src/HOL/Multivariate_Analysis/Integration.thy
author paulson
Mon Jul 20 23:12:50 2015 +0100 (2015-07-20)
changeset 60762 bf0c76ccee8d
parent 60621 bfb14ff43491
child 60800 7d04351c795a
permissions -rw-r--r--
new material for multivariate analysis, etc.
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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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  "~~/src/HOL/Library/Indicator_Function"
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begin
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lemma cSup_abs_le: (* TODO: is this really needed? *)
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  fixes S :: "real set"
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  shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
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  by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2 bdd_aboveI)
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lemma cInf_abs_ge: (* TODO: is this really needed? *)
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  fixes S :: "real set"
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  shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Inf S\<bar> \<le> a"
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  by (simp add: Inf_real_def) (insert cSup_abs_le [of "uminus ` S"], auto)
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lemma cSup_asclose: (* TODO: is this really needed? *)
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  fixes S :: "real set"
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  assumes S: "S \<noteq> {}"
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    and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
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  shows "\<bar>Sup S - l\<bar> \<le> e"
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proof -
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  have th: "\<And>(x::real) l e. \<bar>x - l\<bar> \<le> e \<longleftrightarrow> l - e \<le> x \<and> x \<le> l + e"
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    by arith
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  have "bdd_above S"
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    using b by (auto intro!: bdd_aboveI[of _ "l + e"])
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  with S b show ?thesis
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    unfolding th by (auto intro!: cSup_upper2 cSup_least)
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qed
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lemma cInf_asclose: (* TODO: is this really needed? *)
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  fixes S :: "real set"
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  assumes S: "S \<noteq> {}"
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    and b: "\<forall>x\<in>S. \<bar>x - l\<bar> \<le> e"
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  shows "\<bar>Inf S - l\<bar> \<le> e"
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proof -
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  have "\<bar>- Sup (uminus ` S) - l\<bar> =  \<bar>Sup (uminus ` S) - (-l)\<bar>"
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    by auto
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  also have "\<dots> \<le> e"
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    apply (rule cSup_asclose)
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    using abs_minus_add_cancel b by (auto simp add: S)
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  finally have "\<bar>- Sup (uminus ` S) - l\<bar> \<le> e" .
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  then show ?thesis
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    by (simp add: Inf_real_def)
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qed
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lemma cSup_finite_ge_iff:
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  fixes S :: "real set"
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  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Sup S \<longleftrightarrow> (\<exists>x\<in>S. a \<le> x)"
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  by (metis cSup_eq_Max Max_ge_iff)
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lemma cSup_finite_le_iff:
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  fixes S :: "real set"
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  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Sup S \<longleftrightarrow> (\<forall>x\<in>S. a \<ge> x)"
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  by (metis cSup_eq_Max Max_le_iff)
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lemma cInf_finite_ge_iff:
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  fixes S :: "real set"
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  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
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  by (metis cInf_eq_Min Min_ge_iff)
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lemma cInf_finite_le_iff:
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  fixes S :: "real set"
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  shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists>x\<in>S. a \<ge> x)"
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  by (metis cInf_eq_Min Min_le_iff)
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(*declare not_less[simp] not_le[simp]*)
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lemmas scaleR_simps = scaleR_zero_left scaleR_minus_left scaleR_left_diff_distrib
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  scaleR_zero_right scaleR_minus_right scaleR_right_diff_distrib scaleR_eq_0_iff
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  scaleR_cancel_left scaleR_cancel_right scaleR_add_right scaleR_add_left real_vector_class.scaleR_one
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lemma real_arch_invD:
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  "0 < (e::real) \<Longrightarrow> (\<exists>n::nat. n \<noteq> 0 \<and> 0 < inverse (real n) \<and> inverse (real n) < e)"
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  by (subst(asm) real_arch_inv)
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subsection \<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 "\<And>f. finite f \<Longrightarrow> \<forall>t\<in>f. \<exists>a b. t = cbox a b \<Longrightarrow>
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    \<exists>x. x \<in> s \<inter> interior (\<Union>f) \<Longrightarrow> \<exists>t\<in>f. \<exists>x. \<exists>e>0. ball x e \<subseteq> s \<inter> t"
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  proof -
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    case goal1
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    then show ?case
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    proof (induct rule: finite_induct)
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      case empty
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      obtain x where "x \<in> s \<inter> interior (\<Union>{})"
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        using empty(2) ..
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      then have False
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        unfolding Union_empty interior_empty by auto
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      then show ?case by auto
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    next
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      case (insert i f)
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      obtain x where x: "x \<in> s \<inter> interior (\<Union>insert i f)"
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        using insert(5) ..
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      then obtain e where e: "0 < e \<and> ball x e \<subseteq> s \<inter> interior (\<Union>insert i f)"
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        unfolding open_contains_ball_eq[OF open_Int[OF assms(2) open_interior], rule_format] ..
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      obtain a where "\<exists>b. i = cbox a b"
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        using insert(4)[rule_format,OF insertI1] ..
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      then obtain b where ab: "i = cbox a b" ..
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      show ?case
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      proof (cases "x \<in> i")
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        case False
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        then have "x \<in> UNIV - cbox a b"
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          unfolding ab by auto
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        then obtain d where "0 < d \<and> ball x d \<subseteq> UNIV - cbox a b"
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          unfolding open_contains_ball_eq[OF open_Diff[OF open_UNIV closed_cbox],rule_format] ..
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        then have "0 < d" "ball x (min d e) \<subseteq> UNIV - i"
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          unfolding ab ball_min_Int by auto
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        then have "ball x (min d e) \<subseteq> s \<inter> interior (\<Union>f)"
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          using e unfolding lem1 unfolding  ball_min_Int by auto
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        then have "x \<in> s \<inter> interior (\<Union>f)" using \<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"
immler@56188
   301
            unfolding mem_box by (auto simp add: not_less)
wenzelm@53399
   302
          then have "x\<bullet>k = a\<bullet>k \<or> x\<bullet>k = b\<bullet>k"
immler@56188
   303
            using True unfolding ab and mem_box
hoelzl@50526
   304
              apply (erule_tac x = k in ballE)
wenzelm@49970
   305
              apply auto
wenzelm@49970
   306
              done
wenzelm@53399
   307
          then have "\<exists>x. ball x (e/2) \<subseteq> s \<inter> (\<Union>f)"
wenzelm@53399
   308
          proof (rule disjE)
hoelzl@50526
   309
            let ?z = "x - (e/2) *\<^sub>R k"
hoelzl@50526
   310
            assume as: "x\<bullet>k = a\<bullet>k"
wenzelm@49970
   311
            have "ball ?z (e / 2) \<inter> i = {}"
lp15@60394
   312
            proof (clarsimp simp only: all_not_in_conv [symmetric])
wenzelm@49970
   313
              fix y
lp15@60394
   314
              assume "y \<in> ball ?z (e / 2)" and yi: "y \<in> i"
lp15@60394
   315
              then have "dist ?z y < e/2" by auto
wenzelm@53399
   316
              then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
hoelzl@50526
   317
                using Basis_le_norm[OF k, of "?z - y"] unfolding dist_norm by auto
wenzelm@53399
   318
              then have "y\<bullet>k < a\<bullet>k"
lp15@60394
   319
                using e k
wenzelm@57865
   320
                by (auto simp add: field_simps abs_less_iff as inner_simps)
wenzelm@53399
   321
              then have "y \<notin> i"
immler@56188
   322
                unfolding ab mem_box by (auto intro!: bexI[OF _ k])
wenzelm@53399
   323
              then show False using yi by auto
wenzelm@49970
   324
            qed
wenzelm@49970
   325
            moreover
wenzelm@49970
   326
            have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
wenzelm@53399
   327
              apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
wenzelm@49970
   328
            proof
wenzelm@49970
   329
              fix y
wenzelm@53399
   330
              assume as: "y \<in> ball ?z (e/2)"
hoelzl@50526
   331
              have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y - (e / 2) *\<^sub>R k)"
hoelzl@50526
   332
                apply (rule order_trans,rule norm_triangle_sub[of "x - y" "(e/2) *\<^sub>R k"])
hoelzl@50526
   333
                unfolding norm_scaleR norm_Basis[OF k]
wenzelm@49970
   334
                apply auto
wenzelm@49970
   335
                done
wenzelm@49970
   336
              also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2"
wenzelm@49970
   337
                apply (rule add_strict_left_mono)
lp15@60394
   338
                using as e
lp15@60394
   339
                apply (auto simp add: field_simps dist_norm)
wenzelm@49970
   340
                done
wenzelm@53399
   341
              finally show "y \<in> ball x e"
wenzelm@49970
   342
                unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
wenzelm@49970
   343
            qed
wenzelm@49970
   344
            ultimately show ?thesis
wenzelm@49970
   345
              apply (rule_tac x="?z" in exI)
wenzelm@49970
   346
              unfolding Union_insert
wenzelm@49970
   347
              apply auto
wenzelm@49970
   348
              done
wenzelm@49970
   349
          next
hoelzl@50526
   350
            let ?z = "x + (e/2) *\<^sub>R k"
hoelzl@50526
   351
            assume as: "x\<bullet>k = b\<bullet>k"
wenzelm@49970
   352
            have "ball ?z (e / 2) \<inter> i = {}"
lp15@60394
   353
            proof (clarsimp simp only: all_not_in_conv [symmetric])
wenzelm@49970
   354
              fix y
lp15@60394
   355
              assume "y \<in> ball ?z (e / 2)" and yi: "y \<in> i"
lp15@60394
   356
              then have "dist ?z y < e/2"
wenzelm@53408
   357
                by auto
wenzelm@53399
   358
              then have "\<bar>(?z - y) \<bullet> k\<bar> < e/2"
wenzelm@53399
   359
                using Basis_le_norm[OF k, of "?z - y"]
wenzelm@53399
   360
                unfolding dist_norm by auto
wenzelm@53399
   361
              then have "y\<bullet>k > b\<bullet>k"
lp15@60394
   362
                using e k
wenzelm@53399
   363
                by (auto simp add:field_simps inner_simps inner_Basis as)
wenzelm@53399
   364
              then have "y \<notin> i"
immler@56188
   365
                unfolding ab mem_box by (auto intro!: bexI[OF _ k])
wenzelm@53399
   366
              then show False using yi by auto
wenzelm@49970
   367
            qed
wenzelm@49970
   368
            moreover
wenzelm@49970
   369
            have "ball ?z (e/2) \<subseteq> s \<inter> (\<Union>insert i f)"
wenzelm@49970
   370
              apply (rule order_trans[OF _ e[THEN conjunct2, unfolded lem1]])
wenzelm@49970
   371
            proof
wenzelm@49970
   372
              fix y
wenzelm@49970
   373
              assume as: "y\<in> ball ?z (e/2)"
hoelzl@50526
   374
              have "norm (x - y) \<le> \<bar>e\<bar> / 2 + norm (x - y + (e / 2) *\<^sub>R k)"
wenzelm@53399
   375
                apply (rule order_trans,rule norm_triangle_sub[of "x - y" "- (e/2) *\<^sub>R k"])
wenzelm@49970
   376
                unfolding norm_scaleR
hoelzl@50526
   377
                apply (auto simp: k)
wenzelm@49970
   378
                done
wenzelm@49970
   379
              also have "\<dots> < \<bar>e\<bar> / 2 + \<bar>e\<bar> / 2"
wenzelm@49970
   380
                apply (rule add_strict_left_mono)
wenzelm@49970
   381
                using as unfolding mem_ball dist_norm
wenzelm@49970
   382
                using e apply (auto simp add: field_simps)
wenzelm@49970
   383
                done
wenzelm@53399
   384
              finally show "y \<in> ball x e"
wenzelm@53399
   385
                unfolding mem_ball dist_norm using e by (auto simp add:field_simps)
wenzelm@49970
   386
            qed
wenzelm@49970
   387
            ultimately show ?thesis
wenzelm@49970
   388
              apply (rule_tac x="?z" in exI)
wenzelm@49970
   389
              unfolding Union_insert
wenzelm@49970
   390
              apply auto
wenzelm@49970
   391
              done
wenzelm@53399
   392
          qed
wenzelm@53408
   393
          then obtain x where "ball x (e / 2) \<subseteq> s \<inter> \<Union>f" ..
wenzelm@53399
   394
          then have "x \<in> s \<inter> interior (\<Union>f)"
wenzelm@53408
   395
            unfolding lem1[where U="\<Union>f", symmetric]
lp15@60394
   396
            using centre_in_ball e by auto
lp15@60615
   397
          then show ?thesis
lp15@60384
   398
            using insert.hyps(3) insert.prems(1) by blast
wenzelm@49970
   399
        qed
wenzelm@49970
   400
      qed
wenzelm@49970
   401
    qed
wenzelm@49970
   402
  qed
lp15@60394
   403
  from this[OF assms(1,3)] x
wenzelm@53408
   404
  obtain t x e where "t \<in> f" "0 < e" "ball x e \<subseteq> s \<inter> t"
wenzelm@53408
   405
    by blast
wenzelm@53408
   406
  then have "x \<in> s" "x \<in> interior t"
wenzelm@53399
   407
    using open_subset_interior[OF open_ball, of x e t]
wenzelm@53408
   408
    by auto
wenzelm@53399
   409
  then show False
wenzelm@60420
   410
    using \<open>t \<in> f\<close> assms(4) by auto
wenzelm@60420
   411
qed
wenzelm@60420
   412
wenzelm@60420
   413
subsection \<open>Bounds on intervals where they exist.\<close>
immler@56188
   414
immler@56188
   415
definition interval_upperbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
immler@56188
   416
  where "interval_upperbound s = (\<Sum>i\<in>Basis. (SUP x:s. x\<bullet>i) *\<^sub>R i)"
immler@56188
   417
immler@56188
   418
definition interval_lowerbound :: "('a::euclidean_space) set \<Rightarrow> 'a"
immler@56188
   419
   where "interval_lowerbound s = (\<Sum>i\<in>Basis. (INF x:s. x\<bullet>i) *\<^sub>R i)"
immler@56188
   420
immler@56188
   421
lemma interval_upperbound[simp]:
immler@56188
   422
  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
immler@56188
   423
    interval_upperbound (cbox a b) = (b::'a::euclidean_space)"
immler@56188
   424
  unfolding interval_upperbound_def euclidean_representation_setsum cbox_def SUP_def
immler@56188
   425
  by (safe intro!: cSup_eq) auto
immler@56188
   426
immler@56188
   427
lemma interval_lowerbound[simp]:
immler@56188
   428
  "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow>
immler@56188
   429
    interval_lowerbound (cbox a b) = (a::'a::euclidean_space)"
immler@56188
   430
  unfolding interval_lowerbound_def euclidean_representation_setsum cbox_def INF_def
immler@56188
   431
  by (safe intro!: cInf_eq) auto
immler@56188
   432
immler@56188
   433
lemmas interval_bounds = interval_upperbound interval_lowerbound
immler@56188
   434
immler@56188
   435
lemma
immler@56188
   436
  fixes X::"real set"
immler@56188
   437
  shows interval_upperbound_real[simp]: "interval_upperbound X = Sup X"
immler@56188
   438
    and interval_lowerbound_real[simp]: "interval_lowerbound X = Inf X"
immler@56188
   439
  by (auto simp: interval_upperbound_def interval_lowerbound_def SUP_def INF_def)
immler@56188
   440
immler@56188
   441
lemma interval_bounds'[simp]:
immler@56188
   442
  assumes "cbox a b \<noteq> {}"
immler@56188
   443
  shows "interval_upperbound (cbox a b) = b"
immler@56188
   444
    and "interval_lowerbound (cbox a b) = a"
immler@56188
   445
  using assms unfolding box_ne_empty by auto
wenzelm@53399
   446
hoelzl@59425
   447
lp15@60615
   448
lemma interval_upperbound_Times:
hoelzl@59425
   449
  assumes "A \<noteq> {}" and "B \<noteq> {}"
hoelzl@59425
   450
  shows "interval_upperbound (A \<times> B) = (interval_upperbound A, interval_upperbound B)"
hoelzl@59425
   451
proof-
hoelzl@59425
   452
  from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
hoelzl@59425
   453
  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
   454
      by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
hoelzl@59425
   455
  moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
hoelzl@59425
   456
  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
   457
      by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
hoelzl@59425
   458
  ultimately show ?thesis unfolding interval_upperbound_def
hoelzl@59425
   459
      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
hoelzl@59425
   460
qed
hoelzl@59425
   461
lp15@60615
   462
lemma interval_lowerbound_Times:
hoelzl@59425
   463
  assumes "A \<noteq> {}" and "B \<noteq> {}"
hoelzl@59425
   464
  shows "interval_lowerbound (A \<times> B) = (interval_lowerbound A, interval_lowerbound B)"
hoelzl@59425
   465
proof-
hoelzl@59425
   466
  from assms have fst_image_times': "A = fst ` (A \<times> B)" by simp
hoelzl@59425
   467
  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
   468
      by (subst (2) fst_image_times') (simp del: fst_image_times add: o_def inner_Pair_0)
hoelzl@59425
   469
  moreover from assms have snd_image_times': "B = snd ` (A \<times> B)" by simp
hoelzl@59425
   470
  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
   471
      by (subst (2) snd_image_times') (simp del: snd_image_times add: o_def inner_Pair_0)
hoelzl@59425
   472
  ultimately show ?thesis unfolding interval_lowerbound_def
hoelzl@59425
   473
      by (subst setsum_Basis_prod_eq) (auto simp add: setsum_prod)
hoelzl@59425
   474
qed
hoelzl@59425
   475
wenzelm@60420
   476
subsection \<open>Content (length, area, volume...) of an interval.\<close>
himmelma@35172
   477
immler@56188
   478
definition "content (s::('a::euclidean_space) set) =
immler@56188
   479
  (if s = {} then 0 else (\<Prod>i\<in>Basis. (interval_upperbound s)\<bullet>i - (interval_lowerbound s)\<bullet>i))"
immler@56188
   480
immler@56188
   481
lemma interval_not_empty: "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i \<Longrightarrow> cbox a b \<noteq> {}"
immler@56188
   482
  unfolding box_eq_empty unfolding not_ex not_less by auto
immler@56188
   483
immler@56188
   484
lemma content_cbox:
immler@56188
   485
  fixes a :: "'a::euclidean_space"
hoelzl@50526
   486
  assumes "\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i"
immler@56188
   487
  shows "content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
wenzelm@49970
   488
  using interval_not_empty[OF assms]
immler@54777
   489
  unfolding content_def
lp15@60384
   490
  by auto
immler@56188
   491
immler@56188
   492
lemma content_cbox':
immler@56188
   493
  fixes a :: "'a::euclidean_space"
immler@56188
   494
  assumes "cbox a b \<noteq> {}"
immler@56188
   495
  shows "content (cbox a b) = (\<Prod>i\<in>Basis. b\<bullet>i - a\<bullet>i)"
lp15@60384
   496
    using assms box_ne_empty(1) content_cbox by blast
wenzelm@49970
   497
wenzelm@53408
   498
lemma content_real: "a \<le> b \<Longrightarrow> content {a..b} = b - a"
immler@56188
   499
  by (auto simp: interval_upperbound_def interval_lowerbound_def SUP_def INF_def content_def)
immler@56188
   500
hoelzl@50104
   501
lemma content_singleton[simp]: "content {a} = 0"
hoelzl@50104
   502
proof -
immler@56188
   503
  have "content (cbox a a) = 0"
immler@56188
   504
    by (subst content_cbox) (auto simp: ex_in_conv)
immler@56188
   505
  then show ?thesis by (simp add: cbox_sing)
immler@56188
   506
qed
immler@56188
   507
lp15@60615
   508
lemma content_unit[iff]: "content(cbox 0 (One::'a::euclidean_space)) = 1"
immler@56188
   509
 proof -
immler@56188
   510
   have *: "\<forall>i\<in>Basis. (0::'a)\<bullet>i \<le> (One::'a)\<bullet>i"
immler@56188
   511
    by auto
immler@56188
   512
  have "0 \<in> cbox 0 (One::'a)"
immler@56188
   513
    unfolding mem_box by auto
immler@56188
   514
  then show ?thesis
haftmann@57418
   515
     unfolding content_def interval_bounds[OF *] using setprod.neutral_const by auto
immler@56188
   516
 qed
wenzelm@49970
   517
wenzelm@49970
   518
lemma content_pos_le[intro]:
immler@56188
   519
  fixes a::"'a::euclidean_space"
immler@56188
   520
  shows "0 \<le> content (cbox a b)"
immler@56188
   521
proof (cases "cbox a b = {}")
immler@56188
   522
  case False
immler@56188
   523
  then have *: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
immler@56188
   524
    unfolding box_ne_empty .
immler@56188
   525
  have "0 \<le> (\<Prod>i\<in>Basis. interval_upperbound (cbox a b) \<bullet> i - interval_lowerbound (cbox a b) \<bullet> i)"
immler@56188
   526
    apply (rule setprod_nonneg)
immler@56188
   527
    unfolding interval_bounds[OF *]
immler@56188
   528
    using *
immler@56188
   529
    apply auto
immler@56188
   530
    done
immler@56188
   531
  also have "\<dots> = content (cbox a b)" using False by (simp add: content_def)
immler@56188
   532
  finally show ?thesis .
immler@56188
   533
qed (simp add: content_def)
wenzelm@49970
   534
lp15@60615
   535
corollary content_nonneg [simp]:
lp15@60615
   536
  fixes a::"'a::euclidean_space"
lp15@60615
   537
  shows "~ content (cbox a b) < 0"
lp15@60615
   538
using not_le by blast
lp15@60615
   539
wenzelm@49970
   540
lemma content_pos_lt:
immler@56188
   541
  fixes a :: "'a::euclidean_space"
hoelzl@50526
   542
  assumes "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
immler@56188
   543
  shows "0 < content (cbox a b)"
immler@54777
   544
  using assms
immler@56188
   545
  by (auto simp: content_def box_eq_empty intro!: setprod_pos)
wenzelm@49970
   546
wenzelm@53408
   547
lemma content_eq_0:
immler@56188
   548
  "content (cbox a b) = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i)"
immler@56188
   549
  by (auto simp: content_def box_eq_empty intro!: setprod_pos bexI)
himmelma@35172
   550
wenzelm@53408
   551
lemma cond_cases: "(P \<Longrightarrow> Q x) \<Longrightarrow> (\<not> P \<Longrightarrow> Q y) \<Longrightarrow> Q (if P then x else y)"
wenzelm@53399
   552
  by auto
himmelma@35172
   553
immler@56188
   554
lemma content_cbox_cases:
immler@56188
   555
  "content (cbox a (b::'a::euclidean_space)) =
hoelzl@50526
   556
    (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
   557
  by (auto simp: not_le content_eq_0 intro: less_imp_le content_cbox)
immler@56188
   558
immler@56188
   559
lemma content_eq_0_interior: "content (cbox a b) = 0 \<longleftrightarrow> interior(cbox a b) = {}"
immler@56188
   560
  unfolding content_eq_0 interior_cbox box_eq_empty
wenzelm@53408
   561
  by auto
himmelma@35172
   562
wenzelm@53399
   563
lemma content_pos_lt_eq:
immler@56188
   564
  "0 < content (cbox a (b::'a::euclidean_space)) \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
lp15@60394
   565
proof (rule iffI)
immler@56188
   566
  assume "0 < content (cbox a b)"
immler@56188
   567
  then have "content (cbox a b) \<noteq> 0" by auto
wenzelm@53399
   568
  then show "\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i"
wenzelm@49970
   569
    unfolding content_eq_0 not_ex not_le by fastforce
lp15@60394
   570
next
lp15@60394
   571
  assume "\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i"
lp15@60394
   572
  then show "0 < content (cbox a b)"
lp15@60394
   573
    by (metis content_pos_lt)
wenzelm@49970
   574
qed
wenzelm@49970
   575
wenzelm@53399
   576
lemma content_empty [simp]: "content {} = 0"
wenzelm@53399
   577
  unfolding content_def by auto
himmelma@35172
   578
paulson@60762
   579
lemma content_real_if [simp]: "content {a..b} = (if a \<le> b then b - a else 0)"
paulson@60762
   580
  by (simp add: content_real)
paulson@60762
   581
wenzelm@49698
   582
lemma content_subset:
immler@56188
   583
  assumes "cbox a b \<subseteq> cbox c d"
immler@56188
   584
  shows "content (cbox a b) \<le> content (cbox c d)"
immler@56188
   585
proof (cases "cbox a b = {}")
immler@56188
   586
  case True
immler@56188
   587
  then show ?thesis
immler@56188
   588
    using content_pos_le[of c d] by auto
immler@56188
   589
next
immler@56188
   590
  case False
immler@56188
   591
  then have ab_ne: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
immler@56188
   592
    unfolding box_ne_empty by auto
immler@56188
   593
  then have ab_ab: "a\<in>cbox a b" "b\<in>cbox a b"
immler@56188
   594
    unfolding mem_box by auto
immler@56188
   595
  have "cbox c d \<noteq> {}" using assms False by auto
immler@56188
   596
  then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
immler@56188
   597
    using assms unfolding box_ne_empty by auto
lp15@60394
   598
  have "\<And>i. i \<in> Basis \<Longrightarrow> 0 \<le> b \<bullet> i - a \<bullet> i"
lp15@60394
   599
    using ab_ne by (metis diff_le_iff(1))
lp15@60394
   600
  moreover
lp15@60394
   601
  have "\<And>i. i \<in> Basis \<Longrightarrow> b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
lp15@60394
   602
    using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2)]
lp15@60394
   603
          assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(1)]
lp15@60394
   604
      by (metis diff_mono)
lp15@60394
   605
  ultimately show ?thesis
lp15@60394
   606
    unfolding content_def interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
wenzelm@60420
   607
    by (simp add: setprod_mono if_not_P[OF False] if_not_P[OF \<open>cbox c d \<noteq> {}\<close>])
immler@56188
   608
qed
immler@56188
   609
immler@56188
   610
lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
nipkow@44890
   611
  unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
himmelma@35172
   612
hoelzl@59425
   613
lemma content_times[simp]: "content (A \<times> B) = content A * content B"
hoelzl@59425
   614
proof (cases "A \<times> B = {}")
hoelzl@59425
   615
  let ?ub1 = "interval_upperbound" and ?lb1 = "interval_lowerbound"
hoelzl@59425
   616
  let ?ub2 = "interval_upperbound" and ?lb2 = "interval_lowerbound"
hoelzl@59425
   617
  assume nonempty: "A \<times> B \<noteq> {}"
lp15@60615
   618
  hence "content (A \<times> B) = (\<Prod>i\<in>Basis. (?ub1 A, ?ub2 B) \<bullet> i - (?lb1 A, ?lb2 B) \<bullet> i)"
hoelzl@59425
   619
      unfolding content_def by (simp add: interval_upperbound_Times interval_lowerbound_Times)
hoelzl@59425
   620
  also have "... = content A * content B" unfolding content_def using nonempty
hoelzl@59425
   621
    apply (subst Basis_prod_def, subst setprod.union_disjoint, force, force, force, simp)
hoelzl@59425
   622
    apply (subst (1 2) setprod.reindex, auto intro: inj_onI)
hoelzl@59425
   623
    done
hoelzl@59425
   624
  finally show ?thesis .
hoelzl@59425
   625
qed (auto simp: content_def)
hoelzl@59425
   626
lp15@60615
   627
lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
lp15@60615
   628
  by (simp add: cbox_Pair_eq)
lp15@60615
   629
lp15@60615
   630
lemma content_cbox_pair_eq0_D:
lp15@60615
   631
   "content (cbox (a,c) (b,d)) = 0 \<Longrightarrow> content (cbox a b) = 0 \<or> content (cbox c d) = 0"
lp15@60615
   632
  by (simp add: content_Pair)
lp15@60615
   633
lp15@60615
   634
lemma content_eq_0_gen:
lp15@60615
   635
  fixes s :: "'a::euclidean_space set"
lp15@60615
   636
  assumes "bounded s"
lp15@60615
   637
  shows "content s = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. \<exists>v. \<forall>x \<in> s. x \<bullet> i = v)"  (is "_ = ?rhs")
lp15@60615
   638
proof safe
lp15@60615
   639
  assume "content s = 0" then show ?rhs
lp15@60615
   640
    apply (clarsimp simp: ex_in_conv content_def split: split_if_asm)
lp15@60615
   641
    apply (rule_tac x=a in bexI)
lp15@60615
   642
    apply (rule_tac x="interval_lowerbound s \<bullet> a" in exI)
lp15@60615
   643
    apply (clarsimp simp: interval_upperbound_def interval_lowerbound_def)
lp15@60615
   644
    apply (drule cSUP_eq_cINF_D)
lp15@60615
   645
    apply (auto simp: bounded_inner_imp_bdd_above [OF assms]  bounded_inner_imp_bdd_below [OF assms])
lp15@60615
   646
    done
lp15@60615
   647
next
lp15@60615
   648
  fix i a
lp15@60615
   649
  assume "i \<in> Basis" "\<forall>x\<in>s. x \<bullet> i = a"
lp15@60615
   650
  then show "content s = 0"
lp15@60615
   651
    apply (clarsimp simp: content_def)
lp15@60615
   652
    apply (rule_tac x=i in bexI)
lp15@60615
   653
    apply (auto simp: interval_upperbound_def interval_lowerbound_def)
lp15@60615
   654
    done
lp15@60615
   655
qed
lp15@60615
   656
lp15@60615
   657
lemma content_0_subset_gen:
lp15@60615
   658
  fixes a :: "'a::euclidean_space"
lp15@60615
   659
  assumes "content t = 0" "s \<subseteq> t" "bounded t" shows "content s = 0"
lp15@60615
   660
proof -
lp15@60615
   661
  have "bounded s"
lp15@60615
   662
    using assms by (metis bounded_subset)
lp15@60615
   663
  then show ?thesis
lp15@60615
   664
    using assms
lp15@60615
   665
    by (auto simp: content_eq_0_gen)
lp15@60615
   666
qed
lp15@60615
   667
lp15@60615
   668
lemma content_0_subset: "\<lbrakk>content(cbox a b) = 0; s \<subseteq> cbox a b\<rbrakk> \<Longrightarrow> content s = 0"
lp15@60615
   669
  by (simp add: content_0_subset_gen bounded_cbox)
lp15@60615
   670
wenzelm@49698
   671
wenzelm@60420
   672
subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
himmelma@35172
   673
wenzelm@53408
   674
definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
wenzelm@53399
   675
wenzelm@53399
   676
lemma gaugeI:
wenzelm@53399
   677
  assumes "\<And>x. x \<in> g x"
wenzelm@53399
   678
    and "\<And>x. open (g x)"
wenzelm@53399
   679
  shows "gauge g"
himmelma@35172
   680
  using assms unfolding gauge_def by auto
himmelma@35172
   681
wenzelm@53399
   682
lemma gaugeD[dest]:
wenzelm@53399
   683
  assumes "gauge d"
wenzelm@53399
   684
  shows "x \<in> d x"
wenzelm@53399
   685
    and "open (d x)"
wenzelm@49698
   686
  using assms unfolding gauge_def by auto
himmelma@35172
   687
himmelma@35172
   688
lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
wenzelm@53399
   689
  unfolding gauge_def by auto
wenzelm@53399
   690
wenzelm@53399
   691
lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
wenzelm@53399
   692
  unfolding gauge_def by auto
himmelma@35172
   693
lp15@60466
   694
lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)"
wenzelm@49698
   695
  by (rule gauge_ball) auto
himmelma@35172
   696
wenzelm@53408
   697
lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
wenzelm@53399
   698
  unfolding gauge_def by auto
himmelma@35172
   699
wenzelm@49698
   700
lemma gauge_inters:
wenzelm@53399
   701
  assumes "finite s"
wenzelm@53399
   702
    and "\<forall>d\<in>s. gauge (f d)"
wenzelm@60585
   703
  shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> s})"
wenzelm@49698
   704
proof -
wenzelm@53399
   705
  have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
wenzelm@53399
   706
    by auto
wenzelm@49698
   707
  show ?thesis
wenzelm@53399
   708
    unfolding gauge_def unfolding *
wenzelm@49698
   709
    using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
wenzelm@49698
   710
qed
wenzelm@49698
   711
wenzelm@53399
   712
lemma gauge_existence_lemma:
wenzelm@53408
   713
  "(\<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
   714
  by (metis zero_less_one)
wenzelm@49698
   715
himmelma@35172
   716
wenzelm@60420
   717
subsection \<open>Divisions.\<close>
himmelma@35172
   718
wenzelm@53408
   719
definition division_of (infixl "division'_of" 40)
wenzelm@53408
   720
where
wenzelm@53399
   721
  "s division_of i \<longleftrightarrow>
wenzelm@53399
   722
    finite s \<and>
immler@56188
   723
    (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = cbox a b)) \<and>
wenzelm@53399
   724
    (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
wenzelm@53399
   725
    (\<Union>s = i)"
himmelma@35172
   726
wenzelm@49698
   727
lemma division_ofD[dest]:
wenzelm@49698
   728
  assumes "s division_of i"
wenzelm@53408
   729
  shows "finite s"
wenzelm@53408
   730
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   731
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
immler@56188
   732
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
   733
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
wenzelm@53408
   734
    and "\<Union>s = i"
wenzelm@49698
   735
  using assms unfolding division_of_def by auto
himmelma@35172
   736
himmelma@35172
   737
lemma division_ofI:
wenzelm@53408
   738
  assumes "finite s"
wenzelm@53408
   739
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   740
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
immler@56188
   741
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
   742
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53399
   743
    and "\<Union>s = i"
wenzelm@53399
   744
  shows "s division_of i"
wenzelm@53399
   745
  using assms unfolding division_of_def by auto
himmelma@35172
   746
himmelma@35172
   747
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
himmelma@35172
   748
  unfolding division_of_def by auto
himmelma@35172
   749
immler@56188
   750
lemma division_of_self[intro]: "cbox a b \<noteq> {} \<Longrightarrow> {cbox a b} division_of (cbox a b)"
himmelma@35172
   751
  unfolding division_of_def by auto
himmelma@35172
   752
wenzelm@53399
   753
lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
wenzelm@53399
   754
  unfolding division_of_def by auto
himmelma@35172
   755
wenzelm@49698
   756
lemma division_of_sing[simp]:
immler@56188
   757
  "s division_of cbox a (a::'a::euclidean_space) \<longleftrightarrow> s = {cbox a a}"
wenzelm@53399
   758
  (is "?l = ?r")
wenzelm@49698
   759
proof
wenzelm@49698
   760
  assume ?r
wenzelm@53399
   761
  moreover
lp15@60384
   762
  { fix k
lp15@60384
   763
    assume "s = {{a}}" "k\<in>s"
lp15@60384
   764
    then have "\<exists>x y. k = cbox x y"
wenzelm@50945
   765
      apply (rule_tac x=a in exI)+
lp15@60384
   766
      apply (force simp: cbox_sing)
wenzelm@50945
   767
      done
wenzelm@49698
   768
  }
wenzelm@53399
   769
  ultimately show ?l
immler@56188
   770
    unfolding division_of_def cbox_sing by auto
wenzelm@49698
   771
next
wenzelm@49698
   772
  assume ?l
immler@56188
   773
  note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
wenzelm@53399
   774
  {
wenzelm@53399
   775
    fix x
wenzelm@53399
   776
    assume x: "x \<in> s" have "x = {a}"
wenzelm@53408
   777
      using *(2)[rule_format,OF x] by auto
wenzelm@53399
   778
  }
wenzelm@53408
   779
  moreover have "s \<noteq> {}"
wenzelm@53408
   780
    using *(4) by auto
wenzelm@53408
   781
  ultimately show ?r
immler@56188
   782
    unfolding cbox_sing by auto
wenzelm@49698
   783
qed
himmelma@35172
   784
himmelma@35172
   785
lemma elementary_empty: obtains p where "p division_of {}"
himmelma@35172
   786
  unfolding division_of_trivial by auto
himmelma@35172
   787
immler@56188
   788
lemma elementary_interval: obtains p where "p division_of (cbox a b)"
wenzelm@49698
   789
  by (metis division_of_trivial division_of_self)
himmelma@35172
   790
himmelma@35172
   791
lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
himmelma@35172
   792
  unfolding division_of_def by auto
himmelma@35172
   793
himmelma@35172
   794
lemma forall_in_division:
immler@56188
   795
  "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
   796
  unfolding division_of_def by fastforce
himmelma@35172
   797
wenzelm@53399
   798
lemma division_of_subset:
wenzelm@53399
   799
  assumes "p division_of (\<Union>p)"
wenzelm@53399
   800
    and "q \<subseteq> p"
wenzelm@53399
   801
  shows "q division_of (\<Union>q)"
wenzelm@53408
   802
proof (rule division_ofI)
wenzelm@53408
   803
  note * = division_ofD[OF assms(1)]
wenzelm@49698
   804
  show "finite q"
lp15@60384
   805
    using "*"(1) assms(2) infinite_super by auto
wenzelm@53399
   806
  {
wenzelm@53399
   807
    fix k
wenzelm@49698
   808
    assume "k \<in> q"
wenzelm@53408
   809
    then have kp: "k \<in> p"
wenzelm@53408
   810
      using assms(2) by auto
wenzelm@53408
   811
    show "k \<subseteq> \<Union>q"
wenzelm@60420
   812
      using \<open>k \<in> q\<close> by auto
immler@56188
   813
    show "\<exists>a b. k = cbox a b"
wenzelm@53408
   814
      using *(4)[OF kp] by auto
wenzelm@53408
   815
    show "k \<noteq> {}"
wenzelm@53408
   816
      using *(3)[OF kp] by auto
wenzelm@53399
   817
  }
wenzelm@49698
   818
  fix k1 k2
wenzelm@49698
   819
  assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
wenzelm@53408
   820
  then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
wenzelm@53399
   821
    using assms(2) by auto
wenzelm@53399
   822
  show "interior k1 \<inter> interior k2 = {}"
wenzelm@53408
   823
    using *(5)[OF **] by auto
wenzelm@49698
   824
qed auto
wenzelm@49698
   825
wenzelm@49698
   826
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@49698
   827
  unfolding division_of_def by auto
himmelma@35172
   828
wenzelm@49970
   829
lemma division_of_content_0:
immler@56188
   830
  assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
wenzelm@49970
   831
  shows "\<forall>k\<in>d. content k = 0"
wenzelm@49970
   832
  unfolding forall_in_division[OF assms(2)]
lp15@60384
   833
  by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
wenzelm@49970
   834
wenzelm@49970
   835
lemma division_inter:
immler@56188
   836
  fixes s1 s2 :: "'a::euclidean_space set"
wenzelm@53408
   837
  assumes "p1 division_of s1"
wenzelm@53408
   838
    and "p2 division_of s2"
wenzelm@49970
   839
  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
   840
  (is "?A' division_of _")
wenzelm@49970
   841
proof -
wenzelm@49970
   842
  let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
wenzelm@53408
   843
  have *: "?A' = ?A" by auto
wenzelm@53399
   844
  show ?thesis
wenzelm@53399
   845
    unfolding *
wenzelm@49970
   846
  proof (rule division_ofI)
wenzelm@53399
   847
    have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
wenzelm@53399
   848
      by auto
wenzelm@53399
   849
    moreover have "finite (p1 \<times> p2)"
wenzelm@53399
   850
      using assms unfolding division_of_def by auto
wenzelm@49970
   851
    ultimately show "finite ?A" by auto
wenzelm@53399
   852
    have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
wenzelm@53399
   853
      by auto
wenzelm@49970
   854
    show "\<Union>?A = s1 \<inter> s2"
wenzelm@49970
   855
      apply (rule set_eqI)
wenzelm@49970
   856
      unfolding * and Union_image_eq UN_iff
wenzelm@49970
   857
      using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
wenzelm@49970
   858
      apply auto
wenzelm@49970
   859
      done
wenzelm@53399
   860
    {
wenzelm@53399
   861
      fix k
wenzelm@53399
   862
      assume "k \<in> ?A"
wenzelm@53408
   863
      then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
wenzelm@53399
   864
        by auto
wenzelm@53408
   865
      then show "k \<noteq> {}"
wenzelm@53408
   866
        by auto
wenzelm@49970
   867
      show "k \<subseteq> s1 \<inter> s2"
wenzelm@49970
   868
        using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
wenzelm@49970
   869
        unfolding k by auto
immler@56188
   870
      obtain a1 b1 where k1: "k1 = cbox a1 b1"
wenzelm@53408
   871
        using division_ofD(4)[OF assms(1) k(2)] by blast
immler@56188
   872
      obtain a2 b2 where k2: "k2 = cbox a2 b2"
wenzelm@53408
   873
        using division_ofD(4)[OF assms(2) k(3)] by blast
immler@56188
   874
      show "\<exists>a b. k = cbox a b"
wenzelm@53408
   875
        unfolding k k1 k2 unfolding inter_interval by auto
wenzelm@53408
   876
    }
wenzelm@49970
   877
    fix k1 k2
wenzelm@53408
   878
    assume "k1 \<in> ?A"
wenzelm@53408
   879
    then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
wenzelm@53408
   880
      by auto
wenzelm@53408
   881
    assume "k2 \<in> ?A"
wenzelm@53408
   882
    then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
wenzelm@53408
   883
      by auto
wenzelm@49970
   884
    assume "k1 \<noteq> k2"
wenzelm@53399
   885
    then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
wenzelm@53399
   886
      unfolding k1 k2 by auto
wenzelm@53408
   887
    have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
wenzelm@53408
   888
      interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
wenzelm@53408
   889
      interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
wenzelm@53408
   890
      interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
wenzelm@49970
   891
    show "interior k1 \<inter> interior k2 = {}"
wenzelm@49970
   892
      unfolding k1 k2
wenzelm@49970
   893
      apply (rule *)
lp15@60384
   894
      using assms division_ofD(5) k1 k2(2) k2(3) th apply auto
wenzelm@53399
   895
      done
wenzelm@49970
   896
  qed
wenzelm@49970
   897
qed
wenzelm@49970
   898
wenzelm@49970
   899
lemma division_inter_1:
wenzelm@53408
   900
  assumes "d division_of i"
immler@56188
   901
    and "cbox a (b::'a::euclidean_space) \<subseteq> i"
immler@56188
   902
  shows "{cbox a b \<inter> k | k. k \<in> d \<and> cbox a b \<inter> k \<noteq> {}} division_of (cbox a b)"
immler@56188
   903
proof (cases "cbox a b = {}")
wenzelm@49970
   904
  case True
wenzelm@53399
   905
  show ?thesis
wenzelm@53399
   906
    unfolding True and division_of_trivial by auto
wenzelm@49970
   907
next
wenzelm@49970
   908
  case False
immler@56188
   909
  have *: "cbox a b \<inter> i = cbox a b" using assms(2) by auto
wenzelm@53399
   910
  show ?thesis
wenzelm@53399
   911
    using division_inter[OF division_of_self[OF False] assms(1)]
wenzelm@53399
   912
    unfolding * by auto
wenzelm@49970
   913
qed
wenzelm@49970
   914
wenzelm@49970
   915
lemma elementary_inter:
immler@56188
   916
  fixes s t :: "'a::euclidean_space set"
wenzelm@53408
   917
  assumes "p1 division_of s"
wenzelm@53408
   918
    and "p2 division_of t"
himmelma@35172
   919
  shows "\<exists>p. p division_of (s \<inter> t)"
lp15@60384
   920
using assms division_inter by blast
wenzelm@49970
   921
wenzelm@49970
   922
lemma elementary_inters:
wenzelm@53408
   923
  assumes "finite f"
wenzelm@53408
   924
    and "f \<noteq> {}"
immler@56188
   925
    and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)"
wenzelm@60585
   926
  shows "\<exists>p. p division_of (\<Inter>f)"
wenzelm@49970
   927
  using assms
wenzelm@49970
   928
proof (induct f rule: finite_induct)
wenzelm@49970
   929
  case (insert x f)
wenzelm@49970
   930
  show ?case
wenzelm@49970
   931
  proof (cases "f = {}")
wenzelm@49970
   932
    case True
wenzelm@53399
   933
    then show ?thesis
wenzelm@53399
   934
      unfolding True using insert by auto
wenzelm@49970
   935
  next
wenzelm@49970
   936
    case False
wenzelm@53408
   937
    obtain p where "p division_of \<Inter>f"
wenzelm@53408
   938
      using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
wenzelm@53408
   939
    moreover obtain px where "px division_of x"
wenzelm@53408
   940
      using insert(5)[rule_format,OF insertI1] ..
wenzelm@49970
   941
    ultimately show ?thesis
lp15@60384
   942
      by (simp add: elementary_inter Inter_insert)
wenzelm@49970
   943
  qed
wenzelm@49970
   944
qed auto
himmelma@35172
   945
himmelma@35172
   946
lemma division_disjoint_union:
wenzelm@53408
   947
  assumes "p1 division_of s1"
wenzelm@53408
   948
    and "p2 division_of s2"
wenzelm@53408
   949
    and "interior s1 \<inter> interior s2 = {}"
wenzelm@50945
   950
  shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
wenzelm@50945
   951
proof (rule division_ofI)
wenzelm@53408
   952
  note d1 = division_ofD[OF assms(1)]
wenzelm@53408
   953
  note d2 = division_ofD[OF assms(2)]
wenzelm@53408
   954
  show "finite (p1 \<union> p2)"
wenzelm@53408
   955
    using d1(1) d2(1) by auto
wenzelm@53408
   956
  show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
wenzelm@53408
   957
    using d1(6) d2(6) by auto
wenzelm@50945
   958
  {
wenzelm@50945
   959
    fix k1 k2
wenzelm@50945
   960
    assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
wenzelm@50945
   961
    moreover
wenzelm@50945
   962
    let ?g="interior k1 \<inter> interior k2 = {}"
wenzelm@50945
   963
    {
wenzelm@50945
   964
      assume as: "k1\<in>p1" "k2\<in>p2"
wenzelm@50945
   965
      have ?g
wenzelm@50945
   966
        using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
wenzelm@50945
   967
        using assms(3) by blast
wenzelm@50945
   968
    }
wenzelm@50945
   969
    moreover
wenzelm@50945
   970
    {
wenzelm@50945
   971
      assume as: "k1\<in>p2" "k2\<in>p1"
wenzelm@50945
   972
      have ?g
wenzelm@50945
   973
        using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
wenzelm@50945
   974
        using assms(3) by blast
wenzelm@50945
   975
    }
wenzelm@53399
   976
    ultimately show ?g
wenzelm@53399
   977
      using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
wenzelm@50945
   978
  }
wenzelm@50945
   979
  fix k
wenzelm@50945
   980
  assume k: "k \<in> p1 \<union> p2"
wenzelm@53408
   981
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
   982
    using k d1(2) d2(2) by auto
wenzelm@53408
   983
  show "k \<noteq> {}"
wenzelm@53408
   984
    using k d1(3) d2(3) by auto
immler@56188
   985
  show "\<exists>a b. k = cbox a b"
wenzelm@53408
   986
    using k d1(4) d2(4) by auto
wenzelm@50945
   987
qed
himmelma@35172
   988
himmelma@35172
   989
lemma partial_division_extend_1:
immler@56188
   990
  fixes a b c d :: "'a::euclidean_space"
immler@56188
   991
  assumes incl: "cbox c d \<subseteq> cbox a b"
immler@56188
   992
    and nonempty: "cbox c d \<noteq> {}"
immler@56188
   993
  obtains p where "p division_of (cbox a b)" "cbox c d \<in> p"
hoelzl@50526
   994
proof
wenzelm@53408
   995
  let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
immler@56188
   996
    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
   997
  def p \<equiv> "?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
hoelzl@50526
   998
immler@56188
   999
  show "cbox c d \<in> p"
hoelzl@50526
  1000
    unfolding p_def
immler@56188
  1001
    by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
wenzelm@50945
  1002
  {
wenzelm@50945
  1003
    fix i :: 'a
wenzelm@50945
  1004
    assume "i \<in> Basis"
hoelzl@50526
  1005
    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
  1006
      unfolding box_eq_empty subset_box by (auto simp: not_le)
wenzelm@50945
  1007
  }
hoelzl@50526
  1008
  note ord = this
hoelzl@50526
  1009
immler@56188
  1010
  show "p division_of (cbox a b)"
hoelzl@50526
  1011
  proof (rule division_ofI)
wenzelm@53399
  1012
    show "finite p"
wenzelm@53399
  1013
      unfolding p_def by (auto intro!: finite_PiE)
wenzelm@50945
  1014
    {
wenzelm@50945
  1015
      fix k
wenzelm@50945
  1016
      assume "k \<in> p"
wenzelm@53015
  1017
      then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
hoelzl@50526
  1018
        by (auto simp: p_def)
immler@56188
  1019
      then show "\<exists>a b. k = cbox a b"
wenzelm@53408
  1020
        by auto
immler@56188
  1021
      have "k \<subseteq> cbox a b \<and> k \<noteq> {}"
immler@56188
  1022
      proof (simp add: k box_eq_empty subset_box not_less, safe)
wenzelm@53374
  1023
        fix i :: 'a
wenzelm@53374
  1024
        assume i: "i \<in> Basis"
wenzelm@50945
  1025
        with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
hoelzl@50526
  1026
          by (auto simp: PiE_iff)
wenzelm@53374
  1027
        with i ord[of i]
wenzelm@50945
  1028
        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
  1029
          by auto
hoelzl@50526
  1030
      qed
immler@56188
  1031
      then show "k \<noteq> {}" "k \<subseteq> cbox a b"
wenzelm@53408
  1032
        by auto
wenzelm@50945
  1033
      {
wenzelm@53408
  1034
        fix l
wenzelm@53408
  1035
        assume "l \<in> p"
wenzelm@53015
  1036
        then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
wenzelm@50945
  1037
          by (auto simp: p_def)
wenzelm@50945
  1038
        assume "l \<noteq> k"
wenzelm@50945
  1039
        have "\<exists>i\<in>Basis. f i \<noteq> g i"
wenzelm@50945
  1040
        proof (rule ccontr)
wenzelm@53408
  1041
          assume "\<not> ?thesis"
wenzelm@50945
  1042
          with f g have "f = g"
wenzelm@50945
  1043
            by (auto simp: PiE_iff extensional_def intro!: ext)
wenzelm@60420
  1044
          with \<open>l \<noteq> k\<close> show False
wenzelm@50945
  1045
            by (simp add: l k)
wenzelm@50945
  1046
        qed
wenzelm@53408
  1047
        then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
wenzelm@53408
  1048
        then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
lp15@60384
  1049
                  "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
wenzelm@50945
  1050
          using f g by (auto simp: PiE_iff)
wenzelm@53408
  1051
        with * ord[of i] show "interior l \<inter> interior k = {}"
immler@56188
  1052
          by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i])
wenzelm@50945
  1053
      }
wenzelm@60420
  1054
      note \<open>k \<subseteq> cbox a b\<close>
wenzelm@50945
  1055
    }
hoelzl@50526
  1056
    moreover
wenzelm@50945
  1057
    {
immler@56188
  1058
      fix x assume x: "x \<in> cbox a b"
hoelzl@50526
  1059
      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
  1060
      proof
wenzelm@53408
  1061
        fix i :: 'a
wenzelm@53408
  1062
        assume "i \<in> Basis"
wenzelm@53399
  1063
        with x ord[of i]
hoelzl@50526
  1064
        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
  1065
            (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
immler@56188
  1066
          by (auto simp: cbox_def)
hoelzl@50526
  1067
        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
  1068
          by auto
hoelzl@50526
  1069
      qed
wenzelm@53408
  1070
      then obtain f where
wenzelm@53408
  1071
        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
  1072
        unfolding bchoice_iff ..
wenzelm@53374
  1073
      moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
hoelzl@50526
  1074
        by auto
hoelzl@50526
  1075
      moreover from f have "x \<in> ?B (restrict f Basis)"
immler@56188
  1076
        by (auto simp: mem_box)
hoelzl@50526
  1077
      ultimately have "\<exists>k\<in>p. x \<in> k"
wenzelm@53408
  1078
        unfolding p_def by blast
wenzelm@53408
  1079
    }
immler@56188
  1080
    ultimately show "\<Union>p = cbox a b"
hoelzl@50526
  1081
      by auto
hoelzl@50526
  1082
  qed
hoelzl@50526
  1083
qed
himmelma@35172
  1084
wenzelm@50945
  1085
lemma partial_division_extend_interval:
immler@56188
  1086
  assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
immler@56188
  1087
  obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
wenzelm@50945
  1088
proof (cases "p = {}")
wenzelm@50945
  1089
  case True
immler@56188
  1090
  obtain q where "q division_of (cbox a b)"
wenzelm@53408
  1091
    by (rule elementary_interval)
wenzelm@53399
  1092
  then show ?thesis
lp15@60384
  1093
    using True that by blast
wenzelm@50945
  1094
next
wenzelm@50945
  1095
  case False
wenzelm@50945
  1096
  note p = division_ofD[OF assms(1)]
lp15@60428
  1097
  have div_cbox: "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q"
wenzelm@50945
  1098
  proof
wenzelm@50945
  1099
    case goal1
immler@56188
  1100
    obtain c d where k: "k = cbox c d"
wenzelm@53408
  1101
      using p(4)[OF goal1] by blast
immler@56188
  1102
    have *: "cbox c d \<subseteq> cbox a b" "cbox c d \<noteq> {}"
immler@54775
  1103
      using p(2,3)[OF goal1, unfolded k] using assms(2)
immler@54776
  1104
      by (blast intro: order.trans)+
immler@56188
  1105
    obtain q where "q division_of cbox a b" "cbox c d \<in> q"
wenzelm@53408
  1106
      by (rule partial_division_extend_1[OF *])
wenzelm@53408
  1107
    then show ?case
wenzelm@53408
  1108
      unfolding k by auto
wenzelm@50945
  1109
  qed
immler@56188
  1110
  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
  1111
    using bchoice[OF div_cbox] by blast
lp15@60394
  1112
  { fix x
wenzelm@53408
  1113
    assume x: "x \<in> p"
lp15@60394
  1114
    have "q x division_of \<Union>q x"
wenzelm@50945
  1115
      apply (rule division_ofI)
wenzelm@50945
  1116
      using division_ofD[OF q(1)[OF x]]
wenzelm@50945
  1117
      apply auto
lp15@60394
  1118
      done }
lp15@60394
  1119
  then have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
lp15@60394
  1120
    by (meson Diff_subset division_of_subset)
wenzelm@60585
  1121
  then have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
wenzelm@50945
  1122
    apply -
lp15@60394
  1123
    apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
lp15@60394
  1124
    apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
wenzelm@50945
  1125
    done
wenzelm@53408
  1126
  then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
lp15@60394
  1127
  have "d \<union> p division_of cbox a b"
wenzelm@50945
  1128
  proof -
lp15@60394
  1129
    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
  1130
    have cbox_eq: "cbox a b = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
lp15@60394
  1131
    proof (rule te[OF False], clarify)
wenzelm@50945
  1132
      fix i
wenzelm@53408
  1133
      assume i: "i \<in> p"
immler@56188
  1134
      show "\<Union>(q i - {i}) \<union> i = cbox a b"
wenzelm@50945
  1135
        using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
wenzelm@50945
  1136
    qed
lp15@60428
  1137
    { fix k
wenzelm@53408
  1138
      assume k: "k \<in> p"
lp15@60428
  1139
      have *: "\<And>u t s. t \<inter> s = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<inter> t = {}"
wenzelm@53408
  1140
        by auto
lp15@60428
  1141
      have "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<inter> interior k = {}"
lp15@60428
  1142
      proof (rule *[OF inter_interior_unions_intervals])
wenzelm@50945
  1143
        note qk=division_ofD[OF q(1)[OF k]]
immler@56188
  1144
        show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b"
wenzelm@53408
  1145
          using qk by auto
wenzelm@50945
  1146
        show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
wenzelm@50945
  1147
          using qk(5) using q(2)[OF k] by auto
lp15@60428
  1148
        show "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<subseteq> interior (\<Union>(q k - {k}))"
lp15@60428
  1149
          apply (rule interior_mono)+
wenzelm@53408
  1150
          using k
wenzelm@53408
  1151
          apply auto
wenzelm@53408
  1152
          done
lp15@60428
  1153
      qed } note [simp] = this
lp15@60428
  1154
    show "d \<union> p division_of (cbox a b)"
lp15@60428
  1155
      unfolding cbox_eq
lp15@60428
  1156
      apply (rule division_disjoint_union[OF d assms(1)])
lp15@60428
  1157
      apply (rule inter_interior_unions_intervals)
lp15@60428
  1158
      apply (rule p open_interior ballI)+
lp15@60615
  1159
      apply simp_all
lp15@60428
  1160
      done
lp15@60394
  1161
  qed
lp15@60394
  1162
  then show ?thesis
lp15@60394
  1163
    by (meson Un_upper2 that)
wenzelm@50945
  1164
qed
himmelma@35172
  1165
wenzelm@53399
  1166
lemma elementary_bounded[dest]:
immler@56188
  1167
  fixes s :: "'a::euclidean_space set"
wenzelm@53408
  1168
  shows "p division_of s \<Longrightarrow> bounded s"
immler@56189
  1169
  unfolding division_of_def by (metis bounded_Union bounded_cbox)
wenzelm@53399
  1170
immler@56188
  1171
lemma elementary_subset_cbox:
immler@56188
  1172
  "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> cbox a (b::'a::euclidean_space)"
immler@56188
  1173
  by (meson elementary_bounded bounded_subset_cbox)
wenzelm@50945
  1174
wenzelm@50945
  1175
lemma division_union_intervals_exists:
immler@56188
  1176
  fixes a b :: "'a::euclidean_space"
immler@56188
  1177
  assumes "cbox a b \<noteq> {}"
immler@56188
  1178
  obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
immler@56188
  1179
proof (cases "cbox c d = {}")
wenzelm@50945
  1180
  case True
wenzelm@50945
  1181
  show ?thesis
wenzelm@50945
  1182
    apply (rule that[of "{}"])
wenzelm@50945
  1183
    unfolding True
wenzelm@50945
  1184
    using assms
wenzelm@50945
  1185
    apply auto
wenzelm@50945
  1186
    done
wenzelm@50945
  1187
next
wenzelm@50945
  1188
  case False
wenzelm@50945
  1189
  show ?thesis
immler@56188
  1190
  proof (cases "cbox a b \<inter> cbox c d = {}")
wenzelm@50945
  1191
    case True
wenzelm@50945
  1192
    show ?thesis
immler@56188
  1193
      apply (rule that[of "{cbox c d}"])
lp15@60428
  1194
      apply (subst insert_is_Un)
wenzelm@50945
  1195
      apply (rule division_disjoint_union)
lp15@60428
  1196
      using \<open>cbox c d \<noteq> {}\<close> True assms interior_subset
wenzelm@50945
  1197
      apply auto
wenzelm@50945
  1198
      done
wenzelm@50945
  1199
  next
wenzelm@50945
  1200
    case False
immler@56188
  1201
    obtain u v where uv: "cbox a b \<inter> cbox c d = cbox u v"
wenzelm@50945
  1202
      unfolding inter_interval by auto
lp15@60428
  1203
    have uv_sub: "cbox u v \<subseteq> cbox c d" using uv by auto
immler@56188
  1204
    obtain p where "p division_of cbox c d" "cbox u v \<in> p"
lp15@60428
  1205
      by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
wenzelm@53408
  1206
    note p = this division_ofD[OF this(1)]
lp15@60428
  1207
    have "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = interior(cbox u v \<inter> \<Union>(p - {cbox u v}))"
lp15@60428
  1208
      apply (rule arg_cong[of _ _ interior])
lp15@60428
  1209
      using p(8) uv by auto
lp15@60428
  1210
    also have "\<dots> = {}"
lp15@60428
  1211
      unfolding interior_inter
lp15@60428
  1212
      apply (rule inter_interior_unions_intervals)
lp15@60428
  1213
      using p(6) p(7)[OF p(2)] p(3)
lp15@60428
  1214
      apply auto
lp15@60428
  1215
      done
lp15@60428
  1216
    finally have [simp]: "interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}" by simp
lp15@60615
  1217
    have cbe: "cbox a b \<union> cbox c d = cbox a b \<union> \<Union>(p - {cbox u v})"
wenzelm@53399
  1218
      using p(8) unfolding uv[symmetric] by auto
wenzelm@50945
  1219
    show ?thesis
immler@56188
  1220
      apply (rule that[of "p - {cbox u v}"])
lp15@60428
  1221
      apply (simp add: cbe)
lp15@60428
  1222
      apply (subst insert_is_Un)
wenzelm@50945
  1223
      apply (rule division_disjoint_union)
lp15@60428
  1224
      apply (simp_all add: assms division_of_self)
lp15@60428
  1225
      by (metis Diff_subset division_of_subset p(1) p(8))
wenzelm@50945
  1226
  qed
wenzelm@50945
  1227
qed
himmelma@35172
  1228
wenzelm@53399
  1229
lemma division_of_unions:
wenzelm@53399
  1230
  assumes "finite f"
wenzelm@53408
  1231
    and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@53399
  1232
    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
  1233
  shows "\<Union>f division_of \<Union>\<Union>f"
lp15@60384
  1234
  using assms
lp15@60384
  1235
  by (auto intro!: division_ofI)
wenzelm@53399
  1236
wenzelm@53399
  1237
lemma elementary_union_interval:
immler@56188
  1238
  fixes a b :: "'a::euclidean_space"
wenzelm@53399
  1239
  assumes "p division_of \<Union>p"
immler@56188
  1240
  obtains q where "q division_of (cbox a b \<union> \<Union>p)"
wenzelm@53399
  1241
proof -
wenzelm@53399
  1242
  note assm = division_ofD[OF assms]
wenzelm@53408
  1243
  have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
wenzelm@53399
  1244
    by auto
wenzelm@53399
  1245
  have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
wenzelm@53399
  1246
    by auto
wenzelm@53399
  1247
  {
wenzelm@53399
  1248
    presume "p = {} \<Longrightarrow> thesis"
immler@56188
  1249
      "cbox a b = {} \<Longrightarrow> thesis"
immler@56188
  1250
      "cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis"
immler@56188
  1251
      "p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis"
wenzelm@53399
  1252
    then show thesis by auto
wenzelm@53399
  1253
  next
wenzelm@53399
  1254
    assume as: "p = {}"
immler@56188
  1255
    obtain p where "p division_of (cbox a b)"
wenzelm@53408
  1256
      by (rule elementary_interval)
wenzelm@53399
  1257
    then show thesis
lp15@60384
  1258
      using as that by auto
wenzelm@53399
  1259
  next
immler@56188
  1260
    assume as: "cbox a b = {}"
wenzelm@53399
  1261
    show thesis
lp15@60384
  1262
      using as assms that by auto
wenzelm@53399
  1263
  next
immler@56188
  1264
    assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}"
wenzelm@53399
  1265
    show thesis
immler@56188
  1266
      apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
wenzelm@53399
  1267
      unfolding finite_insert
wenzelm@53399
  1268
      apply (rule assm(1)) unfolding Union_insert
wenzelm@53399
  1269
      using assm(2-4) as
wenzelm@53399
  1270
      apply -
immler@54775
  1271
      apply (fast dest: assm(5))+
wenzelm@53399
  1272
      done
wenzelm@53399
  1273
  next
immler@56188
  1274
    assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}"
immler@56188
  1275
    have "\<forall>k\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
lp15@60615
  1276
    proof
wenzelm@53399
  1277
      case goal1
immler@56188
  1278
      from assm(4)[OF this] obtain c d where "k = cbox c d" by blast
wenzelm@53399
  1279
      then show ?case
lp15@60384
  1280
        by (meson as(3) division_union_intervals_exists)
wenzelm@53399
  1281
    qed
immler@56188
  1282
    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
  1283
    note q = division_ofD[OF this[rule_format]]
immler@56188
  1284
    let ?D = "\<Union>{insert (cbox a b) (q k) | k. k \<in> p}"
lp15@60615
  1285
    show thesis
lp15@60428
  1286
    proof (rule that[OF division_ofI])
immler@56188
  1287
      have *: "{insert (cbox a b) (q k) |k. k \<in> p} = (\<lambda>k. insert (cbox a b) (q k)) ` p"
wenzelm@53399
  1288
        by auto
wenzelm@53399
  1289
      show "finite ?D"
lp15@60384
  1290
        using "*" assm(1) q(1) by auto
immler@56188
  1291
      show "\<Union>?D = cbox a b \<union> \<Union>p"
wenzelm@53399
  1292
        unfolding * lem1
immler@56188
  1293
        unfolding lem2[OF as(1), of "cbox a b", symmetric]
wenzelm@53399
  1294
        using q(6)
wenzelm@53399
  1295
        by auto
wenzelm@53399
  1296
      fix k
wenzelm@53408
  1297
      assume k: "k \<in> ?D"
immler@56188
  1298
      then show "k \<subseteq> cbox a b \<union> \<Union>p"
wenzelm@53408
  1299
        using q(2) by auto
wenzelm@53399
  1300
      show "k \<noteq> {}"
wenzelm@53408
  1301
        using q(3) k by auto
immler@56188
  1302
      show "\<exists>a b. k = cbox a b"
wenzelm@53408
  1303
        using q(4) k by auto
wenzelm@53399
  1304
      fix k'
wenzelm@53408
  1305
      assume k': "k' \<in> ?D" "k \<noteq> k'"
immler@56188
  1306
      obtain x where x: "k \<in> insert (cbox a b) (q x)" "x\<in>p"
wenzelm@53408
  1307
        using k by auto
immler@56188
  1308
      obtain x' where x': "k'\<in>insert (cbox a b) (q x')" "x'\<in>p"
wenzelm@53399
  1309
        using k' by auto
wenzelm@53399
  1310
      show "interior k \<inter> interior k' = {}"
wenzelm@53399
  1311
      proof (cases "x = x'")
wenzelm@53399
  1312
        case True
wenzelm@53399
  1313
        show ?thesis
lp15@60384
  1314
          using True k' q(5) x' x by auto
wenzelm@53399
  1315
      next
wenzelm@53399
  1316
        case False
wenzelm@53399
  1317
        {
immler@56188
  1318
          presume "k = cbox a b \<Longrightarrow> ?thesis"
immler@56188
  1319
            and "k' = cbox a b \<Longrightarrow> ?thesis"
immler@56188
  1320
            and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis"
wenzelm@53399
  1321
          then show ?thesis by auto
wenzelm@53399
  1322
        next
immler@56188
  1323
          assume as': "k  = cbox a b"
wenzelm@53399
  1324
          show ?thesis
lp15@60384
  1325
            using as' k' q(5) x' 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'(2) q(5) x by auto
wenzelm@53399
  1330
        }
immler@56188
  1331
        assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b"
immler@56188
  1332
        obtain c d where k: "k = cbox c d"
wenzelm@53408
  1333
          using q(4)[OF x(2,1)] by blast
immler@56188
  1334
        have "interior k \<inter> interior (cbox a b) = {}"
lp15@60384
  1335
          using as' k'(2) q(5) x by auto
wenzelm@53399
  1336
        then have "interior k \<subseteq> interior x"
lp15@60384
  1337
        using interior_subset_union_intervals
lp15@60384
  1338
          by (metis as(2) k q(2) x interior_subset_union_intervals)
wenzelm@53399
  1339
        moreover
immler@56188
  1340
        obtain c d where c_d: "k' = cbox c d"
wenzelm@53408
  1341
          using q(4)[OF x'(2,1)] by blast
immler@56188
  1342
        have "interior k' \<inter> interior (cbox a b) = {}"
lp15@60384
  1343
          using as'(2) q(5) x' by auto
wenzelm@53399
  1344
        then have "interior k' \<subseteq> interior x'"
lp15@60384
  1345
          by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
wenzelm@53399
  1346
        ultimately show ?thesis
wenzelm@53399
  1347
          using assm(5)[OF x(2) x'(2) False] by auto
wenzelm@53399
  1348
      qed
wenzelm@53399
  1349
    qed
wenzelm@53399
  1350
  }
wenzelm@53399
  1351
qed
himmelma@35172
  1352
himmelma@35172
  1353
lemma elementary_unions_intervals:
wenzelm@53399
  1354
  assumes fin: "finite f"
immler@56188
  1355
    and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a (b::'a::euclidean_space)"
wenzelm@53399
  1356
  obtains p where "p division_of (\<Union>f)"
wenzelm@53399
  1357
proof -
wenzelm@53399
  1358
  have "\<exists>p. p division_of (\<Union>f)"
wenzelm@53399
  1359
  proof (induct_tac f rule:finite_subset_induct)
himmelma@35172
  1360
    show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
wenzelm@53399
  1361
  next
wenzelm@53399
  1362
    fix x F
wenzelm@53399
  1363
    assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
wenzelm@53408
  1364
    from this(3) obtain p where p: "p division_of \<Union>F" ..
immler@56188
  1365
    from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
wenzelm@53399
  1366
    have *: "\<Union>F = \<Union>p"
wenzelm@53399
  1367
      using division_ofD[OF p] by auto
wenzelm@53399
  1368
    show "\<exists>p. p division_of \<Union>insert x F"
wenzelm@53399
  1369
      using elementary_union_interval[OF p[unfolded *], of a b]
lp15@59765
  1370
      unfolding Union_insert x * by metis
wenzelm@53408
  1371
  qed (insert assms, auto)
wenzelm@53399
  1372
  then show ?thesis
lp15@60384
  1373
    using that by auto
wenzelm@53399
  1374
qed
wenzelm@53399
  1375
wenzelm@53399
  1376
lemma elementary_union:
immler@56188
  1377
  fixes s t :: "'a::euclidean_space set"
lp15@60384
  1378
  assumes "ps division_of s" "pt division_of t"
himmelma@35172
  1379
  obtains p where "p division_of (s \<union> t)"
wenzelm@53399
  1380
proof -
lp15@60384
  1381
  have *: "s \<union> t = \<Union>ps \<union> \<Union>pt"
wenzelm@53399
  1382
    using assms unfolding division_of_def by auto
wenzelm@53399
  1383
  show ?thesis
wenzelm@53408
  1384
    apply (rule elementary_unions_intervals[of "ps \<union> pt"])
lp15@60384
  1385
    using assms apply auto
lp15@60384
  1386
    by (simp add: * that)
wenzelm@53399
  1387
qed
wenzelm@53399
  1388
wenzelm@53399
  1389
lemma partial_division_extend:
immler@56188
  1390
  fixes t :: "'a::euclidean_space set"
wenzelm@53399
  1391
  assumes "p division_of s"
wenzelm@53399
  1392
    and "q division_of t"
wenzelm@53399
  1393
    and "s \<subseteq> t"
wenzelm@53399
  1394
  obtains r where "p \<subseteq> r" and "r division_of t"
wenzelm@53399
  1395
proof -
himmelma@35172
  1396
  note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
immler@56188
  1397
  obtain a b where ab: "t \<subseteq> cbox a b"
immler@56188
  1398
    using elementary_subset_cbox[OF assms(2)] by auto
immler@56188
  1399
  obtain r1 where "p \<subseteq> r1" "r1 division_of (cbox a b)"
lp15@60384
  1400
    using assms
lp15@60384
  1401
    by (metis ab dual_order.trans partial_division_extend_interval divp(6))
wenzelm@53399
  1402
  note r1 = this division_ofD[OF this(2)]
wenzelm@53408
  1403
  obtain p' where "p' division_of \<Union>(r1 - p)"
wenzelm@53399
  1404
    apply (rule elementary_unions_intervals[of "r1 - p"])
wenzelm@53399
  1405
    using r1(3,6)
wenzelm@53399
  1406
    apply auto
wenzelm@53399
  1407
    done
wenzelm@53399
  1408
  then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
lp15@60384
  1409
    by (metis assms(2) divq(6) elementary_inter)
wenzelm@53399
  1410
  {
wenzelm@53399
  1411
    fix x
wenzelm@53399
  1412
    assume x: "x \<in> t" "x \<notin> s"
wenzelm@53399
  1413
    then have "x\<in>\<Union>r1"
wenzelm@53399
  1414
      unfolding r1 using ab by auto
wenzelm@53408
  1415
    then obtain r where r: "r \<in> r1" "x \<in> r"
wenzelm@53408
  1416
      unfolding Union_iff ..
wenzelm@53399
  1417
    moreover
wenzelm@53399
  1418
    have "r \<notin> p"
wenzelm@53399
  1419
    proof
wenzelm@53399
  1420
      assume "r \<in> p"
wenzelm@53399
  1421
      then have "x \<in> s" using divp(2) r by auto
wenzelm@53399
  1422
      then show False using x by auto
wenzelm@53399
  1423
    qed
wenzelm@53399
  1424
    ultimately have "x\<in>\<Union>(r1 - p)" by auto
wenzelm@53399
  1425
  }
wenzelm@53399
  1426
  then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
wenzelm@53399
  1427
    unfolding divp divq using assms(3) by auto
wenzelm@53399
  1428
  show ?thesis
wenzelm@53399
  1429
    apply (rule that[of "p \<union> r2"])
wenzelm@53399
  1430
    unfolding *
wenzelm@53399
  1431
    defer
wenzelm@53399
  1432
    apply (rule division_disjoint_union)
wenzelm@53399
  1433
    unfolding divp(6)
wenzelm@53399
  1434
    apply(rule assms r2)+
wenzelm@53399
  1435
  proof -
wenzelm@53399
  1436
    have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
wenzelm@53399
  1437
    proof (rule inter_interior_unions_intervals)
immler@56188
  1438
      show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b"
wenzelm@53399
  1439
        using r1 by auto
wenzelm@53399
  1440
      have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
wenzelm@53399
  1441
        by auto
wenzelm@53399
  1442
      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
wenzelm@53399
  1443
      proof
wenzelm@53399
  1444
        fix m x
wenzelm@53399
  1445
        assume as: "m \<in> r1 - p"
wenzelm@53399
  1446
        have "interior m \<inter> interior (\<Union>p) = {}"
wenzelm@53399
  1447
        proof (rule inter_interior_unions_intervals)
immler@56188
  1448
          show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b"
wenzelm@53399
  1449
            using divp by auto
wenzelm@53399
  1450
          show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
lp15@60384
  1451
            by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
wenzelm@53399
  1452
        qed
wenzelm@53399
  1453
        then show "interior s \<inter> interior m = {}"
wenzelm@53399
  1454
          unfolding divp by auto
wenzelm@53399
  1455
      qed
wenzelm@53399
  1456
    qed
wenzelm@53399
  1457
    then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
wenzelm@53399
  1458
      using interior_subset by auto
wenzelm@53399
  1459
  qed auto
wenzelm@53399
  1460
qed
wenzelm@53399
  1461
himmelma@35172
  1462
wenzelm@60420
  1463
subsection \<open>Tagged (partial) divisions.\<close>
himmelma@35172
  1464
wenzelm@53408
  1465
definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
wenzelm@53408
  1466
  where "s tagged_partial_division_of i \<longleftrightarrow>
wenzelm@53408
  1467
    finite s \<and>
immler@56188
  1468
    (\<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
  1469
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1470
      interior k1 \<inter> interior k2 = {})"
wenzelm@53408
  1471
wenzelm@53408
  1472
lemma tagged_partial_division_ofD[dest]:
wenzelm@53408
  1473
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1474
  shows "finite s"
wenzelm@53408
  1475
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1476
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1477
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1478
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
wenzelm@53408
  1479
      (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1480
  using assms unfolding tagged_partial_division_of_def by blast+
wenzelm@53408
  1481
wenzelm@53408
  1482
definition tagged_division_of (infixr "tagged'_division'_of" 40)
wenzelm@53408
  1483
  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
  1484
huffman@44167
  1485
lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
himmelma@35172
  1486
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1487
himmelma@35172
  1488
lemma tagged_division_of:
wenzelm@53408
  1489
  "s tagged_division_of i \<longleftrightarrow>
wenzelm@53408
  1490
    finite s \<and>
immler@56188
  1491
    (\<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
  1492
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1493
      interior k1 \<inter> interior k2 = {}) \<and>
wenzelm@53408
  1494
    (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1495
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1496
wenzelm@53408
  1497
lemma tagged_division_ofI:
wenzelm@53408
  1498
  assumes "finite s"
wenzelm@53408
  1499
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1500
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1501
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1502
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1503
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1504
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1505
  shows "s tagged_division_of i"
wenzelm@53408
  1506
  unfolding tagged_division_of
lp15@60384
  1507
  using assms
lp15@60384
  1508
  apply auto
lp15@60384
  1509
  apply fastforce+
wenzelm@53408
  1510
  done
wenzelm@53408
  1511
lp15@60384
  1512
lemma tagged_division_ofD[dest]:  (*FIXME USE A LOCALE*)
wenzelm@53408
  1513
  assumes "s tagged_division_of i"
wenzelm@53408
  1514
  shows "finite s"
wenzelm@53408
  1515
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1516
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1517
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1518
    and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1519
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1520
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
wenzelm@53408
  1521
  using assms unfolding tagged_division_of by blast+
wenzelm@53408
  1522
wenzelm@53408
  1523
lemma division_of_tagged_division:
wenzelm@53408
  1524
  assumes "s tagged_division_of i"
wenzelm@53408
  1525
  shows "(snd ` s) division_of i"
wenzelm@53408
  1526
proof (rule division_ofI)
wenzelm@53408
  1527
  note assm = tagged_division_ofD[OF assms]
wenzelm@53408
  1528
  show "\<Union>(snd ` s) = i" "finite (snd ` s)"
wenzelm@53408
  1529
    using assm by auto
wenzelm@53408
  1530
  fix k
wenzelm@53408
  1531
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1532
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1533
    by auto
immler@56188
  1534
  then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = cbox a b"
wenzelm@53408
  1535
    using assm by fastforce+
wenzelm@53408
  1536
  fix k'
wenzelm@53408
  1537
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1538
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1539
    by auto
wenzelm@53408
  1540
  then show "interior k \<inter> interior k' = {}"
lp15@60384
  1541
    using assm(5) k'(2) xk by blast
himmelma@35172
  1542
qed
himmelma@35172
  1543
wenzelm@53408
  1544
lemma partial_division_of_tagged_division:
wenzelm@53408
  1545
  assumes "s tagged_partial_division_of i"
himmelma@35172
  1546
  shows "(snd ` s) division_of \<Union>(snd ` s)"
wenzelm@53408
  1547
proof (rule division_ofI)
wenzelm@53408
  1548
  note assm = tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1549
  show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
wenzelm@53408
  1550
    using assm by auto
wenzelm@53408
  1551
  fix k
wenzelm@53408
  1552
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1553
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1554
    by auto
immler@56188
  1555
  then show "k \<noteq> {}" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>(snd ` s)"
wenzelm@53408
  1556
    using assm by auto
wenzelm@53408
  1557
  fix k'
wenzelm@53408
  1558
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1559
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1560
    by auto
wenzelm@53408
  1561
  then show "interior k \<inter> interior k' = {}"
lp15@60384
  1562
    using assm(5) k'(2) xk by auto
himmelma@35172
  1563
qed
himmelma@35172
  1564
wenzelm@53408
  1565
lemma tagged_partial_division_subset:
wenzelm@53408
  1566
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1567
    and "t \<subseteq> s"
himmelma@35172
  1568
  shows "t tagged_partial_division_of i"
wenzelm@53408
  1569
  using assms
wenzelm@53408
  1570
  unfolding tagged_partial_division_of_def
wenzelm@53408
  1571
  using finite_subset[OF assms(2)]
wenzelm@53408
  1572
  by blast
wenzelm@53408
  1573
wenzelm@53408
  1574
lemma setsum_over_tagged_division_lemma:
wenzelm@53408
  1575
  assumes "p tagged_division_of i"
immler@56188
  1576
    and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> content (cbox u v) = 0 \<Longrightarrow> d (cbox u v) = 0"
himmelma@35172
  1577
  shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
wenzelm@53408
  1578
proof -
wenzelm@53408
  1579
  have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
wenzelm@53408
  1580
    unfolding o_def by (rule ext) auto
hoelzl@57129
  1581
  note assm = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1582
  show ?thesis
wenzelm@53408
  1583
    unfolding *
haftmann@57418
  1584
  proof (rule setsum.reindex_nontrivial[symmetric])
wenzelm@53408
  1585
    show "finite p"
wenzelm@53408
  1586
      using assm by auto
wenzelm@53408
  1587
    fix x y
hoelzl@57129
  1588
    assume "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
immler@56188
  1589
    obtain a b where ab: "snd x = cbox a b"
wenzelm@60420
  1590
      using assm(4)[of "fst x" "snd x"] \<open>x\<in>p\<close> by auto
wenzelm@53408
  1591
    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
wenzelm@60420
  1592
      by (metis pair_collapse \<open>x\<in>p\<close> \<open>snd x = snd y\<close> \<open>x \<noteq> y\<close>)+
wenzelm@60420
  1593
    with \<open>x\<in>p\<close> \<open>y\<in>p\<close> have "interior (snd x) \<inter> interior (snd y) = {}"
hoelzl@57129
  1594
      by (intro assm(5)[of "fst x" _ "fst y"]) auto
immler@56188
  1595
    then have "content (cbox a b) = 0"
wenzelm@60420
  1596
      unfolding \<open>snd x = snd y\<close>[symmetric] ab content_eq_0_interior by auto
immler@56188
  1597
    then have "d (cbox a b) = 0"
wenzelm@60420
  1598
      using assm(2)[of "fst x" "snd x"] \<open>x\<in>p\<close> ab[symmetric] by (intro assms(2)) auto
wenzelm@53408
  1599
    then show "d (snd x) = 0"
wenzelm@53408
  1600
      unfolding ab by auto
wenzelm@53408
  1601
  qed
wenzelm@53408
  1602
qed
wenzelm@53408
  1603
wenzelm@53408
  1604
lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
wenzelm@53408
  1605
  by auto
himmelma@35172
  1606
himmelma@35172
  1607
lemma tagged_division_of_empty: "{} tagged_division_of {}"
himmelma@35172
  1608
  unfolding tagged_division_of by auto
himmelma@35172
  1609
wenzelm@53408
  1610
lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1611
  unfolding tagged_partial_division_of_def by auto
himmelma@35172
  1612
wenzelm@53408
  1613
lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1614
  unfolding tagged_division_of by auto
himmelma@35172
  1615
immler@56188
  1616
lemma tagged_division_of_self: "x \<in> cbox a b \<Longrightarrow> {(x,cbox a b)} tagged_division_of (cbox a b)"
wenzelm@53408
  1617
  by (rule tagged_division_ofI) auto
himmelma@35172
  1618
immler@56188
  1619
lemma tagged_division_of_self_real: "x \<in> {a .. b::real} \<Longrightarrow> {(x,{a .. b})} tagged_division_of {a .. b}"
immler@56188
  1620
  unfolding box_real[symmetric]
immler@56188
  1621
  by (rule tagged_division_of_self)
immler@56188
  1622
himmelma@35172
  1623
lemma tagged_division_union:
wenzelm@53408
  1624
  assumes "p1 tagged_division_of s1"
wenzelm@53408
  1625
    and "p2 tagged_division_of s2"
wenzelm@53408
  1626
    and "interior s1 \<inter> interior s2 = {}"
himmelma@35172
  1627
  shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
wenzelm@53408
  1628
proof (rule tagged_division_ofI)
wenzelm@53408
  1629
  note p1 = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1630
  note p2 = tagged_division_ofD[OF assms(2)]
wenzelm@53408
  1631
  show "finite (p1 \<union> p2)"
wenzelm@53408
  1632
    using p1(1) p2(1) by auto
wenzelm@53408
  1633
  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
wenzelm@53408
  1634
    using p1(6) p2(6) by blast
wenzelm@53408
  1635
  fix x k
wenzelm@53408
  1636
  assume xk: "(x, k) \<in> p1 \<union> p2"
immler@56188
  1637
  show "x \<in> k" "\<exists>a b. k = cbox a b"
wenzelm@53408
  1638
    using xk p1(2,4) p2(2,4) by auto
wenzelm@53408
  1639
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
  1640
    using xk p1(3) p2(3) by blast
wenzelm@53408
  1641
  fix x' k'
wenzelm@53408
  1642
  assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1643
  have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1644
    using assms(3) interior_mono by blast
wenzelm@53408
  1645
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1646
    apply (cases "(x, k) \<in> p1")
lp15@60384
  1647
    apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2))
lp15@60384
  1648
    by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
wenzelm@53408
  1649
qed
himmelma@35172
  1650
himmelma@35172
  1651
lemma tagged_division_unions:
wenzelm@53408
  1652
  assumes "finite iset"
wenzelm@53408
  1653
    and "\<forall>i\<in>iset. pfn i tagged_division_of i"
wenzelm@53408
  1654
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
himmelma@35172
  1655
  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
wenzelm@53408
  1656
proof (rule tagged_division_ofI)
himmelma@35172
  1657
  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
wenzelm@53408
  1658
  show "finite (\<Union>(pfn ` iset))"
wenzelm@53408
  1659
    apply (rule finite_Union)
wenzelm@53408
  1660
    using assms
wenzelm@53408
  1661
    apply auto
wenzelm@53408
  1662
    done
wenzelm@53408
  1663
  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
  1664
    by blast
wenzelm@53408
  1665
  also have "\<dots> = \<Union>iset"
wenzelm@53408
  1666
    using assm(6) by auto
wenzelm@53399
  1667
  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
wenzelm@53408
  1668
  fix x k
wenzelm@53408
  1669
  assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
wenzelm@53408
  1670
  then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
wenzelm@53408
  1671
    by auto
immler@56188
  1672
  show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
wenzelm@53408
  1673
    using assm(2-4)[OF i] using i(1) by auto
wenzelm@53408
  1674
  fix x' k'
wenzelm@53408
  1675
  assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1676
  then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
wenzelm@53408
  1677
    by auto
wenzelm@53408
  1678
  have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1679
    using i(1) i'(1)
wenzelm@53408
  1680
    using assms(3)[rule_format] interior_mono
wenzelm@53408
  1681
    by blast
wenzelm@53408
  1682
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1683
    apply (cases "i = i'")
lp15@60384
  1684
    using assm(5) i' i(2) xk'(2) apply blast
lp15@60384
  1685
    using "*" assm(3) i' i by auto
himmelma@35172
  1686
qed
himmelma@35172
  1687
himmelma@35172
  1688
lemma tagged_partial_division_of_union_self:
wenzelm@53408
  1689
  assumes "p tagged_partial_division_of s"
himmelma@35172
  1690
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1691
  apply (rule tagged_division_ofI)
wenzelm@53408
  1692
  using tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1693
  apply auto
wenzelm@53408
  1694
  done
wenzelm@53408
  1695
wenzelm@53408
  1696
lemma tagged_division_of_union_self:
wenzelm@53408
  1697
  assumes "p tagged_division_of s"
wenzelm@53408
  1698
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1699
  apply (rule tagged_division_ofI)
wenzelm@53408
  1700
  using tagged_division_ofD[OF assms]
wenzelm@53408
  1701
  apply auto
wenzelm@53408
  1702
  done
wenzelm@53408
  1703
himmelma@35172
  1704
wenzelm@60420
  1705
subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close>
himmelma@35172
  1706
wenzelm@53408
  1707
definition fine  (infixr "fine" 46)
wenzelm@53408
  1708
  where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
wenzelm@53408
  1709
wenzelm@53408
  1710
lemma fineI:
wenzelm@53408
  1711
  assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1712
  shows "d fine s"
wenzelm@53408
  1713
  using assms unfolding fine_def by auto
wenzelm@53408
  1714
wenzelm@53408
  1715
lemma fineD[dest]:
wenzelm@53408
  1716
  assumes "d fine s"
wenzelm@53408
  1717
  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1718
  using assms unfolding fine_def by auto
himmelma@35172
  1719
himmelma@35172
  1720
lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
himmelma@35172
  1721
  unfolding fine_def by auto
himmelma@35172
  1722
himmelma@35172
  1723
lemma fine_inters:
wenzelm@60585
  1724
 "(\<lambda>x. \<Inter>{f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
himmelma@35172
  1725
  unfolding fine_def by blast
himmelma@35172
  1726
wenzelm@53408
  1727
lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
himmelma@35172
  1728
  unfolding fine_def by blast
himmelma@35172
  1729
wenzelm@53408
  1730
lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
himmelma@35172
  1731
  unfolding fine_def by auto
himmelma@35172
  1732
wenzelm@53408
  1733
lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
himmelma@35172
  1734
  unfolding fine_def by blast
himmelma@35172
  1735
wenzelm@53408
  1736
wenzelm@60420
  1737
subsection \<open>Gauge integral. Define on compact intervals first, then use a limit.\<close>
himmelma@35172
  1738
wenzelm@53408
  1739
definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
wenzelm@53408
  1740
  where "(f has_integral_compact_interval y) i \<longleftrightarrow>
wenzelm@53408
  1741
    (\<forall>e>0. \<exists>d. gauge d \<and>
wenzelm@53408
  1742
      (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
wenzelm@53408
  1743
        norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1744
wenzelm@53408
  1745
definition has_integral ::
immler@56188
  1746
    "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
wenzelm@53408
  1747
  (infixr "has'_integral" 46)
wenzelm@53408
  1748
  where "(f has_integral y) i \<longleftrightarrow>
immler@56188
  1749
    (if \<exists>a b. i = cbox a b
wenzelm@53408
  1750
     then (f has_integral_compact_interval y) i
immler@56188
  1751
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1752
      (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) (cbox a b) \<and>
wenzelm@53408
  1753
        norm (z - y) < e)))"
himmelma@35172
  1754
himmelma@35172
  1755
lemma has_integral:
immler@56188
  1756
  "(f has_integral y) (cbox a b) \<longleftrightarrow>
wenzelm@53408
  1757
    (\<forall>e>0. \<exists>d. gauge d \<and>
immler@56188
  1758
      (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
wenzelm@53408
  1759
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1760
  unfolding has_integral_def has_integral_compact_interval_def
wenzelm@53408
  1761
  by auto
wenzelm@53408
  1762
immler@56188
  1763
lemma has_integral_real:
immler@56188
  1764
  "(f has_integral y) {a .. b::real} \<longleftrightarrow>
immler@56188
  1765
    (\<forall>e>0. \<exists>d. gauge d \<and>
immler@56188
  1766
      (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
immler@56188
  1767
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
immler@56188
  1768
  unfolding box_real[symmetric]
immler@56188
  1769
  by (rule has_integral)
immler@56188
  1770
wenzelm@53408
  1771
lemma has_integralD[dest]:
immler@56188
  1772
  assumes "(f has_integral y) (cbox a b)"
wenzelm@53408
  1773
    and "e > 0"
wenzelm@53408
  1774
  obtains d where "gauge d"
immler@56188
  1775
    and "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d fine p \<Longrightarrow>
wenzelm@53408
  1776
      norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
himmelma@35172
  1777
  using assms unfolding has_integral by auto
himmelma@35172
  1778
himmelma@35172
  1779
lemma has_integral_alt:
wenzelm@53408
  1780
  "(f has_integral y) i \<longleftrightarrow>
immler@56188
  1781
    (if \<exists>a b. i = cbox a b
wenzelm@53408
  1782
     then (f has_integral y) i
immler@56188
  1783
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1784
      (\<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
  1785
  unfolding has_integral
wenzelm@53408
  1786
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1787
  by auto
himmelma@35172
  1788
himmelma@35172
  1789
lemma has_integral_altD:
wenzelm@53408
  1790
  assumes "(f has_integral y) i"
immler@56188
  1791
    and "\<not> (\<exists>a b. i = cbox a b)"
wenzelm@53408
  1792
    and "e>0"
wenzelm@53408
  1793
  obtains B where "B > 0"
immler@56188
  1794
    and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1795
      (\<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
  1796
  using assms
wenzelm@53408
  1797
  unfolding has_integral
wenzelm@53408
  1798
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1799
  by auto
wenzelm@53408
  1800
wenzelm@53408
  1801
definition integrable_on (infixr "integrable'_on" 46)
wenzelm@53408
  1802
  where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
wenzelm@53408
  1803
wenzelm@53408
  1804
definition "integral i f = (SOME y. (f has_integral y) i)"
himmelma@35172
  1805
wenzelm@53409
  1806
lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
wenzelm@53409
  1807
  unfolding integrable_on_def integral_def by (rule someI_ex)
himmelma@35172
  1808
himmelma@35172
  1809
lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
himmelma@35172
  1810
  unfolding integrable_on_def by auto
himmelma@35172
  1811
wenzelm@53409
  1812
lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
himmelma@35172
  1813
  by auto
himmelma@35172
  1814
himmelma@35172
  1815
lemma setsum_content_null:
immler@56188
  1816
  assumes "content (cbox a b) = 0"
immler@56188
  1817
    and "p tagged_division_of (cbox a b)"
himmelma@35172
  1818
  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
haftmann@57418
  1819
proof (rule setsum.neutral, rule)
wenzelm@53409
  1820
  fix y
wenzelm@53409
  1821
  assume y: "y \<in> p"
wenzelm@53409
  1822
  obtain x k where xk: "y = (x, k)"
wenzelm@53409
  1823
    using surj_pair[of y] by blast
himmelma@35172
  1824
  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
immler@56188
  1825
  from this(2) obtain c d where k: "k = cbox c d" by blast
wenzelm@53409
  1826
  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
wenzelm@53409
  1827
    unfolding xk by auto
wenzelm@53409
  1828
  also have "\<dots> = 0"
wenzelm@53409
  1829
    using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
wenzelm@53409
  1830
    unfolding assms(1) k
wenzelm@53409
  1831
    by auto
himmelma@35172
  1832
  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
himmelma@35172
  1833
qed
himmelma@35172
  1834
wenzelm@53409
  1835
wenzelm@60420
  1836
subsection \<open>Some basic combining lemmas.\<close>
himmelma@35172
  1837
himmelma@35172
  1838
lemma tagged_division_unions_exists:
wenzelm@53409
  1839
  assumes "finite iset"
wenzelm@53409
  1840
    and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
wenzelm@53409
  1841
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
wenzelm@53409
  1842
    and "\<Union>iset = i"
wenzelm@53409
  1843
   obtains p where "p tagged_division_of i" and "d fine p"
wenzelm@53409
  1844
proof -
wenzelm@53409
  1845
  obtain pfn where pfn:
wenzelm@53409
  1846
    "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
wenzelm@53409
  1847
    "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
wenzelm@53409
  1848
    using bchoice[OF assms(2)] by auto
wenzelm@53409
  1849
  show thesis
wenzelm@53409
  1850
    apply (rule_tac p="\<Union>(pfn ` iset)" in that)
lp15@60384
  1851
    using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
lp15@60384
  1852
    by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
himmelma@35172
  1853
qed
himmelma@35172
  1854
wenzelm@53409
  1855
wenzelm@60420
  1856
subsection \<open>The set we're concerned with must be closed.\<close>
himmelma@35172
  1857
wenzelm@53409
  1858
lemma division_of_closed:
immler@56189
  1859
  fixes i :: "'n::euclidean_space set"
wenzelm@53409
  1860
  shows "s division_of i \<Longrightarrow> closed i"
nipkow@44890
  1861
  unfolding division_of_def by fastforce
himmelma@35172
  1862
wenzelm@60420
  1863
subsection \<open>General bisection principle for intervals; might be useful elsewhere.\<close>
himmelma@35172
  1864
wenzelm@53409
  1865
lemma interval_bisection_step:
immler@56188
  1866
  fixes type :: "'a::euclidean_space"
wenzelm@53409
  1867
  assumes "P {}"
wenzelm@53409
  1868
    and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
immler@56188
  1869
    and "\<not> P (cbox a (b::'a))"
immler@56188
  1870
  obtains c d where "\<not> P (cbox c d)"
wenzelm@53409
  1871
    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
  1872
proof -
immler@56188
  1873
  have "cbox a b \<noteq> {}"
immler@54776
  1874
    using assms(1,3) by metis
wenzelm@53409
  1875
  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
immler@56188
  1876
    by (force simp: mem_box)
lp15@60428
  1877
  { fix f
lp15@60428
  1878
    have "\<lbrakk>finite f;
lp15@60428
  1879
           \<And>s. s\<in>f \<Longrightarrow> P s;
lp15@60428
  1880
           \<And>s. s\<in>f \<Longrightarrow> \<exists>a b. s = cbox a b;
lp15@60428
  1881
           \<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
  1882
    proof (induct f rule: finite_induct)
wenzelm@53409
  1883
      case empty
wenzelm@53409
  1884
      show ?case
wenzelm@53409
  1885
        using assms(1) by auto
wenzelm@53409
  1886
    next
wenzelm@53409
  1887
      case (insert x f)
wenzelm@53409
  1888
      show ?case
wenzelm@53409
  1889
        unfolding Union_insert
wenzelm@53409
  1890
        apply (rule assms(2)[rule_format])
lp15@60384
  1891
        using inter_interior_unions_intervals [of f "interior x"]
lp15@60384
  1892
        apply (auto simp: insert)
lp15@60428
  1893
        by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
lp15@60428
  1894
    qed
lp15@60428
  1895
  } note UN_cases = this
immler@56188
  1896
  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
  1897
    (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
hoelzl@50526
  1898
  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
  1899
  {
immler@56188
  1900
    presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False"
wenzelm@53409
  1901
    then show thesis
wenzelm@53409
  1902
      unfolding atomize_not not_all
lp15@60384
  1903
      by (blast intro: that)
wenzelm@53409
  1904
  }
immler@56188
  1905
  assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)"
wenzelm@60585
  1906
  have "P (\<Union>?A)"
lp15@60428
  1907
  proof (rule UN_cases)
immler@56188
  1908
    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
  1909
      (\<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
  1910
    have "?A \<subseteq> ?B"
wenzelm@53409
  1911
    proof
wenzelm@53409
  1912
      case goal1
lp15@60615
  1913
      then obtain c d
lp15@60428
  1914
        where x:  "x = cbox c d"
lp15@60428
  1915
                  "\<And>i. i \<in> Basis \<Longrightarrow>
lp15@60428
  1916
                        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60428
  1917
                        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
wenzelm@53409
  1918
      show "x \<in> ?B"
lp15@60428
  1919
        unfolding image_iff x
wenzelm@53409
  1920
        apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
lp15@60428
  1921
        apply (rule arg_cong2 [where f = cbox])
lp15@60428
  1922
        using x(2) ab
lp15@60428
  1923
        apply (auto simp add: euclidean_eq_iff[where 'a='a])
lp15@60428
  1924
        by fastforce
wenzelm@53409
  1925
    qed
wenzelm@53409
  1926
    then show "finite ?A"
wenzelm@53409
  1927
      by (rule finite_subset) auto
lp15@60428
  1928
  next
wenzelm@53409
  1929
    fix s
wenzelm@53409
  1930
    assume "s \<in> ?A"
lp15@60428
  1931
    then obtain c d
lp15@60428
  1932
      where s: "s = cbox c d"
lp15@60428
  1933
               "\<And>i. i \<in> Basis \<Longrightarrow>
lp15@60428
  1934
                     c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60428
  1935
                     c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
wenzelm@53409
  1936
      by blast
wenzelm@53409
  1937
    show "P s"
wenzelm@53409
  1938
      unfolding s
wenzelm@53409
  1939
      apply (rule as[rule_format])
lp15@60394
  1940
      using ab s(2) by force
immler@56188
  1941
    show "\<exists>a b. s = cbox a b"
wenzelm@53409
  1942
      unfolding s by auto
wenzelm@53409
  1943
    fix t
wenzelm@53409
  1944
    assume "t \<in> ?A"
wenzelm@53409
  1945
    then obtain e f where t:
immler@56188
  1946
      "t = cbox e f"
wenzelm@53409
  1947
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  1948
        e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  1949
        e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
wenzelm@53409
  1950
      by blast
wenzelm@53409
  1951
    assume "s \<noteq> t"
wenzelm@53409
  1952
    then have "\<not> (c = e \<and> d = f)"
wenzelm@53409
  1953
      unfolding s t by auto
wenzelm@53409
  1954
    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
  1955
      unfolding euclidean_eq_iff[where 'a='a] by auto
wenzelm@53409
  1956
    then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
lp15@60394
  1957
      using s(2) t(2) apply fastforce
wenzelm@60420
  1958
      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
  1959
    have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
wenzelm@53409
  1960
      by auto
wenzelm@53409
  1961
    show "interior s \<inter> interior t = {}"
immler@56188
  1962
      unfolding s t interior_cbox
wenzelm@53409
  1963
    proof (rule *)
wenzelm@53409
  1964
      fix x
immler@54775
  1965
      assume "x \<in> box c d" "x \<in> box e f"
wenzelm@53409
  1966
      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
  1967
        unfolding mem_box using i'
lp15@60394
  1968
        by force+
lp15@60394
  1969
      show False  using s(2)[OF i']
lp15@60394
  1970
      proof safe
wenzelm@53409
  1971
        assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
wenzelm@53409
  1972
        show False
wenzelm@53409
  1973
          using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
wenzelm@53409
  1974
      next
wenzelm@53409
  1975
        assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
wenzelm@53409
  1976
        show False
wenzelm@53409
  1977
          using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
wenzelm@53409
  1978
      qed
wenzelm@53409
  1979
    qed
wenzelm@53409
  1980
  qed
wenzelm@60585
  1981
  also have "\<Union>?A = cbox a b"
wenzelm@53409
  1982
  proof (rule set_eqI,rule)
wenzelm@53409
  1983
    fix x
wenzelm@53409
  1984
    assume "x \<in> \<Union>?A"
wenzelm@53409
  1985
    then obtain c d where x:
immler@56188
  1986
      "x \<in> cbox c d"
wenzelm@53409
  1987
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  1988
        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60615
  1989
        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
lp15@60394
  1990
      by blast
immler@56188
  1991
    show "x\<in>cbox a b"
immler@56188
  1992
      unfolding mem_box
wenzelm@53409
  1993
    proof safe
wenzelm@53409
  1994
      fix i :: 'a
wenzelm@53409
  1995
      assume i: "i \<in> Basis"
wenzelm@53409
  1996
      then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
immler@56188
  1997
        using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
wenzelm@53409
  1998
    qed
wenzelm@53409
  1999
  next
wenzelm@53409
  2000
    fix x
immler@56188
  2001
    assume x: "x \<in> cbox a b"
wenzelm@53409
  2002
    have "\<forall>i\<in>Basis.
wenzelm@53409
  2003
      \<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
  2004
      (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
immler@56188
  2005
      unfolding mem_box
hoelzl@50526
  2006
    proof
wenzelm@53409
  2007
      fix i :: 'a
wenzelm@53409
  2008
      assume i: "i \<in> Basis"
hoelzl@50526
  2009
      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
  2010
        using x[unfolded mem_box,THEN bspec, OF i] by auto
wenzelm@53409
  2011
      then show "\<exists>c d. ?P i c d"
wenzelm@53409
  2012
        by blast
hoelzl@50526
  2013
    qed
wenzelm@53409
  2014
    then show "x\<in>\<Union>?A"
hoelzl@50526
  2015
      unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
lp15@60384
  2016
      apply auto
immler@56188
  2017
      apply (rule_tac x="cbox xa xaa" in exI)
immler@56188
  2018
      unfolding mem_box
wenzelm@53409
  2019
      apply auto
wenzelm@53409
  2020
      done
wenzelm@53409
  2021
  qed
wenzelm@53409
  2022
  finally show False
wenzelm@53409
  2023
    using assms by auto
wenzelm@53409
  2024
qed
wenzelm@53409
  2025
wenzelm@53409
  2026
lemma interval_bisection:
immler@56188
  2027
  fixes type :: "'a::euclidean_space"
wenzelm@53409
  2028
  assumes "P {}"
wenzelm@53409
  2029
    and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
immler@56188
  2030
    and "\<not> P (cbox a (b::'a))"
immler@56188
  2031
  obtains x where "x \<in> cbox a b"
immler@56188
  2032
    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
  2033
proof -
immler@56188
  2034
  have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and>
hoelzl@50526
  2035
    (\<forall>i\<in>Basis. fst x\<bullet>i \<le> fst y\<bullet>i \<and> fst y\<bullet>i \<le> snd y\<bullet>i \<and> snd y\<bullet>i \<le> snd x\<bullet>i \<and>
wenzelm@53409
  2036
       2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))"
wenzelm@53409
  2037
  proof
wenzelm@53409
  2038
    case goal1
lp15@60394
  2039
    show ?case
wenzelm@53409
  2040
    proof -
immler@56188
  2041
      presume "\<not> P (cbox (fst x) (snd x)) \<Longrightarrow> ?thesis"
immler@56188
  2042
      then show ?thesis by (cases "P (cbox (fst x) (snd x))") auto
wenzelm@53409
  2043
    next
immler@56188
  2044
      assume as: "\<not> P (cbox (fst x) (snd x))"
immler@56188
  2045
      obtain c d where "\<not> P (cbox c d)"
wenzelm@53409
  2046
        "\<forall>i\<in>Basis.
wenzelm@53409
  2047
           fst x \<bullet> i \<le> c \<bullet> i \<and>
wenzelm@53409
  2048
           c \<bullet> i \<le> d \<bullet> i \<and>
wenzelm@53409
  2049
           d \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@53409
  2050
           2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
wenzelm@53409
  2051
        by (rule interval_bisection_step[of P, OF assms(1-2) as])
wenzelm@53409
  2052
      then show ?thesis
wenzelm@53409
  2053
        apply -
wenzelm@53409
  2054
        apply (rule_tac x="(c,d)" in exI)
wenzelm@53409
  2055
        apply auto
wenzelm@53409
  2056
        done
wenzelm@53409
  2057
    qed
wenzelm@53409
  2058
  qed
wenzelm@55751
  2059
  then obtain f where f:
wenzelm@55751
  2060
    "\<forall>x.
immler@56188
  2061
      \<not> P (cbox (fst x) (snd x)) \<longrightarrow>
immler@56188
  2062
      \<not> P (cbox (fst (f x)) (snd (f x))) \<and>
wenzelm@55751
  2063
        (\<forall>i\<in>Basis.
wenzelm@55751
  2064
            fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and>
wenzelm@55751
  2065
            fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and>
wenzelm@55751
  2066
            snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@55751
  2067
            2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)"
wenzelm@53409
  2068
    apply -
wenzelm@53409
  2069
    apply (drule choice)
wenzelm@55751
  2070
    apply blast
wenzelm@55751
  2071
    done
wenzelm@53409
  2072
  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)"
wenzelm@53409
  2073
  def A \<equiv> "\<lambda>n. fst(AB n)"
wenzelm@53409
  2074
  def B \<equiv> "\<lambda>n. snd(AB n)"
wenzelm@53409
  2075
  note ab_def = A_def B_def AB_def
immler@56188
  2076
  have "A 0 = a" "B 0 = b" "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and>
wenzelm@53399
  2077
    (\<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
  2078
    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
  2079
  proof -
wenzelm@53409
  2080
    show "A 0 = a" "B 0 = b"
wenzelm@53409
  2081
      unfolding ab_def by auto
wenzelm@53409
  2082
    case goal3
wenzelm@53409
  2083
    note S = ab_def funpow.simps o_def id_apply
wenzelm@53409
  2084
    show ?case
wenzelm@53409
  2085
    proof (induct n)
wenzelm@53409
  2086
      case 0
wenzelm@53409
  2087
      then show ?case
wenzelm@53409
  2088
        unfolding S
wenzelm@53409
  2089
        apply (rule f[rule_format]) using assms(3)
wenzelm@53409
  2090
        apply auto
wenzelm@53409
  2091
        done
wenzelm@53409
  2092
    next
wenzelm@53409
  2093
      case (Suc n)
wenzelm@53409
  2094
      show ?case
wenzelm@53409
  2095
        unfolding S
wenzelm@53409
  2096
        apply (rule f[rule_format])
wenzelm@53409
  2097
        using Suc
wenzelm@53409
  2098
        unfolding S
wenzelm@53409
  2099
        apply auto
wenzelm@53409
  2100
        done
wenzelm@53409
  2101
    qed
wenzelm@53409
  2102
  qed
wenzelm@53409
  2103
  note AB = this(1-2) conjunctD2[OF this(3),rule_format]
wenzelm@53409
  2104
immler@56188
  2105
  have interv: "\<And>e. 0 < e \<Longrightarrow> \<exists>n. \<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e"
wenzelm@53409
  2106
  proof -
wenzelm@53409
  2107
    case goal1
wenzelm@53409
  2108
    obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
wenzelm@53409
  2109
      using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] ..
wenzelm@53409
  2110
    show ?case
lp15@60396
  2111
    proof (rule exI [where x=n], clarify)
wenzelm@53409
  2112
      fix x y
immler@56188
  2113
      assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
wenzelm@53409
  2114
      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis"
wenzelm@53409
  2115
        unfolding dist_norm by(rule norm_le_l1)
hoelzl@50526
  2116
      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
wenzelm@53409
  2117
      proof (rule setsum_mono)
wenzelm@53409
  2118
        fix i :: 'a
wenzelm@53409
  2119
        assume i: "i \<in> Basis"
wenzelm@53409
  2120
        show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
immler@56188
  2121
          using xy[unfolded mem_box,THEN bspec, OF i]
wenzelm@53409
  2122
          by (auto simp: inner_diff_left)
wenzelm@53409
  2123
      qed
wenzelm@53409
  2124
      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
wenzelm@53409
  2125
        unfolding setsum_divide_distrib
wenzelm@53409
  2126
      proof (rule setsum_mono)
wenzelm@53409
  2127
        case goal1
wenzelm@53409
  2128
        then show ?case
wenzelm@53409
  2129
        proof (induct n)
wenzelm@53409
  2130
          case 0
wenzelm@53409
  2131
          then show ?case
wenzelm@53409
  2132
            unfolding AB by auto
wenzelm@53409
  2133
        next
wenzelm@53409
  2134
          case (Suc n)
wenzelm@53409
  2135
          have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
hoelzl@37489
  2136
            using AB(4)[of i n] using goal1 by auto
wenzelm@53409
  2137
          also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
wenzelm@53409
  2138
            using Suc by (auto simp add:field_simps)
wenzelm@53409
  2139
          finally show ?case .
wenzelm@53409
  2140
        qed
wenzelm@53409
  2141
      qed
wenzelm@53409
  2142
      also have "\<dots> < e"
wenzelm@53409
  2143
        using n using goal1 by (auto simp add:field_simps)
wenzelm@53409
  2144
      finally show "dist x y < e" .
wenzelm@53409
  2145
    qed
wenzelm@53409
  2146
  qed
wenzelm@53409
  2147
  {
wenzelm@53409
  2148
    fix n m :: nat
immler@56188
  2149
    assume "m \<le> n" then have "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)"
hoelzl@54411
  2150
    proof (induction rule: inc_induct)
wenzelm@53409
  2151
      case (step i)
wenzelm@53409
  2152
      show ?case
immler@56188
  2153
        using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
wenzelm@53409
  2154
    qed simp
wenzelm@53409
  2155
  } note ABsubset = this
immler@56188
  2156
  have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)"
immler@56188
  2157
    by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
immler@54776
  2158
      (metis nat.exhaust AB(1-3) assms(1,3))
immler@56188
  2159
  then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
wenzelm@53409
  2160
    by blast
wenzelm@53409
  2161
  show thesis
wenzelm@53409
  2162
  proof (rule that[rule_format, of x0])
immler@56188
  2163
    show "x0\<in>cbox a b"
wenzelm@53409
  2164
      using x0[of 0] unfolding AB .
wenzelm@53409
  2165
    fix e :: real
wenzelm@53409
  2166
    assume "e > 0"
wenzelm@53409
  2167
    from interv[OF this] obtain n
immler@56188
  2168
      where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" ..
lp15@60396
  2169
    have "\<not> P (cbox (A n) (B n))"
lp15@60396
  2170
      apply (cases "0 < n")
lp15@60396
  2171
      using AB(3)[of "n - 1"] assms(3) AB(1-2)
lp15@60396
  2172
      apply auto
lp15@60396
  2173
      done
lp15@60396
  2174
    moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
lp15@60396
  2175
      using n using x0[of n] by auto
lp15@60396
  2176
    moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
lp15@60396
  2177
      unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
lp15@60396
  2178
    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
  2179
      apply (rule_tac x="A n" in exI)
wenzelm@53409
  2180
      apply (rule_tac x="B n" in exI)
lp15@60396
  2181
      apply (auto simp: x0)
lp15@60396
  2182
      done
wenzelm@53409
  2183
  qed
wenzelm@53409
  2184
qed
wenzelm@53409
  2185
himmelma@35172
  2186
wenzelm@60420
  2187
subsection \<open>Cousin's lemma.\<close>
himmelma@35172
  2188
wenzelm@53409
  2189
lemma fine_division_exists:
immler@56188
  2190
  fixes a b :: "'a::euclidean_space"
wenzelm@53409
  2191
  assumes "gauge g"
immler@56188
  2192
  obtains p where "p tagged_division_of (cbox a b)" "g fine p"
immler@56188
  2193
proof -
immler@56188
  2194
  presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
immler@56188
  2195
  then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
wenzelm@53410
  2196
    by blast
wenzelm@53409
  2197
  then show thesis ..
wenzelm@53409
  2198
next
immler@56188
  2199
  assume as: "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
wenzelm@55751
  2200
  obtain x where x:
lp15@60428
  2201
      "x \<in> (cbox a b)"
lp15@60428
  2202
      "\<And>e. 0 < e \<Longrightarrow>
lp15@60428
  2203
        \<exists>c d.
lp15@60428
  2204
          x \<in> cbox c d \<and>
lp15@60428
  2205
          cbox c d \<subseteq> ball x e \<and>
lp15@60428
  2206
          cbox c d \<subseteq> (cbox a b) \<and>
lp15@60428
  2207
          \<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
lp15@60428
  2208
    apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ as])
lp15@60428
  2209
    apply (simp add: fine_def)
lp15@60428
  2210
    apply (metis tagged_division_union fine_union)
lp15@60428
  2211
    apply (auto simp: )
lp15@60428
  2212
    done
wenzelm@53410
  2213
  obtain e where e: "e > 0" "ball x e \<subseteq> g x"
wenzelm@53409
  2214
    using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
lp15@60615
  2215
  from x(2)[OF e(1)]
lp15@60396
  2216
  obtain c d where c_d: "x \<in> cbox c d"
lp15@60396
  2217
                        "cbox c d \<subseteq> ball x e"
lp15@60396
  2218
                        "cbox c d \<subseteq> cbox a b"
lp15@60396
  2219
                        "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
wenzelm@53410
  2220
    by blast
immler@56188
  2221
  have "g fine {(x, cbox c d)}"
wenzelm@53409
  2222
    unfolding fine_def using e using c_d(2) by auto
wenzelm@53410
  2223
  then show False
wenzelm@53410
  2224
    using tagged_division_of_self[OF c_d(1)] using c_d by auto
wenzelm@53409
  2225
qed
wenzelm@53409
  2226
immler@56188
  2227
lemma fine_division_exists_real:
immler@56188
  2228
  fixes a b :: real
immler@56188
  2229
  assumes "gauge g"
immler@56188
  2230
  obtains p where "p tagged_division_of {a .. b}" "g fine p"
immler@56188
  2231
  by (metis assms box_real(2) fine_division_exists)
himmelma@35172
  2232
wenzelm@60420
  2233
subsection \<open>Basic theorems about integrals.\<close>
himmelma@35172
  2234
wenzelm@53409
  2235
lemma has_integral_unique:
immler@56188
  2236
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2237
  assumes "(f has_integral k1) i"
wenzelm@53410
  2238
    and "(f has_integral k2) i"
wenzelm@53409
  2239
  shows "k1 = k2"
wenzelm@53410
  2240
proof (rule ccontr)
wenzelm@53842
  2241
  let ?e = "norm (k1 - k2) / 2"
wenzelm@53410
  2242
  assume as:"k1 \<noteq> k2"
wenzelm@53410
  2243
  then have e: "?e > 0"
wenzelm@53410
  2244
    by auto
wenzelm@53410
  2245
  have lem: "\<And>f::'n \<Rightarrow> 'a.  \<And>a b k1 k2.
immler@56188
  2246
    (f has_integral k1) (cbox a b) \<Longrightarrow> (f has_integral k2) (cbox a b) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
wenzelm@53410
  2247
  proof -
wenzelm@53410
  2248
    case goal1
wenzelm@53410
  2249
    let ?e = "norm (k1 - k2) / 2"
wenzelm@53410
  2250
    from goal1(3) have e: "?e > 0" by auto
wenzelm@55751
  2251
    obtain d1 where d1:
wenzelm@55751
  2252
        "gauge d1"
immler@56188
  2253
        "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
wenzelm@55751
  2254
          d1 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k1) < norm (k1 - k2) / 2"
wenzelm@55751
  2255
      by (rule has_integralD[OF goal1(1) e]) blast
wenzelm@55751
  2256
    obtain d2 where d2:
wenzelm@55751
  2257
        "gauge d2"
immler@56188
  2258
        "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
wenzelm@55751
  2259
          d2 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k2) < norm (k1 - k2) / 2"
wenzelm@55751
  2260
      by (rule has_integralD[OF goal1(2) e]) blast
wenzelm@55751
  2261
    obtain p where p:
immler@56188
  2262
        "p tagged_division_of cbox a b"
wenzelm@55751
  2263
        "(\<lambda>x. d1 x \<inter> d2 x) fine p"
wenzelm@55751
  2264
      by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)]])
wenzelm@53410
  2265
    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
wenzelm@53410
  2266
    have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
wenzelm@53410
  2267
      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
wenzelm@53410
  2268
      by (auto simp add:algebra_simps norm_minus_commute)
himmelma@35172
  2269
    also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2270
      apply (rule add_strict_mono)
wenzelm@53410
  2271
      apply (rule_tac[!] d2(2) d1(2))
wenzelm@53410
  2272
      using p unfolding fine_def
wenzelm@53410
  2273
      apply auto
wenzelm@53410
  2274
      done
himmelma@35172
  2275
    finally show False by auto
wenzelm@53410
  2276
  qed
wenzelm@53410
  2277
  {
immler@56188
  2278
    presume "\<not> (\<exists>a b. i = cbox a b) \<Longrightarrow> False"
wenzelm@53410
  2279
    then show False
lp15@60396
  2280
      using as assms lem by blast
wenzelm@53410
  2281
  }
immler@56188
  2282
  assume as: "\<not> (\<exists>a b. i = cbox a b)"
wenzelm@55751
  2283
  obtain B1 where B1:
wenzelm@55751
  2284
      "0 < B1"
immler@56188
  2285
      "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2286
        \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
wenzelm@55751
  2287
          norm (z - k1) < norm (k1 - k2) / 2"
wenzelm@55751
  2288
    by (rule has_integral_altD[OF assms(1) as,OF e]) blast
wenzelm@55751
  2289
  obtain B2 where B2:
wenzelm@55751
  2290
      "0 < B2"
immler@56188
  2291
      "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2292
        \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
wenzelm@55751
  2293
          norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@55751
  2294
    by (rule has_integral_altD[OF assms(2) as,OF e]) blast
immler@56188
  2295
  have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> cbox a b"
immler@56188
  2296
    apply (rule bounded_subset_cbox)
wenzelm@53410
  2297
    using bounded_Un bounded_ball
wenzelm@53410
  2298
    apply auto
wenzelm@53410
  2299
    done
immler@56188
  2300
  then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
wenzelm@53410
  2301
    by blast
wenzelm@53410
  2302
  obtain w where w:
immler@56188
  2303
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) (cbox a b)"
wenzelm@53410
  2304
    "norm (w - k1) < norm (k1 - k2) / 2"
wenzelm@53410
  2305
    using B1(2)[OF ab(1)] by blast
wenzelm@53410
  2306
  obtain z where z:
immler@56188
  2307
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b)"
wenzelm@53410
  2308
    "norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@53410
  2309
    using B2(2)[OF ab(2)] by blast
wenzelm@53410
  2310
  have "z = w"
wenzelm@53410
  2311
    using lem[OF w(1) z(1)] by auto
wenzelm@53410
  2312
  then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
wenzelm@53410
  2313
    using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
wenzelm@53410
  2314
    by (auto simp add: norm_minus_commute)
wenzelm@53410
  2315
  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2316
    apply (rule add_strict_mono)
wenzelm@53410
  2317
    apply (rule_tac[!] z(2) w(2))
wenzelm@53410
  2318
    done
wenzelm@53410
  2319
  finally show False by auto
wenzelm@53410
  2320
qed
wenzelm@53410
  2321
wenzelm@53410
  2322
lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
wenzelm@53410
  2323
  unfolding integral_def
wenzelm@53410
  2324
  by (rule some_equality) (auto intro: has_integral_unique)
wenzelm@53410
  2325
wenzelm@53410
  2326
lemma has_integral_is_0:
immler@56188
  2327
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2328
  assumes "\<forall>x\<in>s. f x = 0"
wenzelm@53410
  2329
  shows "(f has_integral 0) s"
wenzelm@53410
  2330
proof -
wenzelm@53410
  2331
  have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
immler@56188
  2332
    (\<forall>x\<in>cbox a b. f(x) = 0) \<Longrightarrow> (f has_integral 0) (cbox a b)"
wenzelm@53410
  2333
    unfolding has_integral
lp15@60396
  2334
  proof clarify
wenzelm@53410
  2335
    fix a b e
wenzelm@53410
  2336
    fix f :: "'n \<Rightarrow> 'a"
immler@56188
  2337
    assume as: "\<forall>x\<in>cbox a b. f x = 0" "0 < (e::real)"
lp15@60396
  2338
    have "\<And>p. p tagged_division_of cbox a b \<Longrightarrow> (\<lambda>x. ball x 1) fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - 0) < e"
wenzelm@53410
  2339
    proof -
wenzelm@53410
  2340
      case goal1
wenzelm@53410
  2341
      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0"
haftmann@57418
  2342
      proof (rule setsum.neutral, rule)
wenzelm@53410
  2343
        fix x
wenzelm@53410
  2344
        assume x: "x \<in> p"
wenzelm@53410
  2345
        have "f (fst x) = 0"
wenzelm@53410
  2346
          using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
wenzelm@53410
  2347
        then show "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0"
wenzelm@53410
  2348
          apply (subst surjective_pairing[of x])
wenzelm@53410
  2349
          unfolding split_conv
wenzelm@53410
  2350
          apply auto
wenzelm@53410
  2351
          done
wenzelm@53410
  2352
      qed
wenzelm@53410
  2353
      then show ?case
wenzelm@53410
  2354
        using as by auto
lp15@60396
  2355
    qed
lp15@60396
  2356
    then show "\<exists>d. gauge d \<and>
lp15@60396
  2357
                   (\<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
  2358
      by auto
wenzelm@53410
  2359
  qed
wenzelm@53410
  2360
  {
immler@56188
  2361
    presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
lp15@60396
  2362
    with assms lem show ?thesis
lp15@60396
  2363
      by blast
wenzelm@53410
  2364
  }
wenzelm@53410
  2365
  have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
wenzelm@53410
  2366
    apply (rule ext)
wenzelm@53410
  2367
    using assms
wenzelm@53410
  2368
    apply auto
wenzelm@53410
  2369
    done
immler@56188
  2370
  assume "\<not> (\<exists>a b. s = cbox a b)"
wenzelm@53410
  2371
  then show ?thesis
lp15@60396
  2372
    using lem
lp15@60396
  2373
    by (subst has_integral_alt) (force simp add: *)
wenzelm@53410
  2374
qed
himmelma@35172
  2375
immler@56188
  2376
lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) s"
wenzelm@53410
  2377
  by (rule has_integral_is_0) auto
himmelma@35172
  2378
himmelma@35172
  2379
lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
himmelma@35172
  2380
  using has_integral_unique[OF has_integral_0] by auto
himmelma@35172
  2381
wenzelm@53410
  2382
lemma has_integral_linear:
immler@56188
  2383
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2384
  assumes "(f has_integral y) s"
wenzelm@53410
  2385
    and "bounded_linear h"
wenzelm@53410
  2386
  shows "((h o f) has_integral ((h y))) s"
wenzelm@53410
  2387
proof -
wenzelm@53410
  2388
  interpret bounded_linear h
wenzelm@53410
  2389
    using assms(2) .
wenzelm@53410
  2390
  from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
wenzelm@53410
  2391
    by blast
wenzelm@53410
  2392
  have lem: "\<And>(f :: 'n \<Rightarrow> 'a) y a b.
immler@56188
  2393
    (f has_integral y) (cbox a b) \<Longrightarrow> ((h o f) has_integral h y) (cbox a b)"
lp15@60396
  2394
  unfolding has_integral
lp15@60396
  2395
  proof clarify
wenzelm@53410
  2396
    case goal1
wenzelm@53410
  2397
    from pos_bounded
wenzelm@53410
  2398
    obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
wenzelm@53410
  2399
      by blast
lp15@60428
  2400
    have "e / B > 0" using goal1(2) B by simp
lp15@60615
  2401
    then obtain g
lp15@60428
  2402
      where g: "gauge g"
lp15@60428
  2403
               "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> g fine p \<Longrightarrow>
lp15@60428
  2404
                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
lp15@60428
  2405
        using goal1(1) by auto
lp15@60428
  2406
    { fix p
immler@56188
  2407
      assume as: "p tagged_division_of (cbox a b)" "g fine p"
lp15@60428
  2408
      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
  2409
        by auto
lp15@60428
  2410
      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
  2411
        unfolding o_def unfolding scaleR[symmetric] hc by simp
wenzelm@53410
  2412
      also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
wenzelm@53410
  2413
        using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
lp15@60428
  2414
      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
  2415
      then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e"
lp15@60428
  2416
        apply (simp add: diff[symmetric])
wenzelm@53410
  2417
        apply (rule le_less_trans[OF B(2)])
wenzelm@53410
  2418
        using g(2)[OF as] B(1)
wenzelm@53410
  2419
        apply (auto simp add: field_simps)
wenzelm@53410
  2420
        done
lp15@60428
  2421
    }
lp15@60428
  2422
    with g show ?case
lp15@60428
  2423
      by (rule_tac x=g in exI) auto
wenzelm@53410
  2424
  qed
wenzelm@53410
  2425
  {
immler@56188
  2426
    presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
wenzelm@53410
  2427
    then show ?thesis
lp15@60396
  2428
      using assms(1) lem by blast
wenzelm@53410
  2429
  }
immler@56188
  2430
  assume as: "\<not> (\<exists>a b. s = cbox a b)"
wenzelm@53410
  2431
  then show ?thesis
lp15@60396
  2432
  proof (subst has_integral_alt, clarsimp)
wenzelm@53410
  2433
    fix e :: real
wenzelm@53410
  2434
    assume e: "e > 0"
nipkow@56541
  2435
    have *: "0 < e/B" using e B(1) by simp
wenzelm@53410
  2436
    obtain M where M:
wenzelm@53410
  2437
      "M > 0"
immler@56188
  2438
      "\<And>a b. ball 0 M \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2439
        \<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
  2440
      using has_integral_altD[OF assms(1) as *] by blast
immler@56188
  2441
    show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  2442
      (\<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)"
lp15@60396
  2443
    proof (rule_tac x=M in exI, clarsimp simp add: M)
wenzelm@53410
  2444
      case goal1
wenzelm@53410
  2445
      obtain z where z:
immler@56188
  2446
        "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b)"
wenzelm@53410
  2447
        "norm (z - y) < e / B"
wenzelm@53410
  2448
        using M(2)[OF goal1(1)] by blast
wenzelm@53410
  2449
      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
  2450
        using zero by auto
wenzelm@53410
  2451
      show ?case
wenzelm@53410
  2452
        apply (rule_tac x="h z" in exI)
lp15@60396
  2453
        apply (simp add: "*" lem z(1))
lp15@60396
  2454
        by (metis B diff le_less_trans pos_less_divide_eq z(2))
wenzelm@53410
  2455
    qed
wenzelm@53410
  2456
  qed
wenzelm@53410
  2457
qed
wenzelm@53410
  2458
lp15@60615
  2459
lemma has_integral_scaleR_left:
hoelzl@57447
  2460
  "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) s"
hoelzl@57447
  2461
  using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
hoelzl@57447
  2462
hoelzl@57447
  2463
lemma has_integral_mult_left:
hoelzl@57447
  2464
  fixes c :: "_ :: {real_normed_algebra}"
hoelzl@57447
  2465
  shows "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) s"
hoelzl@57447
  2466
  using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
hoelzl@57447
  2467
lp15@60615
  2468
corollary integral_mult_left:
lp15@60615
  2469
  fixes c:: "'a::real_normed_algebra"
lp15@60615
  2470
  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x * c) = integral s f * c"
lp15@60615
  2471
  by (blast intro:  has_integral_mult_left)
lp15@60615
  2472
paulson@60762
  2473
lemma has_integral_mult_right:
paulson@60762
  2474
  fixes c :: "'a :: real_normed_algebra"
paulson@60762
  2475
  shows "(f has_integral y) i \<Longrightarrow> ((\<lambda>x. c * f x) has_integral (c * y)) i"
paulson@60762
  2476
  using has_integral_linear[OF _ bounded_linear_mult_right] by (simp add: comp_def)