src/HOL/Multivariate_Analysis/Integration.thy
author paulson <lp15@cam.ac.uk>
Tue Jun 30 13:56:16 2015 +0100 (2015-06-30)
changeset 60615 e5fa1d5d3952
parent 60585 48fdff264eb2
child 60621 bfb14ff43491
permissions -rw-r--r--
Useful lemmas. The theorem concerning swapping the variables in a double integral.
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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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   300
          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
wenzelm@49698
   579
lemma content_subset:
immler@56188
   580
  assumes "cbox a b \<subseteq> cbox c d"
immler@56188
   581
  shows "content (cbox a b) \<le> content (cbox c d)"
immler@56188
   582
proof (cases "cbox a b = {}")
immler@56188
   583
  case True
immler@56188
   584
  then show ?thesis
immler@56188
   585
    using content_pos_le[of c d] by auto
immler@56188
   586
next
immler@56188
   587
  case False
immler@56188
   588
  then have ab_ne: "\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i"
immler@56188
   589
    unfolding box_ne_empty by auto
immler@56188
   590
  then have ab_ab: "a\<in>cbox a b" "b\<in>cbox a b"
immler@56188
   591
    unfolding mem_box by auto
immler@56188
   592
  have "cbox c d \<noteq> {}" using assms False by auto
immler@56188
   593
  then have cd_ne: "\<forall>i\<in>Basis. c \<bullet> i \<le> d \<bullet> i"
immler@56188
   594
    using assms unfolding box_ne_empty by auto
lp15@60394
   595
  have "\<And>i. i \<in> Basis \<Longrightarrow> 0 \<le> b \<bullet> i - a \<bullet> i"
lp15@60394
   596
    using ab_ne by (metis diff_le_iff(1))
lp15@60394
   597
  moreover
lp15@60394
   598
  have "\<And>i. i \<in> Basis \<Longrightarrow> b \<bullet> i - a \<bullet> i \<le> d \<bullet> i - c \<bullet> i"
lp15@60394
   599
    using assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(2)]
lp15@60394
   600
          assms[unfolded subset_eq mem_box,rule_format,OF ab_ab(1)]
lp15@60394
   601
      by (metis diff_mono)
lp15@60394
   602
  ultimately show ?thesis
lp15@60394
   603
    unfolding content_def interval_bounds[OF ab_ne] interval_bounds[OF cd_ne]
wenzelm@60420
   604
    by (simp add: setprod_mono if_not_P[OF False] if_not_P[OF \<open>cbox c d \<noteq> {}\<close>])
immler@56188
   605
qed
immler@56188
   606
immler@56188
   607
lemma content_lt_nz: "0 < content (cbox a b) \<longleftrightarrow> content (cbox a b) \<noteq> 0"
nipkow@44890
   608
  unfolding content_pos_lt_eq content_eq_0 unfolding not_ex not_le by fastforce
himmelma@35172
   609
hoelzl@59425
   610
lemma content_times[simp]: "content (A \<times> B) = content A * content B"
hoelzl@59425
   611
proof (cases "A \<times> B = {}")
hoelzl@59425
   612
  let ?ub1 = "interval_upperbound" and ?lb1 = "interval_lowerbound"
hoelzl@59425
   613
  let ?ub2 = "interval_upperbound" and ?lb2 = "interval_lowerbound"
hoelzl@59425
   614
  assume nonempty: "A \<times> B \<noteq> {}"
lp15@60615
   615
  hence "content (A \<times> B) = (\<Prod>i\<in>Basis. (?ub1 A, ?ub2 B) \<bullet> i - (?lb1 A, ?lb2 B) \<bullet> i)"
hoelzl@59425
   616
      unfolding content_def by (simp add: interval_upperbound_Times interval_lowerbound_Times)
hoelzl@59425
   617
  also have "... = content A * content B" unfolding content_def using nonempty
hoelzl@59425
   618
    apply (subst Basis_prod_def, subst setprod.union_disjoint, force, force, force, simp)
hoelzl@59425
   619
    apply (subst (1 2) setprod.reindex, auto intro: inj_onI)
hoelzl@59425
   620
    done
hoelzl@59425
   621
  finally show ?thesis .
hoelzl@59425
   622
qed (auto simp: content_def)
hoelzl@59425
   623
lp15@60615
   624
lemma content_Pair: "content (cbox (a,c) (b,d)) = content (cbox a b) * content (cbox c d)"
lp15@60615
   625
  by (simp add: cbox_Pair_eq)
lp15@60615
   626
lp15@60615
   627
lemma content_cbox_pair_eq0_D:
lp15@60615
   628
   "content (cbox (a,c) (b,d)) = 0 \<Longrightarrow> content (cbox a b) = 0 \<or> content (cbox c d) = 0"
lp15@60615
   629
  by (simp add: content_Pair)
lp15@60615
   630
lp15@60615
   631
lemma content_eq_0_gen:
lp15@60615
   632
  fixes s :: "'a::euclidean_space set"
lp15@60615
   633
  assumes "bounded s"
lp15@60615
   634
  shows "content s = 0 \<longleftrightarrow> (\<exists>i\<in>Basis. \<exists>v. \<forall>x \<in> s. x \<bullet> i = v)"  (is "_ = ?rhs")
lp15@60615
   635
proof safe
lp15@60615
   636
  assume "content s = 0" then show ?rhs
lp15@60615
   637
    apply (clarsimp simp: ex_in_conv content_def split: split_if_asm)
lp15@60615
   638
    apply (rule_tac x=a in bexI)
lp15@60615
   639
    apply (rule_tac x="interval_lowerbound s \<bullet> a" in exI)
lp15@60615
   640
    apply (clarsimp simp: interval_upperbound_def interval_lowerbound_def)
lp15@60615
   641
    apply (drule cSUP_eq_cINF_D)
lp15@60615
   642
    apply (auto simp: bounded_inner_imp_bdd_above [OF assms]  bounded_inner_imp_bdd_below [OF assms])
lp15@60615
   643
    done
lp15@60615
   644
next
lp15@60615
   645
  fix i a
lp15@60615
   646
  assume "i \<in> Basis" "\<forall>x\<in>s. x \<bullet> i = a"
lp15@60615
   647
  then show "content s = 0"
lp15@60615
   648
    apply (clarsimp simp: content_def)
lp15@60615
   649
    apply (rule_tac x=i in bexI)
lp15@60615
   650
    apply (auto simp: interval_upperbound_def interval_lowerbound_def)
lp15@60615
   651
    done
lp15@60615
   652
qed
lp15@60615
   653
lp15@60615
   654
lemma content_0_subset_gen:
lp15@60615
   655
  fixes a :: "'a::euclidean_space"
lp15@60615
   656
  assumes "content t = 0" "s \<subseteq> t" "bounded t" shows "content s = 0"
lp15@60615
   657
proof -
lp15@60615
   658
  have "bounded s"
lp15@60615
   659
    using assms by (metis bounded_subset)
lp15@60615
   660
  then show ?thesis
lp15@60615
   661
    using assms
lp15@60615
   662
    by (auto simp: content_eq_0_gen)
lp15@60615
   663
qed
lp15@60615
   664
lp15@60615
   665
lemma content_0_subset: "\<lbrakk>content(cbox a b) = 0; s \<subseteq> cbox a b\<rbrakk> \<Longrightarrow> content s = 0"
lp15@60615
   666
  by (simp add: content_0_subset_gen bounded_cbox)
lp15@60615
   667
wenzelm@49698
   668
wenzelm@60420
   669
subsection \<open>The notion of a gauge --- simply an open set containing the point.\<close>
himmelma@35172
   670
wenzelm@53408
   671
definition "gauge d \<longleftrightarrow> (\<forall>x. x \<in> d x \<and> open (d x))"
wenzelm@53399
   672
wenzelm@53399
   673
lemma gaugeI:
wenzelm@53399
   674
  assumes "\<And>x. x \<in> g x"
wenzelm@53399
   675
    and "\<And>x. open (g x)"
wenzelm@53399
   676
  shows "gauge g"
himmelma@35172
   677
  using assms unfolding gauge_def by auto
himmelma@35172
   678
wenzelm@53399
   679
lemma gaugeD[dest]:
wenzelm@53399
   680
  assumes "gauge d"
wenzelm@53399
   681
  shows "x \<in> d x"
wenzelm@53399
   682
    and "open (d x)"
wenzelm@49698
   683
  using assms unfolding gauge_def by auto
himmelma@35172
   684
himmelma@35172
   685
lemma gauge_ball_dependent: "\<forall>x. 0 < e x \<Longrightarrow> gauge (\<lambda>x. ball x (e x))"
wenzelm@53399
   686
  unfolding gauge_def by auto
wenzelm@53399
   687
wenzelm@53399
   688
lemma gauge_ball[intro]: "0 < e \<Longrightarrow> gauge (\<lambda>x. ball x e)"
wenzelm@53399
   689
  unfolding gauge_def by auto
himmelma@35172
   690
lp15@60466
   691
lemma gauge_trivial[intro!]: "gauge (\<lambda>x. ball x 1)"
wenzelm@49698
   692
  by (rule gauge_ball) auto
himmelma@35172
   693
wenzelm@53408
   694
lemma gauge_inter[intro]: "gauge d1 \<Longrightarrow> gauge d2 \<Longrightarrow> gauge (\<lambda>x. d1 x \<inter> d2 x)"
wenzelm@53399
   695
  unfolding gauge_def by auto
himmelma@35172
   696
wenzelm@49698
   697
lemma gauge_inters:
wenzelm@53399
   698
  assumes "finite s"
wenzelm@53399
   699
    and "\<forall>d\<in>s. gauge (f d)"
wenzelm@60585
   700
  shows "gauge (\<lambda>x. \<Inter>{f d x | d. d \<in> s})"
wenzelm@49698
   701
proof -
wenzelm@53399
   702
  have *: "\<And>x. {f d x |d. d \<in> s} = (\<lambda>d. f d x) ` s"
wenzelm@53399
   703
    by auto
wenzelm@49698
   704
  show ?thesis
wenzelm@53399
   705
    unfolding gauge_def unfolding *
wenzelm@49698
   706
    using assms unfolding Ball_def Inter_iff mem_Collect_eq gauge_def by auto
wenzelm@49698
   707
qed
wenzelm@49698
   708
wenzelm@53399
   709
lemma gauge_existence_lemma:
wenzelm@53408
   710
  "(\<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
   711
  by (metis zero_less_one)
wenzelm@49698
   712
himmelma@35172
   713
wenzelm@60420
   714
subsection \<open>Divisions.\<close>
himmelma@35172
   715
wenzelm@53408
   716
definition division_of (infixl "division'_of" 40)
wenzelm@53408
   717
where
wenzelm@53399
   718
  "s division_of i \<longleftrightarrow>
wenzelm@53399
   719
    finite s \<and>
immler@56188
   720
    (\<forall>k\<in>s. k \<subseteq> i \<and> k \<noteq> {} \<and> (\<exists>a b. k = cbox a b)) \<and>
wenzelm@53399
   721
    (\<forall>k1\<in>s. \<forall>k2\<in>s. k1 \<noteq> k2 \<longrightarrow> interior(k1) \<inter> interior(k2) = {}) \<and>
wenzelm@53399
   722
    (\<Union>s = i)"
himmelma@35172
   723
wenzelm@49698
   724
lemma division_ofD[dest]:
wenzelm@49698
   725
  assumes "s division_of i"
wenzelm@53408
   726
  shows "finite s"
wenzelm@53408
   727
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   728
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
immler@56188
   729
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
   730
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior(k1) \<inter> interior(k2) = {}"
wenzelm@53408
   731
    and "\<Union>s = i"
wenzelm@49698
   732
  using assms unfolding division_of_def by auto
himmelma@35172
   733
himmelma@35172
   734
lemma division_ofI:
wenzelm@53408
   735
  assumes "finite s"
wenzelm@53408
   736
    and "\<And>k. k \<in> s \<Longrightarrow> k \<subseteq> i"
wenzelm@53408
   737
    and "\<And>k. k \<in> s \<Longrightarrow> k \<noteq> {}"
immler@56188
   738
    and "\<And>k. k \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
   739
    and "\<And>k1 k2. k1 \<in> s \<Longrightarrow> k2 \<in> s \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53399
   740
    and "\<Union>s = i"
wenzelm@53399
   741
  shows "s division_of i"
wenzelm@53399
   742
  using assms unfolding division_of_def by auto
himmelma@35172
   743
himmelma@35172
   744
lemma division_of_finite: "s division_of i \<Longrightarrow> finite s"
himmelma@35172
   745
  unfolding division_of_def by auto
himmelma@35172
   746
immler@56188
   747
lemma division_of_self[intro]: "cbox a b \<noteq> {} \<Longrightarrow> {cbox a b} division_of (cbox a b)"
himmelma@35172
   748
  unfolding division_of_def by auto
himmelma@35172
   749
wenzelm@53399
   750
lemma division_of_trivial[simp]: "s division_of {} \<longleftrightarrow> s = {}"
wenzelm@53399
   751
  unfolding division_of_def by auto
himmelma@35172
   752
wenzelm@49698
   753
lemma division_of_sing[simp]:
immler@56188
   754
  "s division_of cbox a (a::'a::euclidean_space) \<longleftrightarrow> s = {cbox a a}"
wenzelm@53399
   755
  (is "?l = ?r")
wenzelm@49698
   756
proof
wenzelm@49698
   757
  assume ?r
wenzelm@53399
   758
  moreover
lp15@60384
   759
  { fix k
lp15@60384
   760
    assume "s = {{a}}" "k\<in>s"
lp15@60384
   761
    then have "\<exists>x y. k = cbox x y"
wenzelm@50945
   762
      apply (rule_tac x=a in exI)+
lp15@60384
   763
      apply (force simp: cbox_sing)
wenzelm@50945
   764
      done
wenzelm@49698
   765
  }
wenzelm@53399
   766
  ultimately show ?l
immler@56188
   767
    unfolding division_of_def cbox_sing by auto
wenzelm@49698
   768
next
wenzelm@49698
   769
  assume ?l
immler@56188
   770
  note * = conjunctD4[OF this[unfolded division_of_def cbox_sing]]
wenzelm@53399
   771
  {
wenzelm@53399
   772
    fix x
wenzelm@53399
   773
    assume x: "x \<in> s" have "x = {a}"
wenzelm@53408
   774
      using *(2)[rule_format,OF x] by auto
wenzelm@53399
   775
  }
wenzelm@53408
   776
  moreover have "s \<noteq> {}"
wenzelm@53408
   777
    using *(4) by auto
wenzelm@53408
   778
  ultimately show ?r
immler@56188
   779
    unfolding cbox_sing by auto
wenzelm@49698
   780
qed
himmelma@35172
   781
himmelma@35172
   782
lemma elementary_empty: obtains p where "p division_of {}"
himmelma@35172
   783
  unfolding division_of_trivial by auto
himmelma@35172
   784
immler@56188
   785
lemma elementary_interval: obtains p where "p division_of (cbox a b)"
wenzelm@49698
   786
  by (metis division_of_trivial division_of_self)
himmelma@35172
   787
himmelma@35172
   788
lemma division_contains: "s division_of i \<Longrightarrow> \<forall>x\<in>i. \<exists>k\<in>s. x \<in> k"
himmelma@35172
   789
  unfolding division_of_def by auto
himmelma@35172
   790
himmelma@35172
   791
lemma forall_in_division:
immler@56188
   792
  "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
   793
  unfolding division_of_def by fastforce
himmelma@35172
   794
wenzelm@53399
   795
lemma division_of_subset:
wenzelm@53399
   796
  assumes "p division_of (\<Union>p)"
wenzelm@53399
   797
    and "q \<subseteq> p"
wenzelm@53399
   798
  shows "q division_of (\<Union>q)"
wenzelm@53408
   799
proof (rule division_ofI)
wenzelm@53408
   800
  note * = division_ofD[OF assms(1)]
wenzelm@49698
   801
  show "finite q"
lp15@60384
   802
    using "*"(1) assms(2) infinite_super by auto
wenzelm@53399
   803
  {
wenzelm@53399
   804
    fix k
wenzelm@49698
   805
    assume "k \<in> q"
wenzelm@53408
   806
    then have kp: "k \<in> p"
wenzelm@53408
   807
      using assms(2) by auto
wenzelm@53408
   808
    show "k \<subseteq> \<Union>q"
wenzelm@60420
   809
      using \<open>k \<in> q\<close> by auto
immler@56188
   810
    show "\<exists>a b. k = cbox a b"
wenzelm@53408
   811
      using *(4)[OF kp] by auto
wenzelm@53408
   812
    show "k \<noteq> {}"
wenzelm@53408
   813
      using *(3)[OF kp] by auto
wenzelm@53399
   814
  }
wenzelm@49698
   815
  fix k1 k2
wenzelm@49698
   816
  assume "k1 \<in> q" "k2 \<in> q" "k1 \<noteq> k2"
wenzelm@53408
   817
  then have **: "k1 \<in> p" "k2 \<in> p" "k1 \<noteq> k2"
wenzelm@53399
   818
    using assms(2) by auto
wenzelm@53399
   819
  show "interior k1 \<inter> interior k2 = {}"
wenzelm@53408
   820
    using *(5)[OF **] by auto
wenzelm@49698
   821
qed auto
wenzelm@49698
   822
wenzelm@49698
   823
lemma division_of_union_self[intro]: "p division_of s \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@49698
   824
  unfolding division_of_def by auto
himmelma@35172
   825
wenzelm@49970
   826
lemma division_of_content_0:
immler@56188
   827
  assumes "content (cbox a b) = 0" "d division_of (cbox a b)"
wenzelm@49970
   828
  shows "\<forall>k\<in>d. content k = 0"
wenzelm@49970
   829
  unfolding forall_in_division[OF assms(2)]
lp15@60384
   830
  by (metis antisym_conv assms content_pos_le content_subset division_ofD(2))
wenzelm@49970
   831
wenzelm@49970
   832
lemma division_inter:
immler@56188
   833
  fixes s1 s2 :: "'a::euclidean_space set"
wenzelm@53408
   834
  assumes "p1 division_of s1"
wenzelm@53408
   835
    and "p2 division_of s2"
wenzelm@49970
   836
  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
   837
  (is "?A' division_of _")
wenzelm@49970
   838
proof -
wenzelm@49970
   839
  let ?A = "{s. s \<in>  (\<lambda>(k1,k2). k1 \<inter> k2) ` (p1 \<times> p2) \<and> s \<noteq> {}}"
wenzelm@53408
   840
  have *: "?A' = ?A" by auto
wenzelm@53399
   841
  show ?thesis
wenzelm@53399
   842
    unfolding *
wenzelm@49970
   843
  proof (rule division_ofI)
wenzelm@53399
   844
    have "?A \<subseteq> (\<lambda>(x, y). x \<inter> y) ` (p1 \<times> p2)"
wenzelm@53399
   845
      by auto
wenzelm@53399
   846
    moreover have "finite (p1 \<times> p2)"
wenzelm@53399
   847
      using assms unfolding division_of_def by auto
wenzelm@49970
   848
    ultimately show "finite ?A" by auto
wenzelm@53399
   849
    have *: "\<And>s. \<Union>{x\<in>s. x \<noteq> {}} = \<Union>s"
wenzelm@53399
   850
      by auto
wenzelm@49970
   851
    show "\<Union>?A = s1 \<inter> s2"
wenzelm@49970
   852
      apply (rule set_eqI)
wenzelm@49970
   853
      unfolding * and Union_image_eq UN_iff
wenzelm@49970
   854
      using division_ofD(6)[OF assms(1)] and division_ofD(6)[OF assms(2)]
wenzelm@49970
   855
      apply auto
wenzelm@49970
   856
      done
wenzelm@53399
   857
    {
wenzelm@53399
   858
      fix k
wenzelm@53399
   859
      assume "k \<in> ?A"
wenzelm@53408
   860
      then obtain k1 k2 where k: "k = k1 \<inter> k2" "k1 \<in> p1" "k2 \<in> p2" "k \<noteq> {}"
wenzelm@53399
   861
        by auto
wenzelm@53408
   862
      then show "k \<noteq> {}"
wenzelm@53408
   863
        by auto
wenzelm@49970
   864
      show "k \<subseteq> s1 \<inter> s2"
wenzelm@49970
   865
        using division_ofD(2)[OF assms(1) k(2)] and division_ofD(2)[OF assms(2) k(3)]
wenzelm@49970
   866
        unfolding k by auto
immler@56188
   867
      obtain a1 b1 where k1: "k1 = cbox a1 b1"
wenzelm@53408
   868
        using division_ofD(4)[OF assms(1) k(2)] by blast
immler@56188
   869
      obtain a2 b2 where k2: "k2 = cbox a2 b2"
wenzelm@53408
   870
        using division_ofD(4)[OF assms(2) k(3)] by blast
immler@56188
   871
      show "\<exists>a b. k = cbox a b"
wenzelm@53408
   872
        unfolding k k1 k2 unfolding inter_interval by auto
wenzelm@53408
   873
    }
wenzelm@49970
   874
    fix k1 k2
wenzelm@53408
   875
    assume "k1 \<in> ?A"
wenzelm@53408
   876
    then obtain x1 y1 where k1: "k1 = x1 \<inter> y1" "x1 \<in> p1" "y1 \<in> p2" "k1 \<noteq> {}"
wenzelm@53408
   877
      by auto
wenzelm@53408
   878
    assume "k2 \<in> ?A"
wenzelm@53408
   879
    then obtain x2 y2 where k2: "k2 = x2 \<inter> y2" "x2 \<in> p1" "y2 \<in> p2" "k2 \<noteq> {}"
wenzelm@53408
   880
      by auto
wenzelm@49970
   881
    assume "k1 \<noteq> k2"
wenzelm@53399
   882
    then have th: "x1 \<noteq> x2 \<or> y1 \<noteq> y2"
wenzelm@53399
   883
      unfolding k1 k2 by auto
wenzelm@53408
   884
    have *: "interior x1 \<inter> interior x2 = {} \<or> interior y1 \<inter> interior y2 = {} \<Longrightarrow>
wenzelm@53408
   885
      interior (x1 \<inter> y1) \<subseteq> interior x1 \<Longrightarrow> interior (x1 \<inter> y1) \<subseteq> interior y1 \<Longrightarrow>
wenzelm@53408
   886
      interior (x2 \<inter> y2) \<subseteq> interior x2 \<Longrightarrow> interior (x2 \<inter> y2) \<subseteq> interior y2 \<Longrightarrow>
wenzelm@53408
   887
      interior (x1 \<inter> y1) \<inter> interior (x2 \<inter> y2) = {}" by auto
wenzelm@49970
   888
    show "interior k1 \<inter> interior k2 = {}"
wenzelm@49970
   889
      unfolding k1 k2
wenzelm@49970
   890
      apply (rule *)
lp15@60384
   891
      using assms division_ofD(5) k1 k2(2) k2(3) th apply auto
wenzelm@53399
   892
      done
wenzelm@49970
   893
  qed
wenzelm@49970
   894
qed
wenzelm@49970
   895
wenzelm@49970
   896
lemma division_inter_1:
wenzelm@53408
   897
  assumes "d division_of i"
immler@56188
   898
    and "cbox a (b::'a::euclidean_space) \<subseteq> i"
immler@56188
   899
  shows "{cbox a b \<inter> k | k. k \<in> d \<and> cbox a b \<inter> k \<noteq> {}} division_of (cbox a b)"
immler@56188
   900
proof (cases "cbox a b = {}")
wenzelm@49970
   901
  case True
wenzelm@53399
   902
  show ?thesis
wenzelm@53399
   903
    unfolding True and division_of_trivial by auto
wenzelm@49970
   904
next
wenzelm@49970
   905
  case False
immler@56188
   906
  have *: "cbox a b \<inter> i = cbox a b" using assms(2) by auto
wenzelm@53399
   907
  show ?thesis
wenzelm@53399
   908
    using division_inter[OF division_of_self[OF False] assms(1)]
wenzelm@53399
   909
    unfolding * by auto
wenzelm@49970
   910
qed
wenzelm@49970
   911
wenzelm@49970
   912
lemma elementary_inter:
immler@56188
   913
  fixes s t :: "'a::euclidean_space set"
wenzelm@53408
   914
  assumes "p1 division_of s"
wenzelm@53408
   915
    and "p2 division_of t"
himmelma@35172
   916
  shows "\<exists>p. p division_of (s \<inter> t)"
lp15@60384
   917
using assms division_inter by blast
wenzelm@49970
   918
wenzelm@49970
   919
lemma elementary_inters:
wenzelm@53408
   920
  assumes "finite f"
wenzelm@53408
   921
    and "f \<noteq> {}"
immler@56188
   922
    and "\<forall>s\<in>f. \<exists>p. p division_of (s::('a::euclidean_space) set)"
wenzelm@60585
   923
  shows "\<exists>p. p division_of (\<Inter>f)"
wenzelm@49970
   924
  using assms
wenzelm@49970
   925
proof (induct f rule: finite_induct)
wenzelm@49970
   926
  case (insert x f)
wenzelm@49970
   927
  show ?case
wenzelm@49970
   928
  proof (cases "f = {}")
wenzelm@49970
   929
    case True
wenzelm@53399
   930
    then show ?thesis
wenzelm@53399
   931
      unfolding True using insert by auto
wenzelm@49970
   932
  next
wenzelm@49970
   933
    case False
wenzelm@53408
   934
    obtain p where "p division_of \<Inter>f"
wenzelm@53408
   935
      using insert(3)[OF False insert(5)[unfolded ball_simps,THEN conjunct2]] ..
wenzelm@53408
   936
    moreover obtain px where "px division_of x"
wenzelm@53408
   937
      using insert(5)[rule_format,OF insertI1] ..
wenzelm@49970
   938
    ultimately show ?thesis
lp15@60384
   939
      by (simp add: elementary_inter Inter_insert)
wenzelm@49970
   940
  qed
wenzelm@49970
   941
qed auto
himmelma@35172
   942
himmelma@35172
   943
lemma division_disjoint_union:
wenzelm@53408
   944
  assumes "p1 division_of s1"
wenzelm@53408
   945
    and "p2 division_of s2"
wenzelm@53408
   946
    and "interior s1 \<inter> interior s2 = {}"
wenzelm@50945
   947
  shows "(p1 \<union> p2) division_of (s1 \<union> s2)"
wenzelm@50945
   948
proof (rule division_ofI)
wenzelm@53408
   949
  note d1 = division_ofD[OF assms(1)]
wenzelm@53408
   950
  note d2 = division_ofD[OF assms(2)]
wenzelm@53408
   951
  show "finite (p1 \<union> p2)"
wenzelm@53408
   952
    using d1(1) d2(1) by auto
wenzelm@53408
   953
  show "\<Union>(p1 \<union> p2) = s1 \<union> s2"
wenzelm@53408
   954
    using d1(6) d2(6) by auto
wenzelm@50945
   955
  {
wenzelm@50945
   956
    fix k1 k2
wenzelm@50945
   957
    assume as: "k1 \<in> p1 \<union> p2" "k2 \<in> p1 \<union> p2" "k1 \<noteq> k2"
wenzelm@50945
   958
    moreover
wenzelm@50945
   959
    let ?g="interior k1 \<inter> interior k2 = {}"
wenzelm@50945
   960
    {
wenzelm@50945
   961
      assume as: "k1\<in>p1" "k2\<in>p2"
wenzelm@50945
   962
      have ?g
wenzelm@50945
   963
        using interior_mono[OF d1(2)[OF as(1)]] interior_mono[OF d2(2)[OF as(2)]]
wenzelm@50945
   964
        using assms(3) by blast
wenzelm@50945
   965
    }
wenzelm@50945
   966
    moreover
wenzelm@50945
   967
    {
wenzelm@50945
   968
      assume as: "k1\<in>p2" "k2\<in>p1"
wenzelm@50945
   969
      have ?g
wenzelm@50945
   970
        using interior_mono[OF d1(2)[OF as(2)]] interior_mono[OF d2(2)[OF as(1)]]
wenzelm@50945
   971
        using assms(3) by blast
wenzelm@50945
   972
    }
wenzelm@53399
   973
    ultimately show ?g
wenzelm@53399
   974
      using d1(5)[OF _ _ as(3)] and d2(5)[OF _ _ as(3)] by auto
wenzelm@50945
   975
  }
wenzelm@50945
   976
  fix k
wenzelm@50945
   977
  assume k: "k \<in> p1 \<union> p2"
wenzelm@53408
   978
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
   979
    using k d1(2) d2(2) by auto
wenzelm@53408
   980
  show "k \<noteq> {}"
wenzelm@53408
   981
    using k d1(3) d2(3) by auto
immler@56188
   982
  show "\<exists>a b. k = cbox a b"
wenzelm@53408
   983
    using k d1(4) d2(4) by auto
wenzelm@50945
   984
qed
himmelma@35172
   985
himmelma@35172
   986
lemma partial_division_extend_1:
immler@56188
   987
  fixes a b c d :: "'a::euclidean_space"
immler@56188
   988
  assumes incl: "cbox c d \<subseteq> cbox a b"
immler@56188
   989
    and nonempty: "cbox c d \<noteq> {}"
immler@56188
   990
  obtains p where "p division_of (cbox a b)" "cbox c d \<in> p"
hoelzl@50526
   991
proof
wenzelm@53408
   992
  let ?B = "\<lambda>f::'a\<Rightarrow>'a \<times> 'a.
immler@56188
   993
    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
   994
  def p \<equiv> "?B ` (Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)})"
hoelzl@50526
   995
immler@56188
   996
  show "cbox c d \<in> p"
hoelzl@50526
   997
    unfolding p_def
immler@56188
   998
    by (auto simp add: box_eq_empty cbox_def intro!: image_eqI[where x="\<lambda>(i::'a)\<in>Basis. (c, d)"])
wenzelm@50945
   999
  {
wenzelm@50945
  1000
    fix i :: 'a
wenzelm@50945
  1001
    assume "i \<in> Basis"
hoelzl@50526
  1002
    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
  1003
      unfolding box_eq_empty subset_box by (auto simp: not_le)
wenzelm@50945
  1004
  }
hoelzl@50526
  1005
  note ord = this
hoelzl@50526
  1006
immler@56188
  1007
  show "p division_of (cbox a b)"
hoelzl@50526
  1008
  proof (rule division_ofI)
wenzelm@53399
  1009
    show "finite p"
wenzelm@53399
  1010
      unfolding p_def by (auto intro!: finite_PiE)
wenzelm@50945
  1011
    {
wenzelm@50945
  1012
      fix k
wenzelm@50945
  1013
      assume "k \<in> p"
wenzelm@53015
  1014
      then obtain f where f: "f \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and k: "k = ?B f"
hoelzl@50526
  1015
        by (auto simp: p_def)
immler@56188
  1016
      then show "\<exists>a b. k = cbox a b"
wenzelm@53408
  1017
        by auto
immler@56188
  1018
      have "k \<subseteq> cbox a b \<and> k \<noteq> {}"
immler@56188
  1019
      proof (simp add: k box_eq_empty subset_box not_less, safe)
wenzelm@53374
  1020
        fix i :: 'a
wenzelm@53374
  1021
        assume i: "i \<in> Basis"
wenzelm@50945
  1022
        with f have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
hoelzl@50526
  1023
          by (auto simp: PiE_iff)
wenzelm@53374
  1024
        with i ord[of i]
wenzelm@50945
  1025
        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
  1026
          by auto
hoelzl@50526
  1027
      qed
immler@56188
  1028
      then show "k \<noteq> {}" "k \<subseteq> cbox a b"
wenzelm@53408
  1029
        by auto
wenzelm@50945
  1030
      {
wenzelm@53408
  1031
        fix l
wenzelm@53408
  1032
        assume "l \<in> p"
wenzelm@53015
  1033
        then obtain g where g: "g \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}" and l: "l = ?B g"
wenzelm@50945
  1034
          by (auto simp: p_def)
wenzelm@50945
  1035
        assume "l \<noteq> k"
wenzelm@50945
  1036
        have "\<exists>i\<in>Basis. f i \<noteq> g i"
wenzelm@50945
  1037
        proof (rule ccontr)
wenzelm@53408
  1038
          assume "\<not> ?thesis"
wenzelm@50945
  1039
          with f g have "f = g"
wenzelm@50945
  1040
            by (auto simp: PiE_iff extensional_def intro!: ext)
wenzelm@60420
  1041
          with \<open>l \<noteq> k\<close> show False
wenzelm@50945
  1042
            by (simp add: l k)
wenzelm@50945
  1043
        qed
wenzelm@53408
  1044
        then obtain i where *: "i \<in> Basis" "f i \<noteq> g i" ..
wenzelm@53408
  1045
        then have "f i = (a, c) \<or> f i = (c, d) \<or> f i = (d, b)"
lp15@60384
  1046
                  "g i = (a, c) \<or> g i = (c, d) \<or> g i = (d, b)"
wenzelm@50945
  1047
          using f g by (auto simp: PiE_iff)
wenzelm@53408
  1048
        with * ord[of i] show "interior l \<inter> interior k = {}"
immler@56188
  1049
          by (auto simp add: l k interior_cbox disjoint_interval intro!: bexI[of _ i])
wenzelm@50945
  1050
      }
wenzelm@60420
  1051
      note \<open>k \<subseteq> cbox a b\<close>
wenzelm@50945
  1052
    }
hoelzl@50526
  1053
    moreover
wenzelm@50945
  1054
    {
immler@56188
  1055
      fix x assume x: "x \<in> cbox a b"
hoelzl@50526
  1056
      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
  1057
      proof
wenzelm@53408
  1058
        fix i :: 'a
wenzelm@53408
  1059
        assume "i \<in> Basis"
wenzelm@53399
  1060
        with x ord[of i]
hoelzl@50526
  1061
        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
  1062
            (d \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
immler@56188
  1063
          by (auto simp: cbox_def)
hoelzl@50526
  1064
        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
  1065
          by auto
hoelzl@50526
  1066
      qed
wenzelm@53408
  1067
      then obtain f where
wenzelm@53408
  1068
        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
  1069
        unfolding bchoice_iff ..
wenzelm@53374
  1070
      moreover from f have "restrict f Basis \<in> Basis \<rightarrow>\<^sub>E {(a, c), (c, d), (d, b)}"
hoelzl@50526
  1071
        by auto
hoelzl@50526
  1072
      moreover from f have "x \<in> ?B (restrict f Basis)"
immler@56188
  1073
        by (auto simp: mem_box)
hoelzl@50526
  1074
      ultimately have "\<exists>k\<in>p. x \<in> k"
wenzelm@53408
  1075
        unfolding p_def by blast
wenzelm@53408
  1076
    }
immler@56188
  1077
    ultimately show "\<Union>p = cbox a b"
hoelzl@50526
  1078
      by auto
hoelzl@50526
  1079
  qed
hoelzl@50526
  1080
qed
himmelma@35172
  1081
wenzelm@50945
  1082
lemma partial_division_extend_interval:
immler@56188
  1083
  assumes "p division_of (\<Union>p)" "(\<Union>p) \<subseteq> cbox a b"
immler@56188
  1084
  obtains q where "p \<subseteq> q" "q division_of cbox a (b::'a::euclidean_space)"
wenzelm@50945
  1085
proof (cases "p = {}")
wenzelm@50945
  1086
  case True
immler@56188
  1087
  obtain q where "q division_of (cbox a b)"
wenzelm@53408
  1088
    by (rule elementary_interval)
wenzelm@53399
  1089
  then show ?thesis
lp15@60384
  1090
    using True that by blast
wenzelm@50945
  1091
next
wenzelm@50945
  1092
  case False
wenzelm@50945
  1093
  note p = division_ofD[OF assms(1)]
lp15@60428
  1094
  have div_cbox: "\<forall>k\<in>p. \<exists>q. q division_of cbox a b \<and> k \<in> q"
wenzelm@50945
  1095
  proof
wenzelm@50945
  1096
    case goal1
immler@56188
  1097
    obtain c d where k: "k = cbox c d"
wenzelm@53408
  1098
      using p(4)[OF goal1] by blast
immler@56188
  1099
    have *: "cbox c d \<subseteq> cbox a b" "cbox c d \<noteq> {}"
immler@54775
  1100
      using p(2,3)[OF goal1, unfolded k] using assms(2)
immler@54776
  1101
      by (blast intro: order.trans)+
immler@56188
  1102
    obtain q where "q division_of cbox a b" "cbox c d \<in> q"
wenzelm@53408
  1103
      by (rule partial_division_extend_1[OF *])
wenzelm@53408
  1104
    then show ?case
wenzelm@53408
  1105
      unfolding k by auto
wenzelm@50945
  1106
  qed
immler@56188
  1107
  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
  1108
    using bchoice[OF div_cbox] by blast
lp15@60394
  1109
  { fix x
wenzelm@53408
  1110
    assume x: "x \<in> p"
lp15@60394
  1111
    have "q x division_of \<Union>q x"
wenzelm@50945
  1112
      apply (rule division_ofI)
wenzelm@50945
  1113
      using division_ofD[OF q(1)[OF x]]
wenzelm@50945
  1114
      apply auto
lp15@60394
  1115
      done }
lp15@60394
  1116
  then have "\<And>x. x \<in> p \<Longrightarrow> \<exists>d. d division_of \<Union>(q x - {x})"
lp15@60394
  1117
    by (meson Diff_subset division_of_subset)
wenzelm@60585
  1118
  then have "\<exists>d. d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)"
wenzelm@50945
  1119
    apply -
lp15@60394
  1120
    apply (rule elementary_inters [OF finite_imageI[OF p(1)]])
lp15@60394
  1121
    apply (auto simp: False elementary_inters [OF finite_imageI[OF p(1)]])
wenzelm@50945
  1122
    done
wenzelm@53408
  1123
  then obtain d where d: "d division_of \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p)" ..
lp15@60394
  1124
  have "d \<union> p division_of cbox a b"
wenzelm@50945
  1125
  proof -
lp15@60394
  1126
    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
  1127
    have cbox_eq: "cbox a b = \<Inter>((\<lambda>i. \<Union>(q i - {i})) ` p) \<union> \<Union>p"
lp15@60394
  1128
    proof (rule te[OF False], clarify)
wenzelm@50945
  1129
      fix i
wenzelm@53408
  1130
      assume i: "i \<in> p"
immler@56188
  1131
      show "\<Union>(q i - {i}) \<union> i = cbox a b"
wenzelm@50945
  1132
        using division_ofD(6)[OF q(1)[OF i]] using q(2)[OF i] by auto
wenzelm@50945
  1133
    qed
lp15@60428
  1134
    { fix k
wenzelm@53408
  1135
      assume k: "k \<in> p"
lp15@60428
  1136
      have *: "\<And>u t s. t \<inter> s = {} \<Longrightarrow> u \<subseteq> s \<Longrightarrow> u \<inter> t = {}"
wenzelm@53408
  1137
        by auto
lp15@60428
  1138
      have "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<inter> interior k = {}"
lp15@60428
  1139
      proof (rule *[OF inter_interior_unions_intervals])
wenzelm@50945
  1140
        note qk=division_ofD[OF q(1)[OF k]]
immler@56188
  1141
        show "finite (q k - {k})" "open (interior k)" "\<forall>t\<in>q k - {k}. \<exists>a b. t = cbox a b"
wenzelm@53408
  1142
          using qk by auto
wenzelm@50945
  1143
        show "\<forall>t\<in>q k - {k}. interior k \<inter> interior t = {}"
wenzelm@50945
  1144
          using qk(5) using q(2)[OF k] by auto
lp15@60428
  1145
        show "interior (\<Inter>i\<in>p. \<Union>(q i - {i})) \<subseteq> interior (\<Union>(q k - {k}))"
lp15@60428
  1146
          apply (rule interior_mono)+
wenzelm@53408
  1147
          using k
wenzelm@53408
  1148
          apply auto
wenzelm@53408
  1149
          done
lp15@60428
  1150
      qed } note [simp] = this
lp15@60428
  1151
    show "d \<union> p division_of (cbox a b)"
lp15@60428
  1152
      unfolding cbox_eq
lp15@60428
  1153
      apply (rule division_disjoint_union[OF d assms(1)])
lp15@60428
  1154
      apply (rule inter_interior_unions_intervals)
lp15@60428
  1155
      apply (rule p open_interior ballI)+
lp15@60615
  1156
      apply simp_all
lp15@60428
  1157
      done
lp15@60394
  1158
  qed
lp15@60394
  1159
  then show ?thesis
lp15@60394
  1160
    by (meson Un_upper2 that)
wenzelm@50945
  1161
qed
himmelma@35172
  1162
wenzelm@53399
  1163
lemma elementary_bounded[dest]:
immler@56188
  1164
  fixes s :: "'a::euclidean_space set"
wenzelm@53408
  1165
  shows "p division_of s \<Longrightarrow> bounded s"
immler@56189
  1166
  unfolding division_of_def by (metis bounded_Union bounded_cbox)
wenzelm@53399
  1167
immler@56188
  1168
lemma elementary_subset_cbox:
immler@56188
  1169
  "p division_of s \<Longrightarrow> \<exists>a b. s \<subseteq> cbox a (b::'a::euclidean_space)"
immler@56188
  1170
  by (meson elementary_bounded bounded_subset_cbox)
wenzelm@50945
  1171
wenzelm@50945
  1172
lemma division_union_intervals_exists:
immler@56188
  1173
  fixes a b :: "'a::euclidean_space"
immler@56188
  1174
  assumes "cbox a b \<noteq> {}"
immler@56188
  1175
  obtains p where "(insert (cbox a b) p) division_of (cbox a b \<union> cbox c d)"
immler@56188
  1176
proof (cases "cbox c d = {}")
wenzelm@50945
  1177
  case True
wenzelm@50945
  1178
  show ?thesis
wenzelm@50945
  1179
    apply (rule that[of "{}"])
wenzelm@50945
  1180
    unfolding True
wenzelm@50945
  1181
    using assms
wenzelm@50945
  1182
    apply auto
wenzelm@50945
  1183
    done
wenzelm@50945
  1184
next
wenzelm@50945
  1185
  case False
wenzelm@50945
  1186
  show ?thesis
immler@56188
  1187
  proof (cases "cbox a b \<inter> cbox c d = {}")
wenzelm@50945
  1188
    case True
wenzelm@50945
  1189
    show ?thesis
immler@56188
  1190
      apply (rule that[of "{cbox c d}"])
lp15@60428
  1191
      apply (subst insert_is_Un)
wenzelm@50945
  1192
      apply (rule division_disjoint_union)
lp15@60428
  1193
      using \<open>cbox c d \<noteq> {}\<close> True assms interior_subset
wenzelm@50945
  1194
      apply auto
wenzelm@50945
  1195
      done
wenzelm@50945
  1196
  next
wenzelm@50945
  1197
    case False
immler@56188
  1198
    obtain u v where uv: "cbox a b \<inter> cbox c d = cbox u v"
wenzelm@50945
  1199
      unfolding inter_interval by auto
lp15@60428
  1200
    have uv_sub: "cbox u v \<subseteq> cbox c d" using uv by auto
immler@56188
  1201
    obtain p where "p division_of cbox c d" "cbox u v \<in> p"
lp15@60428
  1202
      by (rule partial_division_extend_1[OF uv_sub False[unfolded uv]])
wenzelm@53408
  1203
    note p = this division_ofD[OF this(1)]
lp15@60428
  1204
    have "interior (cbox a b \<inter> \<Union>(p - {cbox u v})) = interior(cbox u v \<inter> \<Union>(p - {cbox u v}))"
lp15@60428
  1205
      apply (rule arg_cong[of _ _ interior])
lp15@60428
  1206
      using p(8) uv by auto
lp15@60428
  1207
    also have "\<dots> = {}"
lp15@60428
  1208
      unfolding interior_inter
lp15@60428
  1209
      apply (rule inter_interior_unions_intervals)
lp15@60428
  1210
      using p(6) p(7)[OF p(2)] p(3)
lp15@60428
  1211
      apply auto
lp15@60428
  1212
      done
lp15@60428
  1213
    finally have [simp]: "interior (cbox a b) \<inter> interior (\<Union>(p - {cbox u v})) = {}" by simp
lp15@60615
  1214
    have cbe: "cbox a b \<union> cbox c d = cbox a b \<union> \<Union>(p - {cbox u v})"
wenzelm@53399
  1215
      using p(8) unfolding uv[symmetric] by auto
wenzelm@50945
  1216
    show ?thesis
immler@56188
  1217
      apply (rule that[of "p - {cbox u v}"])
lp15@60428
  1218
      apply (simp add: cbe)
lp15@60428
  1219
      apply (subst insert_is_Un)
wenzelm@50945
  1220
      apply (rule division_disjoint_union)
lp15@60428
  1221
      apply (simp_all add: assms division_of_self)
lp15@60428
  1222
      by (metis Diff_subset division_of_subset p(1) p(8))
wenzelm@50945
  1223
  qed
wenzelm@50945
  1224
qed
himmelma@35172
  1225
wenzelm@53399
  1226
lemma division_of_unions:
wenzelm@53399
  1227
  assumes "finite f"
wenzelm@53408
  1228
    and "\<And>p. p \<in> f \<Longrightarrow> p division_of (\<Union>p)"
wenzelm@53399
  1229
    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
  1230
  shows "\<Union>f division_of \<Union>\<Union>f"
lp15@60384
  1231
  using assms
lp15@60384
  1232
  by (auto intro!: division_ofI)
wenzelm@53399
  1233
wenzelm@53399
  1234
lemma elementary_union_interval:
immler@56188
  1235
  fixes a b :: "'a::euclidean_space"
wenzelm@53399
  1236
  assumes "p division_of \<Union>p"
immler@56188
  1237
  obtains q where "q division_of (cbox a b \<union> \<Union>p)"
wenzelm@53399
  1238
proof -
wenzelm@53399
  1239
  note assm = division_ofD[OF assms]
wenzelm@53408
  1240
  have lem1: "\<And>f s. \<Union>\<Union>(f ` s) = \<Union>((\<lambda>x. \<Union>(f x)) ` s)"
wenzelm@53399
  1241
    by auto
wenzelm@53399
  1242
  have lem2: "\<And>f s. f \<noteq> {} \<Longrightarrow> \<Union>{s \<union> t |t. t \<in> f} = s \<union> \<Union>f"
wenzelm@53399
  1243
    by auto
wenzelm@53399
  1244
  {
wenzelm@53399
  1245
    presume "p = {} \<Longrightarrow> thesis"
immler@56188
  1246
      "cbox a b = {} \<Longrightarrow> thesis"
immler@56188
  1247
      "cbox a b \<noteq> {} \<Longrightarrow> interior (cbox a b) = {} \<Longrightarrow> thesis"
immler@56188
  1248
      "p \<noteq> {} \<Longrightarrow> interior (cbox a b)\<noteq>{} \<Longrightarrow> cbox a b \<noteq> {} \<Longrightarrow> thesis"
wenzelm@53399
  1249
    then show thesis by auto
wenzelm@53399
  1250
  next
wenzelm@53399
  1251
    assume as: "p = {}"
immler@56188
  1252
    obtain p where "p division_of (cbox a b)"
wenzelm@53408
  1253
      by (rule elementary_interval)
wenzelm@53399
  1254
    then show thesis
lp15@60384
  1255
      using as that by auto
wenzelm@53399
  1256
  next
immler@56188
  1257
    assume as: "cbox a b = {}"
wenzelm@53399
  1258
    show thesis
lp15@60384
  1259
      using as assms that by auto
wenzelm@53399
  1260
  next
immler@56188
  1261
    assume as: "interior (cbox a b) = {}" "cbox a b \<noteq> {}"
wenzelm@53399
  1262
    show thesis
immler@56188
  1263
      apply (rule that[of "insert (cbox a b) p"],rule division_ofI)
wenzelm@53399
  1264
      unfolding finite_insert
wenzelm@53399
  1265
      apply (rule assm(1)) unfolding Union_insert
wenzelm@53399
  1266
      using assm(2-4) as
wenzelm@53399
  1267
      apply -
immler@54775
  1268
      apply (fast dest: assm(5))+
wenzelm@53399
  1269
      done
wenzelm@53399
  1270
  next
immler@56188
  1271
    assume as: "p \<noteq> {}" "interior (cbox a b) \<noteq> {}" "cbox a b \<noteq> {}"
immler@56188
  1272
    have "\<forall>k\<in>p. \<exists>q. (insert (cbox a b) q) division_of (cbox a b \<union> k)"
lp15@60615
  1273
    proof
wenzelm@53399
  1274
      case goal1
immler@56188
  1275
      from assm(4)[OF this] obtain c d where "k = cbox c d" by blast
wenzelm@53399
  1276
      then show ?case
lp15@60384
  1277
        by (meson as(3) division_union_intervals_exists)
wenzelm@53399
  1278
    qed
immler@56188
  1279
    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
  1280
    note q = division_ofD[OF this[rule_format]]
immler@56188
  1281
    let ?D = "\<Union>{insert (cbox a b) (q k) | k. k \<in> p}"
lp15@60615
  1282
    show thesis
lp15@60428
  1283
    proof (rule that[OF division_ofI])
immler@56188
  1284
      have *: "{insert (cbox a b) (q k) |k. k \<in> p} = (\<lambda>k. insert (cbox a b) (q k)) ` p"
wenzelm@53399
  1285
        by auto
wenzelm@53399
  1286
      show "finite ?D"
lp15@60384
  1287
        using "*" assm(1) q(1) by auto
immler@56188
  1288
      show "\<Union>?D = cbox a b \<union> \<Union>p"
wenzelm@53399
  1289
        unfolding * lem1
immler@56188
  1290
        unfolding lem2[OF as(1), of "cbox a b", symmetric]
wenzelm@53399
  1291
        using q(6)
wenzelm@53399
  1292
        by auto
wenzelm@53399
  1293
      fix k
wenzelm@53408
  1294
      assume k: "k \<in> ?D"
immler@56188
  1295
      then show "k \<subseteq> cbox a b \<union> \<Union>p"
wenzelm@53408
  1296
        using q(2) by auto
wenzelm@53399
  1297
      show "k \<noteq> {}"
wenzelm@53408
  1298
        using q(3) k by auto
immler@56188
  1299
      show "\<exists>a b. k = cbox a b"
wenzelm@53408
  1300
        using q(4) k by auto
wenzelm@53399
  1301
      fix k'
wenzelm@53408
  1302
      assume k': "k' \<in> ?D" "k \<noteq> k'"
immler@56188
  1303
      obtain x where x: "k \<in> insert (cbox a b) (q x)" "x\<in>p"
wenzelm@53408
  1304
        using k by auto
immler@56188
  1305
      obtain x' where x': "k'\<in>insert (cbox a b) (q x')" "x'\<in>p"
wenzelm@53399
  1306
        using k' by auto
wenzelm@53399
  1307
      show "interior k \<inter> interior k' = {}"
wenzelm@53399
  1308
      proof (cases "x = x'")
wenzelm@53399
  1309
        case True
wenzelm@53399
  1310
        show ?thesis
lp15@60384
  1311
          using True k' q(5) x' x by auto
wenzelm@53399
  1312
      next
wenzelm@53399
  1313
        case False
wenzelm@53399
  1314
        {
immler@56188
  1315
          presume "k = cbox a b \<Longrightarrow> ?thesis"
immler@56188
  1316
            and "k' = cbox a b \<Longrightarrow> ?thesis"
immler@56188
  1317
            and "k \<noteq> cbox a b \<Longrightarrow> k' \<noteq> cbox a b \<Longrightarrow> ?thesis"
wenzelm@53399
  1318
          then show ?thesis by auto
wenzelm@53399
  1319
        next
immler@56188
  1320
          assume as': "k  = cbox a b"
wenzelm@53399
  1321
          show ?thesis
lp15@60384
  1322
            using as' k' q(5) x' by auto
wenzelm@53399
  1323
        next
immler@56188
  1324
          assume as': "k' = cbox a b"
wenzelm@53399
  1325
          show ?thesis
lp15@60384
  1326
            using as' k'(2) q(5) x by auto
wenzelm@53399
  1327
        }
immler@56188
  1328
        assume as': "k \<noteq> cbox a b" "k' \<noteq> cbox a b"
immler@56188
  1329
        obtain c d where k: "k = cbox c d"
wenzelm@53408
  1330
          using q(4)[OF x(2,1)] by blast
immler@56188
  1331
        have "interior k \<inter> interior (cbox a b) = {}"
lp15@60384
  1332
          using as' k'(2) q(5) x by auto
wenzelm@53399
  1333
        then have "interior k \<subseteq> interior x"
lp15@60384
  1334
        using interior_subset_union_intervals
lp15@60384
  1335
          by (metis as(2) k q(2) x interior_subset_union_intervals)
wenzelm@53399
  1336
        moreover
immler@56188
  1337
        obtain c d where c_d: "k' = cbox c d"
wenzelm@53408
  1338
          using q(4)[OF x'(2,1)] by blast
immler@56188
  1339
        have "interior k' \<inter> interior (cbox a b) = {}"
lp15@60384
  1340
          using as'(2) q(5) x' by auto
wenzelm@53399
  1341
        then have "interior k' \<subseteq> interior x'"
lp15@60384
  1342
          by (metis as(2) c_d interior_subset_union_intervals q(2) x'(1) x'(2))
wenzelm@53399
  1343
        ultimately show ?thesis
wenzelm@53399
  1344
          using assm(5)[OF x(2) x'(2) False] by auto
wenzelm@53399
  1345
      qed
wenzelm@53399
  1346
    qed
wenzelm@53399
  1347
  }
wenzelm@53399
  1348
qed
himmelma@35172
  1349
himmelma@35172
  1350
lemma elementary_unions_intervals:
wenzelm@53399
  1351
  assumes fin: "finite f"
immler@56188
  1352
    and "\<And>s. s \<in> f \<Longrightarrow> \<exists>a b. s = cbox a (b::'a::euclidean_space)"
wenzelm@53399
  1353
  obtains p where "p division_of (\<Union>f)"
wenzelm@53399
  1354
proof -
wenzelm@53399
  1355
  have "\<exists>p. p division_of (\<Union>f)"
wenzelm@53399
  1356
  proof (induct_tac f rule:finite_subset_induct)
himmelma@35172
  1357
    show "\<exists>p. p division_of \<Union>{}" using elementary_empty by auto
wenzelm@53399
  1358
  next
wenzelm@53399
  1359
    fix x F
wenzelm@53399
  1360
    assume as: "finite F" "x \<notin> F" "\<exists>p. p division_of \<Union>F" "x\<in>f"
wenzelm@53408
  1361
    from this(3) obtain p where p: "p division_of \<Union>F" ..
immler@56188
  1362
    from assms(2)[OF as(4)] obtain a b where x: "x = cbox a b" by blast
wenzelm@53399
  1363
    have *: "\<Union>F = \<Union>p"
wenzelm@53399
  1364
      using division_ofD[OF p] by auto
wenzelm@53399
  1365
    show "\<exists>p. p division_of \<Union>insert x F"
wenzelm@53399
  1366
      using elementary_union_interval[OF p[unfolded *], of a b]
lp15@59765
  1367
      unfolding Union_insert x * by metis
wenzelm@53408
  1368
  qed (insert assms, auto)
wenzelm@53399
  1369
  then show ?thesis
lp15@60384
  1370
    using that by auto
wenzelm@53399
  1371
qed
wenzelm@53399
  1372
wenzelm@53399
  1373
lemma elementary_union:
immler@56188
  1374
  fixes s t :: "'a::euclidean_space set"
lp15@60384
  1375
  assumes "ps division_of s" "pt division_of t"
himmelma@35172
  1376
  obtains p where "p division_of (s \<union> t)"
wenzelm@53399
  1377
proof -
lp15@60384
  1378
  have *: "s \<union> t = \<Union>ps \<union> \<Union>pt"
wenzelm@53399
  1379
    using assms unfolding division_of_def by auto
wenzelm@53399
  1380
  show ?thesis
wenzelm@53408
  1381
    apply (rule elementary_unions_intervals[of "ps \<union> pt"])
lp15@60384
  1382
    using assms apply auto
lp15@60384
  1383
    by (simp add: * that)
wenzelm@53399
  1384
qed
wenzelm@53399
  1385
wenzelm@53399
  1386
lemma partial_division_extend:
immler@56188
  1387
  fixes t :: "'a::euclidean_space set"
wenzelm@53399
  1388
  assumes "p division_of s"
wenzelm@53399
  1389
    and "q division_of t"
wenzelm@53399
  1390
    and "s \<subseteq> t"
wenzelm@53399
  1391
  obtains r where "p \<subseteq> r" and "r division_of t"
wenzelm@53399
  1392
proof -
himmelma@35172
  1393
  note divp = division_ofD[OF assms(1)] and divq = division_ofD[OF assms(2)]
immler@56188
  1394
  obtain a b where ab: "t \<subseteq> cbox a b"
immler@56188
  1395
    using elementary_subset_cbox[OF assms(2)] by auto
immler@56188
  1396
  obtain r1 where "p \<subseteq> r1" "r1 division_of (cbox a b)"
lp15@60384
  1397
    using assms
lp15@60384
  1398
    by (metis ab dual_order.trans partial_division_extend_interval divp(6))
wenzelm@53399
  1399
  note r1 = this division_ofD[OF this(2)]
wenzelm@53408
  1400
  obtain p' where "p' division_of \<Union>(r1 - p)"
wenzelm@53399
  1401
    apply (rule elementary_unions_intervals[of "r1 - p"])
wenzelm@53399
  1402
    using r1(3,6)
wenzelm@53399
  1403
    apply auto
wenzelm@53399
  1404
    done
wenzelm@53399
  1405
  then obtain r2 where r2: "r2 division_of (\<Union>(r1 - p)) \<inter> (\<Union>q)"
lp15@60384
  1406
    by (metis assms(2) divq(6) elementary_inter)
wenzelm@53399
  1407
  {
wenzelm@53399
  1408
    fix x
wenzelm@53399
  1409
    assume x: "x \<in> t" "x \<notin> s"
wenzelm@53399
  1410
    then have "x\<in>\<Union>r1"
wenzelm@53399
  1411
      unfolding r1 using ab by auto
wenzelm@53408
  1412
    then obtain r where r: "r \<in> r1" "x \<in> r"
wenzelm@53408
  1413
      unfolding Union_iff ..
wenzelm@53399
  1414
    moreover
wenzelm@53399
  1415
    have "r \<notin> p"
wenzelm@53399
  1416
    proof
wenzelm@53399
  1417
      assume "r \<in> p"
wenzelm@53399
  1418
      then have "x \<in> s" using divp(2) r by auto
wenzelm@53399
  1419
      then show False using x by auto
wenzelm@53399
  1420
    qed
wenzelm@53399
  1421
    ultimately have "x\<in>\<Union>(r1 - p)" by auto
wenzelm@53399
  1422
  }
wenzelm@53399
  1423
  then have *: "t = \<Union>p \<union> (\<Union>(r1 - p) \<inter> \<Union>q)"
wenzelm@53399
  1424
    unfolding divp divq using assms(3) by auto
wenzelm@53399
  1425
  show ?thesis
wenzelm@53399
  1426
    apply (rule that[of "p \<union> r2"])
wenzelm@53399
  1427
    unfolding *
wenzelm@53399
  1428
    defer
wenzelm@53399
  1429
    apply (rule division_disjoint_union)
wenzelm@53399
  1430
    unfolding divp(6)
wenzelm@53399
  1431
    apply(rule assms r2)+
wenzelm@53399
  1432
  proof -
wenzelm@53399
  1433
    have "interior s \<inter> interior (\<Union>(r1-p)) = {}"
wenzelm@53399
  1434
    proof (rule inter_interior_unions_intervals)
immler@56188
  1435
      show "finite (r1 - p)" and "open (interior s)" and "\<forall>t\<in>r1-p. \<exists>a b. t = cbox a b"
wenzelm@53399
  1436
        using r1 by auto
wenzelm@53399
  1437
      have *: "\<And>s. (\<And>x. x \<in> s \<Longrightarrow> False) \<Longrightarrow> s = {}"
wenzelm@53399
  1438
        by auto
wenzelm@53399
  1439
      show "\<forall>t\<in>r1-p. interior s \<inter> interior t = {}"
wenzelm@53399
  1440
      proof
wenzelm@53399
  1441
        fix m x
wenzelm@53399
  1442
        assume as: "m \<in> r1 - p"
wenzelm@53399
  1443
        have "interior m \<inter> interior (\<Union>p) = {}"
wenzelm@53399
  1444
        proof (rule inter_interior_unions_intervals)
immler@56188
  1445
          show "finite p" and "open (interior m)" and "\<forall>t\<in>p. \<exists>a b. t = cbox a b"
wenzelm@53399
  1446
            using divp by auto
wenzelm@53399
  1447
          show "\<forall>t\<in>p. interior m \<inter> interior t = {}"
lp15@60384
  1448
            by (metis DiffD1 DiffD2 as r1(1) r1(7) set_rev_mp)
wenzelm@53399
  1449
        qed
wenzelm@53399
  1450
        then show "interior s \<inter> interior m = {}"
wenzelm@53399
  1451
          unfolding divp by auto
wenzelm@53399
  1452
      qed
wenzelm@53399
  1453
    qed
wenzelm@53399
  1454
    then show "interior s \<inter> interior (\<Union>(r1-p) \<inter> (\<Union>q)) = {}"
wenzelm@53399
  1455
      using interior_subset by auto
wenzelm@53399
  1456
  qed auto
wenzelm@53399
  1457
qed
wenzelm@53399
  1458
himmelma@35172
  1459
wenzelm@60420
  1460
subsection \<open>Tagged (partial) divisions.\<close>
himmelma@35172
  1461
wenzelm@53408
  1462
definition tagged_partial_division_of (infixr "tagged'_partial'_division'_of" 40)
wenzelm@53408
  1463
  where "s tagged_partial_division_of i \<longleftrightarrow>
wenzelm@53408
  1464
    finite s \<and>
immler@56188
  1465
    (\<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
  1466
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1467
      interior k1 \<inter> interior k2 = {})"
wenzelm@53408
  1468
wenzelm@53408
  1469
lemma tagged_partial_division_ofD[dest]:
wenzelm@53408
  1470
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1471
  shows "finite s"
wenzelm@53408
  1472
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1473
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1474
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1475
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow>
wenzelm@53408
  1476
      (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow> interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1477
  using assms unfolding tagged_partial_division_of_def by blast+
wenzelm@53408
  1478
wenzelm@53408
  1479
definition tagged_division_of (infixr "tagged'_division'_of" 40)
wenzelm@53408
  1480
  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
  1481
huffman@44167
  1482
lemma tagged_division_of_finite: "s tagged_division_of i \<Longrightarrow> finite s"
himmelma@35172
  1483
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1484
himmelma@35172
  1485
lemma tagged_division_of:
wenzelm@53408
  1486
  "s tagged_division_of i \<longleftrightarrow>
wenzelm@53408
  1487
    finite s \<and>
immler@56188
  1488
    (\<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
  1489
    (\<forall>x1 k1 x2 k2. (x1, k1) \<in> s \<and> (x2, k2) \<in> s \<and> (x1, k1) \<noteq> (x2, k2) \<longrightarrow>
wenzelm@53408
  1490
      interior k1 \<inter> interior k2 = {}) \<and>
wenzelm@53408
  1491
    (\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1492
  unfolding tagged_division_of_def tagged_partial_division_of_def by auto
himmelma@35172
  1493
wenzelm@53408
  1494
lemma tagged_division_ofI:
wenzelm@53408
  1495
  assumes "finite s"
wenzelm@53408
  1496
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1497
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1498
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1499
    and "\<And>x1 k1 x2 k2. (x1,k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1500
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1501
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
himmelma@35172
  1502
  shows "s tagged_division_of i"
wenzelm@53408
  1503
  unfolding tagged_division_of
lp15@60384
  1504
  using assms
lp15@60384
  1505
  apply auto
lp15@60384
  1506
  apply fastforce+
wenzelm@53408
  1507
  done
wenzelm@53408
  1508
lp15@60384
  1509
lemma tagged_division_ofD[dest]:  (*FIXME USE A LOCALE*)
wenzelm@53408
  1510
  assumes "s tagged_division_of i"
wenzelm@53408
  1511
  shows "finite s"
wenzelm@53408
  1512
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> x \<in> k"
wenzelm@53408
  1513
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> i"
immler@56188
  1514
    and "\<And>x k. (x,k) \<in> s \<Longrightarrow> \<exists>a b. k = cbox a b"
wenzelm@53408
  1515
    and "\<And>x1 k1 x2 k2. (x1, k1) \<in> s \<Longrightarrow> (x2, k2) \<in> s \<Longrightarrow> (x1, k1) \<noteq> (x2, k2) \<Longrightarrow>
wenzelm@53408
  1516
      interior k1 \<inter> interior k2 = {}"
wenzelm@53408
  1517
    and "(\<Union>{k. \<exists>x. (x,k) \<in> s} = i)"
wenzelm@53408
  1518
  using assms unfolding tagged_division_of by blast+
wenzelm@53408
  1519
wenzelm@53408
  1520
lemma division_of_tagged_division:
wenzelm@53408
  1521
  assumes "s tagged_division_of i"
wenzelm@53408
  1522
  shows "(snd ` s) division_of i"
wenzelm@53408
  1523
proof (rule division_ofI)
wenzelm@53408
  1524
  note assm = tagged_division_ofD[OF assms]
wenzelm@53408
  1525
  show "\<Union>(snd ` s) = i" "finite (snd ` s)"
wenzelm@53408
  1526
    using assm by auto
wenzelm@53408
  1527
  fix k
wenzelm@53408
  1528
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1529
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1530
    by auto
immler@56188
  1531
  then show "k \<subseteq> i" "k \<noteq> {}" "\<exists>a b. k = cbox a b"
wenzelm@53408
  1532
    using assm by fastforce+
wenzelm@53408
  1533
  fix k'
wenzelm@53408
  1534
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1535
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1536
    by auto
wenzelm@53408
  1537
  then show "interior k \<inter> interior k' = {}"
lp15@60384
  1538
    using assm(5) k'(2) xk by blast
himmelma@35172
  1539
qed
himmelma@35172
  1540
wenzelm@53408
  1541
lemma partial_division_of_tagged_division:
wenzelm@53408
  1542
  assumes "s tagged_partial_division_of i"
himmelma@35172
  1543
  shows "(snd ` s) division_of \<Union>(snd ` s)"
wenzelm@53408
  1544
proof (rule division_ofI)
wenzelm@53408
  1545
  note assm = tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1546
  show "finite (snd ` s)" "\<Union>(snd ` s) = \<Union>(snd ` s)"
wenzelm@53408
  1547
    using assm by auto
wenzelm@53408
  1548
  fix k
wenzelm@53408
  1549
  assume k: "k \<in> snd ` s"
wenzelm@53408
  1550
  then obtain xk where xk: "(xk, k) \<in> s"
wenzelm@53408
  1551
    by auto
immler@56188
  1552
  then show "k \<noteq> {}" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>(snd ` s)"
wenzelm@53408
  1553
    using assm by auto
wenzelm@53408
  1554
  fix k'
wenzelm@53408
  1555
  assume k': "k' \<in> snd ` s" "k \<noteq> k'"
wenzelm@53408
  1556
  from this(1) obtain xk' where xk': "(xk', k') \<in> s"
wenzelm@53408
  1557
    by auto
wenzelm@53408
  1558
  then show "interior k \<inter> interior k' = {}"
lp15@60384
  1559
    using assm(5) k'(2) xk by auto
himmelma@35172
  1560
qed
himmelma@35172
  1561
wenzelm@53408
  1562
lemma tagged_partial_division_subset:
wenzelm@53408
  1563
  assumes "s tagged_partial_division_of i"
wenzelm@53408
  1564
    and "t \<subseteq> s"
himmelma@35172
  1565
  shows "t tagged_partial_division_of i"
wenzelm@53408
  1566
  using assms
wenzelm@53408
  1567
  unfolding tagged_partial_division_of_def
wenzelm@53408
  1568
  using finite_subset[OF assms(2)]
wenzelm@53408
  1569
  by blast
wenzelm@53408
  1570
wenzelm@53408
  1571
lemma setsum_over_tagged_division_lemma:
wenzelm@53408
  1572
  assumes "p tagged_division_of i"
immler@56188
  1573
    and "\<And>u v. cbox u v \<noteq> {} \<Longrightarrow> content (cbox u v) = 0 \<Longrightarrow> d (cbox u v) = 0"
himmelma@35172
  1574
  shows "setsum (\<lambda>(x,k). d k) p = setsum d (snd ` p)"
wenzelm@53408
  1575
proof -
wenzelm@53408
  1576
  have *: "(\<lambda>(x,k). d k) = d \<circ> snd"
wenzelm@53408
  1577
    unfolding o_def by (rule ext) auto
hoelzl@57129
  1578
  note assm = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1579
  show ?thesis
wenzelm@53408
  1580
    unfolding *
haftmann@57418
  1581
  proof (rule setsum.reindex_nontrivial[symmetric])
wenzelm@53408
  1582
    show "finite p"
wenzelm@53408
  1583
      using assm by auto
wenzelm@53408
  1584
    fix x y
hoelzl@57129
  1585
    assume "x\<in>p" "y\<in>p" "x\<noteq>y" "snd x = snd y"
immler@56188
  1586
    obtain a b where ab: "snd x = cbox a b"
wenzelm@60420
  1587
      using assm(4)[of "fst x" "snd x"] \<open>x\<in>p\<close> by auto
wenzelm@53408
  1588
    have "(fst x, snd y) \<in> p" "(fst x, snd y) \<noteq> y"
wenzelm@60420
  1589
      by (metis pair_collapse \<open>x\<in>p\<close> \<open>snd x = snd y\<close> \<open>x \<noteq> y\<close>)+
wenzelm@60420
  1590
    with \<open>x\<in>p\<close> \<open>y\<in>p\<close> have "interior (snd x) \<inter> interior (snd y) = {}"
hoelzl@57129
  1591
      by (intro assm(5)[of "fst x" _ "fst y"]) auto
immler@56188
  1592
    then have "content (cbox a b) = 0"
wenzelm@60420
  1593
      unfolding \<open>snd x = snd y\<close>[symmetric] ab content_eq_0_interior by auto
immler@56188
  1594
    then have "d (cbox a b) = 0"
wenzelm@60420
  1595
      using assm(2)[of "fst x" "snd x"] \<open>x\<in>p\<close> ab[symmetric] by (intro assms(2)) auto
wenzelm@53408
  1596
    then show "d (snd x) = 0"
wenzelm@53408
  1597
      unfolding ab by auto
wenzelm@53408
  1598
  qed
wenzelm@53408
  1599
qed
wenzelm@53408
  1600
wenzelm@53408
  1601
lemma tag_in_interval: "p tagged_division_of i \<Longrightarrow> (x, k) \<in> p \<Longrightarrow> x \<in> i"
wenzelm@53408
  1602
  by auto
himmelma@35172
  1603
himmelma@35172
  1604
lemma tagged_division_of_empty: "{} tagged_division_of {}"
himmelma@35172
  1605
  unfolding tagged_division_of by auto
himmelma@35172
  1606
wenzelm@53408
  1607
lemma tagged_partial_division_of_trivial[simp]: "p tagged_partial_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1608
  unfolding tagged_partial_division_of_def by auto
himmelma@35172
  1609
wenzelm@53408
  1610
lemma tagged_division_of_trivial[simp]: "p tagged_division_of {} \<longleftrightarrow> p = {}"
himmelma@35172
  1611
  unfolding tagged_division_of by auto
himmelma@35172
  1612
immler@56188
  1613
lemma tagged_division_of_self: "x \<in> cbox a b \<Longrightarrow> {(x,cbox a b)} tagged_division_of (cbox a b)"
wenzelm@53408
  1614
  by (rule tagged_division_ofI) auto
himmelma@35172
  1615
immler@56188
  1616
lemma tagged_division_of_self_real: "x \<in> {a .. b::real} \<Longrightarrow> {(x,{a .. b})} tagged_division_of {a .. b}"
immler@56188
  1617
  unfolding box_real[symmetric]
immler@56188
  1618
  by (rule tagged_division_of_self)
immler@56188
  1619
himmelma@35172
  1620
lemma tagged_division_union:
wenzelm@53408
  1621
  assumes "p1 tagged_division_of s1"
wenzelm@53408
  1622
    and "p2 tagged_division_of s2"
wenzelm@53408
  1623
    and "interior s1 \<inter> interior s2 = {}"
himmelma@35172
  1624
  shows "(p1 \<union> p2) tagged_division_of (s1 \<union> s2)"
wenzelm@53408
  1625
proof (rule tagged_division_ofI)
wenzelm@53408
  1626
  note p1 = tagged_division_ofD[OF assms(1)]
wenzelm@53408
  1627
  note p2 = tagged_division_ofD[OF assms(2)]
wenzelm@53408
  1628
  show "finite (p1 \<union> p2)"
wenzelm@53408
  1629
    using p1(1) p2(1) by auto
wenzelm@53408
  1630
  show "\<Union>{k. \<exists>x. (x, k) \<in> p1 \<union> p2} = s1 \<union> s2"
wenzelm@53408
  1631
    using p1(6) p2(6) by blast
wenzelm@53408
  1632
  fix x k
wenzelm@53408
  1633
  assume xk: "(x, k) \<in> p1 \<union> p2"
immler@56188
  1634
  show "x \<in> k" "\<exists>a b. k = cbox a b"
wenzelm@53408
  1635
    using xk p1(2,4) p2(2,4) by auto
wenzelm@53408
  1636
  show "k \<subseteq> s1 \<union> s2"
wenzelm@53408
  1637
    using xk p1(3) p2(3) by blast
wenzelm@53408
  1638
  fix x' k'
wenzelm@53408
  1639
  assume xk': "(x', k') \<in> p1 \<union> p2" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1640
  have *: "\<And>a b. a \<subseteq> s1 \<Longrightarrow> b \<subseteq> s2 \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1641
    using assms(3) interior_mono by blast
wenzelm@53408
  1642
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1643
    apply (cases "(x, k) \<in> p1")
lp15@60384
  1644
    apply (meson "*" UnE assms(1) assms(2) p1(5) tagged_division_ofD(3) xk'(1) xk'(2))
lp15@60384
  1645
    by (metis "*" UnE assms(1) assms(2) inf_sup_aci(1) p2(5) tagged_division_ofD(3) xk xk'(1) xk'(2))
wenzelm@53408
  1646
qed
himmelma@35172
  1647
himmelma@35172
  1648
lemma tagged_division_unions:
wenzelm@53408
  1649
  assumes "finite iset"
wenzelm@53408
  1650
    and "\<forall>i\<in>iset. pfn i tagged_division_of i"
wenzelm@53408
  1651
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior(i1) \<inter> interior(i2) = {}"
himmelma@35172
  1652
  shows "\<Union>(pfn ` iset) tagged_division_of (\<Union>iset)"
wenzelm@53408
  1653
proof (rule tagged_division_ofI)
himmelma@35172
  1654
  note assm = tagged_division_ofD[OF assms(2)[rule_format]]
wenzelm@53408
  1655
  show "finite (\<Union>(pfn ` iset))"
wenzelm@53408
  1656
    apply (rule finite_Union)
wenzelm@53408
  1657
    using assms
wenzelm@53408
  1658
    apply auto
wenzelm@53408
  1659
    done
wenzelm@53408
  1660
  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
  1661
    by blast
wenzelm@53408
  1662
  also have "\<dots> = \<Union>iset"
wenzelm@53408
  1663
    using assm(6) by auto
wenzelm@53399
  1664
  finally show "\<Union>{k. \<exists>x. (x, k) \<in> \<Union>(pfn ` iset)} = \<Union>iset" .
wenzelm@53408
  1665
  fix x k
wenzelm@53408
  1666
  assume xk: "(x, k) \<in> \<Union>(pfn ` iset)"
wenzelm@53408
  1667
  then obtain i where i: "i \<in> iset" "(x, k) \<in> pfn i"
wenzelm@53408
  1668
    by auto
immler@56188
  1669
  show "x \<in> k" "\<exists>a b. k = cbox a b" "k \<subseteq> \<Union>iset"
wenzelm@53408
  1670
    using assm(2-4)[OF i] using i(1) by auto
wenzelm@53408
  1671
  fix x' k'
wenzelm@53408
  1672
  assume xk': "(x', k') \<in> \<Union>(pfn ` iset)" "(x, k) \<noteq> (x', k')"
wenzelm@53408
  1673
  then obtain i' where i': "i' \<in> iset" "(x', k') \<in> pfn i'"
wenzelm@53408
  1674
    by auto
wenzelm@53408
  1675
  have *: "\<And>a b. i \<noteq> i' \<Longrightarrow> a \<subseteq> i \<Longrightarrow> b \<subseteq> i' \<Longrightarrow> interior a \<inter> interior b = {}"
wenzelm@53408
  1676
    using i(1) i'(1)
wenzelm@53408
  1677
    using assms(3)[rule_format] interior_mono
wenzelm@53408
  1678
    by blast
wenzelm@53408
  1679
  show "interior k \<inter> interior k' = {}"
wenzelm@53408
  1680
    apply (cases "i = i'")
lp15@60384
  1681
    using assm(5) i' i(2) xk'(2) apply blast
lp15@60384
  1682
    using "*" assm(3) i' i by auto
himmelma@35172
  1683
qed
himmelma@35172
  1684
himmelma@35172
  1685
lemma tagged_partial_division_of_union_self:
wenzelm@53408
  1686
  assumes "p tagged_partial_division_of s"
himmelma@35172
  1687
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1688
  apply (rule tagged_division_ofI)
wenzelm@53408
  1689
  using tagged_partial_division_ofD[OF assms]
wenzelm@53408
  1690
  apply auto
wenzelm@53408
  1691
  done
wenzelm@53408
  1692
wenzelm@53408
  1693
lemma tagged_division_of_union_self:
wenzelm@53408
  1694
  assumes "p tagged_division_of s"
wenzelm@53408
  1695
  shows "p tagged_division_of (\<Union>(snd ` p))"
wenzelm@53408
  1696
  apply (rule tagged_division_ofI)
wenzelm@53408
  1697
  using tagged_division_ofD[OF assms]
wenzelm@53408
  1698
  apply auto
wenzelm@53408
  1699
  done
wenzelm@53408
  1700
himmelma@35172
  1701
wenzelm@60420
  1702
subsection \<open>Fine-ness of a partition w.r.t. a gauge.\<close>
himmelma@35172
  1703
wenzelm@53408
  1704
definition fine  (infixr "fine" 46)
wenzelm@53408
  1705
  where "d fine s \<longleftrightarrow> (\<forall>(x,k) \<in> s. k \<subseteq> d x)"
wenzelm@53408
  1706
wenzelm@53408
  1707
lemma fineI:
wenzelm@53408
  1708
  assumes "\<And>x k. (x, k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1709
  shows "d fine s"
wenzelm@53408
  1710
  using assms unfolding fine_def by auto
wenzelm@53408
  1711
wenzelm@53408
  1712
lemma fineD[dest]:
wenzelm@53408
  1713
  assumes "d fine s"
wenzelm@53408
  1714
  shows "\<And>x k. (x,k) \<in> s \<Longrightarrow> k \<subseteq> d x"
wenzelm@53408
  1715
  using assms unfolding fine_def by auto
himmelma@35172
  1716
himmelma@35172
  1717
lemma fine_inter: "(\<lambda>x. d1 x \<inter> d2 x) fine p \<longleftrightarrow> d1 fine p \<and> d2 fine p"
himmelma@35172
  1718
  unfolding fine_def by auto
himmelma@35172
  1719
himmelma@35172
  1720
lemma fine_inters:
wenzelm@60585
  1721
 "(\<lambda>x. \<Inter>{f d x | d.  d \<in> s}) fine p \<longleftrightarrow> (\<forall>d\<in>s. (f d) fine p)"
himmelma@35172
  1722
  unfolding fine_def by blast
himmelma@35172
  1723
wenzelm@53408
  1724
lemma fine_union: "d fine p1 \<Longrightarrow> d fine p2 \<Longrightarrow> d fine (p1 \<union> p2)"
himmelma@35172
  1725
  unfolding fine_def by blast
himmelma@35172
  1726
wenzelm@53408
  1727
lemma fine_unions: "(\<And>p. p \<in> ps \<Longrightarrow> d fine p) \<Longrightarrow> d fine (\<Union>ps)"
himmelma@35172
  1728
  unfolding fine_def by auto
himmelma@35172
  1729
wenzelm@53408
  1730
lemma fine_subset: "p \<subseteq> q \<Longrightarrow> d fine q \<Longrightarrow> d fine p"
himmelma@35172
  1731
  unfolding fine_def by blast
himmelma@35172
  1732
wenzelm@53408
  1733
wenzelm@60420
  1734
subsection \<open>Gauge integral. Define on compact intervals first, then use a limit.\<close>
himmelma@35172
  1735
wenzelm@53408
  1736
definition has_integral_compact_interval (infixr "has'_integral'_compact'_interval" 46)
wenzelm@53408
  1737
  where "(f has_integral_compact_interval y) i \<longleftrightarrow>
wenzelm@53408
  1738
    (\<forall>e>0. \<exists>d. gauge d \<and>
wenzelm@53408
  1739
      (\<forall>p. p tagged_division_of i \<and> d fine p \<longrightarrow>
wenzelm@53408
  1740
        norm (setsum (\<lambda>(x,k). content k *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1741
wenzelm@53408
  1742
definition has_integral ::
immler@56188
  1743
    "('n::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> 'n set \<Rightarrow> bool"
wenzelm@53408
  1744
  (infixr "has'_integral" 46)
wenzelm@53408
  1745
  where "(f has_integral y) i \<longleftrightarrow>
immler@56188
  1746
    (if \<exists>a b. i = cbox a b
wenzelm@53408
  1747
     then (f has_integral_compact_interval y) i
immler@56188
  1748
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1749
      (\<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral_compact_interval z) (cbox a b) \<and>
wenzelm@53408
  1750
        norm (z - y) < e)))"
himmelma@35172
  1751
himmelma@35172
  1752
lemma has_integral:
immler@56188
  1753
  "(f has_integral y) (cbox a b) \<longleftrightarrow>
wenzelm@53408
  1754
    (\<forall>e>0. \<exists>d. gauge d \<and>
immler@56188
  1755
      (\<forall>p. p tagged_division_of (cbox a b) \<and> d fine p \<longrightarrow>
wenzelm@53408
  1756
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
wenzelm@53408
  1757
  unfolding has_integral_def has_integral_compact_interval_def
wenzelm@53408
  1758
  by auto
wenzelm@53408
  1759
immler@56188
  1760
lemma has_integral_real:
immler@56188
  1761
  "(f has_integral y) {a .. b::real} \<longleftrightarrow>
immler@56188
  1762
    (\<forall>e>0. \<exists>d. gauge d \<and>
immler@56188
  1763
      (\<forall>p. p tagged_division_of {a .. b} \<and> d fine p \<longrightarrow>
immler@56188
  1764
        norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f x) p - y) < e))"
immler@56188
  1765
  unfolding box_real[symmetric]
immler@56188
  1766
  by (rule has_integral)
immler@56188
  1767
wenzelm@53408
  1768
lemma has_integralD[dest]:
immler@56188
  1769
  assumes "(f has_integral y) (cbox a b)"
wenzelm@53408
  1770
    and "e > 0"
wenzelm@53408
  1771
  obtains d where "gauge d"
immler@56188
  1772
    and "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> d fine p \<Longrightarrow>
wenzelm@53408
  1773
      norm (setsum (\<lambda>(x,k). content(k) *\<^sub>R f(x)) p - y) < e"
himmelma@35172
  1774
  using assms unfolding has_integral by auto
himmelma@35172
  1775
himmelma@35172
  1776
lemma has_integral_alt:
wenzelm@53408
  1777
  "(f has_integral y) i \<longleftrightarrow>
immler@56188
  1778
    (if \<exists>a b. i = cbox a b
wenzelm@53408
  1779
     then (f has_integral y) i
immler@56188
  1780
     else (\<forall>e>0. \<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1781
      (\<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
  1782
  unfolding has_integral
wenzelm@53408
  1783
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1784
  by auto
himmelma@35172
  1785
himmelma@35172
  1786
lemma has_integral_altD:
wenzelm@53408
  1787
  assumes "(f has_integral y) i"
immler@56188
  1788
    and "\<not> (\<exists>a b. i = cbox a b)"
wenzelm@53408
  1789
    and "e>0"
wenzelm@53408
  1790
  obtains B where "B > 0"
immler@56188
  1791
    and "\<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  1792
      (\<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
  1793
  using assms
wenzelm@53408
  1794
  unfolding has_integral
wenzelm@53408
  1795
  unfolding has_integral_compact_interval_def has_integral_def
wenzelm@53408
  1796
  by auto
wenzelm@53408
  1797
wenzelm@53408
  1798
definition integrable_on (infixr "integrable'_on" 46)
wenzelm@53408
  1799
  where "f integrable_on i \<longleftrightarrow> (\<exists>y. (f has_integral y) i)"
wenzelm@53408
  1800
wenzelm@53408
  1801
definition "integral i f = (SOME y. (f has_integral y) i)"
himmelma@35172
  1802
wenzelm@53409
  1803
lemma integrable_integral[dest]: "f integrable_on i \<Longrightarrow> (f has_integral (integral i f)) i"
wenzelm@53409
  1804
  unfolding integrable_on_def integral_def by (rule someI_ex)
himmelma@35172
  1805
himmelma@35172
  1806
lemma has_integral_integrable[intro]: "(f has_integral i) s \<Longrightarrow> f integrable_on s"
himmelma@35172
  1807
  unfolding integrable_on_def by auto
himmelma@35172
  1808
wenzelm@53409
  1809
lemma has_integral_integral: "f integrable_on s \<longleftrightarrow> (f has_integral (integral s f)) s"
himmelma@35172
  1810
  by auto
himmelma@35172
  1811
himmelma@35172
  1812
lemma setsum_content_null:
immler@56188
  1813
  assumes "content (cbox a b) = 0"
immler@56188
  1814
    and "p tagged_division_of (cbox a b)"
himmelma@35172
  1815
  shows "setsum (\<lambda>(x,k). content k *\<^sub>R f x) p = (0::'a::real_normed_vector)"
haftmann@57418
  1816
proof (rule setsum.neutral, rule)
wenzelm@53409
  1817
  fix y
wenzelm@53409
  1818
  assume y: "y \<in> p"
wenzelm@53409
  1819
  obtain x k where xk: "y = (x, k)"
wenzelm@53409
  1820
    using surj_pair[of y] by blast
himmelma@35172
  1821
  note assm = tagged_division_ofD(3-4)[OF assms(2) y[unfolded xk]]
immler@56188
  1822
  from this(2) obtain c d where k: "k = cbox c d" by blast
wenzelm@53409
  1823
  have "(\<lambda>(x, k). content k *\<^sub>R f x) y = content k *\<^sub>R f x"
wenzelm@53409
  1824
    unfolding xk by auto
wenzelm@53409
  1825
  also have "\<dots> = 0"
wenzelm@53409
  1826
    using content_subset[OF assm(1)[unfolded k]] content_pos_le[of c d]
wenzelm@53409
  1827
    unfolding assms(1) k
wenzelm@53409
  1828
    by auto
himmelma@35172
  1829
  finally show "(\<lambda>(x, k). content k *\<^sub>R f x) y = 0" .
himmelma@35172
  1830
qed
himmelma@35172
  1831
wenzelm@53409
  1832
wenzelm@60420
  1833
subsection \<open>Some basic combining lemmas.\<close>
himmelma@35172
  1834
himmelma@35172
  1835
lemma tagged_division_unions_exists:
wenzelm@53409
  1836
  assumes "finite iset"
wenzelm@53409
  1837
    and "\<forall>i\<in>iset. \<exists>p. p tagged_division_of i \<and> d fine p"
wenzelm@53409
  1838
    and "\<forall>i1\<in>iset. \<forall>i2\<in>iset. i1 \<noteq> i2 \<longrightarrow> interior i1 \<inter> interior i2 = {}"
wenzelm@53409
  1839
    and "\<Union>iset = i"
wenzelm@53409
  1840
   obtains p where "p tagged_division_of i" and "d fine p"
wenzelm@53409
  1841
proof -
wenzelm@53409
  1842
  obtain pfn where pfn:
wenzelm@53409
  1843
    "\<And>x. x \<in> iset \<Longrightarrow> pfn x tagged_division_of x"
wenzelm@53409
  1844
    "\<And>x. x \<in> iset \<Longrightarrow> d fine pfn x"
wenzelm@53409
  1845
    using bchoice[OF assms(2)] by auto
wenzelm@53409
  1846
  show thesis
wenzelm@53409
  1847
    apply (rule_tac p="\<Union>(pfn ` iset)" in that)
lp15@60384
  1848
    using assms(1) assms(3) assms(4) pfn(1) tagged_division_unions apply force
lp15@60384
  1849
    by (metis (mono_tags, lifting) fine_unions imageE pfn(2))
himmelma@35172
  1850
qed
himmelma@35172
  1851
wenzelm@53409
  1852
wenzelm@60420
  1853
subsection \<open>The set we're concerned with must be closed.\<close>
himmelma@35172
  1854
wenzelm@53409
  1855
lemma division_of_closed:
immler@56189
  1856
  fixes i :: "'n::euclidean_space set"
wenzelm@53409
  1857
  shows "s division_of i \<Longrightarrow> closed i"
nipkow@44890
  1858
  unfolding division_of_def by fastforce
himmelma@35172
  1859
wenzelm@60420
  1860
subsection \<open>General bisection principle for intervals; might be useful elsewhere.\<close>
himmelma@35172
  1861
wenzelm@53409
  1862
lemma interval_bisection_step:
immler@56188
  1863
  fixes type :: "'a::euclidean_space"
wenzelm@53409
  1864
  assumes "P {}"
wenzelm@53409
  1865
    and "\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P (s \<union> t)"
immler@56188
  1866
    and "\<not> P (cbox a (b::'a))"
immler@56188
  1867
  obtains c d where "\<not> P (cbox c d)"
wenzelm@53409
  1868
    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
  1869
proof -
immler@56188
  1870
  have "cbox a b \<noteq> {}"
immler@54776
  1871
    using assms(1,3) by metis
wenzelm@53409
  1872
  then have ab: "\<And>i. i\<in>Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
immler@56188
  1873
    by (force simp: mem_box)
lp15@60428
  1874
  { fix f
lp15@60428
  1875
    have "\<lbrakk>finite f;
lp15@60428
  1876
           \<And>s. s\<in>f \<Longrightarrow> P s;
lp15@60428
  1877
           \<And>s. s\<in>f \<Longrightarrow> \<exists>a b. s = cbox a b;
lp15@60428
  1878
           \<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
  1879
    proof (induct f rule: finite_induct)
wenzelm@53409
  1880
      case empty
wenzelm@53409
  1881
      show ?case
wenzelm@53409
  1882
        using assms(1) by auto
wenzelm@53409
  1883
    next
wenzelm@53409
  1884
      case (insert x f)
wenzelm@53409
  1885
      show ?case
wenzelm@53409
  1886
        unfolding Union_insert
wenzelm@53409
  1887
        apply (rule assms(2)[rule_format])
lp15@60384
  1888
        using inter_interior_unions_intervals [of f "interior x"]
lp15@60384
  1889
        apply (auto simp: insert)
lp15@60428
  1890
        by (metis IntI empty_iff insert.hyps(2) insert.prems(3) insert_iff)
lp15@60428
  1891
    qed
lp15@60428
  1892
  } note UN_cases = this
immler@56188
  1893
  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
  1894
    (c\<bullet>i = (a\<bullet>i + b\<bullet>i) / 2) \<and> (d\<bullet>i = b\<bullet>i)}"
hoelzl@50526
  1895
  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
  1896
  {
immler@56188
  1897
    presume "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d) \<Longrightarrow> False"
wenzelm@53409
  1898
    then show thesis
wenzelm@53409
  1899
      unfolding atomize_not not_all
lp15@60384
  1900
      by (blast intro: that)
wenzelm@53409
  1901
  }
immler@56188
  1902
  assume as: "\<forall>c d. ?PP c d \<longrightarrow> P (cbox c d)"
wenzelm@60585
  1903
  have "P (\<Union>?A)"
lp15@60428
  1904
  proof (rule UN_cases)
immler@56188
  1905
    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
  1906
      (\<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
  1907
    have "?A \<subseteq> ?B"
wenzelm@53409
  1908
    proof
wenzelm@53409
  1909
      case goal1
lp15@60615
  1910
      then obtain c d
lp15@60428
  1911
        where x:  "x = cbox c d"
lp15@60428
  1912
                  "\<And>i. i \<in> Basis \<Longrightarrow>
lp15@60428
  1913
                        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60428
  1914
                        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i" by blast
wenzelm@53409
  1915
      show "x \<in> ?B"
lp15@60428
  1916
        unfolding image_iff x
wenzelm@53409
  1917
        apply (rule_tac x="{i. i\<in>Basis \<and> c\<bullet>i = a\<bullet>i}" in bexI)
lp15@60428
  1918
        apply (rule arg_cong2 [where f = cbox])
lp15@60428
  1919
        using x(2) ab
lp15@60428
  1920
        apply (auto simp add: euclidean_eq_iff[where 'a='a])
lp15@60428
  1921
        by fastforce
wenzelm@53409
  1922
    qed
wenzelm@53409
  1923
    then show "finite ?A"
wenzelm@53409
  1924
      by (rule finite_subset) auto
lp15@60428
  1925
  next
wenzelm@53409
  1926
    fix s
wenzelm@53409
  1927
    assume "s \<in> ?A"
lp15@60428
  1928
    then obtain c d
lp15@60428
  1929
      where s: "s = cbox c d"
lp15@60428
  1930
               "\<And>i. i \<in> Basis \<Longrightarrow>
lp15@60428
  1931
                     c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60428
  1932
                     c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
wenzelm@53409
  1933
      by blast
wenzelm@53409
  1934
    show "P s"
wenzelm@53409
  1935
      unfolding s
wenzelm@53409
  1936
      apply (rule as[rule_format])
lp15@60394
  1937
      using ab s(2) by force
immler@56188
  1938
    show "\<exists>a b. s = cbox a b"
wenzelm@53409
  1939
      unfolding s by auto
wenzelm@53409
  1940
    fix t
wenzelm@53409
  1941
    assume "t \<in> ?A"
wenzelm@53409
  1942
    then obtain e f where t:
immler@56188
  1943
      "t = cbox e f"
wenzelm@53409
  1944
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  1945
        e \<bullet> i = a \<bullet> i \<and> f \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
wenzelm@53409
  1946
        e \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> f \<bullet> i = b \<bullet> i"
wenzelm@53409
  1947
      by blast
wenzelm@53409
  1948
    assume "s \<noteq> t"
wenzelm@53409
  1949
    then have "\<not> (c = e \<and> d = f)"
wenzelm@53409
  1950
      unfolding s t by auto
wenzelm@53409
  1951
    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
  1952
      unfolding euclidean_eq_iff[where 'a='a] by auto
wenzelm@53409
  1953
    then have i: "c\<bullet>i \<noteq> e\<bullet>i" "d\<bullet>i \<noteq> f\<bullet>i"
lp15@60394
  1954
      using s(2) t(2) apply fastforce
wenzelm@60420
  1955
      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
  1956
    have *: "\<And>s t. (\<And>a. a \<in> s \<Longrightarrow> a \<in> t \<Longrightarrow> False) \<Longrightarrow> s \<inter> t = {}"
wenzelm@53409
  1957
      by auto
wenzelm@53409
  1958
    show "interior s \<inter> interior t = {}"
immler@56188
  1959
      unfolding s t interior_cbox
wenzelm@53409
  1960
    proof (rule *)
wenzelm@53409
  1961
      fix x
immler@54775
  1962
      assume "x \<in> box c d" "x \<in> box e f"
wenzelm@53409
  1963
      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
  1964
        unfolding mem_box using i'
lp15@60394
  1965
        by force+
lp15@60394
  1966
      show False  using s(2)[OF i']
lp15@60394
  1967
      proof safe
wenzelm@53409
  1968
        assume as: "c \<bullet> i = a \<bullet> i" "d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2"
wenzelm@53409
  1969
        show False
wenzelm@53409
  1970
          using t(2)[OF i'] and i x unfolding as by (fastforce simp add:field_simps)
wenzelm@53409
  1971
      next
wenzelm@53409
  1972
        assume as: "c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2" "d \<bullet> i = b \<bullet> i"
wenzelm@53409
  1973
        show False
wenzelm@53409
  1974
          using t(2)[OF i'] and i x unfolding as by(fastforce simp add:field_simps)
wenzelm@53409
  1975
      qed
wenzelm@53409
  1976
    qed
wenzelm@53409
  1977
  qed
wenzelm@60585
  1978
  also have "\<Union>?A = cbox a b"
wenzelm@53409
  1979
  proof (rule set_eqI,rule)
wenzelm@53409
  1980
    fix x
wenzelm@53409
  1981
    assume "x \<in> \<Union>?A"
wenzelm@53409
  1982
    then obtain c d where x:
immler@56188
  1983
      "x \<in> cbox c d"
wenzelm@53409
  1984
      "\<And>i. i \<in> Basis \<Longrightarrow>
wenzelm@53409
  1985
        c \<bullet> i = a \<bullet> i \<and> d \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<or>
lp15@60615
  1986
        c \<bullet> i = (a \<bullet> i + b \<bullet> i) / 2 \<and> d \<bullet> i = b \<bullet> i"
lp15@60394
  1987
      by blast
immler@56188
  1988
    show "x\<in>cbox a b"
immler@56188
  1989
      unfolding mem_box
wenzelm@53409
  1990
    proof safe
wenzelm@53409
  1991
      fix i :: 'a
wenzelm@53409
  1992
      assume i: "i \<in> Basis"
wenzelm@53409
  1993
      then show "a \<bullet> i \<le> x \<bullet> i" "x \<bullet> i \<le> b \<bullet> i"
immler@56188
  1994
        using x(2)[OF i] x(1)[unfolded mem_box,THEN bspec, OF i] by auto
wenzelm@53409
  1995
    qed
wenzelm@53409
  1996
  next
wenzelm@53409
  1997
    fix x
immler@56188
  1998
    assume x: "x \<in> cbox a b"
wenzelm@53409
  1999
    have "\<forall>i\<in>Basis.
wenzelm@53409
  2000
      \<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
  2001
      (is "\<forall>i\<in>Basis. \<exists>c d. ?P i c d")
immler@56188
  2002
      unfolding mem_box
hoelzl@50526
  2003
    proof
wenzelm@53409
  2004
      fix i :: 'a
wenzelm@53409
  2005
      assume i: "i \<in> Basis"
hoelzl@50526
  2006
      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
  2007
        using x[unfolded mem_box,THEN bspec, OF i] by auto
wenzelm@53409
  2008
      then show "\<exists>c d. ?P i c d"
wenzelm@53409
  2009
        by blast
hoelzl@50526
  2010
    qed
wenzelm@53409
  2011
    then show "x\<in>\<Union>?A"
hoelzl@50526
  2012
      unfolding Union_iff Bex_def mem_Collect_eq choice_Basis_iff
lp15@60384
  2013
      apply auto
immler@56188
  2014
      apply (rule_tac x="cbox xa xaa" in exI)
immler@56188
  2015
      unfolding mem_box
wenzelm@53409
  2016
      apply auto
wenzelm@53409
  2017
      done
wenzelm@53409
  2018
  qed
wenzelm@53409
  2019
  finally show False
wenzelm@53409
  2020
    using assms by auto
wenzelm@53409
  2021
qed
wenzelm@53409
  2022
wenzelm@53409
  2023
lemma interval_bisection:
immler@56188
  2024
  fixes type :: "'a::euclidean_space"
wenzelm@53409
  2025
  assumes "P {}"
wenzelm@53409
  2026
    and "(\<forall>s t. P s \<and> P t \<and> interior(s) \<inter> interior(t) = {} \<longrightarrow> P(s \<union> t))"
immler@56188
  2027
    and "\<not> P (cbox a (b::'a))"
immler@56188
  2028
  obtains x where "x \<in> cbox a b"
immler@56188
  2029
    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
  2030
proof -
immler@56188
  2031
  have "\<forall>x. \<exists>y. \<not> P (cbox (fst x) (snd x)) \<longrightarrow> (\<not> P (cbox (fst y) (snd y)) \<and>
hoelzl@50526
  2032
    (\<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
  2033
       2 * (snd y\<bullet>i - fst y\<bullet>i) \<le> snd x\<bullet>i - fst x\<bullet>i))"
wenzelm@53409
  2034
  proof
wenzelm@53409
  2035
    case goal1
lp15@60394
  2036
    show ?case
wenzelm@53409
  2037
    proof -
immler@56188
  2038
      presume "\<not> P (cbox (fst x) (snd x)) \<Longrightarrow> ?thesis"
immler@56188
  2039
      then show ?thesis by (cases "P (cbox (fst x) (snd x))") auto
wenzelm@53409
  2040
    next
immler@56188
  2041
      assume as: "\<not> P (cbox (fst x) (snd x))"
immler@56188
  2042
      obtain c d where "\<not> P (cbox c d)"
wenzelm@53409
  2043
        "\<forall>i\<in>Basis.
wenzelm@53409
  2044
           fst x \<bullet> i \<le> c \<bullet> i \<and>
wenzelm@53409
  2045
           c \<bullet> i \<le> d \<bullet> i \<and>
wenzelm@53409
  2046
           d \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@53409
  2047
           2 * (d \<bullet> i - c \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i"
wenzelm@53409
  2048
        by (rule interval_bisection_step[of P, OF assms(1-2) as])
wenzelm@53409
  2049
      then show ?thesis
wenzelm@53409
  2050
        apply -
wenzelm@53409
  2051
        apply (rule_tac x="(c,d)" in exI)
wenzelm@53409
  2052
        apply auto
wenzelm@53409
  2053
        done
wenzelm@53409
  2054
    qed
wenzelm@53409
  2055
  qed
wenzelm@55751
  2056
  then obtain f where f:
wenzelm@55751
  2057
    "\<forall>x.
immler@56188
  2058
      \<not> P (cbox (fst x) (snd x)) \<longrightarrow>
immler@56188
  2059
      \<not> P (cbox (fst (f x)) (snd (f x))) \<and>
wenzelm@55751
  2060
        (\<forall>i\<in>Basis.
wenzelm@55751
  2061
            fst x \<bullet> i \<le> fst (f x) \<bullet> i \<and>
wenzelm@55751
  2062
            fst (f x) \<bullet> i \<le> snd (f x) \<bullet> i \<and>
wenzelm@55751
  2063
            snd (f x) \<bullet> i \<le> snd x \<bullet> i \<and>
wenzelm@55751
  2064
            2 * (snd (f x) \<bullet> i - fst (f x) \<bullet> i) \<le> snd x \<bullet> i - fst x \<bullet> i)"
wenzelm@53409
  2065
    apply -
wenzelm@53409
  2066
    apply (drule choice)
wenzelm@55751
  2067
    apply blast
wenzelm@55751
  2068
    done
wenzelm@53409
  2069
  def AB \<equiv> "\<lambda>n. (f ^^ n) (a,b)"
wenzelm@53409
  2070
  def A \<equiv> "\<lambda>n. fst(AB n)"
wenzelm@53409
  2071
  def B \<equiv> "\<lambda>n. snd(AB n)"
wenzelm@53409
  2072
  note ab_def = A_def B_def AB_def
immler@56188
  2073
  have "A 0 = a" "B 0 = b" "\<And>n. \<not> P (cbox (A(Suc n)) (B(Suc n))) \<and>
wenzelm@53399
  2074
    (\<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
  2075
    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
  2076
  proof -
wenzelm@53409
  2077
    show "A 0 = a" "B 0 = b"
wenzelm@53409
  2078
      unfolding ab_def by auto
wenzelm@53409
  2079
    case goal3
wenzelm@53409
  2080
    note S = ab_def funpow.simps o_def id_apply
wenzelm@53409
  2081
    show ?case
wenzelm@53409
  2082
    proof (induct n)
wenzelm@53409
  2083
      case 0
wenzelm@53409
  2084
      then show ?case
wenzelm@53409
  2085
        unfolding S
wenzelm@53409
  2086
        apply (rule f[rule_format]) using assms(3)
wenzelm@53409
  2087
        apply auto
wenzelm@53409
  2088
        done
wenzelm@53409
  2089
    next
wenzelm@53409
  2090
      case (Suc n)
wenzelm@53409
  2091
      show ?case
wenzelm@53409
  2092
        unfolding S
wenzelm@53409
  2093
        apply (rule f[rule_format])
wenzelm@53409
  2094
        using Suc
wenzelm@53409
  2095
        unfolding S
wenzelm@53409
  2096
        apply auto
wenzelm@53409
  2097
        done
wenzelm@53409
  2098
    qed
wenzelm@53409
  2099
  qed
wenzelm@53409
  2100
  note AB = this(1-2) conjunctD2[OF this(3),rule_format]
wenzelm@53409
  2101
immler@56188
  2102
  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
  2103
  proof -
wenzelm@53409
  2104
    case goal1
wenzelm@53409
  2105
    obtain n where n: "(\<Sum>i\<in>Basis. b \<bullet> i - a \<bullet> i) / e < 2 ^ n"
wenzelm@53409
  2106
      using real_arch_pow2[of "(setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis) / e"] ..
wenzelm@53409
  2107
    show ?case
lp15@60396
  2108
    proof (rule exI [where x=n], clarify)
wenzelm@53409
  2109
      fix x y
immler@56188
  2110
      assume xy: "x\<in>cbox (A n) (B n)" "y\<in>cbox (A n) (B n)"
wenzelm@53409
  2111
      have "dist x y \<le> setsum (\<lambda>i. abs((x - y)\<bullet>i)) Basis"
wenzelm@53409
  2112
        unfolding dist_norm by(rule norm_le_l1)
hoelzl@50526
  2113
      also have "\<dots> \<le> setsum (\<lambda>i. B n\<bullet>i - A n\<bullet>i) Basis"
wenzelm@53409
  2114
      proof (rule setsum_mono)
wenzelm@53409
  2115
        fix i :: 'a
wenzelm@53409
  2116
        assume i: "i \<in> Basis"
wenzelm@53409
  2117
        show "\<bar>(x - y) \<bullet> i\<bar> \<le> B n \<bullet> i - A n \<bullet> i"
immler@56188
  2118
          using xy[unfolded mem_box,THEN bspec, OF i]
wenzelm@53409
  2119
          by (auto simp: inner_diff_left)
wenzelm@53409
  2120
      qed
wenzelm@53409
  2121
      also have "\<dots> \<le> setsum (\<lambda>i. b\<bullet>i - a\<bullet>i) Basis / 2^n"
wenzelm@53409
  2122
        unfolding setsum_divide_distrib
wenzelm@53409
  2123
      proof (rule setsum_mono)
wenzelm@53409
  2124
        case goal1
wenzelm@53409
  2125
        then show ?case
wenzelm@53409
  2126
        proof (induct n)
wenzelm@53409
  2127
          case 0
wenzelm@53409
  2128
          then show ?case
wenzelm@53409
  2129
            unfolding AB by auto
wenzelm@53409
  2130
        next
wenzelm@53409
  2131
          case (Suc n)
wenzelm@53409
  2132
          have "B (Suc n) \<bullet> i - A (Suc n) \<bullet> i \<le> (B n \<bullet> i - A n \<bullet> i) / 2"
hoelzl@37489
  2133
            using AB(4)[of i n] using goal1 by auto
wenzelm@53409
  2134
          also have "\<dots> \<le> (b \<bullet> i - a \<bullet> i) / 2 ^ Suc n"
wenzelm@53409
  2135
            using Suc by (auto simp add:field_simps)
wenzelm@53409
  2136
          finally show ?case .
wenzelm@53409
  2137
        qed
wenzelm@53409
  2138
      qed
wenzelm@53409
  2139
      also have "\<dots> < e"
wenzelm@53409
  2140
        using n using goal1 by (auto simp add:field_simps)
wenzelm@53409
  2141
      finally show "dist x y < e" .
wenzelm@53409
  2142
    qed
wenzelm@53409
  2143
  qed
wenzelm@53409
  2144
  {
wenzelm@53409
  2145
    fix n m :: nat
immler@56188
  2146
    assume "m \<le> n" then have "cbox (A n) (B n) \<subseteq> cbox (A m) (B m)"
hoelzl@54411
  2147
    proof (induction rule: inc_induct)
wenzelm@53409
  2148
      case (step i)
wenzelm@53409
  2149
      show ?case
immler@56188
  2150
        using AB(4) by (intro order_trans[OF step.IH] subset_box_imp) auto
wenzelm@53409
  2151
    qed simp
wenzelm@53409
  2152
  } note ABsubset = this
immler@56188
  2153
  have "\<exists>a. \<forall>n. a\<in> cbox (A n) (B n)"
immler@56188
  2154
    by (rule decreasing_closed_nest[rule_format,OF closed_cbox _ ABsubset interv])
immler@54776
  2155
      (metis nat.exhaust AB(1-3) assms(1,3))
immler@56188
  2156
  then obtain x0 where x0: "\<And>n. x0 \<in> cbox (A n) (B n)"
wenzelm@53409
  2157
    by blast
wenzelm@53409
  2158
  show thesis
wenzelm@53409
  2159
  proof (rule that[rule_format, of x0])
immler@56188
  2160
    show "x0\<in>cbox a b"
wenzelm@53409
  2161
      using x0[of 0] unfolding AB .
wenzelm@53409
  2162
    fix e :: real
wenzelm@53409
  2163
    assume "e > 0"
wenzelm@53409
  2164
    from interv[OF this] obtain n
immler@56188
  2165
      where n: "\<forall>x\<in>cbox (A n) (B n). \<forall>y\<in>cbox (A n) (B n). dist x y < e" ..
lp15@60396
  2166
    have "\<not> P (cbox (A n) (B n))"
lp15@60396
  2167
      apply (cases "0 < n")
lp15@60396
  2168
      using AB(3)[of "n - 1"] assms(3) AB(1-2)
lp15@60396
  2169
      apply auto
lp15@60396
  2170
      done
lp15@60396
  2171
    moreover have "cbox (A n) (B n) \<subseteq> ball x0 e"
lp15@60396
  2172
      using n using x0[of n] by auto
lp15@60396
  2173
    moreover have "cbox (A n) (B n) \<subseteq> cbox a b"
lp15@60396
  2174
      unfolding AB(1-2)[symmetric] by (rule ABsubset) auto
lp15@60396
  2175
    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
  2176
      apply (rule_tac x="A n" in exI)
wenzelm@53409
  2177
      apply (rule_tac x="B n" in exI)
lp15@60396
  2178
      apply (auto simp: x0)
lp15@60396
  2179
      done
wenzelm@53409
  2180
  qed
wenzelm@53409
  2181
qed
wenzelm@53409
  2182
himmelma@35172
  2183
wenzelm@60420
  2184
subsection \<open>Cousin's lemma.\<close>
himmelma@35172
  2185
wenzelm@53409
  2186
lemma fine_division_exists:
immler@56188
  2187
  fixes a b :: "'a::euclidean_space"
wenzelm@53409
  2188
  assumes "gauge g"
immler@56188
  2189
  obtains p where "p tagged_division_of (cbox a b)" "g fine p"
immler@56188
  2190
proof -
immler@56188
  2191
  presume "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p) \<Longrightarrow> False"
immler@56188
  2192
  then obtain p where "p tagged_division_of (cbox a b)" "g fine p"
wenzelm@53410
  2193
    by blast
wenzelm@53409
  2194
  then show thesis ..
wenzelm@53409
  2195
next
immler@56188
  2196
  assume as: "\<not> (\<exists>p. p tagged_division_of (cbox a b) \<and> g fine p)"
wenzelm@55751
  2197
  obtain x where x:
lp15@60428
  2198
      "x \<in> (cbox a b)"
lp15@60428
  2199
      "\<And>e. 0 < e \<Longrightarrow>
lp15@60428
  2200
        \<exists>c d.
lp15@60428
  2201
          x \<in> cbox c d \<and>
lp15@60428
  2202
          cbox c d \<subseteq> ball x e \<and>
lp15@60428
  2203
          cbox c d \<subseteq> (cbox a b) \<and>
lp15@60428
  2204
          \<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
lp15@60428
  2205
    apply (rule interval_bisection[of "\<lambda>s. \<exists>p. p tagged_division_of s \<and> g fine p", OF _ _ as])
lp15@60428
  2206
    apply (simp add: fine_def)
lp15@60428
  2207
    apply (metis tagged_division_union fine_union)
lp15@60428
  2208
    apply (auto simp: )
lp15@60428
  2209
    done
wenzelm@53410
  2210
  obtain e where e: "e > 0" "ball x e \<subseteq> g x"
wenzelm@53409
  2211
    using gaugeD[OF assms, of x] unfolding open_contains_ball by auto
lp15@60615
  2212
  from x(2)[OF e(1)]
lp15@60396
  2213
  obtain c d where c_d: "x \<in> cbox c d"
lp15@60396
  2214
                        "cbox c d \<subseteq> ball x e"
lp15@60396
  2215
                        "cbox c d \<subseteq> cbox a b"
lp15@60396
  2216
                        "\<not> (\<exists>p. p tagged_division_of cbox c d \<and> g fine p)"
wenzelm@53410
  2217
    by blast
immler@56188
  2218
  have "g fine {(x, cbox c d)}"
wenzelm@53409
  2219
    unfolding fine_def using e using c_d(2) by auto
wenzelm@53410
  2220
  then show False
wenzelm@53410
  2221
    using tagged_division_of_self[OF c_d(1)] using c_d by auto
wenzelm@53409
  2222
qed
wenzelm@53409
  2223
immler@56188
  2224
lemma fine_division_exists_real:
immler@56188
  2225
  fixes a b :: real
immler@56188
  2226
  assumes "gauge g"
immler@56188
  2227
  obtains p where "p tagged_division_of {a .. b}" "g fine p"
immler@56188
  2228
  by (metis assms box_real(2) fine_division_exists)
himmelma@35172
  2229
wenzelm@60420
  2230
subsection \<open>Basic theorems about integrals.\<close>
himmelma@35172
  2231
wenzelm@53409
  2232
lemma has_integral_unique:
immler@56188
  2233
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2234
  assumes "(f has_integral k1) i"
wenzelm@53410
  2235
    and "(f has_integral k2) i"
wenzelm@53409
  2236
  shows "k1 = k2"
wenzelm@53410
  2237
proof (rule ccontr)
wenzelm@53842
  2238
  let ?e = "norm (k1 - k2) / 2"
wenzelm@53410
  2239
  assume as:"k1 \<noteq> k2"
wenzelm@53410
  2240
  then have e: "?e > 0"
wenzelm@53410
  2241
    by auto
wenzelm@53410
  2242
  have lem: "\<And>f::'n \<Rightarrow> 'a.  \<And>a b k1 k2.
immler@56188
  2243
    (f has_integral k1) (cbox a b) \<Longrightarrow> (f has_integral k2) (cbox a b) \<Longrightarrow> k1 \<noteq> k2 \<Longrightarrow> False"
wenzelm@53410
  2244
  proof -
wenzelm@53410
  2245
    case goal1
wenzelm@53410
  2246
    let ?e = "norm (k1 - k2) / 2"
wenzelm@53410
  2247
    from goal1(3) have e: "?e > 0" by auto
wenzelm@55751
  2248
    obtain d1 where d1:
wenzelm@55751
  2249
        "gauge d1"
immler@56188
  2250
        "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
wenzelm@55751
  2251
          d1 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k1) < norm (k1 - k2) / 2"
wenzelm@55751
  2252
      by (rule has_integralD[OF goal1(1) e]) blast
wenzelm@55751
  2253
    obtain d2 where d2:
wenzelm@55751
  2254
        "gauge d2"
immler@56188
  2255
        "\<And>p. p tagged_division_of cbox a b \<Longrightarrow>
wenzelm@55751
  2256
          d2 fine p \<Longrightarrow> norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - k2) < norm (k1 - k2) / 2"
wenzelm@55751
  2257
      by (rule has_integralD[OF goal1(2) e]) blast
wenzelm@55751
  2258
    obtain p where p:
immler@56188
  2259
        "p tagged_division_of cbox a b"
wenzelm@55751
  2260
        "(\<lambda>x. d1 x \<inter> d2 x) fine p"
wenzelm@55751
  2261
      by (rule fine_division_exists[OF gauge_inter[OF d1(1) d2(1)]])
wenzelm@53410
  2262
    let ?c = "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
wenzelm@53410
  2263
    have "norm (k1 - k2) \<le> norm (?c - k2) + norm (?c - k1)"
wenzelm@53410
  2264
      using norm_triangle_ineq4[of "k1 - ?c" "k2 - ?c"]
wenzelm@53410
  2265
      by (auto simp add:algebra_simps norm_minus_commute)
himmelma@35172
  2266
    also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2267
      apply (rule add_strict_mono)
wenzelm@53410
  2268
      apply (rule_tac[!] d2(2) d1(2))
wenzelm@53410
  2269
      using p unfolding fine_def
wenzelm@53410
  2270
      apply auto
wenzelm@53410
  2271
      done
himmelma@35172
  2272
    finally show False by auto
wenzelm@53410
  2273
  qed
wenzelm@53410
  2274
  {
immler@56188
  2275
    presume "\<not> (\<exists>a b. i = cbox a b) \<Longrightarrow> False"
wenzelm@53410
  2276
    then show False
lp15@60396
  2277
      using as assms lem by blast
wenzelm@53410
  2278
  }
immler@56188
  2279
  assume as: "\<not> (\<exists>a b. i = cbox a b)"
wenzelm@55751
  2280
  obtain B1 where B1:
wenzelm@55751
  2281
      "0 < B1"
immler@56188
  2282
      "\<And>a b. ball 0 B1 \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2283
        \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
wenzelm@55751
  2284
          norm (z - k1) < norm (k1 - k2) / 2"
wenzelm@55751
  2285
    by (rule has_integral_altD[OF assms(1) as,OF e]) blast
wenzelm@55751
  2286
  obtain B2 where B2:
wenzelm@55751
  2287
      "0 < B2"
immler@56188
  2288
      "\<And>a b. ball 0 B2 \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2289
        \<exists>z. ((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b) \<and>
wenzelm@55751
  2290
          norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@55751
  2291
    by (rule has_integral_altD[OF assms(2) as,OF e]) blast
immler@56188
  2292
  have "\<exists>a b::'n. ball 0 B1 \<union> ball 0 B2 \<subseteq> cbox a b"
immler@56188
  2293
    apply (rule bounded_subset_cbox)
wenzelm@53410
  2294
    using bounded_Un bounded_ball
wenzelm@53410
  2295
    apply auto
wenzelm@53410
  2296
    done
immler@56188
  2297
  then obtain a b :: 'n where ab: "ball 0 B1 \<subseteq> cbox a b" "ball 0 B2 \<subseteq> cbox a b"
wenzelm@53410
  2298
    by blast
wenzelm@53410
  2299
  obtain w where w:
immler@56188
  2300
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral w) (cbox a b)"
wenzelm@53410
  2301
    "norm (w - k1) < norm (k1 - k2) / 2"
wenzelm@53410
  2302
    using B1(2)[OF ab(1)] by blast
wenzelm@53410
  2303
  obtain z where z:
immler@56188
  2304
    "((\<lambda>x. if x \<in> i then f x else 0) has_integral z) (cbox a b)"
wenzelm@53410
  2305
    "norm (z - k2) < norm (k1 - k2) / 2"
wenzelm@53410
  2306
    using B2(2)[OF ab(2)] by blast
wenzelm@53410
  2307
  have "z = w"
wenzelm@53410
  2308
    using lem[OF w(1) z(1)] by auto
wenzelm@53410
  2309
  then have "norm (k1 - k2) \<le> norm (z - k2) + norm (w - k1)"
wenzelm@53410
  2310
    using norm_triangle_ineq4 [of "k1 - w" "k2 - z"]
wenzelm@53410
  2311
    by (auto simp add: norm_minus_commute)
wenzelm@53410
  2312
  also have "\<dots> < norm (k1 - k2) / 2 + norm (k1 - k2) / 2"
wenzelm@53410
  2313
    apply (rule add_strict_mono)
wenzelm@53410
  2314
    apply (rule_tac[!] z(2) w(2))
wenzelm@53410
  2315
    done
wenzelm@53410
  2316
  finally show False by auto
wenzelm@53410
  2317
qed
wenzelm@53410
  2318
wenzelm@53410
  2319
lemma integral_unique [intro]: "(f has_integral y) k \<Longrightarrow> integral k f = y"
wenzelm@53410
  2320
  unfolding integral_def
wenzelm@53410
  2321
  by (rule some_equality) (auto intro: has_integral_unique)
wenzelm@53410
  2322
wenzelm@53410
  2323
lemma has_integral_is_0:
immler@56188
  2324
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2325
  assumes "\<forall>x\<in>s. f x = 0"
wenzelm@53410
  2326
  shows "(f has_integral 0) s"
wenzelm@53410
  2327
proof -
wenzelm@53410
  2328
  have lem: "\<And>a b. \<And>f::'n \<Rightarrow> 'a.
immler@56188
  2329
    (\<forall>x\<in>cbox a b. f(x) = 0) \<Longrightarrow> (f has_integral 0) (cbox a b)"
wenzelm@53410
  2330
    unfolding has_integral
lp15@60396
  2331
  proof clarify
wenzelm@53410
  2332
    fix a b e
wenzelm@53410
  2333
    fix f :: "'n \<Rightarrow> 'a"
immler@56188
  2334
    assume as: "\<forall>x\<in>cbox a b. f x = 0" "0 < (e::real)"
lp15@60396
  2335
    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
  2336
    proof -
wenzelm@53410
  2337
      case goal1
wenzelm@53410
  2338
      have "(\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) = 0"
haftmann@57418
  2339
      proof (rule setsum.neutral, rule)
wenzelm@53410
  2340
        fix x
wenzelm@53410
  2341
        assume x: "x \<in> p"
wenzelm@53410
  2342
        have "f (fst x) = 0"
wenzelm@53410
  2343
          using tagged_division_ofD(2-3)[OF goal1(1), of "fst x" "snd x"] using as x by auto
wenzelm@53410
  2344
        then show "(\<lambda>(x, k). content k *\<^sub>R f x) x = 0"
wenzelm@53410
  2345
          apply (subst surjective_pairing[of x])
wenzelm@53410
  2346
          unfolding split_conv
wenzelm@53410
  2347
          apply auto
wenzelm@53410
  2348
          done
wenzelm@53410
  2349
      qed
wenzelm@53410
  2350
      then show ?case
wenzelm@53410
  2351
        using as by auto
lp15@60396
  2352
    qed
lp15@60396
  2353
    then show "\<exists>d. gauge d \<and>
lp15@60396
  2354
                   (\<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
  2355
      by auto
wenzelm@53410
  2356
  qed
wenzelm@53410
  2357
  {
immler@56188
  2358
    presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
lp15@60396
  2359
    with assms lem show ?thesis
lp15@60396
  2360
      by blast
wenzelm@53410
  2361
  }
wenzelm@53410
  2362
  have *: "(\<lambda>x. if x \<in> s then f x else 0) = (\<lambda>x. 0)"
wenzelm@53410
  2363
    apply (rule ext)
wenzelm@53410
  2364
    using assms
wenzelm@53410
  2365
    apply auto
wenzelm@53410
  2366
    done
immler@56188
  2367
  assume "\<not> (\<exists>a b. s = cbox a b)"
wenzelm@53410
  2368
  then show ?thesis
lp15@60396
  2369
    using lem
lp15@60396
  2370
    by (subst has_integral_alt) (force simp add: *)
wenzelm@53410
  2371
qed
himmelma@35172
  2372
immler@56188
  2373
lemma has_integral_0[simp]: "((\<lambda>x::'n::euclidean_space. 0) has_integral 0) s"
wenzelm@53410
  2374
  by (rule has_integral_is_0) auto
himmelma@35172
  2375
himmelma@35172
  2376
lemma has_integral_0_eq[simp]: "((\<lambda>x. 0) has_integral i) s \<longleftrightarrow> i = 0"
himmelma@35172
  2377
  using has_integral_unique[OF has_integral_0] by auto
himmelma@35172
  2378
wenzelm@53410
  2379
lemma has_integral_linear:
immler@56188
  2380
  fixes f :: "'n::euclidean_space \<Rightarrow> 'a::real_normed_vector"
wenzelm@53410
  2381
  assumes "(f has_integral y) s"
wenzelm@53410
  2382
    and "bounded_linear h"
wenzelm@53410
  2383
  shows "((h o f) has_integral ((h y))) s"
wenzelm@53410
  2384
proof -
wenzelm@53410
  2385
  interpret bounded_linear h
wenzelm@53410
  2386
    using assms(2) .
wenzelm@53410
  2387
  from pos_bounded obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
wenzelm@53410
  2388
    by blast
wenzelm@53410
  2389
  have lem: "\<And>(f :: 'n \<Rightarrow> 'a) y a b.
immler@56188
  2390
    (f has_integral y) (cbox a b) \<Longrightarrow> ((h o f) has_integral h y) (cbox a b)"
lp15@60396
  2391
  unfolding has_integral
lp15@60396
  2392
  proof clarify
wenzelm@53410
  2393
    case goal1
wenzelm@53410
  2394
    from pos_bounded
wenzelm@53410
  2395
    obtain B where B: "0 < B" "\<And>x. norm (h x) \<le> norm x * B"
wenzelm@53410
  2396
      by blast
lp15@60428
  2397
    have "e / B > 0" using goal1(2) B by simp
lp15@60615
  2398
    then obtain g
lp15@60428
  2399
      where g: "gauge g"
lp15@60428
  2400
               "\<And>p. p tagged_division_of (cbox a b) \<Longrightarrow> g fine p \<Longrightarrow>
lp15@60428
  2401
                    norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R f x) - y) < e / B"
lp15@60428
  2402
        using goal1(1) by auto
lp15@60428
  2403
    { fix p
immler@56188
  2404
      assume as: "p tagged_division_of (cbox a b)" "g fine p"
lp15@60428
  2405
      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
  2406
        by auto
lp15@60428
  2407
      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
  2408
        unfolding o_def unfolding scaleR[symmetric] hc by simp
wenzelm@53410
  2409
      also have "\<dots> = h (\<Sum>(x, k)\<in>p. content k *\<^sub>R f x)"
wenzelm@53410
  2410
        using setsum[of "\<lambda>(x,k). content k *\<^sub>R f x" p] using as by auto
lp15@60428
  2411
      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
  2412
      then have "norm ((\<Sum>(x, k)\<in>p. content k *\<^sub>R (h \<circ> f) x) - h y) < e"
lp15@60428
  2413
        apply (simp add: diff[symmetric])
wenzelm@53410
  2414
        apply (rule le_less_trans[OF B(2)])
wenzelm@53410
  2415
        using g(2)[OF as] B(1)
wenzelm@53410
  2416
        apply (auto simp add: field_simps)
wenzelm@53410
  2417
        done
lp15@60428
  2418
    }
lp15@60428
  2419
    with g show ?case
lp15@60428
  2420
      by (rule_tac x=g in exI) auto
wenzelm@53410
  2421
  qed
wenzelm@53410
  2422
  {
immler@56188
  2423
    presume "\<not> (\<exists>a b. s = cbox a b) \<Longrightarrow> ?thesis"
wenzelm@53410
  2424
    then show ?thesis
lp15@60396
  2425
      using assms(1) lem by blast
wenzelm@53410
  2426
  }
immler@56188
  2427
  assume as: "\<not> (\<exists>a b. s = cbox a b)"
wenzelm@53410
  2428
  then show ?thesis
lp15@60396
  2429
  proof (subst has_integral_alt, clarsimp)
wenzelm@53410
  2430
    fix e :: real
wenzelm@53410
  2431
    assume e: "e > 0"
nipkow@56541
  2432
    have *: "0 < e/B" using e B(1) by simp
wenzelm@53410
  2433
    obtain M where M:
wenzelm@53410
  2434
      "M > 0"
immler@56188
  2435
      "\<And>a b. ball 0 M \<subseteq> cbox a b \<Longrightarrow>
immler@56188
  2436
        \<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
  2437
      using has_integral_altD[OF assms(1) as *] by blast
immler@56188
  2438
    show "\<exists>B>0. \<forall>a b. ball 0 B \<subseteq> cbox a b \<longrightarrow>
immler@56188
  2439
      (\<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
  2440
    proof (rule_tac x=M in exI, clarsimp simp add: M)
wenzelm@53410
  2441
      case goal1
wenzelm@53410
  2442
      obtain z where z:
immler@56188
  2443
        "((\<lambda>x. if x \<in> s then f x else 0) has_integral z) (cbox a b)"
wenzelm@53410
  2444
        "norm (z - y) < e / B"
wenzelm@53410
  2445
        using M(2)[OF goal1(1)] by blast
wenzelm@53410
  2446
      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
  2447
        using zero by auto
wenzelm@53410
  2448
      show ?case
wenzelm@53410
  2449
        apply (rule_tac x="h z" in exI)
lp15@60396
  2450
        apply (simp add: "*" lem z(1))
lp15@60396
  2451
        by (metis B diff le_less_trans pos_less_divide_eq z(2))
wenzelm@53410
  2452
    qed
wenzelm@53410
  2453
  qed
wenzelm@53410
  2454
qed
wenzelm@53410
  2455
lp15@60615
  2456
lemma has_integral_scaleR_left:
hoelzl@57447
  2457
  "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x *\<^sub>R c) has_integral (y *\<^sub>R c)) s"
hoelzl@57447
  2458
  using has_integral_linear[OF _ bounded_linear_scaleR_left] by (simp add: comp_def)
hoelzl@57447
  2459
hoelzl@57447
  2460
lemma has_integral_mult_left:
hoelzl@57447
  2461
  fixes c :: "_ :: {real_normed_algebra}"
hoelzl@57447
  2462
  shows "(f has_integral y) s \<Longrightarrow> ((\<lambda>x. f x * c) has_integral (y * c)) s"
hoelzl@57447
  2463
  using has_integral_linear[OF _ bounded_linear_mult_left] by (simp add: comp_def)
hoelzl@57447
  2464
lp15@60615
  2465
corollary integral_mult_left:
lp15@60615
  2466
  fixes c:: "'a::real_normed_algebra"
lp15@60615
  2467
  shows "f integrable_on s \<Longrightarrow> integral s (\<lambda>x. f x * c) = integral s f * c"
lp15@60615
  2468
  by (blast intro:  has_integral_mult_left)
lp15@60615
  2469
wenzelm@53410
  2470
lemma has_integral_cmul: "(f has_integral k) s \<Longrightarrow> ((\<lambda>x. c *\<^sub>R f x) has_integral (c *\<^sub>R k)) s"
wenzelm@53410
  2471
  unfolding o_def[symmetric]
lp15@60396
  2472
  by (metis has_integral_linear bounded_linear_scaleR_right)
himmelma@35172
  2473
hoelzl@50104
  2474
lemma has_integral_cmult_real:
hoelzl@50104
  2475
  fixes c :: real
hoelzl@50104
  2476
  assumes "c \<noteq> 0 \<Longrightarrow> (f has_integral x) A"