src/HOL/Multivariate_Analysis/Integration.thy
author haftmann
Tue Oct 13 09:21:15 2015 +0200 (2015-10-13)
changeset 61424 c3658c18b7bc
parent 61243 44b2d133063e
child 61518 ff12606337e9
permissions -rw-r--r--
prod_case as canonical name for product type eliminator
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(*  Author:     John Harrison
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    Author:     Robert Himmelmann, TU Muenchen (Translation from HOL light); proofs reworked by LCP
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*)
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section \<open>Kurzweil-Henstock Gauge Integration in many dimensions.\<close>
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theory Integration
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imports
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  Derivative
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  Uniform_Limit
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  "~~/src/HOL/Library/Indicator_Function"
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begin
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lemma cSup_abs_le: (* TODO: move to Conditionally_Complete_Lattices.thy? *)
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  fixes S :: "real set"
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  shows "S \<noteq> {} \<Longrightarrow> (\<And>x. x\<in>S \<Longrightarrow> \<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:
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  fixes S :: "real set"
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  shows "S \<noteq> {} \<Longrightarrow> (\<And>x. x\<in>S \<Longrightarrow> \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Inf S\<bar> \<le> a"
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  using cSup_abs_le [of "uminus ` S"]
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  by (fastforce simp add: Inf_real_def)
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lemma cSup_asclose:
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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:
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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 \<open>Sundries\<close>
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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 \<open>?r\<close>
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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 \<open>?r\<close> 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 (blast intro: assms)+
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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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      with Suc.hyps \<open>\<forall>n. R n (Suc n)\<close> assms show ?thesis
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        by blast
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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 (blast intro: assms)+
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    done
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  then show ?thesis by auto
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qed
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subsection \<open>Some useful lemmas about intervals.\<close>
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lemma empty_as_interval: "{} = cbox One (0::'a::euclidean_space)"
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  using nonempty_Basis
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  by (fastforce simp add: set_eq_iff mem_box)
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lemma interior_subset_union_intervals:
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  assumes "i = cbox a b"
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    and "j = cbox c d"
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    and "interior j \<noteq> {}"
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    and "i \<subseteq> j \<union> s"
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    and "interior i \<inter> interior j = {}"
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  shows "interior i \<subseteq> interior s"
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proof -
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  have "box a b \<inter> cbox c d = {}"
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     using inter_interval_mixed_eq_empty[of c d a b] and assms(3,5)
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     unfolding assms(1,2) interior_cbox by auto
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  moreover
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  have "box a b \<subseteq> cbox c d \<union> s"
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    apply (rule order_trans,rule box_subset_cbox)
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    using assms(4) unfolding assms(1,2)
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    apply auto
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    done
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  ultimately
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  show ?thesis
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    unfolding assms interior_cbox
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      by auto (metis IntI UnE empty_iff interior_maximal open_box subsetCE subsetI)
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qed
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lemma inter_interior_unions_intervals:
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  fixes f::"('a::euclidean_space) set set"
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  assumes "finite f"
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    and "open s"
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    and "\<forall>t\<in>f. \<exists>a b. t = cbox a b"
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    and "\<forall>t\<in>f. s \<inter> (interior t) = {}"
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  shows "s \<inter> interior (\<Union>f) = {}"
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proof (clarsimp simp only: all_not_in_conv [symmetric])
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  fix x
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  assume x: "x \<in> s" "x \<in> interior (\<Union>f)"
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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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    using interior_subset
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    by auto (meson Topology_Euclidean_Space.open_ball contra_subsetD interior_maximal mem_ball)
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  have "\<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t"
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    if "finite f" and "\<forall>t\<in>f. \<exists>a b. t = cbox a b" and "\<exists>x. x \<in> s \<inter> interior (\<Union>f)" for f
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    using that
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  proof (induct rule: finite_induct)
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    case empty
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    obtain x where "x \<in> s \<inter> interior (\<Union>{})"
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      using empty(2) ..
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    then have False
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      unfolding Union_empty interior_empty by auto
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    then show ?case by auto
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  next
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    case (insert i f)
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    obtain x where x: "x \<in> s \<inter> interior (\<Union>insert i f)"
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      using insert(5) ..
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    then obtain e where e: "0 < e \<and> ball x e \<subseteq> s \<inter> interior (\<Union>insert i f)"
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      unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior], rule_format] ..
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    obtain a where "\<exists>b. i = cbox a b"
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      using insert(4)[rule_format,OF insertI1] ..
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    then obtain b where ab: "i = cbox a b" ..
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    show ?case
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    proof (cases "x \<in> i")
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      case False
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      then have "x \<in> UNIV - cbox a b"
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        unfolding ab by auto
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      then obtain d where "0 < d \<and> ball x d \<subseteq> UNIV - cbox a b"
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        unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_cbox],rule_format] ..
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      then have "0 < d" "ball x (min d e) \<subseteq> UNIV - i"
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        unfolding ab ball_min_Int by auto
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      then have "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)"
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        using e unfolding lem1 unfolding  ball_min_Int by auto
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      then have "x \<in> s \<inter> interior (\<Union>f)" using \<open>d>0\<close> 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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        using insert.hyps(3) insert.prems(1) by blast
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      then show ?thesis by auto
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    next
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      case True show ?thesis
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      proof (cases "x\<in>box a b")
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        case True
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        then obtain d where "0 < d \<and> ball x d \<subseteq> box a b"
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          unfolding open_contains_ball_eq[OF open_box,rule_format] ..
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        then show ?thesis
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          apply (rule_tac x=i in bexI, rule_tac x=x in exI, rule_tac x="min d e" in exI)
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          unfolding ab
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          using box_subset_cbox[of a b] and e
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          apply fastforce+
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          done
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      next
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        case False
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        then obtain k where "x\<bullet>k \<le> a\<bullet>k \<or> x\<bullet>k \<ge> b\<bullet>k" and k: "k \<in> Basis"
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          unfolding mem_box by (auto simp add: not_less)
wenzelm@61165
   302
        then have "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
wenzelm@61165
   303
          using True unfolding ab and mem_box
wenzelm@61165
   304
            apply (erule_tac x = k in ballE)
wenzelm@61165
   305
            apply auto
wenzelm@49970
   306
            done
wenzelm@61165
   307
        then have "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)"
wenzelm@61165
   308
        proof (rule disjE)
wenzelm@61165
   309
          let ?z = "x - (e/2) *\<^sub>R k"
wenzelm@61165
   310
          assume as: "x\<bullet>k = a\<bullet>k"
wenzelm@61165
   311
          have "ball ?z (e / 2) \<inter> i = {}"
wenzelm@61165
   312
          proof (clarsimp simp only: all_not_in_conv [symmetric])
wenzelm@61165
   313
            fix y
wenzelm@61165
   314
            assume "y \<in> ball ?z (e / 2)" and yi: "y \<in> i"
wenzelm@61165
   315
            then have "dist ?z y < e/2" by auto
wenzelm@61165
   316
            then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
wenzelm@61165
   317
              using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
wenzelm@61165
   318
            then have "y\<bullet>k < a\<bullet>k"
wenzelm@61165
   319
              using e k
wenzelm@61165
   320
              by (auto simp add: field_simps abs_less_iff as inner_simps)
wenzelm@61165
   321
            then have "y \<notin> i"
wenzelm@61165
   322
              unfolding ab mem_box by (auto intro!: bexI[OF _ k])
wenzelm@61165
   323
            then show False using yi by auto
wenzelm@61165
   324
          qed
wenzelm@61165
   325
          moreover
wenzelm@61165
   326
          have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
wenzelm@61165
   327
            apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
wenzelm@61165
   328
          proof
wenzelm@61165
   329
            fix y
wenzelm@61165
   330
            assume as: "y \<in> ball ?z (e/2)"
wenzelm@61165
   331
            have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R k)"
wenzelm@61165
   332
              apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R k"])
wenzelm@61165
   333
              unfolding norm_scaleR norm_Basis[OF k]
wenzelm@49970
   334
              apply auto
wenzelm@49970
   335
              done
wenzelm@61165
   336
            also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2"
wenzelm@61165
   337
              apply (rule add_strict_left_mono)
wenzelm@61165
   338
              using as e
wenzelm@61165
   339
              apply (auto simp add: field_simps dist_norm)
wenzelm@49970
   340
              done
wenzelm@61165
   341
            finally show "y \<in> ball x e"
wenzelm@61165
   342
              unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
wenzelm@61165
   343
          qed
wenzelm@61165
   344
          ultimately show ?thesis
wenzelm@61165
   345
            apply (rule_tac x="?z" in exI)
wenzelm@61165
   346
            unfolding Union_insert
wenzelm@61165
   347
            apply auto
wenzelm@61165
   348
            done
wenzelm@61165
   349
        next
wenzelm@61165
   350
          let ?z = "x + (e/2) *\<^sub>R k"
wenzelm@61165
   351
          assume as: "x\<bullet>k = b\<bullet>k"
wenzelm@61165
   352
          have "ball ?z (e / 2) \<inter> i = {}"
wenzelm@61165
   353
          proof (clarsimp simp only: all_not_in_conv [symmetric])
wenzelm@61165
   354
            fix y
wenzelm@61165
   355
            assume "y \<in> ball ?z (e / 2)" and yi: "y \<in> i"
wenzelm@61165
   356
            then have "dist ?z y < e/2"
wenzelm@61165
   357
              by auto
wenzelm@61165
   358
            then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
wenzelm@61165
   359
              using Basis_le_norm[OF k, of "?z - y"]
wenzelm@61165
   360
              unfolding dist_norm by auto
wenzelm@61165
   361
            then have "y\<bullet>k > b\<bullet>k"
wenzelm@61165
   362
              using e k
wenzelm@61165
   363
              by (auto simp add:field_simps inner_simps inner_Basis as)
wenzelm@61165
   364
            then have "y \<notin> i"
wenzelm@61165
   365
              unfolding ab mem_box by (auto intro!: bexI[OF _ k])
wenzelm@61165
   366
            then show False using yi by auto
wenzelm@61165
   367
          qed
wenzelm@61165
   368
          moreover
wenzelm@61165
   369
          have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
wenzelm@61165
   370
            apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
wenzelm@61165
   371
          proof
wenzelm@61165
   372
            fix y
wenzelm@61165
   373
            assume as: "y\<in> ball ?z (e/2)"
wenzelm@61165
   374
            have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R k)"
wenzelm@61165
   375
              apply (rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R k"])
wenzelm@61165
   376
              unfolding norm_scaleR
wenzelm@61165
   377
              apply (auto simp: k)
wenzelm@49970
   378
              done
wenzelm@61165
   379
            also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2"
wenzelm@61165
   380
              apply (rule add_strict_left_mono)
wenzelm@61165
   381
              using as unfolding mem_ball dist_norm
wenzelm@61165
   382
              using e apply (auto simp add: field_simps)
wenzelm@61165
   383
              done
wenzelm@61165
   384
            finally show "y \<in> ball x e"
wenzelm@61165
   385
              unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
wenzelm@53399
   386
          qed
wenzelm@61165
   387
          ultimately show ?thesis
wenzelm@61165
   388
            apply (rule_tac x="?z" in exI)
wenzelm@61165
   389
            unfolding Union_insert
wenzelm@61165
   390
            apply auto
wenzelm@61165
   391
            done
wenzelm@49970
   392
        qed
wenzelm@61165
   393
        then obtain x where "ball x (e / 2) \<subseteq> s \<inter> \<Union>f" ..
wenzelm@61165
   394
        then have "x \<in> s \<inter> interior (\<Union>f)"
wenzelm@61165
   395
          unfolding lem1[where U="\<Union>f", symmetric]
wenzelm@61165
   396
          using centre_in_ball e by auto
wenzelm@61165
   397
        then show ?thesis
wenzelm@61165
   398
          using insert.hyps(3) insert.prems(1) by blast
wenzelm@49970
   399
      qed
wenzelm@49970
   400
    qed
wenzelm@49970
   401
  qed
lp15@60394
   402
  from this[OF assms(1,3)] x
wenzelm@53408
   403
  obtain t x e where "t \<in> f" "0 < e" "ball x e \<subseteq> s \<inter> t"
wenzelm@53408
   404
    by blast
wenzelm@53408
   405
  then have "x \<in> s" "x \<in> interior t"
wenzelm@53399
   406
    using open_subset_interior[OF open_ball, of x e t]
wenzelm@53408
   407
    by auto
wenzelm@53399
   408
  then show False
wenzelm@60420
   409
    using \<open>t \<in> f\<close> assms(4) by auto
wenzelm@60420
   410
qed
wenzelm@60420
   411
wenzelm@60420
   412
subsection \<open>Bounds on intervals where they exist.\<close>
immler@56188
   413
immler@56188
   414
definition interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
immler@56188
   415
  where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)"
immler@56188
   416
immler@56188
   417
definition interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
immler@56188
   418
   where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x:s. x\<bullet>i) *\<^sub>R i)"
immler@56188
   419
immler@56188
   420
lemma interval_upperbound[simp]:
immler@56188
   421
  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
immler@56188
   422
    interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
immler@56188
   423
  unfolding interval_upperbound_def euclidean_representation_setsum cbox_def SUP_def
immler@56188
   424
  by (safe intro!: cSup_eq) auto
immler@56188
   425
immler@56188
   426
lemma interval_lowerbound[simp]:
immler@56188
   427
  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
immler@56188
   428
    interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
immler@56188
   429
  unfolding interval_lowerbound_def euclidean_representation_setsum cbox_def INF_def
immler@56188
   430
  by (safe intro!: cInf_eq) auto
immler@56188
   431
immler@56188
   432
lemmas interval_bounds = interval_upperbound interval_lowerbound
immler@56188
   433
immler@56188
   434
lemma
immler@56188
   435
  fixes X::"real set"
immler@56188
   436
  shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
immler@56188
   437
    and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
immler@56188
   438
  by (auto simp: interval_upperbound_def interval_lowerbound_def SUP_def INF_def)
immler@56188
   439
immler@56188
   440
lemma interval_bounds'[simp]:
immler@56188
   441
  assumes "cbox a b \<noteq> {}"
immler@56188
   442
  shows "interval_upperbound (cbox a b) = b"
immler@56188
   443
    and "interval_lowerbound (cbox a b) = a"
immler@56188
   444
  using assms unfolding box_ne_empty by auto
wenzelm@53399
   445
hoelzl@59425
   446
lp15@60615
   447
lemma interval_upperbound_Times:
hoelzl@59425
   448
  assumes "A \<noteq> {}" and "B \<noteq> {}"
hoelzl@59425
   449
  shows "interval_upperbound (A \<times> B) = (interval_upperbound A, interval_upperbound B)"
hoelzl@59425
   450
proof-
hoelzl@59425
   451
  from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
hoelzl@59425
   452
  have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:A. x \<bullet> i) *\<^sub>R i)"
hoelzl@59425
   453
      by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
hoelzl@59425
   454
  moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
hoelzl@59425
   455
  have "(\<Sum>i\<in>Basis. (SUP x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (SUP x:B. x \<bullet> i) *\<^sub>R i)"
hoelzl@59425
   456
      by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
hoelzl@59425
   457
  ultimately show ?thesis unfolding interval_upperbound_def
hoelzl@59425
   458
      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
hoelzl@59425
   459
qed
hoelzl@59425
   460
lp15@60615
   461
lemma interval_lowerbound_Times:
hoelzl@59425
   462
  assumes "A \<noteq> {}" and "B \<noteq> {}"
hoelzl@59425
   463
  shows "interval_lowerbound (A \<times> B) = (interval_lowerbound A, interval_lowerbound B)"
hoelzl@59425
   464
proof-
hoelzl@59425
   465
  from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
hoelzl@59425
   466
  have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (i, 0)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:A. x \<bullet> i) *\<^sub>R i)"
hoelzl@59425
   467
      by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
hoelzl@59425
   468
  moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
hoelzl@59425
   469
  have "(\<Sum>i\<in>Basis. (INF x:A \<times> B. x \<bullet> (0, i)) *\<^sub>R i) = (\<Sum>i\<in>Basis. (INF x:B. x \<bullet> i) *\<^sub>R i)"
hoelzl@59425
   470
      by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
hoelzl@59425
   471
  ultimately show ?thesis unfolding interval_lowerbound_def
hoelzl@59425
   472
      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
hoelzl@59425
   473
qed
hoelzl@59425
   474
wenzelm@60420
   475
subsection \<open>Content (length, area, volume...) of an interval.\<close>
himmelma@35172
   476
immler@56188
   477
definition "content (s::('a::euclidean_space) set) =
immler@56188
   478
  (if s = {} then 0 else (\<Prod>i\<in>Basis. (interval_upperbound s)\<bullet>i - (interval_lowerbound s)\<bullet>i))"
immler@56188
   479
immler@56188
   480
lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> cbox a b \<noteq> {}"
immler@56188
   481
  unfolding box_eq_empty unfolding not_ex not_less by auto
immler@56188
   482
immler@56188
   483
lemma content_cbox:
immler@56188
   484
  fixes a :: "'a::euclidean_space"
hoelzl@50526
   485
  assumes "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
immler@56188
   486
  shows "content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
wenzelm@49970
   487
  using interval_not_empty[OF assms]
immler@54777
   488
  unfolding content_def
lp15@60384
   489
  by auto
immler@56188
   490
immler@56188
   491
lemma content_cbox':
immler@56188
   492
  fixes a :: "'a::euclidean_space"
immler@56188
   493
  assumes "cbox a b \<noteq> {}"
immler@56188
   494
  shows "content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
lp15@60384
   495
    using assms box_ne_empty(1) content_cbox by blast
wenzelm@49970
   496
wenzelm@53408
   497
lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
immler@56188
   498
  by (auto simp: interval_upperbound_def interval_lowerbound_def SUP_def INF_def content_def)
immler@56188
   499
paulson@61204
   500
lemma abs_eq_content: "abs (y - x) = (if x\<le>y then content {x .. y} else content {y..x})"
paulson@61204
   501
  by (auto simp: content_real)
paulson@61204
   502
hoelzl@50104
   503
lemma content_singleton[simp]: "content {a} = 0"
hoelzl@50104
   504
proof -
immler@56188
   505
  have "content (cbox a a) = 0"
immler@56188
   506
    by (subst content_cbox) (auto simp: ex_in_conv)
immler@56188
   507
  then show ?thesis by (simp add: cbox_sing)
immler@56188
   508
qed
immler@56188
   509
lp15@60615
   510
lemma content_unit[iff]: "content(cbox 0 (One::'a::euclidean_space)) = 1"
immler@56188
   511
 proof -
immler@56188
   512
   have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i"
immler@56188
   513
    by auto
immler@56188
   514
  have "0 \<in> cbox 0 (One::'a)"
immler@56188
   515
    unfolding mem_box by auto
immler@56188
   516
  then show ?thesis
haftmann@57418
   517
     unfolding content_def interval_bounds[OF *] using setprod.neutral_const by auto
immler@56188
   518
 qed
wenzelm@49970
   519
wenzelm@49970
   520
lemma content_pos_le[intro]:
immler@56188
   521
  fixes a::"'a::euclidean_space"
immler@56188
   522
  shows "0 \<le> content (cbox a b)"
immler@56188
   523
proof (cases "cbox a b = {}")
immler@56188
   524
  case False
immler@56188
   525
  then have *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
immler@56188
   526
    unfolding box_ne_empty .
immler@56188
   527
  have "0 \<le> (\<Prod>i\<in>Basis. interval_upperbound (cbox a b) \<bullet> i - interval_lowerbound (cbox a b) \<bullet> i)"
immler@56188
   528
    apply (rule setprod_nonneg)
immler@56188
   529
    unfolding interval_bounds[OF *]
immler@56188
   530
    using *
immler@56188
   531
    apply auto
immler@56188
   532
    done
immler@56188
   533
  also have "\<dots> = content (cbox a b)" using False by (simp add: content_def)
immler@56188
   534
  finally show ?thesis .
immler@56188
   535
qed (simp add: content_def)
wenzelm@49970
   536
lp15@60615
   537
corollary content_nonneg [simp]:
lp15@60615
   538
  fixes a::"'a::euclidean_space"
lp15@60615
   539
  shows "~ content (cbox a b) < 0"
lp15@60615
   540
using not_le by blast
lp15@60615
   541
wenzelm@49970
   542
lemma content_pos_lt:
immler@56188
   543
  fixes a :: "'a::euclidean_space"
hoelzl@50526
   544
  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
immler@56188
   545
  shows "0 < content (cbox a b)"
immler@54777
   546
  using assms
immler@56188
   547
  by (auto simp: content_def box_eq_empty intro!: setprod_pos)
wenzelm@49970
   548
wenzelm@53408
   549
lemma content_eq_0:
immler@56188
   550
  "content (cbox a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
immler@56188
   551
  by (auto simp: content_def box_eq_empty intro!: setprod_pos bexI)
himmelma@35172
   552
wenzelm@53408
   553
lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
wenzelm@53399
   554
  by auto
himmelma@35172
   555
immler@56188
   556
lemma content_cbox_cases:
immler@56188
   557
  "content (cbox a (b::'a::euclidean_space)) =
hoelzl@50526
   558
    (if \<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i then setprod (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis else 0)"
immler@56188
   559
  by (auto simp: not_le content_eq_0 intro: less_imp_le content_cbox)
immler@56188
   560
immler@56188
   561
lemma content_eq_0_interior: "content (cbox a b) = 0 \<longleftrightarrow> interior(cbox a b) = {}"
immler@56188
   562
  unfolding content_eq_0 interior_cbox box_eq_empty
wenzelm@53408
   563
  by auto
himmelma@35172
   564
wenzelm@53399
   565
lemma content_pos_lt_eq:
immler@56188
   566
  "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
lp15@60394
   567
proof (rule iffI)
immler@56188
   568
  assume "0 < content (cbox a b)"
immler@56188
   569
  then have "content (cbox a b) \<noteq> 0" by auto
wenzelm@53399
   570
  then show "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
wenzelm@49970
   571
    unfolding content_eq_0 not_ex not_le by fastforce
lp15@60394
   572
next
lp15@60394
   573
  assume "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i"
lp15@60394
   574
  then show "0 < content (cbox a b)"
lp15@60394
   575
    by (metis content_pos_lt)
wenzelm@49970
   576
qed
wenzelm@49970
   577
wenzelm@53399
   578
lemma content_empty [simp]: "content {} = 0"
wenzelm@53399
   579
  unfolding content_def by auto
himmelma@35172
   580
paulson@60762
   581
lemma content_real_if [simp]: "content {a..b} = (if a \<le> b then b - a else 0)"
paulson@60762
   582
  by (simp add: content_real)
paulson@60762
   583
wenzelm@49698
   584
lemma content_subset:
immler@56188
   585
  assumes "cbox a b \<subseteq> cbox c d"
immler@56188
   586
  shows "content (cbox a b) \<le> content (cbox c d)"
immler@56188
   587
proof (cases "cbox a b = {}")
immler@56188
   588
  case True
immler@56188
   589
  then show ?thesis
immler@56188
   590
    using content_pos_le[of c d] by auto
immler@56188
   591
next
immler@56188
   592
  case False
immler@56188
   593
  then have ab_ne: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
immler@56188
   594
    unfolding box_ne_empty by auto
immler@56188
   595
  then have ab_ab: "a\<in>cbox a b" "b\<in>cbox a b"
immler@56188
   596
    unfolding mem_box by auto
immler@56188
   597
  have "cbox c d \<noteq> {}" using assms False by auto
immler@56188
   598
  then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
immler@56188
   599
    using assms unfolding box_ne_empty by auto
lp15@60394
   600
  have "\<And>i. i \<in> Basis \<Longrightarrow> 0 \<le> b \<bullet> i - a \<bullet> i"
lp15@60394
   601
    using ab_ne by (metis diff_le_iff(1))
lp15@60394
   602
  moreover
lp15@60394
   603
  have "\<And>i. i \<in> Basis \<Longrightarrow> b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
lp15@60394
   604
    using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2)]
lp15@60394
   605
          assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(1)]
lp15@60394
   606
      by (metis diff_mono)
lp15@60394
   607
  ultimately show ?thesis
lp15@60394
   608
    unfolding content_def interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
wenzelm@60420
   609
    by (simp add: setprod_mono if_not_P[OF False] if_not_P[OF \<open>cbox c d \<noteq> {}\<close>])
immler@56188
   610
qed
immler@56188
   611
immler@56188
   612
lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
nipkow@44890
   613
  unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
himmelma@35172
   614
hoelzl@59425
   615
lemma content_times[simp]: "content (A \<times> B) = content A * content B"
hoelzl@59425
   616
proof (cases "A \<times> B = {}")
hoelzl@59425
   617
  let ?ub1 = "interval_upperbound" and ?lb1 = "interval_lowerbound"
hoelzl@59425
   618
  let ?ub2 = "interval_upperbound" and ?lb2 = "interval_lowerbound"
hoelzl@59425
   619
  assume nonempty: "A \<times> B \<noteq> {}"
lp15@60615
   620
  hence "content (A \<times> B) = (\<Prod>i\<in>Basis. (?ub1 A, ?ub2 B) \<bullet> i - (?lb1 A, ?lb2 B) \<bullet> i)"
hoelzl@59425
   621
      unfolding content_def by (simp add: interval_upperbound_Times interval_lowerbound_Times)
hoelzl@59425
   622
  also have "... = content A * content B" unfolding content_def using nonempty
hoelzl@59425
   623
    apply (subst Basis_prod_def, subst setprod.union_disjoint, force, force, force, simp)
hoelzl@59425
   624
    apply (subst (1 2) setprod.reindex, auto intro: inj_onI)
hoelzl@59425
   625
    done
hoelzl@59425
   626
  finally show ?thesis .
hoelzl@59425
   627
qed (auto simp: content_def)
hoelzl@59425
   628
lp15@60615
   629
lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
lp15@60615
   630
  by (simp add: cbox_Pair_eq)
lp15@60615
   631
lp15@60615
   632
lemma content_cbox_pair_eq0_D:
lp15@60615
   633
   "content (cbox (a,c) (b,d)) = 0 \<Longrightarrow> content (cbox a b) = 0 \<or> content (cbox c d) = 0"
lp15@60615
   634
  by (simp add: content_Pair)
lp15@60615
   635
lp15@60615
   636
lemma content_eq_0_gen:
lp15@60615
   637
  fixes s :: "'a::euclidean_space set"
lp15@60615
   638
  assumes "bounded s"
lp15@60615
   639
  shows "content s = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. \<exists>v. \<forall>x \<in> s. x \<bullet> i = v)"  (is "_ = ?rhs")
lp15@60615
   640
proof safe
lp15@60615
   641
  assume "content s = 0" then show ?rhs
lp15@60615
   642
    apply (clarsimp simp: ex_in_conv content_def split: split_if_asm)
lp15@60615
   643
    apply (rule_tac x=a in bexI)
lp15@60615
   644
    apply (rule_tac x="interval_lowerbound s \<bullet> a" in exI)
lp15@60615
   645
    apply (clarsimp simp: interval_upperbound_def interval_lowerbound_def)
lp15@60615
   646
    apply (drule cSUP_eq_cINF_D)
lp15@60615
   647
    apply (auto simp: bounded_inner_imp_bdd_above [OF assms]  bounded_inner_imp_bdd_below [OF assms])
lp15@60615
   648
    done
lp15@60615
   649
next
lp15@60615
   650
  fix i a
lp15@60615
   651
  assume "i \<in> Basis" "\<forall>x\<in>s. x \<bullet> i = a"
lp15@60615
   652
  then show "content s = 0"
lp15@60615
   653
    apply (clarsimp simp: content_def)
lp15@60615
   654
    apply (rule_tac x=i in bexI)
lp15@60615
   655
    apply (auto simp: interval_upperbound_def interval_lowerbound_def)
lp15@60615
   656
    done
lp15@60615
   657
qed
lp15@60615
   658
lp15@60615
   659
lemma content_0_subset_gen:
lp15@60615
   660
  fixes a :: "'a::euclidean_space"
lp15@60615
   661
  assumes "content t = 0" "s \<subseteq> t" "bounded t" shows "content s = 0"
lp15@60615
   662
proof -
lp15@60615
   663
  have "bounded s"
lp15@60615
   664
    using assms by (metis bounded_subset)
lp15@60615
   665
  then show ?thesis
lp15@60615
   666
    using assms
lp15@60615
   667
    by (auto simp: content_eq_0_gen)
lp15@60615
   668
qed
lp15@60615
   669
lp15@60615
   670
lemma content_0_subset: "\<lbrakk>content(cbox a b) = 0; s \<subseteq> cbox a b\<rbrakk> \<Longrightarrow> content s = 0"
lp15@60615
   671
  by (simp add: content_0_subset_gen bounded_cbox)
lp15@60615
   672
wenzelm@49698
   673
wenzelm@60420
   674
subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
himmelma@35172
   675
wenzelm@53408
   676
definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
wenzelm@53399
   677
wenzelm@53399
   678
lemma gaugeI:
wenzelm@53399
   679
  assumes "\<And>x. x \<in> g x"
wenzelm@53399
   680
    and "\<And>x. open (g x)"
wenzelm@53399
   681
  shows "gauge g"
himmelma@35172
   682
  using assms unfolding gauge_def by auto
himmelma@35172
   683
wenzelm@53399
   684
lemma gaugeD[dest]:
wenzelm@53399
   685
  assumes "gauge d"
wenzelm@53399
   686
  shows "x \<in> d x"
wenzelm@53399
   687
    and "open (d x)"
wenzelm@49698
   688
  using assms unfolding gauge_def by auto
himmelma@35172
   689
himmelma@35172
   690
lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
wenzelm@53399
   691
  unfolding gauge_def by auto
wenzelm@53399
   692
wenzelm@53399
   693
lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
wenzelm@53399
   694
  unfolding gauge_def by auto
himmelma@35172
   695
lp15@60466
   696
lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)"
wenzelm@49698
   697
  by (rule gauge_ball) auto
himmelma@35172
   698
wenzelm@53408
   699
lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
wenzelm@53399
   700
  unfolding gauge_def by auto
himmelma@35172
   701
wenzelm@49698
   702
lemma gauge_inters:
wenzelm@53399
   703
  assumes "finite s"
wenzelm@53399
   704
    and "\<forall>d\<in>s. gauge (f d)"
wenzelm@60585
   705
  shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> s})"
wenzelm@49698
   706
proof -
wenzelm@53399
   707
  have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
wenzelm@53399
   708
    by auto
wenzelm@49698
   709
  show ?thesis
wenzelm@53399
   710
    unfolding gauge_def unfolding *
wenzelm@49698
   711
    using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
wenzelm@49698
   712
qed
wenzelm@49698
   713
wenzelm@53399
   714
lemma gauge_existence_lemma:
wenzelm@53408
   715
  "(\<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
   716
  by (metis zero_less_one)
wenzelm@49698
   717
himmelma@35172
   718
wenzelm@60420
   719
subsection \<open>Divisions.\<close>
himmelma@35172
   720
wenzelm@53408
   721
definition division_of (infixl "division'_of" 40)
wenzelm@53408
   722
where
wenzelm@53399
   723
  "s division_of i \<longleftrightarrow>
wenzelm@53399
   724
    finite s \<and>
immler@56188
   725
    (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = cbox a b)) \<and>
wenzelm@53399
   726
    (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
wenzelm@53399
   727
    (\<Union>s = i)"
himmelma@35172
   728
wenzelm@49698
   729
lemma division_ofD[dest]:
wenzelm@49698
   730
  assumes "s division_of i"
wenzelm@53408
   731
  shows "finite s"
wenzelm@53408
   732
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   733
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
immler@56188
   734
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
   735
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
wenzelm@53408
   736
    and "\<Union>s = i"
wenzelm@49698
   737
  using assms unfolding division_of_def by auto
himmelma@35172
   738
himmelma@35172
   739
lemma division_ofI:
wenzelm@53408
   740
  assumes "finite s"
wenzelm@53408
   741
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   742
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
immler@56188
   743
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
   744
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53399
   745
    and "\<Union>s = i"
wenzelm@53399
   746
  shows "s division_of i"
wenzelm@53399
   747
  using assms unfolding division_of_def by auto
himmelma@35172
   748
himmelma@35172
   749
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
himmelma@35172
   750
  unfolding division_of_def by auto
himmelma@35172
   751
immler@56188
   752
lemma division_of_self[intro]: "cbox a b \<noteq> {} \<Longrightarrow> {cbox a b} division_of (cbox a b)"
himmelma@35172
   753
  unfolding division_of_def by auto
himmelma@35172
   754
wenzelm@53399
   755
lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
wenzelm@53399
   756
  unfolding division_of_def by auto
himmelma@35172
   757
wenzelm@49698
   758
lemma division_of_sing[simp]:
immler@56188
   759
  "s division_of cbox a (a::'a::euclidean_space) \<longleftrightarrow> s = {cbox a a}"
wenzelm@53399
   760
  (is "?l = ?r")
wenzelm@49698
   761
proof
wenzelm@49698
   762
  assume ?r
wenzelm@53399
   763
  moreover
lp15@60384
   764
  { fix k
lp15@60384
   765
    assume "s = {{a}}" "k\<in>s"
lp15@60384
   766
    then have "\<exists>x y. k = cbox x y"
wenzelm@50945
   767
      apply (rule_tac x=a in exI)+
lp15@60384
   768
      apply (force simp: cbox_sing)
wenzelm@50945
   769
      done
wenzelm@49698
   770
  }
wenzelm@53399
   771
  ultimately show ?l
immler@56188
   772
    unfolding division_of_def cbox_sing by auto
wenzelm@49698
   773
next
wenzelm@49698
   774
  assume ?l
immler@56188
   775
  note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
wenzelm@53399
   776
  {
wenzelm@53399
   777
    fix x
wenzelm@53399
   778
    assume x: "x \<in> s" have "x = {a}"
wenzelm@53408
   779
      using *(2)[rule_format,OF x] by auto
wenzelm@53399
   780
  }
wenzelm@53408
   781
  moreover have "s \<noteq> {}"
wenzelm@53408
   782
    using *(4) by auto
wenzelm@53408
   783
  ultimately show ?r
immler@56188
   784
    unfolding cbox_sing by auto
wenzelm@49698
   785
qed
himmelma@35172
   786
himmelma@35172
   787
lemma elementary_empty: obtains p where "p division_of {}"
himmelma@35172
   788
  unfolding division_of_trivial by auto
himmelma@35172
   789
immler@56188
   790
lemma elementary_interval: obtains p where "p division_of (cbox a b)"
wenzelm@49698
   791
  by (metis division_of_trivial division_of_self)
himmelma@35172
   792
himmelma@35172
   793
lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
himmelma@35172
   794
  unfolding division_of_def by auto
himmelma@35172
   795
himmelma@35172
   796
lemma forall_in_division:
immler@56188
   797
  "d division_of i \<Longrightarrow> (\<forall>x\<in>d. P x) \<longleftrightarrow> (\<forall>a b. cbox a b \<in> d \<longrightarrow> P (cbox a b))"
nipkow@44890
   798
  unfolding division_of_def by fastforce
himmelma@35172
   799
wenzelm@53399
   800
lemma division_of_subset:
wenzelm@53399
   801
  assumes "p division_of (\<Union>p)"
wenzelm@53399
   802
    and "q \<subseteq> p"
wenzelm@53399
   803
  shows "q division_of (\<Union>q)"
wenzelm@53408
   804
proof (rule division_ofI)
wenzelm@53408
   805
  note * = division_ofD[OF assms(1)]
wenzelm@49698
   806
  show "finite q"
lp15@60384
   807
    using "*"(1) assms(2) infinite_super by auto
wenzelm@53399
   808
  {
wenzelm@53399
   809
    fix k
wenzelm@49698
   810
    assume "k \<in> q"
wenzelm@53408
   811
    then have kp: "k \<in> p"
wenzelm@53408
   812
      using assms(2) by auto
wenzelm@53408
   813
    show "k \<subseteq> \<Union>q"
wenzelm@60420
   814
      using \<open>k \<in> q\<close> by auto
immler@56188
   815
    show "\<exists>a b. k = cbox a b"
wenzelm@53408
   816
      using *(4)[OF kp] by auto
wenzelm@53408
   817
    show "k \<noteq> {}"
wenzelm@53408
   818
      using *(3)[OF kp] by auto
wenzelm@53399
   819
  }
wenzelm@49698
   820
  fix k1 k2
wenzelm@49698
   821
  assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
wenzelm@53408
   822
  then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
wenzelm@53399
   823
    using assms(2) by auto
wenzelm@53399
   824
  show "interior k1 \<inter> interior k2 = {}"
wenzelm@53408
   825
    using *(5)[OF **] by auto
wenzelm@49698
   826
qed auto
wenzelm@49698
   827
wenzelm@49698
   828
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@49698
   829
  unfolding division_of_def by auto
himmelma@35172
   830
wenzelm@49970
   831
lemma division_of_content_0:
immler@56188
   832
  assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
wenzelm@49970
   833
  shows "\<forall>k\<in>d. content k = 0"
wenzelm@49970
   834
  unfolding forall_in_division[OF assms(2)]
lp15@60384
   835
  by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
wenzelm@49970
   836
wenzelm@49970
   837
lemma division_inter:
immler@56188
   838
  fixes s1 s2 :: "'a::euclidean_space set"
wenzelm@53408
   839
  assumes "p1 division_of s1"
wenzelm@53408
   840
    and "p2 division_of s2"
wenzelm@49970
   841
  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
   842
  (is "?A' division_of _")
wenzelm@49970
   843
proof -
wenzelm@49970
   844
  let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
wenzelm@53408
   845
  have *: "?A' = ?A" by auto
wenzelm@53399
   846
  show ?thesis
wenzelm@53399
   847
    unfolding *
wenzelm@49970
   848
  proof (rule division_ofI)
wenzelm@53399
   849
    have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
wenzelm@53399
   850
      by auto
wenzelm@53399
   851
    moreover have "finite (p1 \<times> p2)"
wenzelm@53399
   852
      using assms unfolding division_of_def by auto
wenzelm@49970
   853
    ultimately show "finite ?A" by auto
wenzelm@53399
   854
    have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
wenzelm@53399
   855
      by auto
wenzelm@49970
   856
    show "\<Union>?A = s1 \<inter> s2"
wenzelm@49970
   857
      apply (rule set_eqI)
wenzelm@49970
   858
      unfolding * and Union_image_eq UN_iff
wenzelm@49970
   859
      using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
wenzelm@49970
   860
      apply auto
wenzelm@49970
   861
      done
wenzelm@53399
   862
    {
wenzelm@53399
   863
      fix k
wenzelm@53399
   864
      assume "k \<in> ?A"
wenzelm@53408
   865
      then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
wenzelm@53399
   866
        by auto
wenzelm@53408
   867
      then show "k \<noteq> {}"
wenzelm@53408
   868
        by auto
wenzelm@49970
   869
      show "k \<subseteq> s1 \<inter> s2"
wenzelm@49970
   870
        using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
wenzelm@49970
   871
        unfolding k by auto
immler@56188
   872
      obtain a1 b1 where k1: "k1 = cbox a1 b1"
wenzelm@53408
   873
        using division_ofD(4)[OF assms(1) k(2)] by blast
immler@56188
   874
      obtain a2 b2 where k2: "k2 = cbox a2 b2"
wenzelm@53408
   875
        using division_ofD(4)[OF assms(2) k(3)] by blast
immler@56188
   876
      show "\<exists>a b. k = cbox a b"
wenzelm@53408
   877
        unfolding k k1 k2 unfolding inter_interval by auto
wenzelm@53408
   878
    }
wenzelm@49970
   879
    fix k1 k2
wenzelm@53408
   880
    assume "k1 \<in> ?A"
wenzelm@53408
   881
    then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
wenzelm@53408
   882
      by auto
wenzelm@53408
   883
    assume "k2 \<in> ?A"
wenzelm@53408
   884
    then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
wenzelm@53408
   885
      by auto
wenzelm@49970
   886
    assume "k1 \<noteq> k2"
wenzelm@53399
   887
    then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
wenzelm@53399
   888
      unfolding k1 k2 by auto
wenzelm@53408
   889
    have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
wenzelm@53408
   890
      interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
wenzelm@53408
   891
      interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
wenzelm@53408
   892
      interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
wenzelm@49970
   893
    show "interior k1 \<inter> interior k2 = {}"
wenzelm@49970
   894
      unfolding k1 k2
wenzelm@49970
   895
      apply (rule *)
lp15@60384
   896
      using assms division_ofD(5) k1 k2(2) k2(3) th apply auto
wenzelm@53399
   897
      done
wenzelm@49970
   898
  qed
wenzelm@49970
   899
qed
wenzelm@49970
   900
wenzelm@49970
   901
lemma division_inter_1:
wenzelm@53408
   902
  assumes "d division_of i"
immler@56188
   903
    and "cbox a (b::'a::euclidean_space) \<subseteq> i"
immler@56188
   904
  shows "{cbox a b \<inter> k | k. k \<in> d \<and> cbox a b \<inter> k \<noteq> {}} division_of (cbox a b)"
immler@56188
   905
proof (cases "cbox a b = {}")
wenzelm@49970
   906
  case True
wenzelm@53399
   907
  show ?thesis
wenzelm@53399
   908
    unfolding True and division_of_trivial by auto
wenzelm@49970
   909
next
wenzelm@49970
   910
  case False
immler@56188
   911
  have *: "cbox a b \<inter> i = cbox a b" using assms(2) by auto
wenzelm@53399
   912
  show ?thesis
wenzelm@53399
   913
    using division_inter[OF division_of_self[OF False] assms(1)]
wenzelm@53399
   914
    unfolding * by auto
wenzelm@49970
   915
qed
wenzelm@49970
   916
wenzelm@49970
   917
lemma elementary_inter:
immler@56188
   918
  fixes s t :: "'a::euclidean_space set"
wenzelm@53408
   919
  assumes "p1 division_of s"
wenzelm@53408
   920
    and "p2 division_of t"
himmelma@35172
   921
  shows "\<exists>p. p division_of (s \<inter> t)"
lp15@60384
   922
using assms division_inter by blast
wenzelm@49970
   923
wenzelm@49970
   924
lemma elementary_inters:
wenzelm@53408
   925
  assumes "finite f"
wenzelm@53408
   926
    and "f \<noteq> {}"
immler@56188
   927
    and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)"
wenzelm@60585
   928
  shows "\<exists>p. p division_of (\<Inter>f)"
wenzelm@49970
   929
  using assms
wenzelm@49970
   930
proof (induct f rule: finite_induct)
wenzelm@49970
   931
  case (insert x f)
wenzelm@49970
   932
  show ?case
wenzelm@49970
   933
  proof (cases "f = {}")
wenzelm@49970
   934
    case True
wenzelm@53399
   935
    then show ?thesis
wenzelm@53399
   936
      unfolding True using insert by auto
wenzelm@49970
   937
  next
wenzelm@49970
   938
    case False
wenzelm@53408
   939
    obtain p where "p division_of \<Inter>f"
wenzelm@53408
   940
      using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
wenzelm@53408
   941
    moreover obtain px where "px division_of x"
wenzelm@53408
   942
      using insert(5)[rule_format,OF insertI1] ..
wenzelm@49970
   943
    ultimately show ?thesis
lp15@60384
   944
      by (simp add: elementary_inter Inter_insert)
wenzelm@49970
   945
  qed
wenzelm@49970
   946
qed auto
himmelma@35172
   947
himmelma@35172
   948
lemma division_disjoint_union:
wenzelm@53408
   949
  assumes "p1 division_of s1"
wenzelm@53408
   950
    and "p2 division_of s2"
wenzelm@53408
   951
    and "interior s1 \<inter> interior s2 = {}"
wenzelm@50945
   952
  shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
wenzelm@50945
   953
proof (rule division_ofI)
wenzelm@53408
   954
  note d1 = division_ofD[OF assms(1)]
wenzelm@53408
   955
  note d2 = division_ofD[OF assms(2)]
wenzelm@53408
   956
  show "finite (p1 \<union> p2)"
wenzelm@53408
   957
    using d1(1) d2(1) by auto
wenzelm@53408
   958
  show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
wenzelm@53408
   959
    using d1(6) d2(6) by auto
wenzelm@50945
   960
  {
wenzelm@50945
   961
    fix k1 k2
wenzelm@50945
   962
    assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
wenzelm@50945
   963
    moreover
wenzelm@50945
   964
    let ?g="interior k1 \<inter> interior k2 = {}"
wenzelm@50945
   965
    {
wenzelm@50945
   966
      assume as: "k1\<in>p1" "k2\<in>p2"
wenzelm@50945
   967
      have ?g
wenzelm@50945
   968
        using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
wenzelm@50945
   969
        using assms(3) by blast
wenzelm@50945
   970
    }
wenzelm@50945
   971
    moreover
wenzelm@50945
   972
    {
wenzelm@50945
   973
      assume as: "k1\<in>p2" "k2\<in>p1"
wenzelm@50945
   974
      have ?g
wenzelm@50945
   975
        using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
wenzelm@50945
   976
        using assms(3) by blast
wenzelm@50945
   977
    }
wenzelm@53399
   978
    ultimately show ?g
wenzelm@53399
   979
      using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
wenzelm@50945
   980
  }
wenzelm@50945
   981
  fix k
wenzelm@50945
   982
  assume k: "k \<in> p1 \<union> p2"
wenzelm@53408
   983
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
   984
    using k d1(2) d2(2) by auto
wenzelm@53408
   985
  show "k \<noteq> {}"
wenzelm@53408
   986
    using k d1(3) d2(3) by auto
immler@56188
   987
  show "\<exists>a b. k = cbox a b"
wenzelm@53408
   988
    using k d1(4) d2(4) by auto
wenzelm@50945
   989
qed
himmelma@35172
   990
himmelma@35172
   991
lemma partial_division_extend_1:
immler@56188
   992
  fixes a b c d :: "'a::euclidean_space"
immler@56188
   993
  assumes incl: "cbox c d \<subseteq> cbox a b"
immler@56188
   994
    and nonempty: "cbox c d \<noteq> {}"
immler@56188
   995
  obtains p where "p division_of (cbox a b)" "cbox c d \<in> p"
hoelzl@50526
   996
proof
wenzelm@53408
   997
  let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
immler@56188
   998
    cbox (\<Sum>i\<in>Basis. (fst (f i) \<bullet> i) *\<^sub>R i) (\<Sum>i\<in>Basis. (snd (f i) \<bullet> i) *\<^sub>R i)"
wenzelm@53015
   999
  def p \<equiv> "?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
hoelzl@50526
  1000
immler@56188
  1001
  show "cbox c d \<in> p"
hoelzl@50526
  1002
    unfolding p_def
immler@56188
  1003
    by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
wenzelm@50945
  1004
  {
wenzelm@50945
  1005
    fix i :: 'a
wenzelm@50945
  1006
    assume "i \<in> Basis"
hoelzl@50526
  1007
    with incl nonempty have "a \<bullet> i \<le> c \<bullet> i" "c \<bullet> i \<le> d \<bullet> i" "d \<bullet> i \<le> b \<bullet> i"
immler@56188
  1008
      unfolding box_eq_empty subset_box by (auto simp: not_le)
wenzelm@50945
  1009
  }
hoelzl@50526
  1010
  note ord = this
hoelzl@50526
  1011
immler@56188
  1012
  show "p division_of (cbox a b)"
hoelzl@50526
  1013
  proof (rule division_ofI)
wenzelm@53399
  1014
    show "finite p"
wenzelm@53399
  1015
      unfolding p_def by (auto intro!: finite_PiE)
wenzelm@50945
  1016
    {
wenzelm@50945
  1017
      fix k
wenzelm@50945
  1018
      assume "k \<in> p"
wenzelm@53015
  1019
      then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
hoelzl@50526
  1020
        by (auto simp: p_def)
immler@56188
  1021
      then show "\<exists>a b. k = cbox a b"
wenzelm@53408
  1022
        by auto
immler@56188
  1023
      have "k \<subseteq> cbox a b \<and> k \<noteq> {}"
immler@56188
  1024
      proof (simp add: k box_eq_empty subset_box not_less, safe)
wenzelm@53374
  1025
        fix i :: 'a
wenzelm@53374
  1026
        assume i: "i \<in> Basis"
wenzelm@50945
  1027
        with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
hoelzl@50526
  1028
          by (auto simp: PiE_iff)
wenzelm@53374
  1029
        with i ord[of i]
wenzelm@50945
  1030
        show "a \<bullet> i \<le> fst (f i) \<bullet> i" "snd (f i) \<bullet> i \<le> b \<bullet> i" "fst (f i) \<bullet> i \<le> snd (f i) \<bullet> i"
immler@54776
  1031
          by auto
hoelzl@50526
  1032
      qed
immler@56188
  1033
      then show "k \<noteq> {}" "k \<subseteq> cbox a b"
wenzelm@53408
  1034
        by auto
wenzelm@50945
  1035
      {
wenzelm@53408
  1036
        fix l
wenzelm@53408
  1037
        assume "l \<in> p"
wenzelm@53015
  1038
        then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
wenzelm@50945
  1039
          by (auto simp: p_def)
wenzelm@50945
  1040
        assume "l \<noteq> k"
wenzelm@50945
  1041
        have "\<exists>i\<in>Basis. f i \<noteq> g i"
wenzelm@50945
  1042
        proof (rule ccontr)
wenzelm@53408
  1043
          assume "\<not> ?thesis"
wenzelm@50945
  1044
          with f g have "f = g"
wenzelm@50945
  1045
            by (auto simp: PiE_iff extensional_def intro!: ext)
wenzelm@60420
  1046
          with \<open>l \<noteq> k\<close> show False
wenzelm@50945
  1047
            by (simp add: l k)
wenzelm@50945
  1048
        qed
wenzelm@53408
  1049
        then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
wenzelm@53408
  1050
        then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
lp15@60384
  1051
                  "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
wenzelm@50945
  1052
          using f g by (auto simp: PiE_iff)
wenzelm@53408
  1053
        with * ord[of i] show "interior l \<inter> interior k = {}"
immler@56188
  1054
          by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i])
wenzelm@50945
  1055
      }
wenzelm@60420
  1056
      note \<open>k \<subseteq> cbox a b\<close>
wenzelm@50945
  1057
    }
hoelzl@50526
  1058
    moreover
wenzelm@50945
  1059
    {
immler@56188
  1060
      fix x assume x: "x \<in> cbox a b"
hoelzl@50526
  1061
      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
  1062
      proof
wenzelm@53408
  1063
        fix i :: 'a
wenzelm@53408
  1064
        assume "i \<in> Basis"
wenzelm@53399
  1065
        with x ord[of i]
hoelzl@50526
  1066
        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
  1067
            (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
immler@56188
  1068
          by (auto simp: cbox_def)
hoelzl@50526
  1069
        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
  1070
          by auto
hoelzl@50526
  1071
      qed
wenzelm@53408
  1072
      then obtain f where
wenzelm@53408
  1073
        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
  1074
        unfolding bchoice_iff ..
wenzelm@53374
  1075
      moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
hoelzl@50526
  1076
        by auto
hoelzl@50526
  1077
      moreover from f have "x \<in> ?B (restrict f Basis)"
immler@56188
  1078
        by (auto simp: mem_box)
hoelzl@50526
  1079
      ultimately have "\<exists>k\<in>p. x \<in> k"
wenzelm@53408
  1080
        unfolding p_def by blast
wenzelm@53408
  1081
    }
immler@56188
  1082
    ultimately show "\<Union>p = cbox a b"
hoelzl@50526
  1083
      by auto
hoelzl@50526
  1084
  qed
hoelzl@50526
  1085
qed
himmelma@35172
  1086
wenzelm@50945
  1087
lemma partial_division_extend_interval:
immler@56188
  1088
  assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
immler@56188
  1089
  obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
wenzelm@50945
  1090
proof (cases "p = {}")
wenzelm@50945
  1091
  case True
immler@56188
  1092
  obtain q where "q division_of (cbox a b)"
wenzelm@53408
  1093
    by (rule elementary_interval)
wenzelm@53399
  1094
  then show ?thesis
lp15@60384
  1095
    using True that by blast
wenzelm@50945
  1096
next
wenzelm@50945
  1097
  case False
wenzelm@50945
  1098
  note p = division_ofD[OF assms(1)]
lp15@60428
  1099
  have div_cbox: "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q"
wenzelm@50945
  1100
  proof
wenzelm@61165
  1101
    fix k
wenzelm@61165
  1102
    assume kp: "k \<in> p"
immler@56188
  1103
    obtain c d where k: "k = cbox c d"
wenzelm@61165
  1104
      using p(4)[OF kp] by blast
immler@56188
  1105
    have *: "cbox c d \<subseteq> cbox a b" "cbox c d \<noteq> {}"
wenzelm@61165
  1106
      using p(2,3)[OF kp, unfolded k] using assms(2)
immler@54776
  1107
      by (blast intro: order.trans)+
immler@56188
  1108
    obtain q where "q division_of cbox a b" "cbox c d \<in> q"
wenzelm@53408
  1109
      by (rule partial_division_extend_1[OF *])
wenzelm@61165
  1110
    then show "\<exists>q. q division_of cbox a b \<and> k \<in> q"
wenzelm@53408
  1111
      unfolding k by auto
wenzelm@50945
  1112
  qed
immler@56188
  1113
  obtain q where q: "\<And>x. x \<in> p \<Longrightarrow> q x division_of cbox a b" "\<And>x. x \<in> p \<Longrightarrow> x \<in> q x"
lp15@60428
  1114
    using bchoice[OF div_cbox] by blast
lp15@60394
  1115
  { fix x
wenzelm@53408
  1116
    assume x: "x \<in> p"
lp15@60394
  1117
    have "q x division_of \<Union>q x"
wenzelm@50945
  1118
      apply (rule division_ofI)
wenzelm@50945
  1119
      using division_ofD[OF q(1)[OF x]]
wenzelm@50945
  1120
      apply auto
lp15@60394
  1121
      done }
lp15@60394
  1122
  then have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
lp15@60394
  1123
    by (meson Diff_subset division_of_subset)
wenzelm@60585
  1124
  then have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
wenzelm@50945
  1125
    apply -
lp15@60394
  1126
    apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
lp15@60394
  1127
    apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
wenzelm@50945
  1128
    done
wenzelm@53408
  1129
  then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
lp15@60394
  1130
  have "d \<union> p division_of cbox a b"
wenzelm@50945
  1131
  proof -
lp15@60394
  1132
    have te: "\<And>s f t. s \<noteq> {} \<Longrightarrow> \<forall>i\<in>s. f i \<union> i = t \<Longrightarrow> t = \<Inter>(f ` s) \<union> \<Union>s" by auto
lp15@60428
  1133
    have cbox_eq: "cbox a b = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
lp15@60394
  1134
    proof (rule te[OF False], clarify)
wenzelm@50945
  1135
      fix i
wenzelm@53408
  1136
      assume i: "i \<in> p"
immler@56188
  1137
      show "\<Union>(q i - {i}) \<union> i = cbox a b"
wenzelm@50945
  1138
        using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
wenzelm@50945
  1139
    qed
lp15@60428
  1140
    { fix k
wenzelm@53408
  1141
      assume k: "k \<in> p"
lp15@60428
  1142
      have *: "\<And>u t s. t \<inter> s = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<inter> t = {}"
wenzelm@53408
  1143
        by auto
lp15@60428
  1144
      have "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<inter> interior k = {}"
lp15@60428
  1145
      proof (rule *[OF inter_interior_unions_intervals])
wenzelm@50945
  1146
        note qk=division_ofD[OF q(1)[OF k]]
immler@56188
  1147
        show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b"
wenzelm@53408
  1148
          using qk by auto
wenzelm@50945
  1149
        show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
wenzelm@50945
  1150
          using qk(5) using q(2)[OF k] by auto
lp15@60428
  1151
        show "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<subseteq> interior (\<Union>(q k - {k}))"
lp15@60428
  1152
          apply (rule interior_mono)+
wenzelm@53408
  1153
          using k
wenzelm@53408
  1154
          apply auto
wenzelm@53408
  1155
          done
lp15@60428
  1156
      qed } note [simp] = this
lp15@60428
  1157
    show "d \<union> p division_of (cbox a b)"
lp15@60428
  1158
      unfolding cbox_eq
lp15@60428
  1159
      apply (rule division_disjoint_union[OF d assms(1)])
lp15@60428
  1160
      apply (rule inter_interior_unions_intervals)
lp15@60428
  1161
      apply (rule p open_interior ballI)+
lp15@60615
  1162
      apply simp_all
lp15@60428
  1163
      done
lp15@60394
  1164
  qed
lp15@60394
  1165
  then show ?thesis
lp15@60394
  1166
    by (meson Un_upper2 that)
wenzelm@50945
  1167
qed
himmelma@35172
  1168
wenzelm@53399
  1169
lemma elementary_bounded[dest]:
immler@56188
  1170
  fixes s :: "'a::euclidean_space set"
wenzelm@53408
  1171
  shows "p division_of s \<Longrightarrow> bounded s"
immler@56189
  1172
  unfolding division_of_def by (metis bounded_Union bounded_cbox)
wenzelm@53399
  1173
immler@56188
  1174
lemma elementary_subset_cbox:
immler@56188
  1175
  "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> cbox a (b::'a::euclidean_space)"
immler@56188
  1176
  by (meson elementary_bounded bounded_subset_cbox)
wenzelm@50945
  1177
wenzelm@50945
  1178
lemma division_union_intervals_exists:
immler@56188
  1179
  fixes a b :: "'a::euclidean_space"
immler@56188
  1180
  assumes "cbox a b \<noteq> {}"
immler@56188
  1181
  obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
immler@56188
  1182
proof (cases "cbox c d = {}")
wenzelm@50945
  1183
  case True
wenzelm@50945
  1184
  show ?thesis
wenzelm@50945
  1185
    apply (rule that[of "{}"])
wenzelm@50945
  1186
    unfolding True
wenzelm@50945
  1187
    using assms
wenzelm@50945
  1188
    apply auto
wenzelm@50945
  1189
    done
wenzelm@50945
  1190
next
wenzelm@50945
  1191
  case False
wenzelm@50945
  1192
  show ?thesis
immler@56188
  1193
  proof (cases "cbox a b \<inter> cbox c d = {}")
wenzelm@50945
  1194
    case True
wenzelm@50945
  1195
    show ?thesis
immler@56188
  1196
      apply (rule that[of "{cbox c d}"])
lp15@60428
  1197
      apply (subst insert_is_Un)
wenzelm@50945
  1198
      apply (rule division_disjoint_union)
lp15@60428
  1199
      using \<open>cbox c d \<noteq> {}\<close> True assms interior_subset
wenzelm@50945
  1200
      apply auto
wenzelm@50945
  1201
      done
wenzelm@50945
  1202
  next
wenzelm@50945
  1203
    case False
immler@56188
  1204
    obtain u v where uv: "cbox a b \<inter> cbox c d = cbox u v"
wenzelm@50945
  1205
      unfolding inter_interval by auto
lp15@60428
  1206
    have uv_sub: "cbox u v \<subseteq> cbox c d" using uv by auto
immler@56188
  1207
    obtain p where "p division_of cbox c d" "cbox u v \<in> p"
lp15@60428
  1208
      by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
wenzelm@53408
  1209
    note p = this division_ofD[OF this(1)]
lp15@60428
  1210
    have "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = interior(cbox u v \<inter> \<Union>(p - {cbox u v}))"
lp15@60428
  1211
      apply (rule arg_cong[of _ _ interior])
lp15@60428
  1212
      using p(8) uv by auto
lp15@60428
  1213
    also have "\<dots> = {}"
lp15@60428
  1214
      unfolding interior_inter
lp15@60428
  1215
      apply (rule inter_interior_unions_intervals)
lp15@60428
  1216
      using p(6) p(7)[OF p(2)] p(3)
lp15@60428
  1217
      apply auto
lp15@60428
  1218
      done
lp15@60428
  1219
    finally have [simp]: "interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}" by simp
lp15@60615
  1220
    have cbe: "cbox a b \<union> cbox c d = cbox a b \<union> \<Union>(p - {cbox u v})"
wenzelm@53399
  1221
      using p(8) unfolding uv[symmetric] by auto
wenzelm@50945
  1222
    show ?thesis
immler@56188
  1223
      apply (rule that[of "p - {cbox u v}"])
lp15@60428
  1224
      apply (simp add: cbe)
lp15@60428
  1225
      apply (subst insert_is_Un)
wenzelm@50945
  1226
      apply (rule division_disjoint_union)
lp15@60428
  1227
      apply (simp_all add: assms division_of_self)
lp15@60428
  1228
      by (metis Diff_subset division_of_subset p(1) p(8))
wenzelm@50945
  1229
  qed
wenzelm@50945
  1230
qed
himmelma@35172
  1231
wenzelm@53399
  1232
lemma division_of_unions:
wenzelm@53399
  1233
  assumes "finite f"
wenzelm@53408
  1234
    and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@53399
  1235
    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
  1236
  shows "\<Union>f division_of \<Union>\<Union>f"
lp15@60384
  1237
  using assms
lp15@60384
  1238
  by (auto intro!: division_ofI)
wenzelm@53399
  1239
wenzelm@53399
  1240
lemma elementary_union_interval:
immler@56188
  1241
  fixes a b :: "'a::euclidean_space"
wenzelm@53399
  1242
  assumes "p division_of \<Union>p"
immler@56188
  1243
  obtains q where "q division_of (cbox a b \<union> \<Union>p)"
wenzelm@53399
  1244
proof -
wenzelm@53399
  1245
  note assm = division_ofD[OF assms]
wenzelm@53408
  1246
  have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
wenzelm@53399
  1247
    by auto
wenzelm@53399
  1248
  have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
wenzelm@53399
  1249
    by auto
wenzelm@53399
  1250
  {
wenzelm@53399
  1251
    presume "p = {} \<Longrightarrow> thesis"
immler@56188
  1252
      "cbox a b = {} \<Longrightarrow> thesis"
immler@56188
  1253
      "cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis"
immler@56188
  1254
      "p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis"
wenzelm@53399
  1255
    then show thesis by auto
wenzelm@53399
  1256
  next
wenzelm@53399
  1257
    assume as: "p = {}"
immler@56188
  1258
    obtain p where "p division_of (cbox a b)"
wenzelm@53408
  1259
      by (rule elementary_interval)
wenzelm@53399
  1260
    then show thesis
lp15@60384
  1261
      using as that by auto
wenzelm@53399
  1262
  next
immler@56188
  1263
    assume as: "cbox a b = {}"
wenzelm@53399
  1264
    show thesis
lp15@60384
  1265
      using as assms that by auto
wenzelm@53399
  1266
  next
immler@56188
  1267
    assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}"
wenzelm@53399
  1268
    show thesis
immler@56188
  1269
      apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
wenzelm@53399
  1270
      unfolding finite_insert
wenzelm@53399
  1271
      apply (rule assm(1)) unfolding Union_insert
wenzelm@53399
  1272
      using assm(2-4) as
wenzelm@53399
  1273
      apply -
immler@54775
  1274
      apply (fast dest: assm(5))+
wenzelm@53399
  1275
      done
wenzelm@53399
  1276
  next
immler@56188
  1277
    assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}"
immler@56188
  1278
    have "\<forall>k\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
lp15@60615
  1279
    proof
wenzelm@61165
  1280
      fix k
wenzelm@61165
  1281
      assume kp: "k \<in> p"
wenzelm@61165
  1282
      from assm(4)[OF kp] obtain c d where "k = cbox c d" by blast
wenzelm@61165
  1283
      then show "\<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
lp15@60384
  1284
        by (meson as(3) division_union_intervals_exists)
wenzelm@53399
  1285
    qed
immler@56188
  1286
    from bchoice[OF this] obtain q where "\<forall>x\<in>p. insert (cbox a b) (q x) division_of (cbox a b) \<union> x" ..
wenzelm@53408
  1287
    note q = division_ofD[OF this[rule_format]]
immler@56188
  1288
    let ?D = "\<Union>{insert (cbox a b) (q k) | k. k \<in> p}"
lp15@60615
  1289
    show thesis
lp15@60428
  1290
    proof (rule that[OF division_ofI])
immler@56188
  1291
      have *: "{insert (cbox a b) (q k) |k. k \<in> p} = (\<lambda>k. insert (cbox a b) (q k)) ` p"
wenzelm@53399
  1292
        by auto
wenzelm@53399
  1293
      show "finite ?D"
lp15@60384
  1294
        using "*" assm(1) q(1) by auto
immler@56188
  1295
      show "\<Union>?D = cbox a b \<union> \<Union>p"
wenzelm@53399
  1296
        unfolding * lem1
immler@56188
  1297
        unfolding lem2[OF as(1), of "cbox a b", symmetric]
wenzelm@53399
  1298
        using q(6)
wenzelm@53399
  1299
        by auto
wenzelm@53399
  1300
      fix k
wenzelm@53408
  1301
      assume k: "k \<in> ?D"
immler@56188
  1302
      then show "k \<subseteq> cbox a b \<union> \<Union>p"
wenzelm@53408
  1303
        using q(2) by auto
wenzelm@53399
  1304
      show "k \<noteq> {}"
wenzelm@53408
  1305
        using q(3) k by auto
immler@56188
  1306
      show "\<exists>a b. k = cbox a b"
wenzelm@53408
  1307
        using q(4) k by auto
wenzelm@53399
  1308
      fix k'
wenzelm@53408
  1309
      assume k': "k' \<in> ?D" "k \<noteq> k'"
immler@56188
  1310
      obtain x where x: "k \<in> insert (cbox a b) (q x)" "x\<in>p"
wenzelm@53408
  1311
        using k by auto
immler@56188
  1312
      obtain x' where x': "k'\<in>insert (cbox a b) (q x')" "x'\<in>p"
wenzelm@53399
  1313
        using k' by auto
wenzelm@53399
  1314
      show "interior k \<inter> interior k' = {}"
wenzelm@53399
  1315
      proof (cases "x = x'")
wenzelm@53399
  1316
        case True
wenzelm@53399
  1317
        show ?thesis
lp15@60384
  1318
          using True k' q(5) x' x by auto
wenzelm@53399
  1319
      next
wenzelm@53399
  1320
        case False
wenzelm@53399
  1321
        {
immler@56188
  1322
          presume "k = cbox a b \<Longrightarrow> ?thesis"
immler@56188
  1323
            and "k' = cbox a b \<Longrightarrow> ?thesis"
immler@56188
  1324
            and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis"
wenzelm@53399
  1325
          then show ?thesis by auto
wenzelm@53399
  1326
        next
immler@56188
  1327
          assume as': "k  = cbox a b"
wenzelm@53399
  1328
          show ?thesis
lp15@60384
  1329
            using as' k' q(5) x' by auto
wenzelm@53399
  1330
        next
immler@56188
  1331
          assume as': "k' = cbox a b"
wenzelm@53399
  1332
          show ?thesis
lp15@60384
  1333
            using as' k'(2) q(5) x by auto
wenzelm@53399
  1334
        }
immler@56188
  1335
        assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b"
immler@56188
  1336
        obtain c d where k: "k = cbox c d"
wenzelm@53408
  1337
          using q(4)[OF x(2,1)] by blast
immler@56188
  1338
        have "interior k \<inter> interior (cbox a b) = {}"
lp15@60384
  1339
          using as' k'(2) q(5) x by auto
wenzelm@53399
  1340
        then have "interior k \<subseteq> interior x"
lp15@60384
  1341
        using interior_subset_union_intervals
lp15@60384
  1342
          by (metis as(2) k q(2) x interior_subset_union_intervals)
wenzelm@53399
  1343
        moreover
immler@56188
  1344
        obtain c d where c_d: "k' = cbox c d"
wenzelm@53408
  1345
          using q(4)[OF x'(2,1)] by blast
immler@56188
  1346
        have "interior k' \<inter> interior (cbox a b) = {}"
lp15@60384
  1347
          using as'(2) q(5) x' by auto
wenzelm@53399
  1348
        then have "interior k' \<subseteq> interior x'"
lp15@60384
  1349
          by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
wenzelm@53399
  1350
        ultimately show ?thesis
wenzelm@53399
  1351
          using assm(5)[OF x(2) x'(2) False] by auto
wenzelm@53399
  1352
      qed
wenzelm@53399
  1353
    qed
wenzelm@53399
  1354
  }
wenzelm@53399
  1355
qed
himmelma@35172
  1356
himmelma@35172
  1357
lemma elementary_unions_intervals:
wenzelm@53399
  1358
  assumes fin: "finite f"
immler@56188
  1359
    and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a (b::'a::euclidean_space)"
wenzelm@53399
  1360
  obtains p where "p division_of (\<Union>f)"
wenzelm@53399
  1361
proof -
wenzelm@53399
  1362
  have "\<exists>p. p division_of (\<Union>f)"
wenzelm@53399
  1363
  proof (induct_tac f rule:finite_subset_induct)
himmelma@35172
  1364
    show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
wenzelm@53399
  1365
  next
wenzelm@53399
  1366
    fix x F
wenzelm@53399
  1367
    assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
wenzelm@53408
  1368
    from this(3) obtain p where p: "p division_of \<Union>F" ..
immler@56188
  1369
    from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
wenzelm@53399
  1370
    have *: "\<Union>F = \<Union>p"
wenzelm@53399
  1371
      using division_ofD[OF p] by auto
wenzelm@53399
  1372
    show "\<exists>p. p division_of \<Union>insert x F"
wenzelm@53399
  1373
      using elementary_union_interval[OF p[unfolded *], of a b]
lp15@59765
  1374
      unfolding Union_insert x * by metis
wenzelm@53408
  1375
  qed (insert assms, auto)
wenzelm@53399
  1376
  then show ?thesis
lp15@60384
  1377
    using that by auto
wenzelm@53399
  1378
qed
wenzelm@53399
  1379
wenzelm@53399
  1380
lemma elementary_union:
immler@56188
  1381
  fixes s t :: "'a::euclidean_space set"
lp15@60384
  1382
  assumes "ps division_of s" "pt division_of t"
himmelma@35172
  1383
  obtains p where "p division_of (s \<union> t)"
wenzelm@53399
  1384
proof -
lp15@60384
  1385
  have *: "s \<union> t = \<Union>ps \<union> \<Union>pt"
wenzelm@53399
  1386
    using assms unfolding division_of_def by auto
wenzelm@53399
  1387
  show ?thesis
wenzelm@53408
  1388
    apply (rule elementary_unions_intervals[of "ps \<union> pt"])
lp15@60384
  1389
    using assms apply auto
lp15@60384
  1390
    by (simp add: * that)
wenzelm@53399
  1391
qed
wenzelm@53399
  1392
wenzelm@53399
  1393
lemma partial_division_extend:
immler@56188
  1394
  fixes t :: "'a::euclidean_space set"
wenzelm@53399
  1395
  assumes "p division_of s"
wenzelm@53399
  1396
    and "q division_of t"
wenzelm@53399
  1397
    and "s \<subseteq> t"
wenzelm@53399
  1398
  obtains r where "p \<subseteq> r" and "r division_of t"
wenzelm@53399
  1399
proof -
himmelma@35172
  1400
  note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
immler@56188
  1401
  obtain a b where ab: "t \<subseteq> cbox a b"
immler@56188
  1402
    using elementary_subset_cbox[OF assms(2)] by auto
immler@56188
  1403
  obtain r1 where "p \<subseteq> r1" "r1 division_of (cbox a b)"
lp15@60384
  1404
    using assms
lp15@60384
  1405
    by (metis ab dual_order.trans partial_division_extend_interval divp(6))
wenzelm@53399
  1406
  note r1 = this division_ofD[OF this(2)]
wenzelm@53408
  1407
  obtain p' where "p' division_of \<Union>(r1 - p)"
wenzelm@53399
  1408
    apply (rule elementary_unions_intervals[of "r1 - p"])
wenzelm@53399
  1409
    using r1(3,6)
wenzelm@53399
  1410
    apply auto
wenzelm@53399
  1411
    done
wenzelm@53399
  1412
  then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
lp15@60384
  1413
    by (metis assms(2) divq(6) elementary_inter)
wenzelm@53399
  1414
  {
wenzelm@53399
  1415
    fix x
wenzelm@53399
  1416
    assume x: "x \<in> t" "x \<notin> s"
wenzelm@53399
  1417
    then have "x\<in>\<Union>r1"
wenzelm@53399
  1418
      unfolding r1 using ab by auto
wenzelm@53408
  1419
    then obtain r where r: "r \<in> r1" "x \<in> r"
wenzelm@53408
  1420
      unfolding Union_iff ..
wenzelm@53399
  1421
    moreover
wenzelm@53399
  1422
    have "r \<notin> p"
wenzelm@53399
  1423
    proof
wenzelm@53399
  1424
      assume "r \<in> p"
wenzelm@53399
  1425
      then have "x \<in> s" using divp(2) r by auto
wenzelm@53399
  1426
      then show False using x by auto
wenzelm@53399
  1427
    qed
wenzelm@53399
  1428
    ultimately have "x\<in>\<Union>(r1 - p)" by auto
wenzelm@53399
  1429
  }
wenzelm@53399
  1430
  then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
wenzelm@53399
  1431
    unfolding divp divq using assms(3) by auto
wenzelm@53399
  1432
  show ?thesis
wenzelm@53399
  1433
    apply (rule that[of "p \<union> r2"])
wenzelm@53399
  1434
    unfolding *
wenzelm@53399
  1435
    defer
wenzelm@53399
  1436
    apply (rule division_disjoint_union)
wenzelm@53399
  1437
    unfolding divp(6)
wenzelm@53399
  1438
    apply(rule assms r2)+
wenzelm@53399
  1439
  proof -
wenzelm@53399
  1440
    have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
wenzelm@53399
  1441
    proof (rule inter_interior_unions_intervals)
immler@56188
  1442
      show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b"
wenzelm@53399
  1443
        using r1 by auto
wenzelm@53399
  1444
      have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
wenzelm@53399
  1445
        by auto
wenzelm@53399
  1446
      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
wenzelm@53399
  1447
      proof
wenzelm@53399
  1448
        fix m x
wenzelm@53399
  1449
        assume as: "m \<in> r1 - p"
wenzelm@53399
  1450
        have "interior m \<inter> interior (\<Union>p) = {}"
wenzelm@53399
  1451
        proof (rule inter_interior_unions_intervals)
immler@56188
  1452
          show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b"
wenzelm@53399
  1453
            using divp by auto
wenzelm@53399
  1454
          show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
lp15@60384
  1455
            by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
wenzelm@53399
  1456
        qed
wenzelm@53399
  1457
        then show "interior s \<inter> interior m = {}"
wenzelm@53399
  1458
          unfolding divp by auto
wenzelm@53399
  1459
      qed
wenzelm@53399
  1460
    qed
wenzelm@53399
  1461
    then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
wenzelm@53399
  1462
      using interior_subset by auto
wenzelm@53399
  1463
  qed auto
wenzelm@53399
  1464
qed
wenzelm@53399
  1465
himmelma@35172
  1466
wenzelm@60420
  1467
subsection \<open>Tagged (partial) divisions.\<close>
himmelma@35172
  1468
wenzelm@53408
  1469
definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
wenzelm@53408
  1470
  where "s tagged_partial_division_of i \<longleftrightarrow>
wenzelm@53408
  1471
    finite s \<and>
immler@56188
  1472
    (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and>
wenzelm@53408
  1473
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1474
      interior k1 \<inter> interior k2 = {})"
wenzelm@53408
  1475
wenzelm@53408
  1476
lemma tagged_partial_division_ofD[dest]:
wenzelm@53408
  1477
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1478
  shows "finite s"
wenzelm@53408
  1479
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1480
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1481
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1482
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
wenzelm@53408
  1483
      (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1484
  using assms unfolding tagged_partial_division_of_def by blast+
wenzelm@53408
  1485
wenzelm@53408
  1486
definition tagged_division_of (infixr "tagged'_division'_of" 40)
wenzelm@53408
  1487
  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
  1488
huffman@44167
  1489
lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
himmelma@35172
  1490
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1491
himmelma@35172
  1492
lemma tagged_division_of:
wenzelm@53408
  1493
  "s tagged_division_of i \<longleftrightarrow>
wenzelm@53408
  1494
    finite s \<and>
immler@56188
  1495
    (\<forall>x k. (x, k) \<in> s \<longrightarrow> x \<in> k \<and> k \<subseteq> i \<and> (\<exists>a b. k = cbox a b)) \<and>
wenzelm@53408
  1496
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1497
      interior k1 \<inter> interior k2 = {}) \<and>
wenzelm@53408
  1498
    (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1499
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1500
wenzelm@53408
  1501
lemma tagged_division_ofI:
wenzelm@53408
  1502
  assumes "finite s"
wenzelm@53408
  1503
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1504
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1505
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1506
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1507
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1508
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1509
  shows "s tagged_division_of i"
wenzelm@53408
  1510
  unfolding tagged_division_of
lp15@60384
  1511
  using assms
lp15@60384
  1512
  apply auto
lp15@60384
  1513
  apply fastforce+
wenzelm@53408
  1514
  done
wenzelm@53408
  1515
lp15@60384
  1516
lemma tagged_division_ofD[dest]:  (*FIXME USE A LOCALE*)
wenzelm@53408
  1517
  assumes "s tagged_division_of i"
wenzelm@53408
  1518
  shows "finite s"
wenzelm@53408
  1519
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1520
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1521
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1522
    and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1523
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1524
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
wenzelm@53408
  1525
  using assms unfolding tagged_division_of by blast+
wenzelm@53408
  1526
wenzelm@53408
  1527
lemma division_of_tagged_division:
wenzelm@53408
  1528
  assumes "s tagged_division_of i"
wenzelm@53408
  1529
  shows "(snd ` s) division_of i"
wenzelm@53408
  1530
proof (rule division_ofI)
wenzelm@53408
  1531
  note assm = tagged_division_ofD[OF assms]
wenzelm@53408
  1532
  show "\<Union>(snd ` s) = i" "finite (snd ` s)"
wenzelm@53408
  1533
    using assm by auto
wenzelm@53408
  1534
  fix k
wenzelm@53408
  1535
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1536
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1537
    by auto
immler@56188
  1538
  then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = cbox a b"
wenzelm@53408
  1539
    using assm by fastforce+
wenzelm@53408
  1540
  fix k'
wenzelm@53408
  1541
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1542
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1543
    by auto
wenzelm@53408
  1544
  then show "interior k \<inter> interior k' = {}"
lp15@60384
  1545
    using assm(5) k'(2) xk by blast
himmelma@35172
  1546
qed
himmelma@35172
  1547
wenzelm@53408
  1548
lemma partial_division_of_tagged_division:
wenzelm@53408
  1549
  assumes "s tagged_partial_division_of i"
himmelma@35172
  1550
  shows "(snd ` s) division_of \<Union>(snd ` s)"
wenzelm@53408
  1551
proof (rule division_ofI)
wenzelm@53408
  1552
  note assm = tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1553
  show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
wenzelm@53408
  1554
    using assm by auto
wenzelm@53408
  1555
  fix k
wenzelm@53408
  1556
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1557
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1558
    by auto
immler@56188
  1559
  then show "k \<noteq> {}" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>(snd ` s)"
wenzelm@53408
  1560
    using assm by auto
wenzelm@53408
  1561
  fix k'
wenzelm@53408
  1562
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1563
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1564
    by auto
wenzelm@53408
  1565
  then show "interior k \<inter> interior k' = {}"
lp15@60384
  1566
    using assm(5) k'(2) xk by auto
himmelma@35172
  1567
qed
himmelma@35172
  1568
wenzelm@53408
  1569
lemma tagged_partial_division_subset:
wenzelm@53408
  1570
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1571
    and "t \<subseteq> s"
himmelma@35172
  1572
  shows "t tagged_partial_division_of i"
wenzelm@53408
  1573
  using assms
wenzelm@53408
  1574
  unfolding tagged_partial_division_of_def
wenzelm@53408
  1575
  using finite_subset[OF assms(2)]
wenzelm@53408
  1576
  by blast
wenzelm@53408
  1577
wenzelm@53408
  1578
lemma setsum_over_tagged_division_lemma:
wenzelm@53408
  1579
  assumes "p tagged_division_of i"
immler@56188
  1580
    and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> content (cbox u v) = 0 \<Longrightarrow> d (cbox u v) = 0"
himmelma@35172
  1581
  shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
wenzelm@53408
  1582
proof -
wenzelm@53408
  1583
  have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
wenzelm@53408
  1584
    unfolding o_def by (rule ext) auto
hoelzl@57129
  1585
  note assm = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1586
  show ?thesis
wenzelm@53408
  1587
    unfolding *
haftmann@57418
  1588
  proof (rule setsum.reindex_nontrivial[symmetric])
wenzelm@53408
  1589
    show "finite p"
wenzelm@53408
  1590
      using assm by auto
wenzelm@53408
  1591
    fix x y
hoelzl@57129
  1592
    assume "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
immler@56188
  1593
    obtain a b where ab: "snd x = cbox a b"
wenzelm@60420
  1594
      using assm(4)[of "fst x" "snd x"] \<open>x\<in>p\<close> by auto
wenzelm@53408
  1595
    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
haftmann@61424
  1596
      by (metis prod.collapse \<open>x\<in>p\<close> \<open>snd x = snd y\<close> \<open>x \<noteq> y\<close>)+
wenzelm@60420
  1597
    with \<open>x\<in>p\<close> \<open>y\<in>p\<close> have "interior (snd x) \<inter> interior (snd y) = {}"
hoelzl@57129
  1598
      by (intro assm(5)[of "fst x" _ "fst y"]) auto
immler@56188
  1599
    then have "content (cbox a b) = 0"
wenzelm@60420
  1600
      unfolding \<open>snd x = snd y\<close>[symmetric] ab content_eq_0_interior by auto
immler@56188
  1601
    then have "d (cbox a b) = 0"
wenzelm@60420
  1602
      using assm(2)[of "fst x" "snd x"] \<open>x\<in>p\<close> ab[symmetric] by (intro assms(2)) auto
wenzelm@53408
  1603
    then show "d (snd x) = 0"
wenzelm@53408
  1604
      unfolding ab by auto
wenzelm@53408
  1605
  qed
wenzelm@53408
  1606
qed
wenzelm@53408
  1607
wenzelm@53408
  1608
lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
wenzelm@53408
  1609
  by auto
himmelma@35172
  1610
himmelma@35172
  1611
lemma tagged_division_of_empty: "{} tagged_division_of {}"
himmelma@35172
  1612
  unfolding tagged_division_of by auto
himmelma@35172
  1613
wenzelm@53408
  1614
lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1615
  unfolding tagged_partial_division_of_def by auto
himmelma@35172
  1616
wenzelm@53408
  1617
lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1618
  unfolding tagged_division_of by auto
himmelma@35172
  1619
immler@56188
  1620
lemma tagged_division_of_self: "x \<in> cbox a b \<Longrightarrow> {(x,cbox a b)} tagged_division_of (cbox a b)"
wenzelm@53408
  1621
  by (rule tagged_division_ofI) auto
himmelma@35172
  1622
immler@56188
  1623
lemma tagged_division_of_self_real: "x \<in> {a .. b::real} \<Longrightarrow> {(x,{a .. b})} tagged_division_of {a .. b}"
immler@56188
  1624
  unfolding box_real[symmetric]
immler@56188
  1625
  by (rule tagged_division_of_self)
immler@56188
  1626
himmelma@35172
  1627
lemma tagged_division_union:
wenzelm@53408
  1628
  assumes "p1 tagged_division_of s1"
wenzelm@53408
  1629
    and "p2 tagged_division_of s2"
wenzelm@53408
  1630
    and "interior s1 \<inter> interior s2 = {}"
himmelma@35172
  1631
  shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
wenzelm@53408
  1632
proof (rule tagged_division_ofI)
wenzelm@53408
  1633
  note p1 = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1634
  note p2 = tagged_division_ofD[OF assms(2)]
wenzelm@53408
  1635
  show "finite (p1 \<union> p2)"
wenzelm@53408
  1636
    using p1(1) p2(1) by auto
wenzelm@53408
  1637
  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
wenzelm@53408
  1638
    using p1(6) p2(6) by blast
wenzelm@53408
  1639
  fix x k
wenzelm@53408
  1640
  assume xk: "(x, k) \<in> p1 \<union> p2"
immler@56188
  1641
  show "x \<in> k" "\<exists>a b. k = cbox a b"
wenzelm@53408
  1642
    using xk p1(2,4) p2(2,4) by auto
wenzelm@53408
  1643
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
  1644
    using xk p1(3) p2(3) by blast
wenzelm@53408
  1645
  fix x' k'
wenzelm@53408
  1646
  assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1647
  have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1648
    using assms(3) interior_mono by blast
wenzelm@53408
  1649
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1650
    apply (cases "(x, k) \<in> p1")
lp15@60384
  1651
    apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2))
lp15@60384
  1652
    by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
wenzelm@53408
  1653
qed
himmelma@35172
  1654
himmelma@35172
  1655
lemma tagged_division_unions:
wenzelm@53408
  1656
  assumes "finite iset"
wenzelm@53408
  1657
    and "\<forall>i\<in>iset. pfn i tagged_division_of i"
wenzelm@53408
  1658
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
himmelma@35172
  1659
  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
wenzelm@53408
  1660
proof (rule tagged_division_ofI)
himmelma@35172
  1661
  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
wenzelm@53408
  1662
  show "finite (\<Union>(pfn ` iset))"
wenzelm@53408
  1663
    apply (rule finite_Union)
wenzelm@53408
  1664
    using assms
wenzelm@53408
  1665
    apply auto
wenzelm@53408
  1666
    done
wenzelm@53408
  1667
  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
  1668
    by blast
wenzelm@53408
  1669
  also have "\<dots> = \<Union>iset"
wenzelm@53408
  1670
    using assm(6) by auto
wenzelm@53399
  1671
  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
wenzelm@53408
  1672
  fix x k
wenzelm@53408
  1673
  assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
wenzelm@53408
  1674
  then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
wenzelm@53408
  1675
    by auto
immler@56188
  1676
  show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
wenzelm@53408
  1677
    using assm(2-4)[OF i] using i(1) by auto
wenzelm@53408
  1678
  fix x' k'
wenzelm@53408
  1679
  assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1680
  then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
wenzelm@53408
  1681
    by auto
wenzelm@53408
  1682
  have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1683
    using i(1) i'(1)
wenzelm@53408
  1684
    using assms(3)[rule_format] interior_mono
wenzelm@53408
  1685
    by blast
wenzelm@53408
  1686
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1687
    apply (cases "i = i'")
lp15@60384
  1688
    using assm(5) i' i(2) xk'(2) apply blast
lp15@60384
  1689
    using "*" assm(3) i' i by auto
himmelma@35172
  1690
qed
himmelma@35172
  1691
himmelma@35172
  1692
lemma tagged_partial_division_of_union_self:
wenzelm@53408
  1693
  assumes "p tagged_partial_division_of s"
himmelma@35172
  1694
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1695
  apply (rule tagged_division_ofI)
wenzelm@53408
  1696
  using tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1697
  apply auto
wenzelm@53408
  1698
  done
wenzelm@53408
  1699
wenzelm@53408
  1700
lemma tagged_division_of_union_self:
wenzelm@53408
  1701
  assumes "p tagged_division_of s"
wenzelm@53408
  1702
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1703
  apply (rule tagged_division_ofI)
wenzelm@53408
  1704
  using tagged_division_ofD[OF assms]
wenzelm@53408
  1705
  apply auto
wenzelm@53408
  1706
  done
wenzelm@53408
  1707
himmelma@35172
  1708
wenzelm@60420
  1709
subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close>
himmelma@35172
  1710
wenzelm@53408
  1711
definition fine  (infixr "fine" 46)
wenzelm@53408
  1712
  where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
wenzelm@53408
  1713
wenzelm@53408
  1714
lemma fineI:
wenzelm@53408
  1715
  assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1716
  shows "d fine s"
wenzelm@53408
  1717
  using assms unfolding fine_def by auto
wenzelm@53408
  1718
wenzelm@53408
  1719
lemma fineD[dest]:
wenzelm@53408
  1720
  assumes "d fine s"
wenzelm@53408
  1721
  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1722
  using assms unfolding fine_def by auto
himmelma@35172
  1723
himmelma@35172
  1724
lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
himmelma@35172
  1725
  unfolding fine_def by auto
himmelma@35172
  1726
himmelma@35172
  1727
lemma fine_inters:
wenzelm@60585
  1728
 "(\<lambda>x. \<Inter>{f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
himmelma@35172
  1729
  unfolding fine_def by blast
himmelma@35172
  1730
wenzelm@53408
  1731
lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
himmelma@35172
  1732
  unfolding fine_def by blast
himmelma@35172
  1733
wenzelm@53408
  1734
lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
himmelma@35172
  1735
  unfolding fine_def by auto
himmelma@35172
  1736
wenzelm@53408
  1737
lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
himmelma@35172
  1738
  unfolding fine_def by blast
himmelma@35172
  1739
wenzelm@53408
  1740
wenzelm@60420
  1741
subsection \<open>Gauge integral. Define on compact intervals first, then use a limit.\<close>
himmelma@35172
  1742
wenzelm@53408
  1743
definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
wenzelm@53408
  1744
  where "(f has_integral_compact_interval y) i \<longleftrightarrow>
wenzelm@53408
  1745
    (\<forall>e>0. \<exists>d. gauge d \<and>
wenzelm@53408
  1746
      (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
wenzelm@53408
  1747
        norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1748
wenzelm@53408
  1749
definition has_integral ::
immler@56188
  1750
    "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
wenzelm@53408
  1751
  (infixr "has'_integral" 46)
wenzelm@53408
  1752
  where "(f has_integral y) i \<longleftrightarrow>
immler@56188
  1753
    (if \<exists>a b. i = cbox a b
wenzelm@53408
  1754
     then (f has_integral_compact_interval y) i
immler@56188
  1755
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1756
      (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) (cbox a b) \<and>
wenzelm@53408
  1757
        norm (z - y) < e)))"
himmelma@35172
  1758
himmelma@35172
  1759
lemma has_integral:
immler@56188
  1760
  "(f has_integral y) (cbox a b) \<longleftrightarrow>
wenzelm@53408
  1761
    (\<forall>e>0. \<exists>d. gauge d \<and>
immler@56188
  1762
      (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
wenzelm@53408
  1763
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1764
  unfolding has_integral_def has_integral_compact_interval_def
wenzelm@53408
  1765
  by auto
wenzelm@53408
  1766
immler@56188
  1767
lemma has_integral_real:
immler@56188
  1768
  "(f has_integral y) {a .. b::real} \<longleftrightarrow>
immler@56188
  1769
    (\<forall>e>0. \<exists>d. gauge d \<and>
immler@56188
  1770
      (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
immler@56188
  1771
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
immler@56188
  1772
  unfolding box_real[symmetric]
immler@56188
  1773
  by (rule has_integral)
immler@56188
  1774
wenzelm@53408
  1775
lemma has_integralD[dest]:
immler@56188
  1776
  assumes "(f has_integral y) (cbox a b)"
wenzelm@53408
  1777
    and "e > 0"
wenzelm@53408
  1778
  obtains d where "gauge d"
immler@56188
  1779
    and "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d fine p \<Longrightarrow>
wenzelm@53408
  1780
      norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
himmelma@35172
  1781
  using assms unfolding has_integral by auto
himmelma@35172
  1782
himmelma@35172
  1783
lemma has_integral_alt:
wenzelm@53408
  1784
  "(f has_integral y) i \<longleftrightarrow>
immler@56188
  1785
    (if \<exists>a b. i = cbox a b
wenzelm@53408
  1786
     then (f has_integral y) i
immler@56188
  1787
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1788
      (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e)))"
wenzelm@53408
  1789
  unfolding has_integral
wenzelm@53408
  1790
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1791
  by auto
himmelma@35172
  1792
himmelma@35172
  1793
lemma has_integral_altD:
wenzelm@53408
  1794
  assumes "(f has_integral y) i"
immler@56188
  1795
    and "\<not> (\<exists>a b. i = cbox a b)"
wenzelm@53408
  1796
    and "e>0"
wenzelm@53408
  1797
  obtains B where "B > 0"
immler@56188
  1798
    and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1799
      (\<exists>z. ((\<lambda>x. if x \<in> i then f(x) else 0) has_integral z) (cbox a b) \<and> norm(z - y) < e)"
wenzelm@53408
  1800
  using assms
wenzelm@53408
  1801
  unfolding has_integral
wenzelm@53408
  1802
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1803
  by auto
wenzelm@53408
  1804
wenzelm@53408
  1805
definition integrable_on (infixr "integrable'_on" 46)
wenzelm@53408
  1806
  where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
wenzelm@53408
  1807
wenzelm@53408
  1808
definition "integral i f = (SOME y. (f has_integral y) i)"
himmelma@35172
  1809
wenzelm@53409
  1810
lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
wenzelm@53409
  1811
  unfolding integrable_on_def integral_def by (rule someI_ex)
himmelma@35172
  1812
himmelma@35172
  1813
lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
himmelma@35172
  1814
  unfolding integrable_on_def by auto
himmelma@35172
  1815
wenzelm@53409
  1816
lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
himmelma@35172
  1817
  by auto
himmelma@35172
  1818
himmelma@35172
  1819
lemma setsum_content_null:
immler@56188
  1820
  assumes "content (cbox a b) = 0"
immler@56188
  1821
    and "p tagged_division_of (cbox a b)"
himmelma@35172
  1822
  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
haftmann@57418
  1823
proof (rule setsum.neutral, rule)
wenzelm@53409
  1824
  fix y
wenzelm@53409
  1825
  assume y: "y \<in> p"
wenzelm@53409
  1826
  obtain x k where xk: "y = (x, k)"
wenzelm@53409
  1827
    using surj_pair[of y] by blast
himmelma@35172
  1828
  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
immler@56188
  1829
  from this(2) obtain c d where k: "k = cbox c d" by blast
wenzelm@53409
  1830
  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
wenzelm@53409
  1831
    unfolding xk by auto
wenzelm@53409
  1832
  also have "\<dots> = 0"
wenzelm@53409
  1833
    using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
wenzelm@53409
  1834
    unfolding assms(1) k
wenzelm@53409
  1835
    by auto
himmelma@35172
  1836
  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
himmelma@35172
  1837
qed
himmelma@35172
  1838
wenzelm@53409
  1839
wenzelm@60420
  1840
subsection \<open>Some basic combining lemmas.\<close>
himmelma@35172
  1841
himmelma@35172
  1842
lemma tagged_division_unions_exists:
wenzelm@53409
  1843
  assumes "finite iset"
wenzelm@53409
  1844
    and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
wenzelm@53409
  1845
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
wenzelm@53409
  1846
    and "\<Union>iset = i"
wenzelm@53409
  1847
   obtains p where "p tagged_division_of i" and "d fine p"
wenzelm@53409
  1848
proof -
wenzelm@53409
  1849
  obtain pfn where pfn:
wenzelm@53409
  1850
    "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
wenzelm@53409
  1851
    "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
wenzelm@53409
  1852
    using bchoice[OF assms(2)] by auto
wenzelm@53409
  1853
  show thesis
wenzelm@53409
  1854
    apply (rule_tac p="\<Union>(pfn ` iset)" in that)
lp15@60384
  1855
    using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
lp15@60384
  1856
    by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
himmelma@35172
  1857
qed
himmelma@35172
  1858
wenzelm@53409
  1859
wenzelm@60420
  1860
subsection \<open>The set we're concerned with must be closed.\<close>
himmelma@35172
  1861
wenzelm@53409
  1862
lemma division_of_closed:
immler@56189
  1863
  fixes i :: "'n::euclidean_space set"
wenzelm@53409
  1864
  shows "s division_of i \<Longrightarrow> closed i"
nipkow@44890
  1865
  unfolding division_of_def by fastforce
himmelma@35172
  1866
wenzelm@60420
  1867
subsection \<open>General bisection principle for intervals; might be useful elsewhere.\<close>
himmelma@35172
  1868
wenzelm@53409
  1869
lemma interval_bisection_step:
immler@56188
  1870
  fixes type :: "'a::euclidean_space"
wenzelm@53409
  1871
  assumes "P {}"
wenzelm@53409
  1872
    and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
immler@56188
  1873
    and "\<not> P (cbox a (b::'a))"
immler@56188
  1874
  obtains c d where "\<not> P (cbox c d)"
wenzelm@53409
  1875
    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
  1876
proof -
immler@56188
  1877
  have "cbox a b \<noteq> {}"
immler@54776
  1878
    using assms(1,3) by metis
wenzelm@53409
  1879
  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
immler@56188
  1880
    by (force simp: mem_box)
lp15@60428
  1881
  { fix f
lp15@60428
  1882
    have "\<lbrakk>finite f;
lp15@60428
  1883
           \<And>s. s\<in>f \<Longrightarrow> P s;
lp15@60428
  1884
           \<And>s. s\<in>f \<Longrightarrow> \<exists>a b. s = cbox a b;
lp15@60428
  1885
           \<And>s t. s\<in>f \<Longrightarrow> t\<in>f \<Longrightarrow> s \<noteq> t \<Longrightarrow> interior s \<inter> interior t = {}\<rbrakk> \<Longrightarrow> P (\<Union>f)"
wenzelm@53409
  1886
    proof (induct f rule: finite_induct)
wenzelm@53409
  1887
      case empty
wenzelm@53409
  1888
      show ?case
wenzelm@53409
  1889
        using assms(1) by auto
wenzelm@53409
  1890
    next
wenzelm@53409
  1891
      case (insert x f)
wenzelm@53409
  1892
      show ?case
wenzelm@53409
  1893
        unfolding Union_insert
wenzelm@53409
  1894
        apply (rule assms(2)[rule_format])
lp15@60384
  1895
        using inter_interior_unions_intervals [of f "interior x"]
lp15@60384
  1896
        apply (auto simp: insert)
lp15@60428
  1897
        by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
lp15@60428
  1898
    qed
lp15@60428
  1899
  } note UN_cases = this
immler@56188
  1900
  let ?A = "{cbox c d | c d::'a. \<forall>i\<in>Basis. (c\<bullet>i = a\<bullet>i) \<and> (d\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<or>
wenzelm@53409
  1901
    (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
hoelzl@50526
  1902
  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
  1903
  {
immler@56188
  1904
    presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False"
wenzelm@53409
  1905
    then show thesis
wenzelm@53409
  1906
      unfolding atomize_not not_all
lp15@60384
  1907
      by (blast intro: that)
wenzelm@53409
  1908
  }
immler@56188
  1909
  assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)"
wenzelm@60585
  1910
  have "P (\<Union>?A)"
lp15@60428
  1911
  proof (rule UN_cases)
immler@56188
  1912
    let ?B = "(\<lambda>s. cbox (\<Sum>i\<in>Basis. (if i \<in> s then a\<bullet>i else (a\<bullet>i + b\<bullet>i) / 2) *\<^sub>R i::'a)
immler@56188
  1913
      (\<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
  1914
    have "?A \<subseteq> ?B"
wenzelm@53409
  1915
    proof
wenzelm@61165
  1916
      fix x
wenzelm@61165
  1917
      assume "x \<in> ?A"
lp15@60615
  1918
      then obtain c d
lp15@60428
  1919
        where x:  "x = cbox c d"
lp15@60428
  1920
                  "\<And>i. i \<in> Basis \<Longrightarrow>
lp15@60428
  1921
                        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60428
  1922
                        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
wenzelm@53409
  1923
      show "x \<in> ?B"
lp15@60428
  1924
        unfolding image_iff x
wenzelm@53409
  1925
        apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
lp15@60428
  1926
        apply (rule arg_cong2 [where f = cbox])
lp15@60428
  1927
        using x(2) ab
lp15@60428
  1928
        apply (auto simp add: euclidean_eq_iff[where 'a='a])
lp15@60428
  1929
        by fastforce
wenzelm@53409
  1930
    qed
wenzelm@53409
  1931
    then show "finite ?A"
wenzelm@53409
  1932
      by (rule finite_subset) auto
lp15@60428
  1933
  next
wenzelm@53409
  1934
    fix s
wenzelm@53409
  1935
    assume "s \<in> ?A"
lp15@60428
  1936
    then obtain c d
lp15@60428
  1937
      where s: "s = cbox c d"
lp15@60428
  1938
               "\<And>i. i \<in> Basis \<Longrightarrow>
lp15@60428
  1939
                     c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60428
  1940
                     c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
wenzelm@53409
  1941
      by blast
wenzelm@53409
  1942
    show "P s"
wenzelm@53409
  1943
      unfolding s
wenzelm@53409
  1944
      apply (rule as[rule_format])
lp15@60394
  1945
      using ab s(2) by force
immler@56188
  1946
    show "\<exists>a b. s = cbox a b"
wenzelm@53409
  1947
      unfolding s by auto
wenzelm@53409
  1948
    fix t
wenzelm@53409
  1949
    assume "t \<in> ?A"
wenzelm@53409
  1950
    then obtain e f where t:
immler@56188
  1951
      "t = cbox e f"
wenzelm@53409
  1952
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  1953
        e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  1954
        e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
wenzelm@53409
  1955
      by blast
wenzelm@53409
  1956
    assume "s \<noteq> t"
wenzelm@53409
  1957
    then have "\<not> (c = e \<and> d = f)"
wenzelm@53409
  1958
      unfolding s t by auto
wenzelm@53409
  1959
    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
  1960
      unfolding euclidean_eq_iff[where 'a='a] by auto
wenzelm@53409
  1961
    then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
lp15@60394
  1962
      using s(2) t(2) apply fastforce
wenzelm@60420
  1963
      using t(2)[OF i'] \<open>c \<bullet> i \<noteq> e \<bullet> i \<or> d \<bullet> i \<noteq> f \<bullet> i\<close> i' s(2) t(2) by fastforce
wenzelm@53409
  1964
    have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
wenzelm@53409
  1965
      by auto
wenzelm@53409
  1966
    show "interior s \<inter> interior t = {}"
immler@56188
  1967
      unfolding s t interior_cbox
wenzelm@53409
  1968
    proof (rule *)
wenzelm@53409
  1969
      fix x
immler@54775
  1970
      assume "x \<in> box c d" "x \<in> box e f"
wenzelm@53409
  1971
      then have x: "c\<bullet>i < d\<bullet>i" "e\<bullet>i < f\<bullet>i" "c\<bullet>i < f\<bullet>i" "e\<bullet>i < d\<bullet>i"
immler@56188
  1972
        unfolding mem_box using i'
lp15@60394
  1973
        by force+
lp15@60394
  1974
      show False  using s(2)[OF i']
lp15@60394
  1975
      proof safe
wenzelm@53409
  1976
        assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
wenzelm@53409
  1977
        show False
wenzelm@53409
  1978
          using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
wenzelm@53409
  1979
      next
wenzelm@53409
  1980
        assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
wenzelm@53409
  1981
        show False
wenzelm@53409
  1982
          using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
wenzelm@53409
  1983
      qed
wenzelm@53409
  1984
    qed
wenzelm@53409
  1985
  qed
wenzelm@60585
  1986
  also have "\<Union>?A = cbox a b"
wenzelm@53409
  1987
  proof (rule set_eqI,rule)
wenzelm@53409
  1988
    fix x
wenzelm@53409
  1989
    assume "x \<in> \<Union>?A"
wenzelm@53409
  1990
    then obtain c d where x:
immler@56188
  1991
      "x \<in> cbox c d"
wenzelm@53409
  1992
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  1993
        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60615
  1994
        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
lp15@60394
  1995
      by blast
immler@56188
  1996
    show "x\<in>cbox a b"
immler@56188
  1997
      unfolding mem_box
wenzelm@53409
  1998
    proof safe
wenzelm@53409
  1999
      fix i :: 'a
wenzelm@53409
  2000
      assume i: "i \<in> Basis"
wenzelm@53409
  2001
      then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
immler@56188
  2002
        using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
wenzelm@53409
  2003
    qed
wenzelm@53409
  2004
  next
wenzelm@53409
  2005
    fix x
immler@56188
  2006
    assume x: "x \<in> cbox a b"
wenzelm@53409
  2007
    have "\<forall>i\<in>Basis.
wenzelm@53409
  2008
      \<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
  2009
      (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
immler@56188
  2010
      unfolding mem_box
hoelzl@50526
  2011
    proof
wenzelm@53409
  2012
      fix i :: 'a
wenzelm@53409
  2013
      assume i: "i \<in> Basis"
hoelzl@50526
  2014
      have "?P i (a\<bullet>i) ((a \<bullet> i + b \<bullet> i) / 2) \<or> ?P i ((a \<bullet> i + b \<bullet> i) / 2) (b\<bullet>i)"
immler@56188
  2015
        using x[unfolded mem_box,THEN bspec, OF i] by auto
wenzelm@53409
  2016
      then show "\<exists>c d. ?P i c d"
wenzelm@53409
  2017
        by blast
hoelzl@50526
  2018
    qed
wenzelm@53409
  2019
    then show "x\<in>\<Union>?A"
hoelzl@50526
  2020
      unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
lp15@60384
  2021
      apply auto
immler@56188
  2022
      apply (rule_tac x="cbox xa xaa" in exI)
immler@56188
  2023
      unfolding mem_box
wenzelm@53409
  2024
      apply auto
wenzelm@53409
  2025
      done
wenzelm@53409
  2026
  qed
wenzelm@53409
  2027
  finally show False
wenzelm@53409
  2028
    using assms by auto
wenzelm@53409
  2029
qed
wenzelm@53409
  2030
wenzelm@53409
  2031
lemma interval_bisection:
immler@56188
  2032
  fixes type :: "'a::euclidean_space"
wenzelm@53409
  2033
  assumes "P {}"
wenzelm@53409
  2034
    and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
immler@56188
  2035
    and "\<not> P (cbox a (b::'a))"
immler@56188
  2036
  obtains x where "x \<in> cbox a b"
immler@56188
  2037
    and "\<forall>e>0. \<exists>c d. x \<in> cbox c d \<and> cbox c d \<subseteq> ball x e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
immler@56188
  2038
proof -
immler@56188
  2039
  have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and>
hoelzl@50526
  2040
    (\<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@61165
  2041
       2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))" (is "\<forall>x. ?P x")
wenzelm@53409
  2042
  proof
wenzelm@61165
  2043
    show "?P x" for x
wenzelm@61165
  2044
    proof (cases "P (cbox (fst x) (snd x))")
wenzelm@61165
  2045
      case True
wenzelm@61165
  2046
      then show ?thesis by auto
wenzelm@53409
  2047
    next
wenzelm@61165
  2048
      case as: False
immler@56188
  2049
      obtain c d where "\<not> P (cbox c d)"
wenzelm@53409
  2050
        "\<forall>i\<in>Basis.
wenzelm@53409
  2051
           fst x \<bullet> i \<le> c \<bullet> i \<and>
wenzelm@53409
  2052
           c \<bullet> i \<le> d \<bullet> i \<and>
wenzelm@53409
  2053
           d \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@53409
  2054
           2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
wenzelm@53409
  2055
        by (rule interval_bisection_step[of P, OF assms(1-2) as])
wenzelm@53409
  2056
      then show ?thesis
wenzelm@53409
  2057
        apply -
wenzelm@53409
  2058
        apply (rule_tac x="(c,d)" in exI)
wenzelm@53409
  2059
        apply auto
wenzelm@53409
  2060
        done
wenzelm@53409
  2061
    qed
wenzelm@53409
  2062
  qed
wenzelm@55751
  2063
  then obtain f where f:
wenzelm@55751
  2064
    "\<forall>x.
immler@56188
  2065
      \<not> P (cbox (fst x) (snd x)) \<longrightarrow>
immler@56188
  2066
      \<not> P (cbox (fst (f x)) (snd (f x))) \<and>
wenzelm@55751
  2067
        (\<forall>i\<in>Basis.
wenzelm@55751
  2068
            fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and>
wenzelm@55751
  2069
            fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and>
wenzelm@55751
  2070
            snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@55751
  2071
            2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)"
wenzelm@53409
  2072
    apply -
wenzelm@53409
  2073
    apply (drule choice)
wenzelm@55751
  2074
    apply blast
wenzelm@55751
  2075
    done
wenzelm@53409
  2076
  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)"
wenzelm@53409
  2077
  def A \<equiv> "\<lambda>n. fst(AB n)"
wenzelm@53409
  2078
  def B \<equiv> "\<lambda>n. snd(AB n)"
wenzelm@53409
  2079
  note ab_def = A_def B_def AB_def
immler@56188
  2080
  have "A 0 = a" "B 0 = b" "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and>
wenzelm@53399
  2081
    (\<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
  2082
    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
  2083
  proof -
wenzelm@53409
  2084
    show "A 0 = a" "B 0 = b"
wenzelm@53409
  2085
      unfolding ab_def by auto
wenzelm@53409
  2086
    note S = ab_def funpow.simps o_def id_apply
wenzelm@61165
  2087
    show "?P n" for n
wenzelm@53409
  2088
    proof (induct n)
wenzelm@53409
  2089
      case 0
wenzelm@53409
  2090
      then show ?case
wenzelm@53409
  2091
        unfolding S
wenzelm@53409
  2092
        apply (rule f[rule_format]) using assms(3)
wenzelm@53409
  2093
        apply auto
wenzelm@53409
  2094
        done
wenzelm@53409
  2095
    next
wenzelm@53409
  2096
      case (Suc n)
wenzelm@53409
  2097
      show ?case
wenzelm@53409
  2098
        unfolding S
wenzelm@53409
  2099
        apply (rule f[rule_format])
wenzelm@53409
  2100
        using Suc
wenzelm@53409
  2101
        unfolding S
wenzelm@53409
  2102
        apply auto
wenzelm@53409
  2103
        done
wenzelm@53409
  2104
    qed
wenzelm@53409
  2105
  qed
wenzelm@53409
  2106
  note AB = this(1-2) conjunctD2[OF this(3),rule_format]
wenzelm@53409
  2107
wenzelm@61165
  2108
  have interv: "\<exists>n. \<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e"
wenzelm@61165
  2109
    if e: "0 < e" for e
wenzelm@53409
  2110
  proof -
wenzelm@53409
  2111
    obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
wenzelm@53409
  2112
      using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] ..
wenzelm@61165
  2113
    show ?thesis
lp15@60396
  2114
    proof (rule exI [where x=n], clarify)
wenzelm@53409
  2115
      fix x y
immler@56188
  2116
      assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
wenzelm@53409
  2117
      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis"
wenzelm@53409
  2118
        unfolding dist_norm by(rule norm_le_l1)
hoelzl@50526
  2119
      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
wenzelm@53409
  2120
      proof (rule setsum_mono)
wenzelm@53409
  2121
        fix i :: 'a
wenzelm@53409
  2122
        assume i: "i \<in> Basis"
wenzelm@53409
  2123
        show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
immler@56188
  2124
          using xy[unfolded mem_box,THEN bspec, OF i]
wenzelm@53409
  2125
          by (auto simp: inner_diff_left)
wenzelm@53409
  2126
      qed
wenzelm@53409
  2127
      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
wenzelm@53409
  2128
        unfolding setsum_divide_distrib
wenzelm@53409
  2129
      proof (rule setsum_mono)
wenzelm@61165
  2130
        show "B n \<bullet> i - A n \<bullet> i \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ n" if i: "i \<in> Basis" for i
wenzelm@53409
  2131
        proof (induct n)
wenzelm@53409
  2132
          case 0
wenzelm@53409
  2133
          then show ?case
wenzelm@53409
  2134
            unfolding AB by auto
wenzelm@53409
  2135
        next
wenzelm@53409
  2136
          case (Suc n)
wenzelm@53409
  2137
          have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
wenzelm@61165
  2138
            using AB(4)[of i n] using i by auto
wenzelm@53409
  2139
          also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
wenzelm@61165
  2140
            using Suc by (auto simp add: field_simps)
wenzelm@53409
  2141
          finally show ?case .
wenzelm@53409
  2142
        qed
wenzelm@53409
  2143
      qed
wenzelm@53409
  2144
      also have "\<dots> < e"
wenzelm@61165
  2145
        using n using e by (auto simp add: field_simps)
wenzelm@53409
  2146
      finally show "dist x y < e" .
wenzelm@53409
  2147
    qed
wenzelm@53409
  2148
  qed
wenzelm@53409
  2149
  {
wenzelm@53409
  2150
    fix n m :: nat
immler@56188
  2151
    assume "m \<le> n" then have "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)"
hoelzl@54411
  2152
    proof (induction rule: inc_induct)
wenzelm@53409
  2153
      case (step i)
wenzelm@53409
  2154
      show ?case
immler@56188
  2155
        using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
wenzelm@53409
  2156
    qed simp
wenzelm@53409
  2157
  } note ABsubset = this
immler@56188
  2158
  have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)"
immler@56188
  2159
    by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
immler@54776
  2160
      (metis nat.exhaust AB(1-3) assms(1,3))
immler@56188
  2161
  then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
wenzelm@53409
  2162
    by blast
wenzelm@53409
  2163
  show thesis
wenzelm@53409
  2164
  proof (rule that[rule_format, of x0])
immler@56188
  2165
    show "x0\<in>cbox a b"
wenzelm@53409
  2166
      using x0[of 0] unfolding AB .
wenzelm@53409
  2167
    fix e :: real
wenzelm@53409
  2168
    assume "e > 0"
wenzelm@53409
  2169
    from interv[OF this] obtain n
immler@56188
  2170
      where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" ..
lp15@60396
  2171
    have "\<not> P (cbox (A n) (B n))"
lp15@60396
  2172
      apply (cases "0 < n")
lp15@60396
  2173
      using AB(3)[of "n - 1"] assms(3) AB(1-2)
lp15@60396
  2174
      apply auto
lp15@60396
  2175
      done
lp15@60396
  2176
    moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
lp15@60396
  2177
      using n using x0[of n] by auto
lp15@60396
  2178
    moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
lp15@60396
  2179
      unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
lp15@60396
  2180
    ultimately show "\<exists>c d. x0 \<in> cbox c d \<and> cbox c d \<subseteq> ball x0 e \<and> cbox c d \<subseteq> cbox a b \<and> \<not> P (cbox c d)"
wenzelm@53409
  2181
      apply (rule_tac x="A n" in exI)
wenzelm@53409
  2182
      apply (rule_tac x="B n" in exI)
lp15@60396
  2183
      apply (auto simp: x0)
lp15@60396
  2184
      done
wenzelm@53409
  2185
  qed
wenzelm@53409
  2186
qed
wenzelm@53409
  2187
himmelma@35172
  2188
wenzelm@60420
  2189
subsection \<open>Cousin's lemma.\<close>
himmelma@35172
  2190
wenzelm@53409
  2191
lemma fine_division_exists:
immler@56188
  2192
  fixes a b :: "'a::euclidean_space"
wenzelm@53409
  2193
  assumes "gauge g"
immler@56188
  2194
  obtains p where "p tagged_division_of (cbox a b)" "g fine p"
immler@56188
  2195
proof -
immler@56188
  2196
  presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
immler@56188
  2197
  then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
wenzelm@53410
  2198
    by blast
wenzelm@53409
  2199
  then show thesis ..
wenzelm@53409
  2200
next
immler@56188
  2201
  assume as: "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
wenzelm@55751
  2202
  obtain x where x:
lp15@60428
  2203
      "x \<in> (cbox a b)"
lp15@60428
  2204
      "\<And>e. 0 < e \<Longrightarrow>
lp15@60428
  2205
        \<exists>c d.
lp15@60428
  2206
          x \<in> cbox c d \<and>
lp15@60428
  2207
          cbox c d \<subseteq> ball x e \<and>
lp15@60428
  2208
          cbox c d \<subseteq> (cbox a b) \<and>
lp15@60428
  2209
          \<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
lp15@60428
  2210
    apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ as])
lp15@60428
  2211
    apply (simp add: fine_def)
lp15@60428
  2212
    apply (metis tagged_division_union fine_union)
lp15@60428
  2213
    apply (auto simp: )
lp15@60428
  2214
    done
wenzelm@53410
  2215
  obtain e where e: "e > 0" "ball x e \<subseteq> g x"
wenzelm@53409
  2216
    using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
lp15@60615
  2217
  from x(2)[OF e(1)]
lp15@60396
  2218
  obtain c d where c_d: "x \<in> cbox c d"
lp15@60396
  2219
                        "cbox c d \<subseteq> ball x e"
lp15@60396
  2220
                        "cbox c d \<subseteq> cbox a b"
lp15@60396
  2221
                        "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
wenzelm@53410
  2222
    by blast
immler@56188
  2223
  have "g fine {(x, cbox c d)}"
wenzelm@53409
  2224
    unfolding fine_def using e using c_d(2) by auto
wenzelm@53410
  2225
  then show False
wenzelm@53410
  2226
    using tagged_division_of_self[OF c_d(1)] using c_d by auto
wenzelm@53409
  2227
qed
wenzelm@53409
  2228
immler@56188
  2229
lemma fine_division_exists_real:
immler@56188
  2230
  fixes a b :: real
immler@56188
  2231
  assumes "gauge g"
immler@56188
  2232
  obtains p where "p tagged_division_of {a .. b}" "g fine p"
immler@56188
  2233
  by (metis assms box_real(2) fine_division_exists)
himmelma@35172
  2234
wenzelm@60420
  2235
subsection \<open>Basic theorems about integrals.\<close>
himmelma@35172
  2236
wenzelm@53409
  2237
lemma has_integral_unique:
immler@56188
  2238
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2239
  assumes "(f has_integral k1) i"
wenzelm@53410
  2240
    and "(f has_integral k2) i"
wenzelm@53409
  2241
  shows "k1 = k2"
wenzelm@53410
  2242
proof (rule ccontr)
wenzelm@53842
  2243
  let ?e = "norm (k1 - k2) / 2"
wenzelm@61165
  2244
  assume as: "k1 \<noteq> k2"
wenzelm@53410
  2245
  then have e: "?e > 0"
wenzelm@53410
  2246
    by auto
wenzelm@61165
  2247
  have lem: False
wenzelm@61165
  2248
    if f_k1: "(f has_integral k1) (cbox a b)"
wenzelm@61165
  2249
    and f_k2: "(f has_integral k2) (cbox a b)"
wenzelm@61165
  2250
    and "k1 \<noteq> k2"
wenzelm@61165
  2251
    for f :: "'n \<Rightarrow> 'a" and a b k1 k2
wenzelm@53410
  2252
  proof -
wenzelm@53410
  2253
    let ?e = "norm (k1 - k2) / 2"
wenzelm@61165
  2254
    from \<open>k1 \<noteq> k2\<close> have e: "?e > 0" by auto
wenzelm@55751
  2255
    obtain d1 where d1:
wenzelm@55751
  2256
        "gauge d1"
immler@56188
  2257
        "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
wenzelm@55751
  2258
          d1 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k1) < norm (k1 - k2) / 2"
wenzelm@61165
  2259
      by (rule has_integralD[OF f_k1 e]) blast
wenzelm@55751
  2260
    obtain d2 where d2:
wenzelm@55751
  2261
        "gauge d2"
immler@56188
  2262
        "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
wenzelm@55751
  2263
          d2 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k2) < norm (k1 - k2) / 2"
wenzelm@61165
  2264
      by (rule has_integralD[OF f_k2 e]) blast
wenzelm@55751
  2265
    obtain p where p:
immler@56188
  2266
        "p tagged_division_of cbox a b"
wenzelm@55751
  2267
        "(\<lambda>x. d1 x \<inter> d2 x) fine p"
wenzelm@55751
  2268
      by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)]])
wenzelm@53410
  2269
    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
wenzelm@53410
  2270
    have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
wenzelm@53410
  2271
      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
wenzelm@53410
  2272
      by (auto simp add:algebra_simps norm_minus_commute)
himmelma@35172
  2273
    also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2274
      apply (rule add_strict_mono)
wenzelm@53410
  2275
      apply (rule_tac[!] d2(2) d1(2))
wenzelm@53410
  2276
      using p unfolding fine_def
wenzelm@53410
  2277
      apply auto
wenzelm@53410
  2278
      done
himmelma@35172
  2279
    finally show False by auto
wenzelm@53410
  2280
  qed
wenzelm@53410
  2281
  {
immler@56188
  2282
    presume "\<not> (\<exists>a b. i = cbox a b) \<Longrightarrow> False"
wenzelm@53410
  2283
    then show False
lp15@60396
  2284
      using as assms lem by blast
wenzelm@53410
  2285
  }
immler@56188
  2286
  assume as: "\<not> (\<exists>a b. i = cbox a b)"
wenzelm@55751
  2287
  obtain B1 where B1:
wenzelm@55751
  2288
      "0 < B1"
immler@56188
  2289
      "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2290
        \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
wenzelm@55751
  2291
          norm (z - k1) < norm (k1 - k2) / 2"
wenzelm@55751
  2292
    by (rule has_integral_altD[OF assms(1) as,OF e]) blast
wenzelm@55751
  2293
  obtain B2 where B2:
wenzelm@55751
  2294
      "0 < B2"
immler@56188
  2295
      "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2296
        \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
wenzelm@55751
  2297
          norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@55751
  2298
    by (rule has_integral_altD[OF assms(2) as,OF e]) blast
immler@56188
  2299
  have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> cbox a b"
immler@56188
  2300
    apply (rule bounded_subset_cbox)
wenzelm@53410
  2301
    using bounded_Un bounded_ball
wenzelm@53410
  2302
    apply auto
wenzelm@53410
  2303
    done
immler@56188
  2304
  then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
wenzelm@53410
  2305
    by blast
wenzelm@53410
  2306
  obtain w where w:
immler@56188
  2307
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) (cbox a b)"
wenzelm@53410
  2308
    "norm (w - k1) < norm (k1 - k2) / 2"
wenzelm@53410
  2309
    using B1(2)[OF ab(1)] by blast
wenzelm@53410
  2310
  obtain z where z:
immler@56188
  2311
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b)"
wenzelm@53410
  2312
    "norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@53410
  2313
    using B2(2)[OF ab(2)] by blast
wenzelm@53410
  2314
  have "z = w"
wenzelm@53410
  2315
    using lem[OF w(1) z(1)] by auto
wenzelm@53410
  2316
  then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
wenzelm@53410
  2317
    using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
wenzelm@53410
  2318
    by (auto simp add: norm_minus_commute)
wenzelm@53410
  2319
  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2320
    apply (rule add_strict_mono)
wenzelm@53410
  2321
    apply (rule_tac[!] z(2) w(2))
wenzelm@53410
  2322
    done
wenzelm@53410
  2323
  finally show False by auto
wenzelm@53410
  2324
qed
wenzelm@53410
  2325
wenzelm@53410
  2326
lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
wenzelm@53410
  2327
  unfolding integral_def
wenzelm@53410
  2328
  by (rule some_equality) (auto intro: has_integral_unique)
wenzelm@53410
  2329
wenzelm@53410
  2330
lemma has_integral_is_0:
immler@56188
  2331
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2332
  assumes "\<forall>x\<in>s. f x = 0"
wenzelm@53410
  2333
  shows "(f has_integral 0) s"
wenzelm@53410
  2334
proof -
wenzelm@53410
  2335
  have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
immler@56188
  2336
    (\<forall>x\<in>cbox a b. f(x) = 0) \<Longrightarrow> (f has_integral 0) (cbox a b)"
wenzelm@53410
  2337
    unfolding has_integral
lp15@60396
  2338
  proof clarify
wenzelm@53410
  2339
    fix a b e
wenzelm@53410
  2340
    fix f :: "'n \<Rightarrow> 'a"
immler@56188
  2341
    assume as: "\<forall>x\<in>cbox a b. f x = 0" "0 < (e::real)"
wenzelm@61165
  2342
    have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
wenzelm@61165
  2343
      if p: "p tagged_division_of cbox a b" for p
wenzelm@53410
  2344
    proof -
wenzelm@53410
  2345
      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0"
haftmann@57418
  2346
      proof (rule setsum.neutral, rule)
wenzelm@53410
  2347
        fix x
wenzelm@53410
  2348
        assume x: "x \<in> p"
wenzelm@53410
  2349
        have "f (fst x) = 0"
wenzelm@61165
  2350
          using tagged_division_ofD(2-3)[OF p, of "fst x" "snd x"] using as x by auto
wenzelm@53410
  2351
        then show "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0"
wenzelm@53410
  2352
          apply (subst surjective_pairing[of x])
wenzelm@53410
  2353
          unfolding split_conv
wenzelm@53410
  2354
          apply auto
wenzelm@53410
  2355
          done
wenzelm@53410
  2356
      qed
wenzelm@61165
  2357
      then show ?thesis
wenzelm@53410
  2358
        using as by auto
lp15@60396
  2359
    qed
lp15@60396
  2360
    then show "\<exists>d. gauge d \<and>
wenzelm@61165
  2361
        (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e)"
lp15@60396
  2362
      by auto
wenzelm@53410
  2363
  qed
wenzelm@53410
  2364
  {
immler@56188
  2365
    presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
lp15@60396
  2366
    with assms lem show ?thesis
lp15@60396
  2367
      by blast
wenzelm@53410
  2368
  }
wenzelm@53410
  2369
  have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
wenzelm@53410
  2370
    apply (rule ext)
wenzelm@53410
  2371
    using assms
wenzelm@53410
  2372
    apply auto
wenzelm@53410
  2373
    done
immler@56188
  2374
  assume "\<not> (\<exists>a b. s = cbox a b)"
wenzelm@53410
  2375
  then show ?thesis
lp15@60396
  2376
    using lem
lp15@60396
  2377
    by (subst has_integral_alt) (force simp add: *)
wenzelm@53410
  2378
qed
himmelma@35172
  2379
immler@56188
  2380
lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) s"
wenzelm@53410
  2381
  by (rule has_integral_is_0) auto
himmelma@35172
  2382
himmelma@35172
  2383
lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
himmelma@35172
  2384
  using has_integral_unique[OF has_integral_0] by auto
himmelma@35172
  2385
wenzelm@53410
  2386
lemma has_integral_linear:
immler@56188
  2387
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2388
  assumes "(f has_integral y) s"
wenzelm@53410
  2389
    and "bounded_linear h"
wenzelm@53410
  2390
  shows "((h o f) has_integral ((h y))) s"
wenzelm@53410
  2391
proof -
wenzelm@53410
  2392
  interpret bounded_linear h
wenzelm@53410
  2393
    using assms(2) .
wenzelm@53410
  2394
  from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
wenzelm@53410
  2395
    by blast
wenzelm@53410
  2396
  have lem: "\<And>(f :: 'n \<Rightarrow> 'a) y a b.
immler@56188
  2397
    (f has_integral y) (cbox a b) \<Longrightarrow> ((h o f) has_integral h y) (cbox a b)"
wenzelm@61165
  2398
    unfolding has_integral
wenzelm@61166
  2399
  proof (clarify, goal_cases)
wenzelm@61167
  2400
    case prems: (1 f y a b e)
wenzelm@53410
  2401
    from pos_bounded
wenzelm@53410
  2402
    obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
wenzelm@53410
  2403
      by blast
wenzelm@61167
  2404
    have "e / B > 0" using prems(2) B by simp
lp15@60615
  2405
    then obtain g
lp15@60428
  2406
      where g: "gauge g"
lp15@60428
  2407
               "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> g fine p \<Longrightarrow>
lp15@60428
  2408
                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
wenzelm@61167
  2409
        using prems(1) by auto
wenzelm@61165
  2410
    {
wenzelm@61165
  2411
      fix p
immler@56188
  2412
      assume as: "p tagged_division_of (cbox a b)" "g fine p"
lp15@60428
  2413
      have hc: "\<And>x k. h ((\<lambda>(x, k). content k *\<^sub>R f x) x) = (\<lambda>(x, k). h (content k *\<^sub>R f x)) x"
wenzelm@53410
  2414
        by auto
lp15@60428
  2415
      then have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = setsum (h \<circ> (\<lambda>(x, k). content k *\<^sub>R f x)) p"
lp15@60428
  2416
        unfolding o_def unfolding scaleR[symmetric] hc by simp
wenzelm@53410
  2417
      also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
wenzelm@53410
  2418
        using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
lp15@60428
  2419
      finally have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)" .
lp15@60428
  2420
      then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e"
lp15@60428
  2421
        apply (simp add: diff[symmetric])
wenzelm@53410
  2422
        apply (rule le_less_trans[OF B(2)])
wenzelm@53410
  2423
        using g(2)[OF as] B(1)
wenzelm@53410
  2424
        apply (auto simp add: field_simps)
wenzelm@53410
  2425
        done
lp15@60428
  2426
    }
lp15@60428
  2427
    with g show ?case
lp15@60428
  2428
      by (rule_tac x=g in exI) auto
wenzelm@53410
  2429
  qed
wenzelm@53410
  2430
  {
immler@56188
  2431
    presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
wenzelm@53410
  2432
    then show ?thesis
lp15@60396
  2433
      using assms(1) lem by blast
wenzelm@53410
  2434
  }
immler@56188
  2435
  assume as: "\<not> (\<exists>a b. s = cbox a b)"
wenzelm@53410
  2436
  then show ?thesis
lp15@60396
  2437
  proof (subst has_integral_alt, clarsimp)
wenzelm@53410
  2438
    fix e :: real
wenzelm@53410
  2439
    assume e: "e > 0"
nipkow@56541
  2440
    have *: "0 < e/B" using e B(1) by simp
wenzelm@53410
  2441
    obtain M where M:
wenzelm@53410
  2442
      "M > 0"
immler@56188
  2443
      "\<And>a b. ball 0 M \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2444
        \<exists>z. ((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b) \<and> norm (z - y) < e / B"
wenzelm@53410
  2445
      using has_integral_altD[OF assms(1) as *] by blast
immler@56188
  2446
    show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  2447
      (\<exists>z. ((\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) has_integral z) (cbox a b) \<and> norm (z - h y) < e)"
wenzelm@61166
  2448
    proof (rule_tac x=M in exI, clarsimp simp add: M, goal_cases)
wenzelm@61167
  2449
      case prems: (1 a b)
wenzelm@53410
  2450
      obtain z where z:
immler@56188
  2451
        "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b)"
wenzelm@53410
  2452
        "norm (z - y) < e / B"
wenzelm@61167
  2453
        using M(2)[OF prems(1)] by blast
wenzelm@53410
  2454
      have *: "(\<lambda>x. if x \<in> s then (h \<circ> f) x else 0) = h \<circ> (\<lambda>x. if x \<in> s then f x else 0)"
lp15@60396
  2455
        using zero by auto
wenzelm@53410
  2456
      show ?case
wenzelm@53410
  2457
        apply (rule_tac x="h z" in exI)
wenzelm@61165
  2458
        apply (simp add: * lem z(1))
wenzelm@61165
  2459
        apply (metis B diff le_less_trans pos_less_divide_eq z(2))
wenzelm@61165
  2460
        done
wenzelm@53410
  2461
    qed
wenzelm@53410
  2462
  qed
wenzelm@53410
  2463
qed
wenzelm@53410
  2464
lp15@60615
  2465
lemma has_integral_scaleR_left:
hoelzl@57447
  2466
  "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) s"
hoelzl@57447
  2467
  using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
hoelzl@57447
  2468
hoelzl@57447
  2469
lemma has_integral_mult_left:
hoelzl@57447
  2470
  fixes c :: "_ :: {real_normed_algebra}"
hoelzl@57447
  2471
  shows "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) s"
hoelzl@57447
  2472
  using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
hoelzl@57447
  2473
lp15@60615
  2474
corollary integral_mult_left:
lp15@60615
  2475
  fixes c:: "'a::real_normed_algebra"
lp15@60615
  2476
  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x * c) = integral s f * c"
lp15@60615
  2477
  by (blast intro:  has_integral_mult_left)
lp15@60615
  2478
paulson@60762
  2479
lemma has_integral_mult_right:
paulson@60762
  2480
  fixes c :: "'a :: real_normed_algebra"
paulson@60762
  2481
  shows "(f has_integral y) i \<Longrightarrow> ((\<lambda>x. c * f x) has_integral (c * y)) i"
paulson@60762
  2482
  using h