
    by blast
wenzelm@53408
  1817
  also have "\<dots> = \<Union>iset"
wenzelm@53408
  1818
    using assm(6) by auto
wenzelm@53399
  1819
  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
wenzelm@53408
  1820
  fix x k
wenzelm@53408
  1821
  assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
wenzelm@53408
  1822
  then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
wenzelm@53408
  1823
    by auto
wenzelm@53408
  1824
  show "x \<in> k" "\<exists>a b. k = {a..b}" "k \<subseteq> \<Union>iset"
wenzelm@53408
  1825
    using assm(2-4)[OF i] using i(1) by auto
wenzelm@53408
  1826
  fix x' k'
wenzelm@53408
  1827
  assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1828
  then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
wenzelm@53408
  1829
    by auto
wenzelm@53408
  1830
  have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1831
    using i(1) i'(1)
wenzelm@53408
  1832
    using assms(3)[rule_format] interior_mono
wenzelm@53408
  1833
    by blast
wenzelm@53408
  1834
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1835
    apply (cases "i = i'")
wenzelm@53408
  1836
    using assm(5)[OF i _ xk'(2)] i'(2)
wenzelm@53408
  1837
    using assm(3)[OF i] assm(3)[OF i']
wenzelm@53408
  1838
    defer
wenzelm@53408
  1839
    apply -
wenzelm@53408
  1840
    apply (rule *)
wenzelm@53408
  1841
    apply auto
wenzelm@53408
  1842
    done
himmelma@35172
  1843
qed
himmelma@35172
  1844
himmelma@35172
  1845
lemma tagged_partial_division_of_union_self:
wenzelm@53408
  1846
  assumes "p tagged_partial_division_of s"
himmelma@35172
  1847
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1848
  apply (rule tagged_division_ofI)
wenzelm@53408
  1849
  using tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1850
  apply auto
wenzelm@53408
  1851
  done
wenzelm@53408
  1852
wenzelm@53408
  1853
lemma tagged_division_of_union_self:
wenzelm@53408
  1854
  assumes "p tagged_division_of s"
wenzelm@53408
  1855
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1856
  apply (rule tagged_division_ofI)
wenzelm@53408
  1857
  using tagged_division_ofD[OF assms]
wenzelm@53408
  1858
  apply auto
wenzelm@53408
  1859
  done
wenzelm@53408
  1860
himmelma@35172
  1861
himmelma@35172
  1862
subsection {* Fine-ness of a partition w.r.t. a gauge. *}
himmelma@35172
  1863
wenzelm@53408
  1864
definition fine  (infixr "fine" 46)
wenzelm@53408
  1865
  where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
wenzelm@53408
  1866
wenzelm@53408
  1867
lemma fineI:
wenzelm@53408
  1868
  assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1869
  shows "d fine s"
wenzelm@53408
  1870
  using assms unfolding fine_def by auto
wenzelm@53408
  1871
wenzelm@53408
  1872
lemma fineD[dest]:
wenzelm@53408
  1873
  assumes "d fine s"
wenzelm@53408
  1874
  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1875
  using assms unfolding fine_def by auto
himmelma@35172
  1876
himmelma@35172
  1877
lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
himmelma@35172
  1878
  unfolding fine_def by auto
himmelma@35172
  1879
himmelma@35172
  1880
lemma fine_inters:
himmelma@35172
  1881
 "(\<lambda>x. \<Inter> {f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
himmelma@35172
  1882
  unfolding fine_def by blast
himmelma@35172
  1883
wenzelm@53408
  1884
lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
himmelma@35172
  1885
  unfolding fine_def by blast
himmelma@35172
  1886
wenzelm@53408
  1887
lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
himmelma@35172
  1888
  unfolding fine_def by auto
himmelma@35172
  1889
wenzelm@53408
  1890
lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
himmelma@35172
  1891
  unfolding fine_def by blast
himmelma@35172
  1892
wenzelm@53408
  1893
himmelma@35172
  1894
subsection {* Gauge integral. Define on compact intervals first, then use a limit. *}
himmelma@35172
  1895
wenzelm@53408
  1896
definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
wenzelm@53408
  1897
  where "(f has_integral_compact_interval y) i \<longleftrightarrow>
wenzelm@53408
  1898
    (\<forall>e>0. \<exists>d. gauge d \<and>
wenzelm@53408
  1899
      (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
wenzelm@53408
  1900
        norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1901
wenzelm@53408
  1902
definition has_integral ::
wenzelm@53408
  1903
    "('n::ordered_euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
wenzelm@53408
  1904
  (infixr "has'_integral" 46)
wenzelm@53408
  1905
  where "(f has_integral y) i \<longleftrightarrow>
wenzelm@53408
  1906
    (if \<exists>a b. i = {a..b}
wenzelm@53408
  1907
     then (f has_integral_compact_interval y) i
wenzelm@53408
  1908
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
wenzelm@53408
  1909
      (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) {a..b} \<and>
wenzelm@53408
  1910
        norm (z - y) < e)))"
himmelma@35172
  1911
himmelma@35172
  1912
lemma has_integral:
wenzelm@53408
  1913
  "(f has_integral y) {a..b} \<longleftrightarrow>
wenzelm@53408
  1914
    (\<forall>e>0. \<exists>d. gauge d \<and>
wenzelm@53408
  1915
      (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow>
wenzelm@53408
  1916
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1917
  unfolding has_integral_def has_integral_compact_interval_def
wenzelm@53408
  1918
  by auto
wenzelm@53408
  1919
wenzelm@53408
  1920
lemma has_integralD[dest]:
wenzelm@53408
  1921
  assumes "(f has_integral y) ({a..b})"
wenzelm@53408
  1922
    and "e > 0"
wenzelm@53408
  1923
  obtains d where "gauge d"
wenzelm@53408
  1924
    and "\<And>p. p tagged_division_of {a..b} \<Longrightarrow> d fine p \<Longrightarrow>
wenzelm@53408
  1925
      norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
himmelma@35172
  1926
  using assms unfolding has_integral by auto
himmelma@35172
  1927
himmelma@35172
  1928
lemma has_integral_alt:
wenzelm@53408
  1929
  "(f has_integral y) i \<longleftrightarrow>
wenzelm@53408
  1930
    (if \<exists>a b. i = {a..b}
wenzelm@53408
  1931
     then (f has_integral y) i
wenzelm@53408
  1932
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
wenzelm@53408
  1933
      (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm (z - y) < e)))"
wenzelm@53408
  1934
  unfolding has_integral
wenzelm@53408
  1935
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1936
  by auto
himmelma@35172
  1937
himmelma@35172
  1938
lemma has_integral_altD:
wenzelm@53408
  1939
  assumes "(f has_integral y) i"
wenzelm@53408
  1940
    and "\<not> (\<exists>a b. i = {a..b})"
wenzelm@53408
  1941
    and "e>0"
wenzelm@53408
  1942
  obtains B where "B > 0"
wenzelm@53408
  1943
    and "\<forall>a b. ball 0 B \<subseteq> {a..b} \<longrightarrow>
wenzelm@53408
  1944
      (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) ({a..b}) \<and> norm(z - y) < e)"
wenzelm@53408
  1945
  using assms
wenzelm@53408
  1946
  unfolding has_integral
wenzelm@53408
  1947
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1948
  by auto
wenzelm@53408
  1949
wenzelm@53408
  1950
definition integrable_on (infixr "integrable'_on" 46)
wenzelm@53408
  1951
  where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
wenzelm@53408
  1952
wenzelm@53408
  1953
definition "integral i f = (SOME y. (f has_integral y) i)"
himmelma@35172
  1954
wenzelm@53409
  1955
lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
wenzelm@53409
  1956
  unfolding integrable_on_def integral_def by (rule someI_ex)
himmelma@35172
  1957
himmelma@35172
  1958
lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
himmelma@35172
  1959
  unfolding integrable_on_def by auto
himmelma@35172
  1960
wenzelm@53409
  1961
lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
himmelma@35172
  1962
  by auto
himmelma@35172
  1963
himmelma@35172
  1964
lemma setsum_content_null:
wenzelm@53409
  1965
  assumes "content {a..b} = 0"
wenzelm@53409
  1966
    and "p tagged_division_of {a..b}"
himmelma@35172
  1967
  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
wenzelm@53409
  1968
proof (rule setsum_0', rule)
wenzelm@53409
  1969
  fix y
wenzelm@53409
  1970
  assume y: "y \<in> p"
wenzelm@53409
  1971
  obtain x k where xk: "y = (x, k)"
wenzelm@53409
  1972
    using surj_pair[of y] by blast
himmelma@35172
  1973
  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
wenzelm@53409
  1974
  from this(2) obtain c d where k: "k = {c..d}" by blast
wenzelm@53409
  1975
  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
wenzelm@53409
  1976
    unfolding xk by auto
wenzelm@53409
  1977
  also have "\<dots> = 0"
wenzelm@53409
  1978
    using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
wenzelm@53409
  1979
    unfolding assms(1) k
wenzelm@53409
  1980
    by auto
himmelma@35172
  1981
  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
himmelma@35172
  1982
qed
himmelma@35172
  1983
wenzelm@53409
  1984
himmelma@35172
  1985
subsection {* Some basic combining lemmas. *}
himmelma@35172
  1986
himmelma@35172
  1987
lemma tagged_division_unions_exists:
wenzelm@53409
  1988
  assumes "finite iset"
wenzelm@53409
  1989
    and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
wenzelm@53409
  1990
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
wenzelm@53409
  1991
    and "\<Union>iset = i"
wenzelm@53409
  1992
   obtains p where "p tagged_division_of i" and "d fine p"
wenzelm@53409
  1993
proof -
wenzelm@53409
  1994
  obtain pfn where pfn:
wenzelm@53409
  1995
    "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
wenzelm@53409
  1996
    "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
wenzelm@53409
  1997
    using bchoice[OF assms(2)] by auto
wenzelm@53409
  1998
  show thesis
wenzelm@53409
  1999
    apply (rule_tac p="\<Union>(pfn ` iset)" in that)
wenzelm@53409
  2000
    unfolding assms(4)[symmetric]
wenzelm@53409
  2001
    apply (rule tagged_division_unions[OF assms(1) _ assms(3)])
wenzelm@53409
  2002
    defer
wenzelm@53409
  2003
    apply (rule fine_unions)
wenzelm@53409
  2004
    using pfn
wenzelm@53409
  2005
    apply auto
wenzelm@53409
  2006
    done
himmelma@35172
  2007
qed
himmelma@35172
  2008
wenzelm@53409
  2009
himmelma@35172
  2010
subsection {* The set we're concerned with must be closed. *}
himmelma@35172
  2011
wenzelm@53409
  2012
lemma division_of_closed:
wenzelm@53409
  2013
  fixes i :: "'n::ordered_euclidean_space set"
wenzelm@53409
  2014
  shows "s division_of i \<Longrightarrow> closed i"
nipkow@44890
  2015
  unfolding division_of_def by fastforce
himmelma@35172
  2016
himmelma@35172
  2017
subsection {* General bisection principle for intervals; might be useful elsewhere. *}
himmelma@35172
  2018
wenzelm@53409
  2019
lemma interval_bisection_step:
wenzelm@53409
  2020
  fixes type :: "'a::ordered_euclidean_space"
wenzelm@53409
  2021
  assumes "P {}"
wenzelm@53409
  2022
    and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
wenzelm@53409
  2023
    and "\<not> P {a..b::'a}"
wenzelm@53409
  2024
  obtains c d where "\<not> P{c..d}"
wenzelm@53409
  2025
    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
  2026
proof -
wenzelm@53409
  2027
  have "{a..b} \<noteq> {}"
wenzelm@53409
  2028
    using assms(1,3) by auto
wenzelm@53409
  2029
  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
wenzelm@53409
  2030
    by (auto simp: interval_eq_empty not_le)
wenzelm@53409
  2031
  {
wenzelm@53409
  2032
    fix f
wenzelm@53409
  2033
    have "finite f \<Longrightarrow>
wenzelm@53409
  2034
      \<forall>s\<in>f. P s \<Longrightarrow>
wenzelm@53409
  2035
      \<forall>s\<in>f. \<exists>a b. s = {a..b} \<Longrightarrow>
wenzelm@53409
  2036
      \<forall>s\<in>f.\<forall>t\<in>f. s \<noteq> t \<longrightarrow> interior s \<inter> interior t = {} \<Longrightarrow> P (\<Union>f)"
wenzelm@53409
  2037
    proof (induct f rule: finite_induct)
wenzelm@53409
  2038
      case empty
wenzelm@53409
  2039
      show ?case
wenzelm@53409
  2040
        using assms(1) by auto
wenzelm@53409
  2041
    next
wenzelm@53409
  2042
      case (insert x f)
wenzelm@53409
  2043
      show ?case
wenzelm@53409
  2044
        unfolding Union_insert
wenzelm@53409
  2045
        apply (rule assms(2)[rule_format])
wenzelm@53409
  2046
        apply rule
wenzelm@53409
  2047
        defer
wenzelm@53409
  2048
        apply rule
wenzelm@53409
  2049
        defer
wenzelm@53409
  2050
        apply (rule inter_interior_unions_intervals)
wenzelm@53409
  2051
        using insert
wenzelm@53409
  2052
        apply auto
wenzelm@53409
  2053
        done
wenzelm@53409
  2054
    qed
wenzelm@53409
  2055
  } note * = this
wenzelm@53409
  2056
  let ?A = "{{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
  2057
    (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
hoelzl@50526
  2058
  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
  2059
  {
wenzelm@53409
  2060
    presume "\<forall>c d. ?PP c d \<longrightarrow> P {c..d} \<Longrightarrow> False"
wenzelm@53409
  2061
    then show thesis
wenzelm@53409
  2062
      unfolding atomize_not not_all
wenzelm@53409
  2063
      apply -
wenzelm@53409
  2064
      apply (erule exE)+
wenzelm@53409
  2065
      apply (rule_tac c=x and d=xa in that)
wenzelm@53409
  2066
      apply auto
wenzelm@53409
  2067
      done
wenzelm@53409
  2068
  }
wenzelm@53409
  2069
  assume as: "\<forall>c d. ?PP c d \<longrightarrow> P {c..d}"
wenzelm@53409
  2070
  have "P (\<Union> ?A)"
wenzelm@53409
  2071
    apply (rule *)
wenzelm@53409
  2072
    apply (rule_tac[2-] ballI)
wenzelm@53409
  2073
    apply (rule_tac[4] ballI)
wenzelm@53409
  2074
    apply (rule_tac[4] impI)
wenzelm@53409
  2075
  proof -
hoelzl@50526
  2076
    let ?B = "(\<lambda>s.{(\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i)::'a ..
hoelzl@50526
  2077
      (\<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
  2078
    have "?A \<subseteq> ?B"
wenzelm@53409
  2079
    proof
wenzelm@53409
  2080
      case goal1
wenzelm@53409
  2081
      then obtain c d where x: "x = {c..d}"
wenzelm@53409
  2082
        "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  2083
          c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  2084
          c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
wenzelm@53409
  2085
      have *: "\<And>a b c d. a = c \<Longrightarrow> b = d \<Longrightarrow> {a..b} = {c..d}"
wenzelm@53409
  2086
        by auto
wenzelm@53409
  2087
      show "x \<in> ?B"
wenzelm@53409
  2088
        unfolding image_iff
wenzelm@53409
  2089
        apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
wenzelm@53409
  2090
        unfolding x
wenzelm@53409
  2091
        apply (rule *)
hoelzl@50526
  2092
        apply (simp_all only: euclidean_eq_iff[where 'a='a] inner_setsum_left_Basis mem_Collect_eq simp_thms
wenzelm@53409
  2093
          cong: ball_cong)
hoelzl@50526
  2094
        apply safe
wenzelm@53409
  2095
      proof -
wenzelm@53409
  2096
        fix i :: 'a
wenzelm@53409
  2097
        assume i: "i \<in> Basis"
wenzelm@53409
  2098
        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
  2099
          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
  2100
          using x(2)[of i] ab[OF i] by (auto simp add:field_simps)
wenzelm@53409
  2101
      qed
wenzelm@53409
  2102
    qed
wenzelm@53409
  2103
    then show "finite ?A"
wenzelm@53409
  2104
      by (rule finite_subset) auto
wenzelm@53409
  2105
    fix s
wenzelm@53409
  2106
    assume "s \<in> ?A"
wenzelm@53409
  2107
    then obtain c d where s:
wenzelm@53409
  2108
      "s = {c..d}"
wenzelm@53409
  2109
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  2110
         c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  2111
         c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
wenzelm@53409
  2112
      by blast
wenzelm@53409
  2113
    show "P s"
wenzelm@53409
  2114
      unfolding s
wenzelm@53409
  2115
      apply (rule as[rule_format])
wenzelm@53409
  2116
    proof -
wenzelm@53409
  2117
      case goal1
wenzelm@53409
  2118
      then show ?case
wenzelm@53409
  2119
        using s(2)[of i] using ab[OF `i \<in> Basis`] by auto
wenzelm@53409
  2120
    qed
wenzelm@53409
  2121
    show "\<exists>a b. s = {a..b}"
wenzelm@53409
  2122
      unfolding s by auto
wenzelm@53409
  2123
    fix t
wenzelm@53409
  2124
    assume "t \<in> ?A"
wenzelm@53409
  2125
    then obtain e f where t:
wenzelm@53409
  2126
      "t = {e..f}"
wenzelm@53409
  2127
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  2128
        e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  2129
        e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
wenzelm@53409
  2130
      by blast
wenzelm@53409
  2131
    assume "s \<noteq> t"
wenzelm@53409
  2132
    then have "\<not> (c = e \<and> d = f)"
wenzelm@53409
  2133
      unfolding s t by auto
wenzelm@53409
  2134
    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
  2135
      unfolding euclidean_eq_iff[where 'a='a] by auto
wenzelm@53409
  2136
    then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
wenzelm@53409
  2137
      apply -
wenzelm@53409
  2138
      apply(erule_tac[!] disjE)
wenzelm@53409
  2139
    proof -
wenzelm@53409
  2140
      assume "c\<bullet>i \<noteq> e\<bullet>i"
wenzelm@53409
  2141
      then show "d\<bullet>i \<noteq> f\<bullet>i"
wenzelm@53409
  2142
        using s(2)[OF i'] t(2)[OF i'] by fastforce
wenzelm@53409
  2143
    next
wenzelm@53409
  2144
      assume "d\<bullet>i \<noteq> f\<bullet>i"
wenzelm@53409
  2145
      then show "c\<bullet>i \<noteq> e\<bullet>i"
wenzelm@53409
  2146
        using s(2)[OF i'] t(2)[OF i'] by fastforce
wenzelm@53409
  2147
    qed
wenzelm@53409
  2148
    have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
wenzelm@53409
  2149
      by auto
wenzelm@53409
  2150
    show "interior s \<inter> interior t = {}"
wenzelm@53409
  2151
      unfolding s t interior_closed_interval
wenzelm@53409
  2152
    proof (rule *)
wenzelm@53409
  2153
      fix x
immler@54775
  2154
      assume "x \<in> box c d" "x \<in> box e f"
wenzelm@53409
  2155
      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"
wenzelm@53409
  2156
        unfolding mem_interval using i'
wenzelm@53409
  2157
        apply -
wenzelm@53409
  2158
        apply (erule_tac[!] x=i in ballE)+
wenzelm@53409
  2159
        apply auto
wenzelm@53409
  2160
        done
wenzelm@53409
  2161
      show False
wenzelm@53409
  2162
        using s(2)[OF i']
wenzelm@53409
  2163
        apply -
wenzelm@53409
  2164
        apply (erule_tac disjE)
wenzelm@53409
  2165
        apply (erule_tac[!] conjE)
wenzelm@53409
  2166
      proof -
wenzelm@53409
  2167
        assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
wenzelm@53409
  2168
        show False
wenzelm@53409
  2169
          using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
wenzelm@53409
  2170
      next
wenzelm@53409
  2171
        assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
wenzelm@53409
  2172
        show False
wenzelm@53409
  2173
          using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
wenzelm@53409
  2174
      qed
wenzelm@53409
  2175
    qed
wenzelm@53409
  2176
  qed
wenzelm@53409
  2177
  also have "\<Union> ?A = {a..b}"
wenzelm@53409
  2178
  proof (rule set_eqI,rule)
wenzelm@53409
  2179
    fix x
wenzelm@53409
  2180
    assume "x \<in> \<Union>?A"
wenzelm@53409
  2181
    then obtain c d where x:
wenzelm@53409
  2182
      "x \<in> {c..d}"
wenzelm@53409
  2183
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  2184
        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  2185
        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
wenzelm@53409
  2186
    show "x\<in>{a..b}"
wenzelm@53409
  2187
      unfolding mem_interval
wenzelm@53409
  2188
    proof safe
wenzelm@53409
  2189
      fix i :: 'a
wenzelm@53409
  2190
      assume i: "i \<in> Basis"
wenzelm@53409
  2191
      then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
wenzelm@53409
  2192
        using x(2)[OF i] x(1)[unfolded mem_interval,THEN bspec, OF i] by auto
wenzelm@53409
  2193
    qed
wenzelm@53409
  2194
  next
wenzelm@53409
  2195
    fix x
wenzelm@53409
  2196
    assume x: "x \<in> {a..b}"
wenzelm@53409
  2197
    have "\<forall>i\<in>Basis.
wenzelm@53409
  2198
      \<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
  2199
      (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
wenzelm@53409
  2200
      unfolding mem_interval
hoelzl@50526
  2201
    proof
wenzelm@53409
  2202
      fix i :: 'a
wenzelm@53409
  2203
      assume i: "i \<in> Basis"
hoelzl@50526
  2204
      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)"
wenzelm@53409
  2205
        using x[unfolded mem_interval,THEN bspec, OF i] by auto
wenzelm@53409
  2206
      then show "\<exists>c d. ?P i c d"
wenzelm@53409
  2207
        by blast
hoelzl@50526
  2208
    qed
wenzelm@53409
  2209
    then show "x\<in>\<Union>?A"
hoelzl@50526
  2210
      unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
wenzelm@53409
  2211
      apply -
wenzelm@53409
  2212
      apply (erule exE)+
wenzelm@53409
  2213
      apply (rule_tac x="{xa..xaa}" in exI)
wenzelm@53409
  2214
      unfolding mem_interval
wenzelm@53409
  2215
      apply auto
wenzelm@53409
  2216
      done
wenzelm@53409
  2217
  qed
wenzelm@53409
  2218
  finally show False
wenzelm@53409
  2219
    using assms by auto
wenzelm@53409
  2220
qed
wenzelm@53409
  2221
wenzelm@53409
  2222
lemma interval_bisection:
wenzelm@53409
  2223
  fixes type :: "'a::ordered_euclidean_space"
wenzelm@53409
  2224
  assumes "P {}"
wenzelm@53409
  2225
    and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
wenzelm@53409
  2226
    and "\<not> P {a..b::'a}"
wenzelm@53409
  2227
  obtains x where "x \<in> {a..b}"
wenzelm@53409
  2228
    and "\<forall>e>0. \<exists>c d. x \<in> {c..d} \<and> {c..d} \<subseteq> ball x e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
wenzelm@53409
  2229
proof -
hoelzl@37489
  2230
  have "\<forall>x. \<exists>y. \<not> P {fst x..snd x} \<longrightarrow> (\<not> P {fst y..snd y} \<and>
hoelzl@50526
  2231
    (\<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
  2232
       2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))"
wenzelm@53409
  2233
  proof
wenzelm@53409
  2234
    case goal1
wenzelm@53409
  2235
    then show ?case
wenzelm@53409
  2236
    proof -
himmelma@35172
  2237
      presume "\<not> P {fst x..snd x} \<Longrightarrow> ?thesis"
wenzelm@53409
  2238
      then show ?thesis by (cases "P {fst x..snd x}") auto
wenzelm@53409
  2239
    next
wenzelm@53409
  2240
      assume as: "\<not> P {fst x..snd x}"
wenzelm@53409
  2241
      obtain c d where "\<not> P {c..d}"
wenzelm@53409
  2242
        "\<forall>i\<in>Basis.
wenzelm@53409
  2243
           fst x \<bullet> i \<le> c \<bullet> i \<and>
wenzelm@53409
  2244
           c \<bullet> i \<le> d \<bullet> i \<and>
wenzelm@53409
  2245
           d \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@53409
  2246
           2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
wenzelm@53409
  2247
        by (rule interval_bisection_step[of P, OF assms(1-2) as])
wenzelm@53409
  2248
      then show ?thesis
wenzelm@53409
  2249
        apply -
wenzelm@53409
  2250
        apply (rule_tac x="(c,d)" in exI)
wenzelm@53409
  2251
        apply auto
wenzelm@53409
  2252
        done
wenzelm@53409
  2253
    qed
wenzelm@53409
  2254
  qed
wenzelm@53409
  2255
  then guess f
wenzelm@53409
  2256
    apply -
wenzelm@53409
  2257
    apply (drule choice)
wenzelm@53409
  2258
    apply (erule exE)
wenzelm@53409
  2259
    done
wenzelm@53409
  2260
  note f = this
wenzelm@53409
  2261
  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)"
wenzelm@53409
  2262
  def A \<equiv> "\<lambda>n. fst(AB n)"
wenzelm@53409
  2263
  def B \<equiv> "\<lambda>n. snd(AB n)"
wenzelm@53409
  2264
  note ab_def = A_def B_def AB_def
himmelma@35172
  2265
  have "A 0 = a" "B 0 = b" "\<And>n. \<not> P {A(Suc n)..B(Suc n)} \<and>
wenzelm@53399
  2266
    (\<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
  2267
    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
  2268
  proof -
wenzelm@53409
  2269
    show "A 0 = a" "B 0 = b"
wenzelm@53409
  2270
      unfolding ab_def by auto
wenzelm@53409
  2271
    case goal3
wenzelm@53409
  2272
    note S = ab_def funpow.simps o_def id_apply
wenzelm@53409
  2273
    show ?case
wenzelm@53409
  2274
    proof (induct n)
wenzelm@53409
  2275
      case 0
wenzelm@53409
  2276
      then show ?case
wenzelm@53409
  2277
        unfolding S
wenzelm@53409
  2278
        apply (rule f[rule_format]) using assms(3)
wenzelm@53409
  2279
        apply auto
wenzelm@53409
  2280
        done
wenzelm@53409
  2281
    next
wenzelm@53409
  2282
      case (Suc n)
wenzelm@53409
  2283
      show ?case
wenzelm@53409
  2284
        unfolding S
wenzelm@53409
  2285
        apply (rule f[rule_format])
wenzelm@53409
  2286
        using Suc
wenzelm@53409
  2287
        unfolding S
wenzelm@53409
  2288
        apply auto
wenzelm@53409
  2289
        done
wenzelm@53409
  2290
    qed
wenzelm@53409
  2291
  qed
wenzelm@53409
  2292
  note AB = this(1-2) conjunctD2[OF this(3),rule_format]
wenzelm@53409
  2293
wenzelm@53409
  2294
  have interv: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e"
wenzelm@53409
  2295
  proof -
wenzelm@53409
  2296
    case goal1
wenzelm@53409
  2297
    obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
wenzelm@53409
  2298
      using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] ..
wenzelm@53409
  2299
    show ?case
wenzelm@53409
  2300
      apply (rule_tac x=n in exI)
wenzelm@53409
  2301
      apply rule
wenzelm@53409
  2302
      apply rule
wenzelm@53409
  2303
    proof -
wenzelm@53409
  2304
      fix x y
wenzelm@53409
  2305
      assume xy: "x\<in>{A n..B n}" "y\<in>{A n..B n}"
wenzelm@53409
  2306
      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis"
wenzelm@53409
  2307
        unfolding dist_norm by(rule norm_le_l1)
hoelzl@50526
  2308
      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
wenzelm@53409
  2309
      proof (rule setsum_mono)
wenzelm@53409
  2310
        fix i :: 'a
wenzelm@53409
  2311
        assume i: "i \<in> Basis"
wenzelm@53409
  2312
        show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
wenzelm@53409
  2313
          using xy[unfolded mem_interval,THEN bspec, OF i]
wenzelm@53409
  2314
          by (auto simp: inner_diff_left)
wenzelm@53409
  2315
      qed
wenzelm@53409
  2316
      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
wenzelm@53409
  2317
        unfolding setsum_divide_distrib
wenzelm@53409
  2318
      proof (rule setsum_mono)
wenzelm@53409
  2319
        case goal1
wenzelm@53409
  2320
        then show ?case
wenzelm@53409
  2321
        proof (induct n)
wenzelm@53409
  2322
          case 0
wenzelm@53409
  2323
          then show ?case
wenzelm@53409
  2324
            unfolding AB by auto
wenzelm@53409
  2325
        next
wenzelm@53409
  2326
          case (Suc n)
wenzelm@53409
  2327
          have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
hoelzl@37489
  2328
            using AB(4)[of i n] using goal1 by auto
wenzelm@53409
  2329
          also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
wenzelm@53409
  2330
            using Suc by (auto simp add:field_simps)
wenzelm@53409
  2331
          finally show ?case .
wenzelm@53409
  2332
        qed
wenzelm@53409
  2333
      qed
wenzelm@53409
  2334
      also have "\<dots> < e"
wenzelm@53409
  2335
        using n using goal1 by (auto simp add:field_simps)
wenzelm@53409
  2336
      finally show "dist x y < e" .
wenzelm@53409
  2337
    qed
wenzelm@53409
  2338
  qed
wenzelm@53409
  2339
  {
wenzelm@53409
  2340
    fix n m :: nat
hoelzl@54411
  2341
    assume "m \<le> n" then have "{A n..B n} \<subseteq> {A m..B m}"
hoelzl@54411
  2342
    proof (induction rule: inc_induct)
wenzelm@53409
  2343
      case (step i)
wenzelm@53409
  2344
      show ?case
hoelzl@54411
  2345
        using AB(4) by (intro order_trans[OF step.IH] subset_interval_imp) auto
wenzelm@53409
  2346
    qed simp
wenzelm@53409
  2347
  } note ABsubset = this
wenzelm@53409
  2348
  have "\<exists>a. \<forall>n. a\<in>{A n..B n}"
wenzelm@53409
  2349
    apply (rule decreasing_closed_nest[rule_format,OF closed_interval _ ABsubset interv])
wenzelm@53409
  2350
  proof -
wenzelm@53409
  2351
    fix n
wenzelm@53409
  2352
    show "{A n..B n} \<noteq> {}"
wenzelm@53409
  2353
      apply (cases "0 < n")
wenzelm@53409
  2354
      using AB(3)[of "n - 1"] assms(1,3) AB(1-2)
wenzelm@53409
  2355
      apply auto
wenzelm@53409
  2356
      done
wenzelm@53409
  2357
  qed auto
wenzelm@53409
  2358
  then obtain x0 where x0: "\<And>n. x0 \<in> {A n..B n}"
wenzelm@53409
  2359
    by blast
wenzelm@53409
  2360
  show thesis
wenzelm@53409
  2361
  proof (rule that[rule_format, of x0])
wenzelm@53409
  2362
    show "x0\<in>{a..b}"
wenzelm@53409
  2363
      using x0[of 0] unfolding AB .
wenzelm@53409
  2364
    fix e :: real
wenzelm@53409
  2365
    assume "e > 0"
wenzelm@53409
  2366
    from interv[OF this] obtain n
wenzelm@53409
  2367
      where n: "\<forall>x\<in>{A n..B n}. \<forall>y\<in>{A n..B n}. dist x y < e" ..
himmelma@35172
  2368
    show "\<exists>c d. x0 \<in> {c..d} \<and> {c..d} \<subseteq> ball x0 e \<and> {c..d} \<subseteq> {a..b} \<and> \<not> P {c..d}"
wenzelm@53409
  2369
      apply (rule_tac x="A n" in exI)
wenzelm@53409
  2370
      apply (rule_tac x="B n" in exI)
wenzelm@53409
  2371
      apply rule
wenzelm@53409
  2372
      apply (rule x0)
wenzelm@53409
  2373
      apply rule
wenzelm@53409
  2374
      defer
wenzelm@53409
  2375
      apply rule
wenzelm@53409
  2376
    proof -
wenzelm@53409
  2377
      show "\<not> P {A n..B n}"
wenzelm@53409
  2378
        apply (cases "0 < n")
wenzelm@53409
  2379
        using AB(3)[of "n - 1"] assms(3) AB(1-2)
wenzelm@53409
  2380
        apply auto
wenzelm@53409
  2381
        done
wenzelm@53409
  2382
      show "{A n..B n} \<subseteq> ball x0 e"
wenzelm@53409
  2383
        using n using x0[of n] by auto
wenzelm@53409
  2384
      show "{A n..B n} \<subseteq> {a..b}"
wenzelm@53409
  2385
        unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
wenzelm@53409
  2386
    qed
wenzelm@53409
  2387
  qed
wenzelm@53409
  2388
qed
wenzelm@53409
  2389
himmelma@35172
  2390
himmelma@35172
  2391
subsection {* Cousin's lemma. *}
himmelma@35172
  2392
wenzelm@53409
  2393
lemma fine_division_exists:
wenzelm@53409
  2394
  fixes a b :: "'a::ordered_euclidean_space"
wenzelm@53409
  2395
  assumes "gauge g"
wenzelm@53409
  2396
  obtains p where "p tagged_division_of {a..b}" "g fine p"
wenzelm@53409
  2397
proof -
wenzelm@53409
  2398
  presume "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p) \<Longrightarrow> False"
wenzelm@53410
  2399
  then obtain p where "p tagged_division_of {a..b}" "g fine p"
wenzelm@53410
  2400
    by blast
wenzelm@53409
  2401
  then show thesis ..
wenzelm@53409
  2402
next
wenzelm@53409
  2403
  assume as: "\<not> (\<exists>p. p tagged_division_of {a..b} \<and> g fine p)"
wenzelm@53410
  2404
  guess x
wenzelm@53410
  2405
    apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p",rule_format,OF _ _ as])
wenzelm@53410
  2406
    apply (rule_tac x="{}" in exI)
wenzelm@53410
  2407
    defer
wenzelm@53410
  2408
    apply (erule conjE exE)+
wenzelm@53409
  2409
  proof -
wenzelm@53410
  2410
    show "{} tagged_division_of {} \<and> g fine {}"
wenzelm@53410
  2411
      unfolding fine_def by auto
wenzelm@53410
  2412
    fix s t p p'
wenzelm@53410
  2413
    assume "p tagged_division_of s" "g fine p" "p' tagged_division_of t" "g fine p'"
wenzelm@53410
  2414
      "interior s \<inter> interior t = {}"
wenzelm@53409
  2415
    then show "\<exists>p. p tagged_division_of s \<union> t \<and> g fine p"
wenzelm@53409
  2416
      apply -
wenzelm@53409
  2417
      apply (rule_tac x="p \<union> p'" in exI)
wenzelm@53409
  2418
      apply rule
wenzelm@53409
  2419
      apply (rule tagged_division_union)
wenzelm@53409
  2420
      prefer 4
wenzelm@53409
  2421
      apply (rule fine_union)
wenzelm@53409
  2422
      apply auto
wenzelm@53409
  2423
      done
wenzelm@53410
  2424
  qed note x = this
wenzelm@53410
  2425
  obtain e where e: "e > 0" "ball x e \<subseteq> g x"
wenzelm@53409
  2426
    using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
wenzelm@53410
  2427
  from x(2)[OF e(1)] obtain c d where c_d:
wenzelm@53410
  2428
    "x \<in> {c..d}"
wenzelm@53410
  2429
    "{c..d} \<subseteq> ball x e"
wenzelm@53410
  2430
    "{c..d} \<subseteq> {a..b}"
wenzelm@53410
  2431
    "\<not> (\<exists>p. p tagged_division_of {c..d} \<and> g fine p)"
wenzelm@53410
  2432
    by blast
wenzelm@53409
  2433
  have "g fine {(x, {c..d})}"
wenzelm@53409
  2434
    unfolding fine_def using e using c_d(2) by auto
wenzelm@53410
  2435
  then show False
wenzelm@53410
  2436
    using tagged_division_of_self[OF c_d(1)] using c_d by auto
wenzelm@53409
  2437
qed
wenzelm@53409
  2438
himmelma@35172
  2439
himmelma@35172
  2440
subsection {* Basic theorems about integrals. *}
himmelma@35172
  2441
wenzelm@53409
  2442
lemma has_integral_unique:
wenzelm@53409
  2443
  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2444
  assumes "(f has_integral k1) i"
wenzelm@53410
  2445
    and "(f has_integral k2) i"
wenzelm@53409
  2446
  shows "k1 = k2"
wenzelm@53410
  2447
proof (rule ccontr)
wenzelm@53842
  2448
  let ?e = "norm (k1 - k2) / 2"
wenzelm@53410
  2449
  assume as:"k1 \<noteq> k2"
wenzelm@53410
  2450
  then have e: "?e > 0"
wenzelm@53410
  2451
    by auto
wenzelm@53410
  2452
  have lem: "\<And>f::'n \<Rightarrow> 'a.  \<And>a b k1 k2.
himmelma@35172
  2453
    (f has_integral k1) ({a..b}) \<Longrightarrow> (f has_integral k2) ({a..b}) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
wenzelm@53410
  2454
  proof -
wenzelm@53410
  2455
    case goal1
wenzelm@53410
  2456
    let ?e = "norm (k1 - k2) / 2"
wenzelm@53410
  2457
    from goal1(3) have e: "?e > 0" by auto
wenzelm@53410
  2458
    guess d1 by (rule has_integralD[OF goal1(1) e]) note d1=this
wenzelm@53410
  2459
    guess d2 by (rule has_integralD[OF goal1(2) e]) note d2=this
wenzelm@53410
  2460
    guess p by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)],of a b]) note p=this
wenzelm@53410
  2461
    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
wenzelm@53410
  2462
    have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
wenzelm@53410
  2463
      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
wenzelm@53410
  2464
      by (auto simp add:algebra_simps norm_minus_commute)
himmelma@35172
  2465
    also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2466
      apply (rule add_strict_mono)
wenzelm@53410
  2467
      apply (rule_tac[!] d2(2) d1(2))
wenzelm@53410
  2468
      using p unfolding fine_def
wenzelm@53410
  2469
      apply auto
wenzelm@53410
  2470
      done
himmelma@35172
  2471
    finally show False by auto
wenzelm@53410
  2472
  qed
wenzelm@53410
  2473
  {
wenzelm@53410
  2474
    presume "\<not> (\<exists>a b. i = {a..b}) \<Longrightarrow> False"
wenzelm@53410
  2475
    then show False
wenzelm@53410
  2476
      apply -
wenzelm@53410
  2477
      apply (cases "\<exists>a b. i = {a..b}")
wenzelm@53410
  2478
      using assms
wenzelm@53410
  2479
      apply (auto simp add:has_integral intro:lem[OF _ _ as])
wenzelm@53410
  2480
      done
wenzelm@53410
  2481
  }
wenzelm@53410
  2482
  assume as: "\<not> (\<exists>a b. i = {a..b})"
wenzelm@53410
  2483
  guess B1 by (rule has_integral_altD[OF assms(1) as,OF e]) note B1=this[rule_format]
wenzelm@53410
  2484
  guess B2 by (rule has_integral_altD[OF assms(2) as,OF e]) note B2=this[rule_format]
wenzelm@53410
  2485
  have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> {a..b}"
wenzelm@53410
  2486
    apply (rule bounded_subset_closed_interval)
wenzelm@53410
  2487
    using bounded_Un bounded_ball
wenzelm@53410
  2488
    apply auto
wenzelm@53410
  2489
    done
wenzelm@53410
  2490
  then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> {a..b}" "ball 0 B2 \<subseteq> {a..b}"
wenzelm@53410
  2491
    by blast
wenzelm@53410
  2492
  obtain w where w:
wenzelm@53410
  2493
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) {a..b}"
wenzelm@53410
  2494
    "norm (w - k1) < norm (k1 - k2) / 2"
wenzelm@53410
  2495
    using B1(2)[OF ab(1)] by blast
wenzelm@53410
  2496
  obtain z where z:
wenzelm@53410
  2497
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) {a..b}"
wenzelm@53410
  2498
    "norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@53410
  2499
    using B2(2)[OF ab(2)] by blast
wenzelm@53410
  2500
  have "z = w"
wenzelm@53410
  2501
    using lem[OF w(1) z(1)] by auto
wenzelm@53410
  2502
  then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
wenzelm@53410
  2503
    using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
wenzelm@53410
  2504
    by (auto simp add: norm_minus_commute)
wenzelm@53410
  2505
  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2506
    apply (rule add_strict_mono)
wenzelm@53410
  2507
    apply (rule_tac[!] z(2) w(2))
wenzelm@53410
  2508
    done
wenzelm@53410
  2509
  finally show False by auto
wenzelm@53410
  2510
qed
wenzelm@53410
  2511
wenzelm@53410
  2512
lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
wenzelm@53410
  2513
  unfolding integral_def
wenzelm@53410
  2514
  by (rule some_equality) (auto intro: has_integral_unique)
wenzelm@53410
  2515
wenzelm@53410
  2516
lemma has_integral_is_0:
wenzelm@53410
  2517
  fixes f :: "'n::ordered_euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2518
  assumes "\<forall>x\<in>s. f x = 0"
wenzelm@53410
  2519
  shows "(f has_integral 0) s"
wenzelm@53410
  2520
proof -
wenzelm@53410
  2521
  have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
wenzelm@53410
  2522
    (\<forall>x\<in>{a..b}. f(x) = 0) \<Longrightarrow> (f has_integral 0) ({a..b})"
wenzelm@53410
  2523
    unfolding has_integral
wenzelm@53410
  2524
    apply rule
wenzelm@53410
  2525
    apply rule
wenzelm@53410
  2526
  proof -
wenzelm@53410
  2527
    fix a b e
wenzelm@53410
  2528
    fix f :: "'n \<Rightarrow> 'a"
wenzelm@53410
  2529
    assume as: "\<forall>x\<in>{a..b}. f x = 0" "0 < (e::real)"
wenzelm@53410
  2530
    show "\<exists>d. gauge d \<and>
wenzelm@53410
  2531
      (\<forall>p. p tagged_division_of {a..b} \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
wenzelm@53410
  2532
      apply (rule_tac x="\<lambda>x. ball x 1" in exI)
wenzelm@53410
  2533
      apply rule
wenzelm@53410
  2534
      apply (rule gaugeI)
wenzelm@53410
  2535
      unfolding centre_in_ball
wenzelm@53410
  2536
      defer
wenzelm@53410
  2537
      apply (rule open_ball)
wenzelm@53410
  2538