src/HOL/Analysis/Topology_Euclidean_Space.thy
author hoelzl
Thu Sep 29 13:02:43 2016 +0200 (2016-09-29)
changeset 63957 c3da799b1b45
parent 63955 51a3d38d2281
child 63967 2aa42596edc3
permissions -rw-r--r--
HOL-Analysis: move gauges and (tagged) divisions to its own theory file
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(*  Author:     L C Paulson, University of Cambridge
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    Author:     Amine Chaieb, University of Cambridge
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    Author:     Robert Himmelmann, TU Muenchen
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    Author:     Brian Huffman, Portland State University
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*)
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section \<open>Elementary topology in Euclidean space.\<close>
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theory Topology_Euclidean_Space
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imports
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  "~~/src/HOL/Library/Indicator_Function"
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  "~~/src/HOL/Library/Countable_Set"
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  "~~/src/HOL/Library/FuncSet"
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  Linear_Algebra
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  Norm_Arith
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begin
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(* FIXME: move elsewhere *)
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definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
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where
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  "support_on s f = {x\<in>s. f x \<noteq> 0}"
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lemma in_support_on: "x \<in> support_on s f \<longleftrightarrow> x \<in> s \<and> f x \<noteq> 0"
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  by (simp add: support_on_def)
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lemma support_on_simps[simp]:
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  "support_on {} f = {}"
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  "support_on (insert x s) f =
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    (if f x = 0 then support_on s f else insert x (support_on s f))"
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  "support_on (s \<union> t) f = support_on s f \<union> support_on t f"
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  "support_on (s \<inter> t) f = support_on s f \<inter> support_on t f"
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  "support_on (s - t) f = support_on s f - support_on t f"
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  "support_on (f ` s) g = f ` (support_on s (g \<circ> f))"
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  unfolding support_on_def by auto
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lemma support_on_cong:
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  "(\<And>x. x \<in> s \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on s f = support_on s g"
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  by (auto simp: support_on_def)
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lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
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  by (auto simp: support_on_def)
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lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
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  by (auto simp: support_on_def)
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lemma finite_support[intro]: "finite s \<Longrightarrow> finite (support_on s f)"
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  unfolding support_on_def by auto
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(* TODO: is supp_setsum really needed? TODO: Generalize to Finite_Set.fold *)
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definition (in comm_monoid_add) supp_setsum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
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where
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  "supp_setsum f s = (\<Sum>x\<in>support_on s f. f x)"
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lemma supp_setsum_empty[simp]: "supp_setsum f {} = 0"
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  unfolding supp_setsum_def by auto
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lemma supp_setsum_insert[simp]:
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  "finite (support_on s f) \<Longrightarrow>
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    supp_setsum f (insert x s) = (if x \<in> s then supp_setsum f s else f x + supp_setsum f s)"
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  by (simp add: supp_setsum_def in_support_on insert_absorb)
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lemma supp_setsum_divide_distrib: "supp_setsum f A / (r::'a::field) = supp_setsum (\<lambda>n. f n / r) A"
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  by (cases "r = 0")
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     (auto simp: supp_setsum_def setsum_divide_distrib intro!: setsum.cong support_on_cong)
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(*END OF SUPPORT, ETC.*)
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lemma image_affinity_interval:
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  fixes c :: "'a::ordered_real_vector"
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  shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) = (if {a..b}={} then {}
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            else if 0 <= m then {m *\<^sub>R a + c .. m  *\<^sub>R b + c}
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            else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
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  apply (case_tac "m=0", force)
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  apply (auto simp: scaleR_left_mono)
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  apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI, auto simp: pos_le_divideR_eq le_diff_eq scaleR_left_mono_neg)
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  apply (metis diff_le_eq inverse_inverse_eq order.not_eq_order_implies_strict pos_le_divideR_eq positive_imp_inverse_positive)
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  apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI, auto simp: not_le neg_le_divideR_eq diff_le_eq)
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  using le_diff_eq scaleR_le_cancel_left_neg
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  apply fastforce
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  done
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lemma countable_PiE:
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  "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"
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  by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
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lemma continuous_on_cases:
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  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>
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    \<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>
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    continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
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  by (rule continuous_on_If) auto
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subsection \<open>Topological Basis\<close>
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context topological_space
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begin
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definition "topological_basis B \<longleftrightarrow>
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  (\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
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lemma topological_basis:
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  "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
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  unfolding topological_basis_def
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  apply safe
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     apply fastforce
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    apply fastforce
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   apply (erule_tac x="x" in allE)
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   apply simp
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   apply (rule_tac x="{x}" in exI)
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  apply auto
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  done
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lemma topological_basis_iff:
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  assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
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  shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
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    (is "_ \<longleftrightarrow> ?rhs")
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proof safe
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  fix O' and x::'a
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  assume H: "topological_basis B" "open O'" "x \<in> O'"
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  then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
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  then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
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  then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
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next
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  assume H: ?rhs
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  show "topological_basis B"
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    using assms unfolding topological_basis_def
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  proof safe
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    fix O' :: "'a set"
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    assume "open O'"
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    with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
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      by (force intro: bchoice simp: Bex_def)
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    then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
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      by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
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  qed
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qed
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lemma topological_basisI:
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  assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
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    and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
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  shows "topological_basis B"
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  using assms by (subst topological_basis_iff) auto
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lemma topological_basisE:
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  fixes O'
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  assumes "topological_basis B"
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    and "open O'"
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    and "x \<in> O'"
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  obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
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proof atomize_elim
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  from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
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    by (simp add: topological_basis_def)
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  with topological_basis_iff assms
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  show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
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    using assms by (simp add: Bex_def)
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qed
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lemma topological_basis_open:
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  assumes "topological_basis B"
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    and "X \<in> B"
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  shows "open X"
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  using assms by (simp add: topological_basis_def)
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lemma topological_basis_imp_subbasis:
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  assumes B: "topological_basis B"
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  shows "open = generate_topology B"
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proof (intro ext iffI)
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  fix S :: "'a set"
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  assume "open S"
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  with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
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    unfolding topological_basis_def by blast
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  then show "generate_topology B S"
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    by (auto intro: generate_topology.intros dest: topological_basis_open)
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next
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  fix S :: "'a set"
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  assume "generate_topology B S"
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  then show "open S"
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    by induct (auto dest: topological_basis_open[OF B])
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qed
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lemma basis_dense:
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  fixes B :: "'a set set"
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    and f :: "'a set \<Rightarrow> 'a"
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  assumes "topological_basis B"
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    and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
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  shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)"
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proof (intro allI impI)
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  fix X :: "'a set"
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  assume "open X" and "X \<noteq> {}"
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  from topological_basisE[OF \<open>topological_basis B\<close> \<open>open X\<close> choosefrom_basis[OF \<open>X \<noteq> {}\<close>]]
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  obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .
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  then show "\<exists>B'\<in>B. f B' \<in> X"
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    by (auto intro!: choosefrom_basis)
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qed
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end
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lemma topological_basis_prod:
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  assumes A: "topological_basis A"
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    and B: "topological_basis B"
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  shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
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  unfolding topological_basis_def
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proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
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  fix S :: "('a \<times> 'b) set"
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  assume "open S"
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  then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
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  proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
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    fix x y
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    assume "(x, y) \<in> S"
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    from open_prod_elim[OF \<open>open S\<close> this]
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    obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
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      by (metis mem_Sigma_iff)
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    moreover
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    from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"
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      by (rule topological_basisE)
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    moreover
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    from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"
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      by (rule topological_basisE)
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    ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
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      by (intro UN_I[of "(A0, B0)"]) auto
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  qed auto
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qed (metis A B topological_basis_open open_Times)
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subsection \<open>Countable Basis\<close>
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locale countable_basis =
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  fixes B :: "'a::topological_space set set"
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  assumes is_basis: "topological_basis B"
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    and countable_basis: "countable B"
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begin
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lemma open_countable_basis_ex:
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  assumes "open X"
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  shows "\<exists>B' \<subseteq> B. X = \<Union>B'"
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  using assms countable_basis is_basis
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  unfolding topological_basis_def by blast
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lemma open_countable_basisE:
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  assumes "open X"
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  obtains B' where "B' \<subseteq> B" "X = \<Union>B'"
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  using assms open_countable_basis_ex
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  by (atomize_elim) simp
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lemma countable_dense_exists:
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  "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
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proof -
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  let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
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  have "countable (?f ` B)" using countable_basis by simp
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  with basis_dense[OF is_basis, of ?f] show ?thesis
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    by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
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qed
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lemma countable_dense_setE:
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  obtains D :: "'a set"
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  where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
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  using countable_dense_exists by blast
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end
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lemma (in first_countable_topology) first_countable_basisE:
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  obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
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    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
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  using first_countable_basis[of x]
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  apply atomize_elim
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  apply (elim exE)
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  apply (rule_tac x="range A" in exI)
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  apply auto
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  done
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lemma (in first_countable_topology) first_countable_basis_Int_stableE:
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  obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
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    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
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    "\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A"
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proof atomize_elim
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  obtain A' where A':
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    "countable A'"
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    "\<And>a. a \<in> A' \<Longrightarrow> x \<in> a"
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    "\<And>a. a \<in> A' \<Longrightarrow> open a"
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    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A'. a \<subseteq> S"
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    by (rule first_countable_basisE) blast
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  define A where [abs_def]:
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    "A = (\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)"
wenzelm@53255
   285
  then show "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and>
immler@51105
   286
        (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)"
immler@51105
   287
  proof (safe intro!: exI[where x=A])
wenzelm@53255
   288
    show "countable A"
wenzelm@53255
   289
      unfolding A_def by (intro countable_image countable_Collect_finite)
wenzelm@53255
   290
    fix a
wenzelm@53255
   291
    assume "a \<in> A"
wenzelm@53255
   292
    then show "x \<in> a" "open a"
wenzelm@53255
   293
      using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
immler@51105
   294
  next
haftmann@52141
   295
    let ?int = "\<lambda>N. \<Inter>(from_nat_into A' ` N)"
wenzelm@53255
   296
    fix a b
wenzelm@53255
   297
    assume "a \<in> A" "b \<in> A"
wenzelm@53255
   298
    then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)"
wenzelm@53255
   299
      by (auto simp: A_def)
wenzelm@53255
   300
    then show "a \<inter> b \<in> A"
wenzelm@53255
   301
      by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])
immler@51105
   302
  next
wenzelm@53255
   303
    fix S
wenzelm@53255
   304
    assume "open S" "x \<in> S"
wenzelm@53255
   305
    then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast
wenzelm@53255
   306
    then show "\<exists>a\<in>A. a \<subseteq> S" using a A'
immler@51105
   307
      by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])
immler@51105
   308
  qed
immler@51105
   309
qed
immler@51105
   310
hoelzl@51473
   311
lemma (in topological_space) first_countableI:
wenzelm@53255
   312
  assumes "countable A"
wenzelm@53255
   313
    and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
wenzelm@53255
   314
    and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
hoelzl@51473
   315
  shows "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
hoelzl@51473
   316
proof (safe intro!: exI[of _ "from_nat_into A"])
wenzelm@53255
   317
  fix i
hoelzl@51473
   318
  have "A \<noteq> {}" using 2[of UNIV] by auto
wenzelm@53255
   319
  show "x \<in> from_nat_into A i" "open (from_nat_into A i)"
wenzelm@60420
   320
    using range_from_nat_into_subset[OF \<open>A \<noteq> {}\<close>] 1 by auto
wenzelm@53255
   321
next
wenzelm@53255
   322
  fix S
wenzelm@53255
   323
  assume "open S" "x\<in>S" from 2[OF this]
wenzelm@53255
   324
  show "\<exists>i. from_nat_into A i \<subseteq> S"
wenzelm@60420
   325
    using subset_range_from_nat_into[OF \<open>countable A\<close>] by auto
hoelzl@51473
   326
qed
hoelzl@51350
   327
hoelzl@50883
   328
instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
hoelzl@50883
   329
proof
hoelzl@50883
   330
  fix x :: "'a \<times> 'b"
wenzelm@55522
   331
  obtain A where A:
wenzelm@55522
   332
      "countable A"
wenzelm@55522
   333
      "\<And>a. a \<in> A \<Longrightarrow> fst x \<in> a"
wenzelm@55522
   334
      "\<And>a. a \<in> A \<Longrightarrow> open a"
wenzelm@55522
   335
      "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
wenzelm@55522
   336
    by (rule first_countable_basisE[of "fst x"]) blast
wenzelm@55522
   337
  obtain B where B:
wenzelm@55522
   338
      "countable B"
wenzelm@55522
   339
      "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
wenzelm@55522
   340
      "\<And>a. a \<in> B \<Longrightarrow> open a"
wenzelm@55522
   341
      "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
wenzelm@55522
   342
    by (rule first_countable_basisE[of "snd x"]) blast
wenzelm@53282
   343
  show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set.
wenzelm@53282
   344
    (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
hoelzl@51473
   345
  proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
wenzelm@53255
   346
    fix a b
wenzelm@53255
   347
    assume x: "a \<in> A" "b \<in> B"
wenzelm@53640
   348
    with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" and "open (a \<times> b)"
wenzelm@53640
   349
      unfolding mem_Times_iff
wenzelm@53640
   350
      by (auto intro: open_Times)
hoelzl@50883
   351
  next
wenzelm@53255
   352
    fix S
wenzelm@53255
   353
    assume "open S" "x \<in> S"
wenzelm@55522
   354
    then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
wenzelm@55522
   355
      by (rule open_prod_elim)
wenzelm@55522
   356
    moreover
wenzelm@55522
   357
    from a'b' A(4)[of a'] B(4)[of b']
wenzelm@55522
   358
    obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
wenzelm@55522
   359
      by auto
wenzelm@55522
   360
    ultimately
wenzelm@55522
   361
    show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"
hoelzl@50883
   362
      by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
hoelzl@50883
   363
  qed (simp add: A B)
hoelzl@50883
   364
qed
hoelzl@50883
   365
hoelzl@50881
   366
class second_countable_topology = topological_space +
wenzelm@53282
   367
  assumes ex_countable_subbasis:
wenzelm@53282
   368
    "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
hoelzl@51343
   369
begin
hoelzl@51343
   370
hoelzl@51343
   371
lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
hoelzl@51343
   372
proof -
wenzelm@53255
   373
  from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
wenzelm@53255
   374
    by blast
hoelzl@51343
   375
  let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
hoelzl@51343
   376
hoelzl@51343
   377
  show ?thesis
hoelzl@51343
   378
  proof (intro exI conjI)
hoelzl@51343
   379
    show "countable ?B"
hoelzl@51343
   380
      by (intro countable_image countable_Collect_finite_subset B)
wenzelm@53255
   381
    {
wenzelm@53255
   382
      fix S
wenzelm@53255
   383
      assume "open S"
hoelzl@51343
   384
      then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
hoelzl@51343
   385
        unfolding B
hoelzl@51343
   386
      proof induct
wenzelm@53255
   387
        case UNIV
wenzelm@53255
   388
        show ?case by (intro exI[of _ "{{}}"]) simp
hoelzl@51343
   389
      next
hoelzl@51343
   390
        case (Int a b)
hoelzl@51343
   391
        then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
hoelzl@51343
   392
          and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
hoelzl@51343
   393
          by blast
hoelzl@51343
   394
        show ?case
hoelzl@51343
   395
          unfolding x y Int_UN_distrib2
hoelzl@51343
   396
          by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
hoelzl@51343
   397
      next
hoelzl@51343
   398
        case (UN K)
hoelzl@51343
   399
        then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
wenzelm@55522
   400
        then obtain k where
wenzelm@55522
   401
            "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> UNION (k ka) Inter = ka"
wenzelm@55522
   402
          unfolding bchoice_iff ..
hoelzl@51343
   403
        then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
hoelzl@51343
   404
          by (intro exI[of _ "UNION K k"]) auto
hoelzl@51343
   405
      next
wenzelm@53255
   406
        case (Basis S)
wenzelm@53255
   407
        then show ?case
hoelzl@51343
   408
          by (intro exI[of _ "{{S}}"]) auto
hoelzl@51343
   409
      qed
hoelzl@51343
   410
      then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
hoelzl@51343
   411
        unfolding subset_image_iff by blast }
hoelzl@51343
   412
    then show "topological_basis ?B"
hoelzl@51343
   413
      unfolding topological_space_class.topological_basis_def
wenzelm@53282
   414
      by (safe intro!: topological_space_class.open_Inter)
hoelzl@51343
   415
         (simp_all add: B generate_topology.Basis subset_eq)
hoelzl@51343
   416
  qed
hoelzl@51343
   417
qed
hoelzl@51343
   418
hoelzl@51343
   419
end
hoelzl@51343
   420
hoelzl@51343
   421
sublocale second_countable_topology <
hoelzl@51343
   422
  countable_basis "SOME B. countable B \<and> topological_basis B"
hoelzl@51343
   423
  using someI_ex[OF ex_countable_basis]
hoelzl@51343
   424
  by unfold_locales safe
immler@50094
   425
hoelzl@50882
   426
instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
hoelzl@50882
   427
proof
hoelzl@50882
   428
  obtain A :: "'a set set" where "countable A" "topological_basis A"
hoelzl@50882
   429
    using ex_countable_basis by auto
hoelzl@50882
   430
  moreover
hoelzl@50882
   431
  obtain B :: "'b set set" where "countable B" "topological_basis B"
hoelzl@50882
   432
    using ex_countable_basis by auto
hoelzl@51343
   433
  ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
hoelzl@51343
   434
    by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
hoelzl@51343
   435
      topological_basis_imp_subbasis)
hoelzl@50882
   436
qed
hoelzl@50882
   437
hoelzl@50883
   438
instance second_countable_topology \<subseteq> first_countable_topology
hoelzl@50883
   439
proof
hoelzl@50883
   440
  fix x :: 'a
wenzelm@63040
   441
  define B :: "'a set set" where "B = (SOME B. countable B \<and> topological_basis B)"
hoelzl@50883
   442
  then have B: "countable B" "topological_basis B"
hoelzl@50883
   443
    using countable_basis is_basis
hoelzl@50883
   444
    by (auto simp: countable_basis is_basis)
wenzelm@53282
   445
  then show "\<exists>A::nat \<Rightarrow> 'a set.
wenzelm@53282
   446
    (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
hoelzl@51473
   447
    by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
hoelzl@51473
   448
       (fastforce simp: topological_space_class.topological_basis_def)+
hoelzl@50883
   449
qed
hoelzl@50883
   450
wenzelm@53255
   451
wenzelm@60420
   452
subsection \<open>Polish spaces\<close>
wenzelm@60420
   453
wenzelm@60420
   454
text \<open>Textbooks define Polish spaces as completely metrizable.
wenzelm@60420
   455
  We assume the topology to be complete for a given metric.\<close>
immler@50087
   456
hoelzl@50881
   457
class polish_space = complete_space + second_countable_topology
immler@50087
   458
wenzelm@60420
   459
subsection \<open>General notion of a topology as a value\<close>
himmelma@33175
   460
wenzelm@53255
   461
definition "istopology L \<longleftrightarrow>
wenzelm@60585
   462
  L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union>K))"
wenzelm@53255
   463
wenzelm@49834
   464
typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
himmelma@33175
   465
  morphisms "openin" "topology"
himmelma@33175
   466
  unfolding istopology_def by blast
himmelma@33175
   467
lp15@62843
   468
lemma istopology_openin[intro]: "istopology(openin U)"
himmelma@33175
   469
  using openin[of U] by blast
himmelma@33175
   470
himmelma@33175
   471
lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
huffman@44170
   472
  using topology_inverse[unfolded mem_Collect_eq] .
himmelma@33175
   473
himmelma@33175
   474
lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
lp15@62843
   475
  using topology_inverse[of U] istopology_openin[of "topology U"] by auto
himmelma@33175
   476
himmelma@33175
   477
lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
wenzelm@53255
   478
proof
wenzelm@53255
   479
  assume "T1 = T2"
wenzelm@53255
   480
  then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
wenzelm@53255
   481
next
wenzelm@53255
   482
  assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
wenzelm@53255
   483
  then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
wenzelm@53255
   484
  then have "topology (openin T1) = topology (openin T2)" by simp
wenzelm@53255
   485
  then show "T1 = T2" unfolding openin_inverse .
himmelma@33175
   486
qed
himmelma@33175
   487
wenzelm@60420
   488
text\<open>Infer the "universe" from union of all sets in the topology.\<close>
himmelma@33175
   489
wenzelm@53640
   490
definition "topspace T = \<Union>{S. openin T S}"
himmelma@33175
   491
wenzelm@60420
   492
subsubsection \<open>Main properties of open sets\<close>
himmelma@33175
   493
himmelma@33175
   494
lemma openin_clauses:
himmelma@33175
   495
  fixes U :: "'a topology"
wenzelm@53282
   496
  shows
wenzelm@53282
   497
    "openin U {}"
wenzelm@53282
   498
    "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
wenzelm@53282
   499
    "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
wenzelm@53282
   500
  using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
himmelma@33175
   501
himmelma@33175
   502
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
himmelma@33175
   503
  unfolding topspace_def by blast
wenzelm@53255
   504
wenzelm@53255
   505
lemma openin_empty[simp]: "openin U {}"
lp15@62843
   506
  by (rule openin_clauses)
himmelma@33175
   507
himmelma@33175
   508
lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
lp15@62843
   509
  by (rule openin_clauses)
lp15@62843
   510
lp15@62843
   511
lemma openin_Union[intro]: "(\<And>S. S \<in> K \<Longrightarrow> openin U S) \<Longrightarrow> openin U (\<Union>K)"
lp15@63075
   512
  using openin_clauses by blast
himmelma@33175
   513
himmelma@33175
   514
lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
himmelma@33175
   515
  using openin_Union[of "{S,T}" U] by auto
himmelma@33175
   516
wenzelm@53255
   517
lemma openin_topspace[intro, simp]: "openin U (topspace U)"
lp15@62843
   518
  by (force simp add: openin_Union topspace_def)
himmelma@33175
   519
wenzelm@49711
   520
lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
wenzelm@49711
   521
  (is "?lhs \<longleftrightarrow> ?rhs")
huffman@36584
   522
proof
wenzelm@49711
   523
  assume ?lhs
wenzelm@49711
   524
  then show ?rhs by auto
huffman@36584
   525
next
huffman@36584
   526
  assume H: ?rhs
huffman@36584
   527
  let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
lp15@62843
   528
  have "openin U ?t" by (force simp add: openin_Union)
huffman@36584
   529
  also have "?t = S" using H by auto
huffman@36584
   530
  finally show "openin U S" .
himmelma@33175
   531
qed
himmelma@33175
   532
wenzelm@49711
   533
wenzelm@60420
   534
subsubsection \<open>Closed sets\<close>
himmelma@33175
   535
himmelma@33175
   536
definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
himmelma@33175
   537
wenzelm@53255
   538
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
wenzelm@53255
   539
  by (metis closedin_def)
wenzelm@53255
   540
wenzelm@53255
   541
lemma closedin_empty[simp]: "closedin U {}"
wenzelm@53255
   542
  by (simp add: closedin_def)
wenzelm@53255
   543
wenzelm@53255
   544
lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
wenzelm@53255
   545
  by (simp add: closedin_def)
wenzelm@53255
   546
himmelma@33175
   547
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
himmelma@33175
   548
  by (auto simp add: Diff_Un closedin_def)
himmelma@33175
   549
wenzelm@60585
   550
lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union>{A - s|s. s\<in>S}"
wenzelm@53255
   551
  by auto
wenzelm@53255
   552
lp15@63955
   553
lemma closedin_Union:
lp15@63955
   554
  assumes "finite S" "\<And>T. T \<in> S \<Longrightarrow> closedin U T"
lp15@63955
   555
    shows "closedin U (\<Union>S)"
lp15@63955
   556
  using assms by induction auto
lp15@63955
   557
wenzelm@53255
   558
lemma closedin_Inter[intro]:
wenzelm@53255
   559
  assumes Ke: "K \<noteq> {}"
paulson@62131
   560
    and Kc: "\<And>S. S \<in>K \<Longrightarrow> closedin U S"
wenzelm@60585
   561
  shows "closedin U (\<Inter>K)"
wenzelm@53255
   562
  using Ke Kc unfolding closedin_def Diff_Inter by auto
himmelma@33175
   563
paulson@62131
   564
lemma closedin_INT[intro]:
paulson@62131
   565
  assumes "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> closedin U (B x)"
paulson@62131
   566
  shows "closedin U (\<Inter>x\<in>A. B x)"
paulson@62131
   567
  apply (rule closedin_Inter)
paulson@62131
   568
  using assms
paulson@62131
   569
  apply auto
paulson@62131
   570
  done
paulson@62131
   571
himmelma@33175
   572
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
himmelma@33175
   573
  using closedin_Inter[of "{S,T}" U] by auto
himmelma@33175
   574
himmelma@33175
   575
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
himmelma@33175
   576
  apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
himmelma@33175
   577
  apply (metis openin_subset subset_eq)
himmelma@33175
   578
  done
himmelma@33175
   579
wenzelm@53255
   580
lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
himmelma@33175
   581
  by (simp add: openin_closedin_eq)
himmelma@33175
   582
wenzelm@53255
   583
lemma openin_diff[intro]:
wenzelm@53255
   584
  assumes oS: "openin U S"
wenzelm@53255
   585
    and cT: "closedin U T"
wenzelm@53255
   586
  shows "openin U (S - T)"
wenzelm@53255
   587
proof -
himmelma@33175
   588
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
himmelma@33175
   589
    by (auto simp add: topspace_def openin_subset)
wenzelm@53282
   590
  then show ?thesis using oS cT
wenzelm@53282
   591
    by (auto simp add: closedin_def)
himmelma@33175
   592
qed
himmelma@33175
   593
wenzelm@53255
   594
lemma closedin_diff[intro]:
wenzelm@53255
   595
  assumes oS: "closedin U S"
wenzelm@53255
   596
    and cT: "openin U T"
wenzelm@53255
   597
  shows "closedin U (S - T)"
wenzelm@53255
   598
proof -
wenzelm@53255
   599
  have "S - T = S \<inter> (topspace U - T)"
wenzelm@53282
   600
    using closedin_subset[of U S] oS cT by (auto simp add: topspace_def)
wenzelm@53255
   601
  then show ?thesis
wenzelm@53255
   602
    using oS cT by (auto simp add: openin_closedin_eq)
wenzelm@53255
   603
qed
wenzelm@53255
   604
himmelma@33175
   605
wenzelm@60420
   606
subsubsection \<open>Subspace topology\<close>
huffman@44170
   607
huffman@44170
   608
definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
huffman@44170
   609
huffman@44170
   610
lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
huffman@44170
   611
  (is "istopology ?L")
wenzelm@53255
   612
proof -
huffman@44170
   613
  have "?L {}" by blast
wenzelm@53255
   614
  {
wenzelm@53255
   615
    fix A B
wenzelm@53255
   616
    assume A: "?L A" and B: "?L B"
wenzelm@53255
   617
    from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V"
wenzelm@53255
   618
      by blast
wenzelm@53255
   619
    have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
wenzelm@53255
   620
      using Sa Sb by blast+
wenzelm@53255
   621
    then have "?L (A \<inter> B)" by blast
wenzelm@53255
   622
  }
himmelma@33175
   623
  moreover
wenzelm@53255
   624
  {
wenzelm@53282
   625
    fix K
wenzelm@53282
   626
    assume K: "K \<subseteq> Collect ?L"
huffman@44170
   627
    have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
lp15@55775
   628
      by blast
himmelma@33175
   629
    from K[unfolded th0 subset_image_iff]
wenzelm@53255
   630
    obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
wenzelm@53255
   631
      by blast
wenzelm@53255
   632
    have "\<Union>K = (\<Union>Sk) \<inter> V"
wenzelm@53255
   633
      using Sk by auto
wenzelm@60585
   634
    moreover have "openin U (\<Union>Sk)"
wenzelm@53255
   635
      using Sk by (auto simp add: subset_eq)
wenzelm@53255
   636
    ultimately have "?L (\<Union>K)" by blast
wenzelm@53255
   637
  }
huffman@44170
   638
  ultimately show ?thesis
haftmann@62343
   639
    unfolding subset_eq mem_Collect_eq istopology_def by auto
himmelma@33175
   640
qed
himmelma@33175
   641
wenzelm@53255
   642
lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
himmelma@33175
   643
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
huffman@44170
   644
  by auto
himmelma@33175
   645
wenzelm@53255
   646
lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
himmelma@33175
   647
  by (auto simp add: topspace_def openin_subtopology)
himmelma@33175
   648
wenzelm@53255
   649
lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
himmelma@33175
   650
  unfolding closedin_def topspace_subtopology
lp15@55775
   651
  by (auto simp add: openin_subtopology)
himmelma@33175
   652
himmelma@33175
   653
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
himmelma@33175
   654
  unfolding openin_subtopology
lp15@55775
   655
  by auto (metis IntD1 in_mono openin_subset)
wenzelm@49711
   656
wenzelm@49711
   657
lemma subtopology_superset:
wenzelm@49711
   658
  assumes UV: "topspace U \<subseteq> V"
himmelma@33175
   659
  shows "subtopology U V = U"
wenzelm@53255
   660
proof -
wenzelm@53255
   661
  {
wenzelm@53255
   662
    fix S
wenzelm@53255
   663
    {
wenzelm@53255
   664
      fix T
wenzelm@53255
   665
      assume T: "openin U T" "S = T \<inter> V"
wenzelm@53255
   666
      from T openin_subset[OF T(1)] UV have eq: "S = T"
wenzelm@53255
   667
        by blast
wenzelm@53255
   668
      have "openin U S"
wenzelm@53255
   669
        unfolding eq using T by blast
wenzelm@53255
   670
    }
himmelma@33175
   671
    moreover
wenzelm@53255
   672
    {
wenzelm@53255
   673
      assume S: "openin U S"
wenzelm@53255
   674
      then have "\<exists>T. openin U T \<and> S = T \<inter> V"
wenzelm@53255
   675
        using openin_subset[OF S] UV by auto
wenzelm@53255
   676
    }
wenzelm@53255
   677
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
wenzelm@53255
   678
      by blast
wenzelm@53255
   679
  }
wenzelm@53255
   680
  then show ?thesis
wenzelm@53255
   681
    unfolding topology_eq openin_subtopology by blast
himmelma@33175
   682
qed
himmelma@33175
   683
himmelma@33175
   684
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
himmelma@33175
   685
  by (simp add: subtopology_superset)
himmelma@33175
   686
himmelma@33175
   687
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
himmelma@33175
   688
  by (simp add: subtopology_superset)
himmelma@33175
   689
lp15@62948
   690
lemma openin_subtopology_empty:
lp15@62948
   691
   "openin (subtopology U {}) s \<longleftrightarrow> s = {}"
lp15@62948
   692
by (metis Int_empty_right openin_empty openin_subtopology)
lp15@62948
   693
lp15@62948
   694
lemma closedin_subtopology_empty:
lp15@62948
   695
   "closedin (subtopology U {}) s \<longleftrightarrow> s = {}"
lp15@62948
   696
by (metis Int_empty_right closedin_empty closedin_subtopology)
lp15@62948
   697
lp15@62948
   698
lemma closedin_subtopology_refl:
lp15@62948
   699
   "closedin (subtopology U u) u \<longleftrightarrow> u \<subseteq> topspace U"
lp15@62948
   700
by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology)
lp15@62948
   701
lp15@62948
   702
lemma openin_imp_subset:
lp15@62948
   703
   "openin (subtopology U s) t \<Longrightarrow> t \<subseteq> s"
lp15@62948
   704
by (metis Int_iff openin_subtopology subsetI)
lp15@62948
   705
lp15@62948
   706
lemma closedin_imp_subset:
lp15@62948
   707
   "closedin (subtopology U s) t \<Longrightarrow> t \<subseteq> s"
lp15@62948
   708
by (simp add: closedin_def topspace_subtopology)
lp15@62948
   709
lp15@62948
   710
lemma openin_subtopology_Un:
lp15@62948
   711
    "openin (subtopology U t) s \<and> openin (subtopology U u) s
lp15@62948
   712
     \<Longrightarrow> openin (subtopology U (t \<union> u)) s"
lp15@62948
   713
by (simp add: openin_subtopology) blast
lp15@62948
   714
wenzelm@53255
   715
wenzelm@60420
   716
subsubsection \<open>The standard Euclidean topology\<close>
himmelma@33175
   717
wenzelm@53255
   718
definition euclidean :: "'a::topological_space topology"
wenzelm@53255
   719
  where "euclidean = topology open"
himmelma@33175
   720
himmelma@33175
   721
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
himmelma@33175
   722
  unfolding euclidean_def
himmelma@33175
   723
  apply (rule cong[where x=S and y=S])
himmelma@33175
   724
  apply (rule topology_inverse[symmetric])
himmelma@33175
   725
  apply (auto simp add: istopology_def)
huffman@44170
   726
  done
himmelma@33175
   727
lp15@63492
   728
lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
himmelma@33175
   729
  apply (simp add: topspace_def)
nipkow@39302
   730
  apply (rule set_eqI)
wenzelm@53255
   731
  apply (auto simp add: open_openin[symmetric])
wenzelm@53255
   732
  done
himmelma@33175
   733
himmelma@33175
   734
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
himmelma@33175
   735
  by (simp add: topspace_euclidean topspace_subtopology)
himmelma@33175
   736
himmelma@33175
   737
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
himmelma@33175
   738
  by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
himmelma@33175
   739
himmelma@33175
   740
lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
himmelma@33175
   741
  by (simp add: open_openin openin_subopen[symmetric])
himmelma@33175
   742
lp15@62948
   743
lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
lp15@62948
   744
  by (metis openin_topspace topspace_euclidean_subtopology)
lp15@62948
   745
wenzelm@60420
   746
text \<open>Basic "localization" results are handy for connectedness.\<close>
huffman@44210
   747
huffman@44210
   748
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
huffman@44210
   749
  by (auto simp add: openin_subtopology open_openin[symmetric])
huffman@44210
   750
lp15@63305
   751
lemma openin_Int_open:
lp15@63305
   752
   "\<lbrakk>openin (subtopology euclidean U) S; open T\<rbrakk>
lp15@63305
   753
        \<Longrightarrow> openin (subtopology euclidean U) (S \<inter> T)"
lp15@63305
   754
by (metis open_Int Int_assoc openin_open)
lp15@63305
   755
huffman@44210
   756
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
huffman@44210
   757
  by (auto simp add: openin_open)
huffman@44210
   758
huffman@44210
   759
lemma open_openin_trans[trans]:
wenzelm@53255
   760
  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
huffman@44210
   761
  by (metis Int_absorb1  openin_open_Int)
huffman@44210
   762
wenzelm@53255
   763
lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
huffman@44210
   764
  by (auto simp add: openin_open)
huffman@44210
   765
huffman@44210
   766
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
huffman@44210
   767
  by (simp add: closedin_subtopology closed_closedin Int_ac)
huffman@44210
   768
wenzelm@53291
   769
lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
huffman@44210
   770
  by (metis closedin_closed)
huffman@44210
   771
huffman@44210
   772
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
huffman@44210
   773
  by (auto simp add: closedin_closed)
huffman@44210
   774
lp15@63928
   775
lemma finite_imp_closedin:
lp15@63928
   776
  fixes S :: "'a::t1_space set"
lp15@63928
   777
  shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (subtopology euclidean T) S"
lp15@63928
   778
    by (simp add: finite_imp_closed closed_subset)
lp15@63928
   779
lp15@63305
   780
lemma closedin_singleton [simp]:
lp15@63305
   781
  fixes a :: "'a::t1_space"
lp15@63305
   782
  shows "closedin (subtopology euclidean U) {a} \<longleftrightarrow> a \<in> U"
lp15@63305
   783
using closedin_subset  by (force intro: closed_subset)
lp15@63305
   784
huffman@44210
   785
lemma openin_euclidean_subtopology_iff:
huffman@44210
   786
  fixes S U :: "'a::metric_space set"
wenzelm@53255
   787
  shows "openin (subtopology euclidean U) S \<longleftrightarrow>
wenzelm@53255
   788
    S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)"
wenzelm@53255
   789
  (is "?lhs \<longleftrightarrow> ?rhs")
huffman@44210
   790
proof
wenzelm@53255
   791
  assume ?lhs
wenzelm@53282
   792
  then show ?rhs
wenzelm@53282
   793
    unfolding openin_open open_dist by blast
huffman@44210
   794
next
wenzelm@63040
   795
  define T where "T = {x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
huffman@44210
   796
  have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
huffman@44210
   797
    unfolding T_def
huffman@44210
   798
    apply clarsimp
huffman@44210
   799
    apply (rule_tac x="d - dist x a" in exI)
huffman@44210
   800
    apply (clarsimp simp add: less_diff_eq)
lp15@55775
   801
    by (metis dist_commute dist_triangle_lt)
wenzelm@53282
   802
  assume ?rhs then have 2: "S = U \<inter> T"
lp15@60141
   803
    unfolding T_def
lp15@55775
   804
    by auto (metis dist_self)
huffman@44210
   805
  from 1 2 show ?lhs
huffman@44210
   806
    unfolding openin_open open_dist by fast
huffman@44210
   807
qed
lp15@61609
   808
lp15@62843
   809
lemma connected_openin:
lp15@61306
   810
      "connected s \<longleftrightarrow>
lp15@61306
   811
       ~(\<exists>e1 e2. openin (subtopology euclidean s) e1 \<and>
lp15@61306
   812
                 openin (subtopology euclidean s) e2 \<and>
lp15@61306
   813
                 s \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and> e1 \<noteq> {} \<and> e2 \<noteq> {})"
lp15@61306
   814
  apply (simp add: connected_def openin_open, safe)
wenzelm@61808
   815
  apply (simp_all, blast+)  \<comment>\<open>slow\<close>
lp15@61306
   816
  done
lp15@61306
   817
lp15@62843
   818
lemma connected_openin_eq:
lp15@61306
   819
      "connected s \<longleftrightarrow>
lp15@61306
   820
       ~(\<exists>e1 e2. openin (subtopology euclidean s) e1 \<and>
lp15@61306
   821
                 openin (subtopology euclidean s) e2 \<and>
lp15@61306
   822
                 e1 \<union> e2 = s \<and> e1 \<inter> e2 = {} \<and>
lp15@61306
   823
                 e1 \<noteq> {} \<and> e2 \<noteq> {})"
lp15@62843
   824
  apply (simp add: connected_openin, safe)
lp15@61306
   825
  apply blast
lp15@61306
   826
  by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)
lp15@61306
   827
lp15@62843
   828
lemma connected_closedin:
lp15@61306
   829
      "connected s \<longleftrightarrow>
lp15@61306
   830
       ~(\<exists>e1 e2.
lp15@61306
   831
             closedin (subtopology euclidean s) e1 \<and>
lp15@61306
   832
             closedin (subtopology euclidean s) e2 \<and>
lp15@61306
   833
             s \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and>
lp15@61306
   834
             e1 \<noteq> {} \<and> e2 \<noteq> {})"
lp15@61306
   835
proof -
lp15@61306
   836
  { fix A B x x'
lp15@61306
   837
    assume s_sub: "s \<subseteq> A \<union> B"
lp15@61306
   838
       and disj: "A \<inter> B \<inter> s = {}"
lp15@61306
   839
       and x: "x \<in> s" "x \<in> B" and x': "x' \<in> s" "x' \<in> A"
lp15@61306
   840
       and cl: "closed A" "closed B"
lp15@61306
   841
    assume "\<forall>e1. (\<forall>T. closed T \<longrightarrow> e1 \<noteq> s \<inter> T) \<or> (\<forall>e2. e1 \<inter> e2 = {} \<longrightarrow> s \<subseteq> e1 \<union> e2 \<longrightarrow> (\<forall>T. closed T \<longrightarrow> e2 \<noteq> s \<inter> T) \<or> e1 = {} \<or> e2 = {})"
lp15@61306
   842
    then have "\<And>C D. s \<inter> C = {} \<or> s \<inter> D = {} \<or> s \<inter> (C \<inter> (s \<inter> D)) \<noteq> {} \<or> \<not> s \<subseteq> s \<inter> (C \<union> D) \<or> \<not> closed C \<or> \<not> closed D"
lp15@61306
   843
      by (metis (no_types) Int_Un_distrib Int_assoc)
lp15@61306
   844
    moreover have "s \<inter> (A \<inter> B) = {}" "s \<inter> (A \<union> B) = s" "s \<inter> B \<noteq> {}"
lp15@61306
   845
      using disj s_sub x by blast+
lp15@61306
   846
    ultimately have "s \<inter> A = {}"
lp15@61306
   847
      using cl by (metis inf.left_commute inf_bot_right order_refl)
lp15@61306
   848
    then have False
lp15@61306
   849
      using x' by blast
lp15@61306
   850
  } note * = this
lp15@61306
   851
  show ?thesis
lp15@61306
   852
    apply (simp add: connected_closed closedin_closed)
lp15@61306
   853
    apply (safe; simp)
lp15@61306
   854
    apply blast
lp15@61306
   855
    apply (blast intro: *)
lp15@61306
   856
    done
lp15@61306
   857
qed
lp15@61306
   858
lp15@62843
   859
lemma connected_closedin_eq:
lp15@61306
   860
      "connected s \<longleftrightarrow>
lp15@61306
   861
           ~(\<exists>e1 e2.
lp15@61306
   862
                 closedin (subtopology euclidean s) e1 \<and>
lp15@61306
   863
                 closedin (subtopology euclidean s) e2 \<and>
lp15@61306
   864
                 e1 \<union> e2 = s \<and> e1 \<inter> e2 = {} \<and>
lp15@61306
   865
                 e1 \<noteq> {} \<and> e2 \<noteq> {})"
lp15@62843
   866
  apply (simp add: connected_closedin, safe)
lp15@61306
   867
  apply blast
lp15@61306
   868
  by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)
lp15@61609
   869
wenzelm@60420
   870
text \<open>These "transitivity" results are handy too\<close>
huffman@44210
   871
wenzelm@53255
   872
lemma openin_trans[trans]:
wenzelm@53255
   873
  "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
wenzelm@53255
   874
    openin (subtopology euclidean U) S"
huffman@44210
   875
  unfolding open_openin openin_open by blast
huffman@44210
   876
huffman@44210
   877
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
huffman@44210
   878
  by (auto simp add: openin_open intro: openin_trans)
huffman@44210
   879
huffman@44210
   880
lemma closedin_trans[trans]:
wenzelm@53255
   881
  "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
wenzelm@53255
   882
    closedin (subtopology euclidean U) S"
huffman@44210
   883
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
huffman@44210
   884
huffman@44210
   885
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
huffman@44210
   886
  by (auto simp add: closedin_closed intro: closedin_trans)
huffman@44210
   887
lp15@62843
   888
lemma openin_subtopology_Int_subset:
lp15@62843
   889
   "\<lbrakk>openin (subtopology euclidean u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (subtopology euclidean v) (v \<inter> S)"
paulson@61518
   890
  by (auto simp: openin_subtopology)
paulson@61518
   891
paulson@61518
   892
lemma openin_open_eq: "open s \<Longrightarrow> (openin (subtopology euclidean s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
paulson@61518
   893
  using open_subset openin_open_trans openin_subset by fastforce
paulson@61518
   894
huffman@44210
   895
wenzelm@60420
   896
subsection \<open>Open and closed balls\<close>
himmelma@33175
   897
wenzelm@53255
   898
definition ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
wenzelm@53255
   899
  where "ball x e = {y. dist x y < e}"
wenzelm@53255
   900
wenzelm@53255
   901
definition cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
wenzelm@53255
   902
  where "cball x e = {y. dist x y \<le> e}"
himmelma@33175
   903
lp15@61762
   904
definition sphere :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
lp15@61762
   905
  where "sphere x e = {y. dist x y = e}"
lp15@61762
   906
huffman@45776
   907
lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
huffman@45776
   908
  by (simp add: ball_def)
huffman@45776
   909
huffman@45776
   910
lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
huffman@45776
   911
  by (simp add: cball_def)
huffman@45776
   912
lp15@61848
   913
lemma mem_sphere [simp]: "y \<in> sphere x e \<longleftrightarrow> dist x y = e"
lp15@61848
   914
  by (simp add: sphere_def)
lp15@61848
   915
paulson@61518
   916
lemma ball_trivial [simp]: "ball x 0 = {}"
paulson@61518
   917
  by (simp add: ball_def)
paulson@61518
   918
paulson@61518
   919
lemma cball_trivial [simp]: "cball x 0 = {x}"
paulson@61518
   920
  by (simp add: cball_def)
paulson@61518
   921
lp15@63469
   922
lemma sphere_trivial [simp]: "sphere x 0 = {x}"
lp15@63469
   923
  by (simp add: sphere_def)
lp15@63469
   924
paulson@61518
   925
lemma mem_ball_0 [simp]:
himmelma@33175
   926
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   927
  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
himmelma@33175
   928
  by (simp add: dist_norm)
himmelma@33175
   929
paulson@61518
   930
lemma mem_cball_0 [simp]:
himmelma@33175
   931
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   932
  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
himmelma@33175
   933
  by (simp add: dist_norm)
himmelma@33175
   934
lp15@63114
   935
lemma mem_sphere_0 [simp]:
lp15@63114
   936
  fixes x :: "'a::real_normed_vector"
lp15@63114
   937
  shows "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
lp15@63114
   938
  by (simp add: dist_norm)
lp15@63114
   939
lp15@63881
   940
lemma sphere_empty [simp]:
lp15@63881
   941
  fixes a :: "'a::metric_space"
lp15@63881
   942
  shows "r < 0 \<Longrightarrow> sphere a r = {}"
lp15@63881
   943
by auto
lp15@63881
   944
paulson@61518
   945
lemma centre_in_ball [simp]: "x \<in> ball x e \<longleftrightarrow> 0 < e"
huffman@45776
   946
  by simp
huffman@45776
   947
paulson@61518
   948
lemma centre_in_cball [simp]: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
huffman@45776
   949
  by simp
huffman@45776
   950
paulson@61518
   951
lemma ball_subset_cball [simp,intro]: "ball x e \<subseteq> cball x e"
wenzelm@53255
   952
  by (simp add: subset_eq)
wenzelm@53255
   953
lp15@61907
   954
lemma sphere_cball [simp,intro]: "sphere z r \<subseteq> cball z r"
lp15@61907
   955
  by force
lp15@61907
   956
wenzelm@53282
   957
lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
wenzelm@53255
   958
  by (simp add: subset_eq)
wenzelm@53255
   959
wenzelm@53282
   960
lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
wenzelm@53255
   961
  by (simp add: subset_eq)
wenzelm@53255
   962
himmelma@33175
   963
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
nipkow@39302
   964
  by (simp add: set_eq_iff) arith
himmelma@33175
   965
himmelma@33175
   966
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
nipkow@39302
   967
  by (simp add: set_eq_iff)
himmelma@33175
   968
lp15@61426
   969
lemma cball_diff_eq_sphere: "cball a r - ball a r =  {x. dist x a = r}"
lp15@61426
   970
  by (auto simp: cball_def ball_def dist_commute)
lp15@61426
   971
lp15@62533
   972
lemma image_add_ball [simp]:
lp15@62533
   973
  fixes a :: "'a::real_normed_vector"
lp15@62533
   974
  shows "op + b ` ball a r = ball (a+b) r"
lp15@62533
   975
apply (intro equalityI subsetI)
lp15@62533
   976
apply (force simp: dist_norm)
lp15@62533
   977
apply (rule_tac x="x-b" in image_eqI)
lp15@62533
   978
apply (auto simp: dist_norm algebra_simps)
lp15@62533
   979
done
lp15@62533
   980
lp15@62533
   981
lemma image_add_cball [simp]:
lp15@62533
   982
  fixes a :: "'a::real_normed_vector"
lp15@62533
   983
  shows "op + b ` cball a r = cball (a+b) r"
lp15@62533
   984
apply (intro equalityI subsetI)
lp15@62533
   985
apply (force simp: dist_norm)
lp15@62533
   986
apply (rule_tac x="x-b" in image_eqI)
lp15@62533
   987
apply (auto simp: dist_norm algebra_simps)
lp15@62533
   988
done
lp15@62533
   989
huffman@54070
   990
lemma open_ball [intro, simp]: "open (ball x e)"
huffman@54070
   991
proof -
huffman@54070
   992
  have "open (dist x -` {..<e})"
hoelzl@56371
   993
    by (intro open_vimage open_lessThan continuous_intros)
huffman@54070
   994
  also have "dist x -` {..<e} = ball x e"
huffman@54070
   995
    by auto
huffman@54070
   996
  finally show ?thesis .
huffman@54070
   997
qed
himmelma@33175
   998
himmelma@33175
   999
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
wenzelm@63170
  1000
  by (simp add: open_dist subset_eq mem_ball Ball_def dist_commute)
himmelma@33175
  1001
lp15@62381
  1002
lemma openI [intro?]: "(\<And>x. x\<in>S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S) \<Longrightarrow> open S"
lp15@62381
  1003
  by (auto simp: open_contains_ball)
lp15@62381
  1004
hoelzl@33714
  1005
lemma openE[elim?]:
wenzelm@53282
  1006
  assumes "open S" "x\<in>S"
hoelzl@33714
  1007
  obtains e where "e>0" "ball x e \<subseteq> S"
hoelzl@33714
  1008
  using assms unfolding open_contains_ball by auto
hoelzl@33714
  1009
lp15@62381
  1010
lemma open_contains_ball_eq: "open S \<Longrightarrow> x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
  1011
  by (metis open_contains_ball subset_eq centre_in_ball)
himmelma@33175
  1012
lp15@62843
  1013
lemma openin_contains_ball:
lp15@62843
  1014
    "openin (subtopology euclidean t) s \<longleftrightarrow>
lp15@62843
  1015
     s \<subseteq> t \<and> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> ball x e \<inter> t \<subseteq> s)"
lp15@62843
  1016
    (is "?lhs = ?rhs")
lp15@62843
  1017
proof
lp15@62843
  1018
  assume ?lhs
lp15@62843
  1019
  then show ?rhs
lp15@62843
  1020
    apply (simp add: openin_open)
lp15@62843
  1021
    apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE)
lp15@62843
  1022
    done
lp15@62843
  1023
next
lp15@62843
  1024
  assume ?rhs
lp15@62843
  1025
  then show ?lhs
lp15@62843
  1026
    apply (simp add: openin_euclidean_subtopology_iff)
lp15@62843
  1027
    by (metis (no_types) Int_iff dist_commute inf.absorb_iff2 mem_ball)
lp15@62843
  1028
qed
lp15@62843
  1029
lp15@62843
  1030
lemma openin_contains_cball:
lp15@62843
  1031
   "openin (subtopology euclidean t) s \<longleftrightarrow>
lp15@62843
  1032
        s \<subseteq> t \<and>
lp15@62843
  1033
        (\<forall>x \<in> s. \<exists>e. 0 < e \<and> cball x e \<inter> t \<subseteq> s)"
lp15@62843
  1034
apply (simp add: openin_contains_ball)
lp15@62843
  1035
apply (rule iffI)
lp15@62843
  1036
apply (auto dest!: bspec)
lp15@62843
  1037
apply (rule_tac x="e/2" in exI)
lp15@62843
  1038
apply force+
lp15@62843
  1039
done
lp15@63075
  1040
himmelma@33175
  1041
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
nipkow@39302
  1042
  unfolding mem_ball set_eq_iff
himmelma@33175
  1043
  apply (simp add: not_less)
wenzelm@52624
  1044
  apply (metis zero_le_dist order_trans dist_self)
wenzelm@52624
  1045
  done
himmelma@33175
  1046
lp15@61694
  1047
lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
himmelma@33175
  1048
hoelzl@50526
  1049
lemma euclidean_dist_l2:
hoelzl@50526
  1050
  fixes x y :: "'a :: euclidean_space"
hoelzl@50526
  1051
  shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
hoelzl@50526
  1052
  unfolding dist_norm norm_eq_sqrt_inner setL2_def
hoelzl@50526
  1053
  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
hoelzl@50526
  1054
eberlm@61531
  1055
lemma eventually_nhds_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>x. x \<in> ball z d) (nhds z)"
eberlm@61531
  1056
  by (rule eventually_nhds_in_open) simp_all
eberlm@61531
  1057
eberlm@61531
  1058
lemma eventually_at_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<in> A) (at z within A)"
eberlm@61531
  1059
  unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
eberlm@61531
  1060
eberlm@61531
  1061
lemma eventually_at_ball': "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
eberlm@61531
  1062
  unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
eberlm@61531
  1063
immler@56189
  1064
wenzelm@60420
  1065
subsection \<open>Boxes\<close>
immler@56189
  1066
hoelzl@57447
  1067
abbreviation One :: "'a::euclidean_space"
hoelzl@57447
  1068
  where "One \<equiv> \<Sum>Basis"
hoelzl@57447
  1069
lp15@63114
  1070
lemma One_non_0: assumes "One = (0::'a::euclidean_space)" shows False
lp15@63114
  1071
proof -
lp15@63114
  1072
  have "dependent (Basis :: 'a set)"
lp15@63114
  1073
    apply (simp add: dependent_finite)
lp15@63114
  1074
    apply (rule_tac x="\<lambda>i. 1" in exI)
lp15@63114
  1075
    using SOME_Basis apply (auto simp: assms)
lp15@63114
  1076
    done
lp15@63114
  1077
  with independent_Basis show False by force
lp15@63114
  1078
qed
lp15@63114
  1079
lp15@63114
  1080
corollary One_neq_0[iff]: "One \<noteq> 0"
lp15@63114
  1081
  by (metis One_non_0)
lp15@63114
  1082
lp15@63114
  1083
corollary Zero_neq_One[iff]: "0 \<noteq> One"
lp15@63114
  1084
  by (metis One_non_0)
lp15@63114
  1085
immler@54775
  1086
definition (in euclidean_space) eucl_less (infix "<e" 50)
immler@54775
  1087
  where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
immler@54775
  1088
immler@54775
  1089
definition box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"
immler@56188
  1090
definition "cbox a b = {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i}"
immler@54775
  1091
immler@54775
  1092
lemma box_def: "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
immler@54775
  1093
  and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b"
immler@56188
  1094
  and mem_box: "x \<in> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i)"
immler@56188
  1095
    "x \<in> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i)"
immler@56188
  1096
  by (auto simp: box_eucl_less eucl_less_def cbox_def)
immler@56188
  1097
lp15@60615
  1098
lemma cbox_Pair_eq: "cbox (a, c) (b, d) = cbox a b \<times> cbox c d"
lp15@60615
  1099
  by (force simp: cbox_def Basis_prod_def)
lp15@60615
  1100
lp15@60615
  1101
lemma cbox_Pair_iff [iff]: "(x, y) \<in> cbox (a, c) (b, d) \<longleftrightarrow> x \<in> cbox a b \<and> y \<in> cbox c d"
lp15@60615
  1102
  by (force simp: cbox_Pair_eq)
lp15@60615
  1103
lp15@60615
  1104
lemma cbox_Pair_eq_0: "cbox (a, c) (b, d) = {} \<longleftrightarrow> cbox a b = {} \<or> cbox c d = {}"
lp15@60615
  1105
  by (force simp: cbox_Pair_eq)
lp15@60615
  1106
lp15@60615
  1107
lemma swap_cbox_Pair [simp]: "prod.swap ` cbox (c, a) (d, b) = cbox (a,c) (b,d)"
lp15@60615
  1108
  by auto
lp15@60615
  1109
immler@56188
  1110
lemma mem_box_real[simp]:
immler@56188
  1111
  "(x::real) \<in> box a b \<longleftrightarrow> a < x \<and> x < b"
immler@56188
  1112
  "(x::real) \<in> cbox a b \<longleftrightarrow> a \<le> x \<and> x \<le> b"
immler@56188
  1113
  by (auto simp: mem_box)
immler@56188
  1114
immler@56188
  1115
lemma box_real[simp]:
immler@56188
  1116
  fixes a b:: real
immler@56188
  1117
  shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
immler@56188
  1118
  by auto
hoelzl@50526
  1119
hoelzl@57447
  1120
lemma box_Int_box:
hoelzl@57447
  1121
  fixes a :: "'a::euclidean_space"
hoelzl@57447
  1122
  shows "box a b \<inter> box c d =
hoelzl@57447
  1123
    box (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
hoelzl@57447
  1124
  unfolding set_eq_iff and Int_iff and mem_box by auto
hoelzl@57447
  1125
immler@50087
  1126
lemma rational_boxes:
wenzelm@61076
  1127
  fixes x :: "'a::euclidean_space"
wenzelm@53291
  1128
  assumes "e > 0"
hoelzl@50526
  1129
  shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
immler@50087
  1130
proof -
wenzelm@63040
  1131
  define e' where "e' = e / (2 * sqrt (real (DIM ('a))))"
wenzelm@53291
  1132
  then have e: "e' > 0"
nipkow@56541
  1133
    using assms by (auto simp: DIM_positive)
hoelzl@50526
  1134
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
immler@50087
  1135
  proof
wenzelm@53255
  1136
    fix i
wenzelm@53255
  1137
    from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e
wenzelm@53255
  1138
    show "?th i" by auto
immler@50087
  1139
  qed
wenzelm@55522
  1140
  from choice[OF this] obtain a where
wenzelm@55522
  1141
    a: "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa - a xa < e'" ..
hoelzl@50526
  1142
  have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
immler@50087
  1143
  proof
wenzelm@53255
  1144
    fix i
wenzelm@53255
  1145
    from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e
wenzelm@53255
  1146
    show "?th i" by auto
immler@50087
  1147
  qed
wenzelm@55522
  1148
  from choice[OF this] obtain b where
wenzelm@55522
  1149
    b: "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa - x \<bullet> xa < e'" ..
hoelzl@50526
  1150
  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
hoelzl@50526
  1151
  show ?thesis
hoelzl@50526
  1152
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
wenzelm@53255
  1153
    fix y :: 'a
wenzelm@53255
  1154
    assume *: "y \<in> box ?a ?b"
wenzelm@53015
  1155
    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
immler@50087
  1156
      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
hoelzl@50526
  1157
    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
immler@50087
  1158
    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
wenzelm@53255
  1159
      fix i :: "'a"
wenzelm@53255
  1160
      assume i: "i \<in> Basis"
wenzelm@53255
  1161
      have "a i < y\<bullet>i \<and> y\<bullet>i < b i"
wenzelm@53255
  1162
        using * i by (auto simp: box_def)
wenzelm@53255
  1163
      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"
wenzelm@53255
  1164
        using a by auto
wenzelm@53255
  1165
      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"
wenzelm@53255
  1166
        using b by auto
wenzelm@53255
  1167
      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'"
wenzelm@53255
  1168
        by auto
hoelzl@50526
  1169
      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
immler@50087
  1170
        unfolding e'_def by (auto simp: dist_real_def)
wenzelm@53015
  1171
      then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
immler@50087
  1172
        by (rule power_strict_mono) auto
wenzelm@53015
  1173
      then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
immler@50087
  1174
        by (simp add: power_divide)
immler@50087
  1175
    qed auto
wenzelm@53255
  1176
    also have "\<dots> = e"
lp15@61609
  1177
      using \<open>0 < e\<close> by simp
wenzelm@53255
  1178
    finally show "y \<in> ball x e"
wenzelm@53255
  1179
      by (auto simp: ball_def)
hoelzl@50526
  1180
  qed (insert a b, auto simp: box_def)
hoelzl@50526
  1181
qed
immler@51103
  1182
hoelzl@50526
  1183
lemma open_UNION_box:
wenzelm@61076
  1184
  fixes M :: "'a::euclidean_space set"
wenzelm@53282
  1185
  assumes "open M"
hoelzl@50526
  1186
  defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
hoelzl@50526
  1187
  defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
wenzelm@53015
  1188
  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
hoelzl@50526
  1189
  shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
wenzelm@52624
  1190
proof -
wenzelm@60462
  1191
  have "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))" if "x \<in> M" for x
wenzelm@60462
  1192
  proof -
wenzelm@52624
  1193
    obtain e where e: "e > 0" "ball x e \<subseteq> M"
wenzelm@60420
  1194
      using openE[OF \<open>open M\<close> \<open>x \<in> M\<close>] by auto
wenzelm@53282
  1195
    moreover obtain a b where ab:
wenzelm@53282
  1196
      "x \<in> box a b"
wenzelm@53282
  1197
      "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
wenzelm@53282
  1198
      "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>"
wenzelm@53282
  1199
      "box a b \<subseteq> ball x e"
wenzelm@52624
  1200
      using rational_boxes[OF e(1)] by metis
wenzelm@60462
  1201
    ultimately show ?thesis
wenzelm@52624
  1202
       by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
wenzelm@52624
  1203
          (auto simp: euclidean_representation I_def a'_def b'_def)
wenzelm@60462
  1204
  qed
wenzelm@52624
  1205
  then show ?thesis by (auto simp: I_def)
wenzelm@52624
  1206
qed
wenzelm@52624
  1207
immler@56189
  1208
lemma box_eq_empty:
immler@56189
  1209
  fixes a :: "'a::euclidean_space"
immler@56189
  1210
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)
immler@56189
  1211
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
immler@56189
  1212
proof -
immler@56189
  1213
  {
immler@56189
  1214
    fix i x
immler@56189
  1215
    assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>box a b"
immler@56189
  1216
    then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"
immler@56189
  1217
      unfolding mem_box by (auto simp: box_def)
immler@56189
  1218
    then have "a\<bullet>i < b\<bullet>i" by auto
immler@56189
  1219
    then have False using as by auto
immler@56189
  1220
  }
immler@56189
  1221
  moreover
immler@56189
  1222
  {
immler@56189
  1223
    assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
immler@56189
  1224
    let ?x = "(1/2) *\<^sub>R (a + b)"
immler@56189
  1225
    {
immler@56189
  1226
      fix i :: 'a
immler@56189
  1227
      assume i: "i \<in> Basis"
immler@56189
  1228
      have "a\<bullet>i < b\<bullet>i"
immler@56189
  1229
        using as[THEN bspec[where x=i]] i by auto
immler@56189
  1230
      then have "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
immler@56189
  1231
        by (auto simp: inner_add_left)
immler@56189
  1232
    }
immler@56189
  1233
    then have "box a b \<noteq> {}"
immler@56189
  1234
      using mem_box(1)[of "?x" a b] by auto
immler@56189
  1235
  }
immler@56189
  1236
  ultimately show ?th1 by blast
immler@56189
  1237
immler@56189
  1238
  {
immler@56189
  1239
    fix i x
immler@56189
  1240
    assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>cbox a b"
immler@56189
  1241
    then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
immler@56189
  1242
      unfolding mem_box by auto
immler@56189
  1243
    then have "a\<bullet>i \<le> b\<bullet>i" by auto
immler@56189
  1244
    then have False using as by auto
immler@56189
  1245
  }
immler@56189
  1246
  moreover
immler@56189
  1247
  {
immler@56189
  1248
    assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
immler@56189
  1249
    let ?x = "(1/2) *\<^sub>R (a + b)"
immler@56189
  1250
    {
immler@56189
  1251
      fix i :: 'a
immler@56189
  1252
      assume i:"i \<in> Basis"
immler@56189
  1253
      have "a\<bullet>i \<le> b\<bullet>i"
immler@56189
  1254
        using as[THEN bspec[where x=i]] i by auto
immler@56189
  1255
      then have "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
immler@56189
  1256
        by (auto simp: inner_add_left)
immler@56189
  1257
    }
immler@56189
  1258
    then have "cbox a b \<noteq> {}"
immler@56189
  1259
      using mem_box(2)[of "?x" a b] by auto
immler@56189
  1260
  }
immler@56189
  1261
  ultimately show ?th2 by blast
immler@56189
  1262
qed
immler@56189
  1263
immler@56189
  1264
lemma box_ne_empty:
immler@56189
  1265
  fixes a :: "'a::euclidean_space"
immler@56189
  1266
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"
immler@56189
  1267
  and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
immler@56189
  1268
  unfolding box_eq_empty[of a b] by fastforce+
immler@56189
  1269
immler@56189
  1270
lemma
immler@56189
  1271
  fixes a :: "'a::euclidean_space"
immler@56189
  1272
  shows cbox_sing: "cbox a a = {a}"
immler@56189
  1273
    and box_sing: "box a a = {}"
immler@56189
  1274
  unfolding set_eq_iff mem_box eq_iff [symmetric]
immler@56189
  1275
  by (auto intro!: euclidean_eqI[where 'a='a])
immler@56189
  1276
     (metis all_not_in_conv nonempty_Basis)
immler@56189
  1277
immler@56189
  1278
lemma subset_box_imp:
immler@56189
  1279
  fixes a :: "'a::euclidean_space"
immler@56189
  1280
  shows "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> cbox a b"
immler@56189
  1281
    and "(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> cbox c d \<subseteq> box a b"
immler@56189
  1282
    and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> cbox a b"
immler@56189
  1283
     and "(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> box c d \<subseteq> box a b"
immler@56189
  1284
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box
wenzelm@58757
  1285
  by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
immler@56189
  1286
immler@56189
  1287
lemma box_subset_cbox:
immler@56189
  1288
  fixes a :: "'a::euclidean_space"
immler@56189
  1289
  shows "box a b \<subseteq> cbox a b"
immler@56189
  1290
  unfolding subset_eq [unfolded Ball_def] mem_box
immler@56189
  1291
  by (fast intro: less_imp_le)
immler@56189
  1292
immler@56189
  1293
lemma subset_box:
immler@56189
  1294
  fixes a :: "'a::euclidean_space"
immler@56189
  1295
  shows "cbox c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1)
immler@56189
  1296
    and "cbox c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2)
immler@56189
  1297
    and "box c d \<subseteq> cbox a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3)
immler@56189
  1298
    and "box c d \<subseteq> box a b \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
immler@56189
  1299
proof -
immler@56189
  1300
  show ?th1
immler@56189
  1301
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1302
    by (auto intro: order_trans)
immler@56189
  1303
  show ?th2
immler@56189
  1304
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1305
    by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
immler@56189
  1306
  {
immler@56189
  1307
    assume as: "box c d \<subseteq> cbox a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
immler@56189
  1308
    then have "box c d \<noteq> {}"
immler@56189
  1309
      unfolding box_eq_empty by auto
immler@56189
  1310
    fix i :: 'a
immler@56189
  1311
    assume i: "i \<in> Basis"
immler@56189
  1312
    (** TODO combine the following two parts as done in the HOL_light version. **)
immler@56189
  1313
    {
immler@56189
  1314
      let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
immler@56189
  1315
      assume as2: "a\<bullet>i > c\<bullet>i"
immler@56189
  1316
      {
immler@56189
  1317
        fix j :: 'a
immler@56189
  1318
        assume j: "j \<in> Basis"
immler@56189
  1319
        then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
immler@56189
  1320
          apply (cases "j = i")
immler@56189
  1321
          using as(2)[THEN bspec[where x=j]] i
immler@56189
  1322
          apply (auto simp add: as2)
immler@56189
  1323
          done
immler@56189
  1324
      }
immler@56189
  1325
      then have "?x\<in>box c d"
immler@56189
  1326
        using i unfolding mem_box by auto
immler@56189
  1327
      moreover
immler@56189
  1328
      have "?x \<notin> cbox a b"
immler@56189
  1329
        unfolding mem_box
immler@56189
  1330
        apply auto
immler@56189
  1331
        apply (rule_tac x=i in bexI)
immler@56189
  1332
        using as(2)[THEN bspec[where x=i]] and as2 i
immler@56189
  1333
        apply auto
immler@56189
  1334
        done
immler@56189
  1335
      ultimately have False using as by auto
immler@56189
  1336
    }
immler@56189
  1337
    then have "a\<bullet>i \<le> c\<bullet>i" by (rule ccontr) auto
immler@56189
  1338
    moreover
immler@56189
  1339
    {
immler@56189
  1340
      let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
immler@56189
  1341
      assume as2: "b\<bullet>i < d\<bullet>i"
immler@56189
  1342
      {
immler@56189
  1343
        fix j :: 'a
immler@56189
  1344
        assume "j\<in>Basis"
immler@56189
  1345
        then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"
immler@56189
  1346
          apply (cases "j = i")
immler@56189
  1347
          using as(2)[THEN bspec[where x=j]]
immler@56189
  1348
          apply (auto simp add: as2)
immler@56189
  1349
          done
immler@56189
  1350
      }
immler@56189
  1351
      then have "?x\<in>box c d"
immler@56189
  1352
        unfolding mem_box by auto
immler@56189
  1353
      moreover
immler@56189
  1354
      have "?x\<notin>cbox a b"
immler@56189
  1355
        unfolding mem_box
immler@56189
  1356
        apply auto
immler@56189
  1357
        apply (rule_tac x=i in bexI)
immler@56189
  1358
        using as(2)[THEN bspec[where x=i]] and as2 using i
immler@56189
  1359
        apply auto
immler@56189
  1360
        done
immler@56189
  1361
      ultimately have False using as by auto
immler@56189
  1362
    }
immler@56189
  1363
    then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto
immler@56189
  1364
    ultimately
immler@56189
  1365
    have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
immler@56189
  1366
  } note part1 = this
immler@56189
  1367
  show ?th3
immler@56189
  1368
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1369
    apply (rule, rule, rule, rule)
immler@56189
  1370
    apply (rule part1)
immler@56189
  1371
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1372
    prefer 4
immler@56189
  1373
    apply auto
immler@56189
  1374
    apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+
immler@56189
  1375
    done
immler@56189
  1376
  {
immler@56189
  1377
    assume as: "box c d \<subseteq> box a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
immler@56189
  1378
    fix i :: 'a
immler@56189
  1379
    assume i:"i\<in>Basis"
immler@56189
  1380
    from as(1) have "box c d \<subseteq> cbox a b"
immler@56189
  1381
      using box_subset_cbox[of a b] by auto
immler@56189
  1382
    then have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
immler@56189
  1383
      using part1 and as(2) using i by auto
immler@56189
  1384
  } note * = this
immler@56189
  1385
  show ?th4
immler@56189
  1386
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1387
    apply (rule, rule, rule, rule)
immler@56189
  1388
    apply (rule *)
immler@56189
  1389
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1390
    prefer 4
immler@56189
  1391
    apply auto
immler@56189
  1392
    apply (erule_tac x=xa in allE, simp)+
immler@56189
  1393
    done
immler@56189
  1394
qed
immler@56189
  1395
lp15@63945
  1396
lemma eq_cbox: "cbox a b = cbox c d \<longleftrightarrow> cbox a b = {} \<and> cbox c d = {} \<or> a = c \<and> b = d"
lp15@63945
  1397
      (is "?lhs = ?rhs")
lp15@63945
  1398
proof
lp15@63945
  1399
  assume ?lhs
lp15@63945
  1400
  then have "cbox a b \<subseteq> cbox c d" "cbox c d \<subseteq> cbox a b"
lp15@63945
  1401
    by auto
lp15@63945
  1402
  then show ?rhs
lp15@63945
  1403
    by (force simp add: subset_box box_eq_empty intro: antisym euclidean_eqI)
lp15@63945
  1404
next
lp15@63945
  1405
  assume ?rhs
lp15@63945
  1406
  then show ?lhs
lp15@63945
  1407
    by force
lp15@63945
  1408
qed
lp15@63945
  1409
lp15@63945
  1410
lemma eq_cbox_box [simp]: "cbox a b = box c d \<longleftrightarrow> cbox a b = {} \<and> box c d = {}"
lp15@63945
  1411
      (is "?lhs = ?rhs")
lp15@63945
  1412
proof
lp15@63945
  1413
  assume ?lhs
lp15@63945
  1414
  then have "cbox a b \<subseteq> box c d" "box c d \<subseteq>cbox a b"
lp15@63945
  1415
    by auto
lp15@63945
  1416
  then show ?rhs
hoelzl@63957
  1417
    apply (simp add: subset_box)
lp15@63945
  1418
    using \<open>cbox a b = box c d\<close> box_ne_empty box_sing
lp15@63945
  1419
    apply (fastforce simp add:)
lp15@63945
  1420
    done
lp15@63945
  1421
next
lp15@63945
  1422
  assume ?rhs
lp15@63945
  1423
  then show ?lhs
lp15@63945
  1424
    by force
lp15@63945
  1425
qed
lp15@63945
  1426
lp15@63945
  1427
lemma eq_box_cbox [simp]: "box a b = cbox c d \<longleftrightarrow> box a b = {} \<and> cbox c d = {}"
lp15@63945
  1428
  by (metis eq_cbox_box)
lp15@63945
  1429
lp15@63945
  1430
lemma eq_box: "box a b = box c d \<longleftrightarrow> box a b = {} \<and> box c d = {} \<or> a = c \<and> b = d"
lp15@63945
  1431
      (is "?lhs = ?rhs")
lp15@63945
  1432
proof
lp15@63945
  1433
  assume ?lhs
lp15@63945
  1434
  then have "box a b \<subseteq> box c d" "box c d \<subseteq> box a b"
lp15@63945
  1435
    by auto
lp15@63945
  1436
  then show ?rhs
lp15@63945
  1437
    apply (simp add: subset_box)
hoelzl@63957
  1438
    using box_ne_empty(2) \<open>box a b = box c d\<close>
lp15@63945
  1439
    apply auto
lp15@63945
  1440
     apply (meson euclidean_eqI less_eq_real_def not_less)+
lp15@63945
  1441
    done
lp15@63945
  1442
next
lp15@63945
  1443
  assume ?rhs
lp15@63945
  1444
  then show ?lhs
lp15@63945
  1445
    by force
lp15@63945
  1446
qed
lp15@63945
  1447
lp15@63945
  1448
lemma Int_interval:
immler@56189
  1449
  fixes a :: "'a::euclidean_space"
immler@56189
  1450
  shows "cbox a b \<inter> cbox c d =
immler@56189
  1451
    cbox (\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)"
immler@56189
  1452
  unfolding set_eq_iff and Int_iff and mem_box
immler@56189
  1453
  by auto
immler@56189
  1454
immler@56189
  1455
lemma disjoint_interval:
immler@56189
  1456
  fixes a::"'a::euclidean_space"
immler@56189
  1457
  shows "cbox a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1)
immler@56189
  1458
    and "cbox a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2)
immler@56189
  1459
    and "box a b \<inter> cbox c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3)
immler@56189
  1460
    and "box a b \<inter> box c d = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
immler@56189
  1461
proof -
immler@56189
  1462
  let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"
immler@56189
  1463
  have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
immler@56189
  1464
      (\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"
immler@56189
  1465
    by blast
immler@56189
  1466
  note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)
immler@56189
  1467
  show ?th1 unfolding * by (intro **) auto
immler@56189
  1468
  show ?th2 unfolding * by (intro **) auto
immler@56189
  1469
  show ?th3 unfolding * by (intro **) auto
immler@56189
  1470
  show ?th4 unfolding * by (intro **) auto
immler@56189
  1471
qed
immler@56189
  1472
hoelzl@57447
  1473
lemma UN_box_eq_UNIV: "(\<Union>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One)) = UNIV"
hoelzl@57447
  1474
proof -
wenzelm@61942
  1475
  have "\<bar>x \<bullet> b\<bar> < real_of_int (\<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil> + 1)"
wenzelm@60462
  1476
    if [simp]: "b \<in> Basis" for x b :: 'a
wenzelm@60462
  1477
  proof -
wenzelm@61942
  1478
    have "\<bar>x \<bullet> b\<bar> \<le> real_of_int \<lceil>\<bar>x \<bullet> b\<bar>\<rceil>"
lp15@61609
  1479
      by (rule le_of_int_ceiling)
wenzelm@61942
  1480
    also have "\<dots> \<le> real_of_int \<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil>"
nipkow@59587
  1481
      by (auto intro!: ceiling_mono)
wenzelm@61942
  1482
    also have "\<dots> < real_of_int (\<lceil>Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)\<rceil> + 1)"
hoelzl@57447
  1483
      by simp
wenzelm@60462
  1484
    finally show ?thesis .
wenzelm@60462
  1485
  qed
wenzelm@60462
  1486
  then have "\<exists>n::nat. \<forall>b\<in>Basis. \<bar>x \<bullet> b\<bar> < real n" for x :: 'a
nipkow@59587
  1487
    by (metis order.strict_trans reals_Archimedean2)
hoelzl@57447
  1488
  moreover have "\<And>x b::'a. \<And>n::nat.  \<bar>x \<bullet> b\<bar> < real n \<longleftrightarrow> - real n < x \<bullet> b \<and> x \<bullet> b < real n"
hoelzl@57447
  1489
    by auto
hoelzl@57447
  1490
  ultimately show ?thesis
hoelzl@57447
  1491
    by (auto simp: box_def inner_setsum_left inner_Basis setsum.If_cases)
hoelzl@57447
  1492
qed
hoelzl@57447
  1493
wenzelm@60420
  1494
text \<open>Intervals in general, including infinite and mixtures of open and closed.\<close>
immler@56189
  1495
immler@56189
  1496
definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
immler@56189
  1497
  (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. ((a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i) \<or> (b\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> a\<bullet>i))) \<longrightarrow> x \<in> s)"
immler@56189
  1498
immler@56189
  1499
lemma is_interval_cbox: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
immler@56189
  1500
  and is_interval_box: "is_interval (box a b)" (is ?th2)
immler@56189
  1501
  unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
immler@56189
  1502
  by (meson order_trans le_less_trans less_le_trans less_trans)+
immler@56189
  1503
lp15@61609
  1504
lemma is_interval_empty [iff]: "is_interval {}"
lp15@61609
  1505
  unfolding is_interval_def  by simp
lp15@61609
  1506
lp15@61609
  1507
lemma is_interval_univ [iff]: "is_interval UNIV"
lp15@61609
  1508
  unfolding is_interval_def  by simp
immler@56189
  1509
immler@56189
  1510
lemma mem_is_intervalI:
immler@56189
  1511
  assumes "is_interval s"
immler@56189
  1512
  assumes "a \<in> s" "b \<in> s"
immler@56189
  1513
  assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i \<or> b \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> a \<bullet> i"
immler@56189
  1514
  shows "x \<in> s"
immler@56189
  1515
  by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])
immler@56189
  1516
immler@56189
  1517
lemma interval_subst:
immler@56189
  1518
  fixes S::"'a::euclidean_space set"
immler@56189
  1519
  assumes "is_interval S"
immler@56189
  1520
  assumes "x \<in> S" "y j \<in> S"
immler@56189
  1521
  assumes "j \<in> Basis"
immler@56189
  1522
  shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"
immler@56189
  1523
  by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)
immler@56189
  1524
immler@56189
  1525
lemma mem_box_componentwiseI:
immler@56189
  1526
  fixes S::"'a::euclidean_space set"
immler@56189
  1527
  assumes "is_interval S"
immler@56189
  1528
  assumes "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> ((\<lambda>x. x \<bullet> i) ` S)"
immler@56189
  1529
  shows "x \<in> S"
immler@56189
  1530
proof -
immler@56189
  1531
  from assms have "\<forall>i \<in> Basis. \<exists>s \<in> S. x \<bullet> i = s \<bullet> i"
immler@56189
  1532
    by auto
immler@56189
  1533
  with finite_Basis obtain s and bs::"'a list" where
immler@56189
  1534
    s: "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i = s i \<bullet> i" "\<And>i. i \<in> Basis \<Longrightarrow> s i \<in> S" and
immler@56189
  1535
    bs: "set bs = Basis" "distinct bs"
immler@56189
  1536
    by (metis finite_distinct_list)
immler@56189
  1537
  from nonempty_Basis s obtain j where j: "j \<in> Basis" "s j \<in> S" by blast
wenzelm@63040
  1538
  define y where
wenzelm@63040
  1539
    "y = rec_list (s j) (\<lambda>j _ Y. (\<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i))"
immler@56189
  1540
  have "x = (\<Sum>i\<in>Basis. (if i \<in> set bs then s i \<bullet> i else s j \<bullet> i) *\<^sub>R i)"
immler@56189
  1541
    using bs by (auto simp add: s(1)[symmetric] euclidean_representation)
immler@56189
  1542
  also have [symmetric]: "y bs = \<dots>"
immler@56189
  1543
    using bs(2) bs(1)[THEN equalityD1]
immler@56189
  1544
    by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
immler@56189
  1545
  also have "y bs \<in> S"
immler@56189
  1546
    using bs(1)[THEN equalityD1]
immler@56189
  1547
    apply (induct bs)
immler@56189
  1548
    apply (auto simp: y_def j)
immler@56189
  1549
    apply (rule interval_subst[OF assms(1)])
immler@56189
  1550
    apply (auto simp: s)
immler@56189
  1551
    done
immler@56189
  1552
  finally show ?thesis .
immler@56189
  1553
qed
immler@56189
  1554
lp15@63007
  1555
lemma cbox01_nonempty [simp]: "cbox 0 One \<noteq> {}"
lp15@63007
  1556
  by (simp add: box_ne_empty inner_Basis inner_setsum_left setsum_nonneg)
lp15@63007
  1557
lp15@63007
  1558
lemma box01_nonempty [simp]: "box 0 One \<noteq> {}"
lp15@63007
  1559
  by (simp add: box_ne_empty inner_Basis inner_setsum_left) (simp add: setsum.remove)
lp15@63075
  1560
himmelma@33175
  1561
wenzelm@60420
  1562
subsection\<open>Connectedness\<close>
himmelma@33175
  1563
himmelma@33175
  1564
lemma connected_local:
wenzelm@53255
  1565
 "connected S \<longleftrightarrow>
wenzelm@53255
  1566
  \<not> (\<exists>e1 e2.
wenzelm@53255
  1567
      openin (subtopology euclidean S) e1 \<and>
wenzelm@53255
  1568
      openin (subtopology euclidean S) e2 \<and>
wenzelm@53255
  1569
      S \<subseteq> e1 \<union> e2 \<and>
wenzelm@53255
  1570
      e1 \<inter> e2 = {} \<and>
wenzelm@53255
  1571
      e1 \<noteq> {} \<and>
wenzelm@53255
  1572
      e2 \<noteq> {})"
wenzelm@53282
  1573
  unfolding connected_def openin_open
lp15@59765
  1574
  by safe blast+
himmelma@33175
  1575
huffman@34105
  1576
lemma exists_diff:
huffman@34105
  1577
  fixes P :: "'a set \<Rightarrow> bool"
wenzelm@60462
  1578
  shows "(\<exists>S. P (- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53255
  1579
proof -
wenzelm@53255
  1580
  {
wenzelm@53255
  1581
    assume "?lhs"
wenzelm@53255
  1582
    then have ?rhs by blast
wenzelm@53255
  1583
  }
himmelma@33175
  1584
  moreover
wenzelm@53255
  1585
  {
wenzelm@53255
  1586
    fix S
wenzelm@53255
  1587
    assume H: "P S"
huffman@34105
  1588
    have "S = - (- S)" by auto
wenzelm@53255
  1589
    with H have "P (- (- S))" by metis
wenzelm@53255
  1590
  }
himmelma@33175
  1591
  ultimately show ?thesis by metis
himmelma@33175
  1592
qed
himmelma@33175
  1593
himmelma@33175
  1594
lemma connected_clopen: "connected S \<longleftrightarrow>
wenzelm@53255
  1595
  (\<forall>T. openin (subtopology euclidean S) T \<and>
wenzelm@53255
  1596
     closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53255
  1597
proof -
wenzelm@53255
  1598
  have "\<not> connected S \<longleftrightarrow>
wenzelm@53255
  1599
    (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
himmelma@33175
  1600
    unfolding connected_def openin_open closedin_closed
lp15@55775
  1601
    by (metis double_complement)
wenzelm@53282
  1602
  then have th0: "connected S \<longleftrightarrow>
wenzelm@53255
  1603
    \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
wenzelm@52624
  1604
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")
wenzelm@52624
  1605
    apply (simp add: closed_def)
wenzelm@52624
  1606
    apply metis
wenzelm@52624
  1607
    done
himmelma@33175
  1608
  have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
himmelma@33175
  1609
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
himmelma@33175
  1610
    unfolding connected_def openin_open closedin_closed by auto
wenzelm@53255
  1611
  {
wenzelm@53255
  1612
    fix e2
wenzelm@53255
  1613
    {
wenzelm@53255
  1614
      fix e1
wenzelm@53282
  1615
      have "?P e2 e1 \<longleftrightarrow> (\<exists>t. closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t \<noteq> S)"
wenzelm@53255
  1616
        by auto
wenzelm@53255
  1617
    }
wenzelm@53255
  1618
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
wenzelm@53255
  1619
      by metis
wenzelm@53255
  1620
  }
wenzelm@53255
  1621
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
wenzelm@53255
  1622
    by blast
wenzelm@53255
  1623
  then show ?thesis
wenzelm@53255
  1624
    unfolding th0 th1 by simp
himmelma@33175
  1625
qed
himmelma@33175
  1626
wenzelm@60420
  1627
subsection\<open>Limit points\<close>
himmelma@33175
  1628
wenzelm@53282
  1629
definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)
wenzelm@53255
  1630
  where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
himmelma@33175
  1631
himmelma@33175
  1632
lemma islimptI:
himmelma@33175
  1633
  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
himmelma@33175
  1634
  shows "x islimpt S"
himmelma@33175
  1635
  using assms unfolding islimpt_def by auto
himmelma@33175
  1636
himmelma@33175
  1637
lemma islimptE:
himmelma@33175
  1638
  assumes "x islimpt S" and "x \<in> T" and "open T"
himmelma@33175
  1639
  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
himmelma@33175
  1640
  using assms unfolding islimpt_def by auto
himmelma@33175
  1641
huffman@44584
  1642
lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
huffman@44584
  1643
  unfolding islimpt_def eventually_at_topological by auto
huffman@44584
  1644
wenzelm@53255
  1645
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
huffman@44584
  1646
  unfolding islimpt_def by fast
himmelma@33175
  1647
himmelma@33175
  1648
lemma islimpt_approachable:
himmelma@33175
  1649
  fixes x :: "'a::metric_space"
himmelma@33175
  1650
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
huffman@44584
  1651
  unfolding islimpt_iff_eventually eventually_at by fast
himmelma@33175
  1652
himmelma@33175
  1653
lemma islimpt_approachable_le:
himmelma@33175
  1654
  fixes x :: "'a::metric_space"
wenzelm@53640
  1655
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
himmelma@33175
  1656
  unfolding islimpt_approachable
huffman@44584
  1657
  using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
huffman@44584
  1658
    THEN arg_cong [where f=Not]]
huffman@44584
  1659
  by (simp add: Bex_def conj_commute conj_left_commute)
himmelma@33175
  1660
huffman@44571
  1661
lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
huffman@44571
  1662
  unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
huffman@44571
  1663
hoelzl@51351
  1664
lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
hoelzl@51351
  1665
  unfolding islimpt_def by blast
hoelzl@51351
  1666
wenzelm@60420
  1667
text \<open>A perfect space has no isolated points.\<close>
huffman@44210
  1668
huffman@44571
  1669
lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
huffman@44571
  1670
  unfolding islimpt_UNIV_iff by (rule not_open_singleton)
himmelma@33175
  1671
himmelma@33175
  1672
lemma perfect_choose_dist:
huffman@44072
  1673
  fixes x :: "'a::{perfect_space, metric_space}"
himmelma@33175
  1674
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
wenzelm@53255
  1675
  using islimpt_UNIV [of x]
wenzelm@53255
  1676
  by (simp add: islimpt_approachable)
himmelma@33175
  1677
himmelma@33175
  1678
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
himmelma@33175
  1679
  unfolding closed_def
himmelma@33175
  1680
  apply (subst open_subopen)
huffman@34105
  1681
  apply (simp add: islimpt_def subset_eq)
wenzelm@52624
  1682
  apply (metis ComplE ComplI)
wenzelm@52624
  1683
  done
himmelma@33175
  1684
himmelma@33175
  1685
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
himmelma@33175
  1686
  unfolding islimpt_def by auto
himmelma@33175
  1687
himmelma@33175
  1688
lemma finite_set_avoid:
himmelma@33175
  1689
  fixes a :: "'a::metric_space"
wenzelm@53255
  1690
  assumes fS: "finite S"
wenzelm@53640
  1691
  shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
wenzelm@53255
  1692
proof (induct rule: finite_induct[OF fS])
wenzelm@53255
  1693
  case 1
wenzelm@53255
  1694
  then show ?case by (auto intro: zero_less_one)
himmelma@33175
  1695
next
himmelma@33175
  1696
  case (2 x F)
wenzelm@60462
  1697
  from 2 obtain d where d: "d > 0" "\<forall>x\<in>F. x \<noteq> a \<longrightarrow> d \<le> dist a x"
wenzelm@53255
  1698
    by blast
wenzelm@53255
  1699
  show ?case
wenzelm@53255
  1700
  proof (cases "x = a")
wenzelm@53255
  1701
    case True
wenzelm@53255
  1702
    then show ?thesis using d by auto
wenzelm@53255
  1703
  next
wenzelm@53255
  1704
    case False
himmelma@33175
  1705
    let ?d = "min d (dist a x)"
wenzelm@53255
  1706
    have dp: "?d > 0"
paulson@62087
  1707
      using False d(1) by auto
wenzelm@60462
  1708
    from d have d': "\<forall>x\<in>F. x \<noteq> a \<longrightarrow> ?d \<le> dist a x"
wenzelm@53255
  1709
      by auto
wenzelm@53255
  1710
    with dp False show ?thesis
wenzelm@53255
  1711
      by (auto intro!: exI[where x="?d"])
wenzelm@53255
  1712
  qed
himmelma@33175
  1713
qed
himmelma@33175
  1714
himmelma@33175
  1715
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
huffman@50897
  1716
  by (simp add: islimpt_iff_eventually eventually_conj_iff)
himmelma@33175
  1717
himmelma@33175
  1718
lemma discrete_imp_closed:
himmelma@33175
  1719
  fixes S :: "'a::metric_space set"
wenzelm@53255
  1720
  assumes e: "0 < e"
wenzelm@53255
  1721
    and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
himmelma@33175
  1722
  shows "closed S"
wenzelm@53255
  1723
proof -
wenzelm@53255
  1724
  {
wenzelm@53255
  1725
    fix x
wenzelm@53255
  1726
    assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
himmelma@33175
  1727
    from e have e2: "e/2 > 0" by arith
wenzelm@53282
  1728
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
wenzelm@53255
  1729
      by blast
himmelma@33175
  1730
    let ?m = "min (e/2) (dist x y) "
wenzelm@53255
  1731
    from e2 y(2) have mp: "?m > 0"
paulson@62087
  1732
      by simp
wenzelm@53282
  1733
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
wenzelm@53255
  1734
      by blast
himmelma@33175
  1735
    have th: "dist z y < e" using z y
himmelma@33175
  1736
      by (intro dist_triangle_lt [where z=x], simp)
himmelma@33175
  1737
    from d[rule_format, OF y(1) z(1) th] y z
himmelma@33175
  1738
    have False by (auto simp add: dist_commute)}
wenzelm@53255
  1739
  then show ?thesis
wenzelm@53255
  1740
    by (metis islimpt_approachable closed_limpt [where 'a='a])
himmelma@33175
  1741
qed
himmelma@33175
  1742
eberlm@61524
  1743
lemma closed_of_nat_image: "closed (of_nat ` A :: 'a :: real_normed_algebra_1 set)"
eberlm@61524
  1744
  by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
eberlm@61524
  1745
eberlm@61524
  1746
lemma closed_of_int_image: "closed (of_int ` A :: 'a :: real_normed_algebra_1 set)"
eberlm@61524
  1747
  by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
eberlm@61524
  1748
eberlm@61524
  1749
lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
eberlm@61524
  1750
  unfolding Nats_def by (rule closed_of_nat_image)
eberlm@61524
  1751
eberlm@61524
  1752
lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
eberlm@61524
  1753
  unfolding Ints_def by (rule closed_of_int_image)
eberlm@61524
  1754
huffman@44210
  1755
wenzelm@60420
  1756
subsection \<open>Interior of a Set\<close>
huffman@44210
  1757
huffman@44519
  1758
definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
huffman@44519
  1759
huffman@44519
  1760
lemma interiorI [intro?]:
huffman@44519
  1761
  assumes "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
  1762
  shows "x \<in> interior S"
huffman@44519
  1763
  using assms unfolding interior_def by fast
huffman@44519
  1764
huffman@44519
  1765
lemma interiorE [elim?]:
huffman@44519
  1766
  assumes "x \<in> interior S"
huffman@44519
  1767
  obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
  1768
  using assms unfolding interior_def by fast
huffman@44519
  1769
huffman@44519
  1770
lemma open_interior [simp, intro]: "open (interior S)"
huffman@44519
  1771
  by (simp add: interior_def open_Union)
huffman@44519
  1772
huffman@44519
  1773
lemma interior_subset: "interior S \<subseteq> S"
huffman@44519
  1774
  by (auto simp add: interior_def)
huffman@44519
  1775
huffman@44519
  1776
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
huffman@44519
  1777
  by (auto simp add: interior_def)
huffman@44519
  1778
huffman@44519
  1779
lemma interior_open: "open S \<Longrightarrow> interior S = S"
huffman@44519
  1780
  by (intro equalityI interior_subset interior_maximal subset_refl)
himmelma@33175
  1781
himmelma@33175
  1782
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
huffman@44519
  1783
  by (metis open_interior interior_open)
huffman@44519
  1784
huffman@44519
  1785
lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
himmelma@33175
  1786
  by (metis interior_maximal interior_subset subset_trans)
himmelma@33175
  1787
huffman@44519
  1788
lemma interior_empty [simp]: "interior {} = {}"
huffman@44519
  1789
  using open_empty by (rule interior_open)
huffman@44519
  1790
huffman@44522
  1791
lemma interior_UNIV [simp]: "interior UNIV = UNIV"
huffman@44522
  1792
  using open_UNIV by (rule interior_open)
huffman@44522
  1793
huffman@44519
  1794
lemma interior_interior [simp]: "interior (interior S) = interior S"
huffman@44519
  1795
  using open_interior by (rule interior_open)
huffman@44519
  1796
huffman@44522
  1797
lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
huffman@44522
  1798
  by (auto simp add: interior_def)
huffman@44519
  1799
huffman@44519
  1800
lemma interior_unique:
huffman@44519
  1801
  assumes "T \<subseteq> S" and "open T"
huffman@44519
  1802
  assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
huffman@44519
  1803
  shows "interior S = T"
huffman@44519
  1804
  by (intro equalityI assms interior_subset open_interior interior_maximal)
huffman@44519
  1805
paulson@61518
  1806
lemma interior_singleton [simp]:
paulson@61518
  1807
      fixes a :: "'a::perfect_space" shows "interior {a} = {}"
paulson@61518
  1808
  apply (rule interior_unique, simp_all)
paulson@61518
  1809
  using not_open_singleton subset_singletonD by fastforce
paulson@61518
  1810
paulson@61518
  1811
lemma interior_Int [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
huffman@44522
  1812
  by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
huffman@44519
  1813
    Int_lower2 interior_maximal interior_subset open_Int open_interior)
huffman@44519
  1814
huffman@44519
  1815
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
huffman@44519
  1816
  using open_contains_ball_eq [where S="interior S"]
huffman@44519
  1817
  by (simp add: open_subset_interior)
himmelma@33175
  1818
eberlm@61531
  1819
lemma eventually_nhds_in_nhd: "x \<in> interior s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
eberlm@61531
  1820
  using interior_subset[of s] by (subst eventually_nhds) blast
eberlm@61531
  1821
himmelma@33175
  1822
lemma interior_limit_point [intro]:
himmelma@33175
  1823
  fixes x :: "'a::perfect_space"
wenzelm@53255
  1824
  assumes x: "x \<in> interior S"
wenzelm@53255
  1825
  shows "x islimpt S"
huffman@44072
  1826
  using x islimpt_UNIV [of x]
huffman@44072
  1827
  unfolding interior_def islimpt_def
huffman@44072
  1828
  apply (clarsimp, rename_tac T T')
huffman@44072
  1829
  apply (drule_tac x="T \<inter> T'" in spec)
huffman@44072
  1830
  apply (auto simp add: open_Int)
huffman@44072
  1831
  done
himmelma@33175
  1832
himmelma@33175
  1833
lemma interior_closed_Un_empty_interior:
wenzelm@53255
  1834
  assumes cS: "closed S"
wenzelm@53255
  1835
    and iT: "interior T = {}"
huffman@44519
  1836
  shows "interior (S \<union> T) = interior S"
himmelma@33175
  1837
proof
huffman@44519
  1838
  show "interior S \<subseteq> interior (S \<union> T)"
wenzelm@53255
  1839
    by (rule interior_mono) (rule Un_upper1)
himmelma@33175
  1840
  show "interior (S \<union> T) \<subseteq> interior S"
himmelma@33175
  1841
  proof
wenzelm@53255
  1842
    fix x
wenzelm@53255
  1843
    assume "x \<in> interior (S \<union> T)"
huffman@44519
  1844
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
himmelma@33175
  1845
    show "x \<in> interior S"
himmelma@33175
  1846
    proof (rule ccontr)
himmelma@33175
  1847
      assume "x \<notin> interior S"
wenzelm@60420
  1848
      with \<open>x \<in> R\<close> \<open>open R\<close> obtain y where "y \<in> R - S"
huffman@44519
  1849
        unfolding interior_def by fast
wenzelm@60420
  1850
      from \<open>open R\<close> \<open>closed S\<close> have "open (R - S)"
wenzelm@53282
  1851
        by (rule open_Diff)
wenzelm@60420
  1852
      from \<open>R \<subseteq> S \<union> T\<close> have "R - S \<subseteq> T"
wenzelm@53282
  1853
        by fast
wenzelm@60420
  1854
      from \<open>y \<in> R - S\<close> \<open>open (R - S)\<close> \<open>R - S \<subseteq> T\<close> \<open>interior T = {}\<close> show False
wenzelm@53282
  1855
        unfolding interior_def by fast
himmelma@33175
  1856
    qed
himmelma@33175
  1857
  qed
himmelma@33175
  1858
qed
himmelma@33175
  1859
huffman@44365
  1860
lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
huffman@44365
  1861
proof (rule interior_unique)
huffman@44365
  1862
  show "interior A \<times> interior B \<subseteq> A \<times> B"
huffman@44365
  1863
    by (intro Sigma_mono interior_subset)
huffman@44365
  1864
  show "open (interior A \<times> interior B)"
huffman@44365
  1865
    by (intro open_Times open_interior)
wenzelm@53255
  1866
  fix T
wenzelm@53255
  1867
  assume "T \<subseteq> A \<times> B" and "open T"
wenzelm@53255
  1868
  then show "T \<subseteq> interior A \<times> interior B"
wenzelm@53282
  1869
  proof safe
wenzelm@53255
  1870
    fix x y
wenzelm@53255
  1871
    assume "(x, y) \<in> T"
huffman@44519
  1872
    then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
wenzelm@60420
  1873
      using \<open>open T\<close> unfolding open_prod_def by fast
wenzelm@53255
  1874
    then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
wenzelm@60420
  1875
      using \<open>T \<subseteq> A \<times> B\<close> by auto
wenzelm@53255
  1876
    then show "x \<in> interior A" and "y \<in> interior B"
huffman@44519
  1877
      by (auto intro: interiorI)
huffman@44519
  1878
  qed
huffman@44365
  1879
qed
huffman@44365
  1880
hoelzl@61245
  1881
lemma interior_Ici:
hoelzl@61245
  1882
  fixes x :: "'a :: {dense_linorder, linorder_topology}"
hoelzl@61245
  1883
  assumes "b < x"
hoelzl@61245
  1884
  shows "interior { x ..} = { x <..}"
hoelzl@61245
  1885
proof (rule interior_unique)
hoelzl@61245
  1886
  fix T assume "T \<subseteq> {x ..}" "open T"
hoelzl@61245
  1887
  moreover have "x \<notin> T"
hoelzl@61245
  1888
  proof
hoelzl@61245
  1889
    assume "x \<in> T"
hoelzl@61245
  1890
    obtain y where "y < x" "{y <.. x} \<subseteq> T"
hoelzl@61245
  1891
      using open_left[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>b < x\<close>] by auto
hoelzl@61245
  1892
    with dense[OF \<open>y < x\<close>] obtain z where "z \<in> T" "z < x"
hoelzl@61245
  1893
      by (auto simp: subset_eq Ball_def)
hoelzl@61245
  1894
    with \<open>T \<subseteq> {x ..}\<close> show False by auto
hoelzl@61245
  1895
  qed
hoelzl@61245
  1896
  ultimately show "T \<subseteq> {x <..}"
hoelzl@61245
  1897
    by (auto simp: subset_eq less_le)
hoelzl@61245
  1898
qed auto
hoelzl@61245
  1899
hoelzl@61245
  1900
lemma interior_Iic:
hoelzl@61245
  1901
  fixes x :: "'a :: {dense_linorder, linorder_topology}"
hoelzl@61245
  1902
  assumes "x < b"
hoelzl@61245
  1903
  shows "interior {.. x} = {..< x}"
hoelzl@61245
  1904
proof (rule interior_unique)
hoelzl@61245
  1905
  fix T assume "T \<subseteq> {.. x}" "open T"
hoelzl@61245
  1906
  moreover have "x \<notin> T"
hoelzl@61245
  1907
  proof
hoelzl@61245
  1908
    assume "x \<in> T"
hoelzl@61245
  1909
    obtain y where "x < y" "{x ..< y} \<subseteq> T"
hoelzl@61245
  1910
      using open_right[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>x < b\<close>] by auto
hoelzl@61245
  1911
    with dense[OF \<open>x < y\<close>] obtain z where "z \<in> T" "x < z"
hoelzl@61245
  1912
      by (auto simp: subset_eq Ball_def less_le)
hoelzl@61245
  1913
    with \<open>T \<subseteq> {.. x}\<close> show False by auto
hoelzl@61245
  1914
  qed
hoelzl@61245
  1915
  ultimately show "T \<subseteq> {..< x}"
hoelzl@61245
  1916
    by (auto simp: subset_eq less_le)
hoelzl@61245
  1917
qed auto
himmelma@33175
  1918
wenzelm@60420
  1919
subsection \<open>Closure of a Set\<close>
himmelma@33175
  1920
himmelma@33175
  1921
definition "closure S = S \<union> {x | x. x islimpt S}"
himmelma@33175
  1922
huffman@44518
  1923
lemma interior_closure: "interior S = - (closure (- S))"
huffman@44518
  1924
  unfolding interior_def closure_def islimpt_def by auto
huffman@44518
  1925
huffman@34105
  1926
lemma closure_interior: "closure S = - interior (- S)"
huffman@44518
  1927
  unfolding interior_closure by simp
himmelma@33175
  1928
himmelma@33175
  1929
lemma closed_closure[simp, intro]: "closed (closure S)"
huffman@44518
  1930
  unfolding closure_interior by (simp add: closed_Compl)
huffman@44518
  1931
huffman@44518
  1932
lemma closure_subset: "S \<subseteq> closure S"
huffman@44518
  1933
  unfolding closure_def by simp
himmelma@33175
  1934
himmelma@33175
  1935
lemma closure_hull: "closure S = closed hull S"
huffman@44519
  1936
  unfolding hull_def closure_interior interior_def by auto
himmelma@33175
  1937
himmelma@33175
  1938
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
huffman@44519
  1939
  unfolding closure_hull using closed_Inter by (rule hull_eq)
huffman@44519
  1940
huffman@44519
  1941
lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
huffman@44519
  1942
  unfolding closure_eq .
huffman@44519
  1943
huffman@44519
  1944
lemma closure_closure [simp]: "closure (closure S) = closure S"
huffman@44518
  1945
  unfolding closure_hull by (rule hull_hull)
himmelma@33175
  1946
huffman@44522
  1947
lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
huffman@44518
  1948
  unfolding closure_hull by (rule hull_mono)
himmelma@33175
  1949
huffman@44519
  1950
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
huffman@44518
  1951
  unfolding closure_hull by (rule hull_minimal)
himmelma@33175
  1952
huffman@44519
  1953
lemma closure_unique:
wenzelm@53255
  1954
  assumes "S \<subseteq> T"
wenzelm@53255
  1955
    and "closed T"
wenzelm@53255
  1956
    and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
huffman@44519
  1957
  shows "closure S = T"
huffman@44519
  1958
  using assms unfolding closure_hull by (rule hull_unique)
huffman@44519
  1959
huffman@44519
  1960
lemma closure_empty [simp]: "closure {} = {}"
huffman@44518
  1961
  using closed_empty by (rule closure_closed)
himmelma@33175
  1962
huffman@44522
  1963
lemma closure_UNIV [simp]: "closure UNIV = UNIV"
huffman@44518
  1964
  using closed_UNIV by (rule closure_closed)
huffman@44518
  1965
huffman@44518
  1966
lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
huffman@44518
  1967
  unfolding closure_interior by simp
himmelma@33175
  1968
lp15@60974
  1969
lemma closure_eq_empty [iff]: "closure S = {} \<longleftrightarrow> S = {}"
himmelma@33175
  1970
  using closure_empty closure_subset[of S]
himmelma@33175
  1971
  by blast
himmelma@33175
  1972
himmelma@33175
  1973
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
  1974
  using closure_eq[of S] closure_subset[of S]
himmelma@33175
  1975
  by simp
himmelma@33175
  1976
lp15@62843
  1977
lemma open_Int_closure_eq_empty:
himmelma@33175
  1978
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
huffman@34105
  1979
  using open_subset_interior[of S "- T"]
huffman@34105
  1980
  using interior_subset[of "- T"]
himmelma@33175
  1981
  unfolding closure_interior
himmelma@33175
  1982
  by auto
himmelma@33175
  1983
lp15@63128
  1984
lemma open_Int_closure_subset:
himmelma@33175
  1985
  "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
himmelma@33175
  1986
proof
himmelma@33175
  1987
  fix x
himmelma@33175
  1988
  assume as: "open S" "x \<in> S \<inter> closure T"
wenzelm@53255
  1989
  {
wenzelm@53282
  1990
    assume *: "x islimpt T"
himmelma@33175
  1991
    have "x islimpt (S \<inter> T)"
himmelma@33175
  1992
    proof (rule islimptI)
himmelma@33175
  1993
      fix A
himmelma@33175
  1994
      assume "x \<in> A" "open A"
himmelma@33175
  1995
      with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
himmelma@33175
  1996
        by (simp_all add: open_Int)
himmelma@33175
  1997
      with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
himmelma@33175
  1998
        by (rule islimptE)
wenzelm@53255
  1999
      then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
himmelma@33175
  2000
        by simp_all
wenzelm@53255
  2001
      then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
himmelma@33175
  2002
    qed
himmelma@33175
  2003
  }
himmelma@33175
  2004
  then show "x \<in> closure (S \<inter> T)" using as
himmelma@33175
  2005
    unfolding closure_def
himmelma@33175
  2006
    by blast
himmelma@33175
  2007
qed
himmelma@33175
  2008
huffman@44519
  2009
lemma closure_complement: "closure (- S) = - interior S"
huffman@44518
  2010
  unfolding closure_interior by simp
himmelma@33175
  2011
huffman@44519
  2012
lemma interior_complement: "interior (- S) = - closure S"
huffman@44518
  2013
  unfolding closure_interior by simp
himmelma@33175
  2014
huffman@44365
  2015
lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
huffman@44519
  2016
proof (rule closure_unique)
huffman@44365
  2017
  show "A \<times> B \<subseteq> closure A \<times> closure B"
huffman@44365
  2018
    by (intro Sigma_mono closure_subset)
huffman@44365
  2019
  show "closed (closure A \<times> closure B)"
huffman@44365
  2020
    by (intro closed_Times closed_closure)
wenzelm@53282
  2021
  fix T
wenzelm@53282
  2022
  assume "A \<times> B \<subseteq> T" and "closed T"
wenzelm@53282
  2023
  then show "closure A \<times> closure B \<subseteq> T"
huffman@44365
  2024
    apply (simp add: closed_def open_prod_def, clarify)
huffman@44365
  2025
    apply (rule ccontr)
huffman@44365
  2026
    apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
huffman@44365
  2027
    apply (simp add: closure_interior interior_def)
huffman@44365
  2028
    apply (drule_tac x=C in spec)
huffman@44365
  2029
    apply (drule_tac x=D in spec)
huffman@44365
  2030
    apply auto
huffman@44365
  2031
    done
huffman@44365
  2032
qed
huffman@44365
  2033
hoelzl@51351
  2034
lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"
hoelzl@51351
  2035
  unfolding closure_def using islimpt_punctured by blast
hoelzl@51351
  2036
lp15@63301
  2037
lemma connected_imp_connected_closure: "connected S \<Longrightarrow> connected (closure S)"
lp15@62843
  2038
    by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)
lp15@61306
  2039
lp15@61306
  2040
lemma limpt_of_limpts:
lp15@61306
  2041
      fixes x :: "'a::metric_space"
lp15@63301
  2042
      shows "x islimpt {y. y islimpt S} \<Longrightarrow> x islimpt S"
lp15@61306
  2043
  apply (clarsimp simp add: islimpt_approachable)
lp15@61306
  2044
  apply (drule_tac x="e/2" in spec)
lp15@61306
  2045
  apply (auto simp: simp del: less_divide_eq_numeral1)
lp15@61306
  2046
  apply (drule_tac x="dist x' x" in spec)
lp15@61306
  2047
  apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1)
lp15@61306
  2048
  apply (erule rev_bexI)
lp15@61306
  2049
  by (metis dist_commute dist_triangle_half_r less_trans less_irrefl)
lp15@61306
  2050
lp15@63301
  2051
lemma closed_limpts:  "closed {x::'a::metric_space. x islimpt S}"
lp15@61306
  2052
  using closed_limpt limpt_of_limpts by blast
lp15@61306
  2053
lp15@61306
  2054
lemma limpt_of_closure:
lp15@61306
  2055
      fixes x :: "'a::metric_space"
lp15@63301
  2056
      shows "x islimpt closure S \<longleftrightarrow> x islimpt S"
lp15@61306
  2057
  by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)
lp15@61306
  2058
lp15@62843
  2059
lemma closedin_limpt:
lp15@63301
  2060
   "closedin (subtopology euclidean T) S \<longleftrightarrow> S \<subseteq> T \<and> (\<forall>x. x islimpt S \<and> x \<in> T \<longrightarrow> x \<in> S)"
lp15@61306
  2061
  apply (simp add: closedin_closed, safe)
lp15@61306
  2062
  apply (simp add: closed_limpt islimpt_subset)
lp15@63301
  2063
  apply (rule_tac x="closure S" in exI)
lp15@61306
  2064
  apply simp
lp15@61306
  2065
  apply (force simp: closure_def)
lp15@61306
  2066
  done
lp15@61306
  2067
paulson@61518
  2068
lemma closedin_closed_eq:
lp15@63301
  2069
    "closed S \<Longrightarrow> (closedin (subtopology euclidean S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S)"
lp15@62843
  2070
  by (meson closedin_limpt closed_subset closedin_closed_trans)
paulson@61518
  2071
lp15@63301
  2072
lemma closedin_subset_trans:
lp15@63301
  2073
   "\<lbrakk>closedin (subtopology euclidean U) S; S \<subseteq> T; T \<subseteq> U\<rbrakk>
lp15@63301
  2074
    \<Longrightarrow> closedin (subtopology euclidean T) S"
lp15@63301
  2075
by (meson closedin_limpt subset_iff)
lp15@63301
  2076
lp15@63469
  2077
lemma openin_subset_trans:
lp15@63469
  2078
    "\<lbrakk>openin (subtopology euclidean U) S; S \<subseteq> T; T \<subseteq> U\<rbrakk>
lp15@63469
  2079
     \<Longrightarrow> openin (subtopology euclidean T) S"
lp15@63469
  2080
  by (auto simp: openin_open)
lp15@63469
  2081
lp15@63301
  2082
lemma closedin_Times:
lp15@63301
  2083
   "\<lbrakk>closedin (subtopology euclidean S) S'; closedin (subtopology euclidean T) T'\<rbrakk>
lp15@63301
  2084
    \<Longrightarrow> closedin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
lp15@63301
  2085
unfolding closedin_closed using closed_Times by blast
lp15@63301
  2086
hoelzl@62083
  2087
lemma bdd_below_closure:
hoelzl@62083
  2088
  fixes A :: "real set"
hoelzl@62083
  2089
  assumes "bdd_below A"
hoelzl@62083
  2090
  shows "bdd_below (closure A)"
hoelzl@62083
  2091
proof -
hoelzl@62083
  2092
  from assms obtain m where "\<And>x. x \<in> A \<Longrightarrow> m \<le> x" unfolding bdd_below_def by auto
hoelzl@62083
  2093
  hence "A \<subseteq> {m..}" by auto
hoelzl@62083
  2094
  hence "closure A \<subseteq> {m..}" using closed_real_atLeast by (rule closure_minimal)
hoelzl@62083
  2095
  thus ?thesis unfolding bdd_below_def by auto
hoelzl@62083
  2096
qed
eberlm@61531
  2097
lp15@61306
  2098
subsection\<open>Connected components, considered as a connectedness relation or a set\<close>
lp15@61306
  2099
lp15@61306
  2100
definition
lp15@61306
  2101
   "connected_component s x y \<equiv> \<exists>t. connected t \<and> t \<subseteq> s \<and> x \<in> t \<and> y \<in> t"
lp15@61306
  2102
lp15@61306
  2103
abbreviation
lp15@61306
  2104
   "connected_component_set s x \<equiv> Collect (connected_component s x)"
lp15@61306
  2105
lp15@61426
  2106
lemma connected_componentI:
lp15@61426
  2107
    "\<lbrakk>connected t; t \<subseteq> s; x \<in> t; y \<in> t\<rbrakk> \<Longrightarrow> connected_component s x y"
lp15@61426
  2108
  by (auto simp: connected_component_def)
lp15@61426
  2109
lp15@61306
  2110
lemma connected_component_in: "connected_component s x y \<Longrightarrow> x \<in> s \<and> y \<in> s"
lp15@61306
  2111
  by (auto simp: connected_component_def)
lp15@61306
  2112
lp15@61306
  2113
lemma connected_component_refl: "x \<in> s \<Longrightarrow> connected_component s x x"
lp15@61306
  2114
  apply (auto simp: connected_component_def)
lp15@61306
  2115
  using connected_sing by blast
lp15@61306
  2116
lp15@61306
  2117
lemma connected_component_refl_eq [simp]: "connected_component s x x \<longleftrightarrow> x \<in> s"
lp15@61306
  2118
  by (auto simp: connected_component_refl) (auto simp: connected_component_def)
lp15@61306
  2119
lp15@61306
  2120
lemma connected_component_sym: "connected_component s x y \<Longrightarrow> connected_component s y x"
lp15@61306
  2121
  by (auto simp: connected_component_def)
lp15@61306
  2122
lp15@61306
  2123
lemma connected_component_trans:
lp15@61306
  2124
    "\<lbrakk>connected_component s x y; connected_component s y z\<rbrakk> \<Longrightarrow> connected_component s x z"
lp15@61306
  2125
  unfolding connected_component_def
lp15@61306
  2126
  by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)
lp15@61306
  2127
lp15@61306
  2128
lemma connected_component_of_subset: "\<lbrakk>connected_component s x y; s \<subseteq> t\<rbrakk> \<Longrightarrow> connected_component t x y"
lp15@61306
  2129
  by (auto simp: connected_component_def)
lp15@61306
  2130
wenzelm@61952
  2131
lemma connected_component_Union: "connected_component_set s x = \<Union>{t. connected t \<and> x \<in> t \<and> t \<subseteq> s}"
lp15@61306
  2132
  by (auto simp: connected_component_def)
lp15@61306
  2133
lp15@61306
  2134
lemma connected_connected_component [iff]: "connected (connected_component_set s x)"
lp15@61306
  2135
  by (auto simp: connected_component_Union intro: connected_Union)
lp15@61306
  2136
lp15@61306
  2137
lemma connected_iff_eq_connected_component_set: "connected s \<longleftrightarrow> (\<forall>x \<in> s. connected_component_set s x = s)"
lp15@61306
  2138
proof (cases "s={}")
lp15@61306
  2139
  case True then show ?thesis by simp
lp15@61306
  2140
next
lp15@61306
  2141
  case False
lp15@61306
  2142
  then obtain x where "x \<in> s" by auto
lp15@61306
  2143
  show ?thesis
lp15@61306
  2144
  proof
lp15@61306
  2145
    assume "connected s"
lp15@61306
  2146
    then show "\<forall>x \<in> s. connected_component_set s x = s"
lp15@61306
  2147
      by (force simp: connected_component_def)
lp15@61306
  2148
  next
lp15@61306
  2149
    assume "\<forall>x \<in> s. connected_component_set s x = s"
lp15@61306
  2150
    then show "connected s"
wenzelm@61808
  2151
      by (metis \<open>x \<in> s\<close> connected_connected_component)
lp15@61306
  2152
  qed
lp15@61306
  2153
qed
lp15@61306
  2154
lp15@61306
  2155
lemma connected_component_subset: "connected_component_set s x \<subseteq> s"
lp15@61306
  2156
  using connected_component_in by blast
lp15@61306
  2157
lp15@61306
  2158
lemma connected_component_eq_self: "\<lbrakk>connected s; x \<in> s\<rbrakk> \<Longrightarrow> connected_component_set s x = s"
lp15@61306
  2159
  by (simp add: connected_iff_eq_connected_component_set)
lp15@61306
  2160
lp15@61306
  2161
lemma connected_iff_connected_component:
lp15@61306
  2162
    "connected s \<longleftrightarrow> (\<forall>x \<in> s. \<forall>y \<in> s. connected_component s x y)"
lp15@61306
  2163
  using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)
lp15@61306
  2164
lp15@61306
  2165
lemma connected_component_maximal:
lp15@61306
  2166
    "\<lbrakk>x \<in> t; connected t; t \<subseteq> s\<rbrakk> \<Longrightarrow> t \<subseteq> (connected_component_set s x)"
lp15@61306
  2167
  using connected_component_eq_self connected_component_of_subset by blast
lp15@61306
  2168
lp15@61306
  2169
lemma connected_component_mono:
lp15@61306
  2170
    "s \<subseteq> t \<Longrightarrow> (connected_component_set s x) \<subseteq> (connected_component_set t x)"
lp15@61306
  2171
  by (simp add: Collect_mono connected_component_of_subset)
lp15@61306
  2172
lp15@61306
  2173
lemma connected_component_eq_empty [simp]: "connected_component_set s x = {} \<longleftrightarrow> (x \<notin> s)"
lp15@61306
  2174
  using connected_component_refl by (fastforce simp: connected_component_in)
lp15@61306
  2175
lp15@61306
  2176
lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"
lp15@61306
  2177
  using connected_component_eq_empty by blast
lp15@61306
  2178
lp15@61306
  2179
lemma connected_component_eq:
lp15@61306
  2180
    "y \<in> connected_component_set s x
lp15@61306
  2181
     \<Longrightarrow> (connected_component_set s y = connected_component_set s x)"
lp15@61306
  2182
  by (metis (no_types, lifting) Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)
lp15@61306
  2183
lp15@61306
  2184
lemma closed_connected_component:
lp15@61306
  2185
  assumes s: "closed s" shows "closed (connected_component_set s x)"
lp15@61306
  2186
proof (cases "x \<in> s")
lp15@61306
  2187
  case False then show ?thesis
lp15@61306
  2188
    by (metis connected_component_eq_empty closed_empty)
lp15@61306
  2189
next
lp15@61306
  2190
  case True
lp15@61306
  2191
  show ?thesis
lp15@61306
  2192
    unfolding closure_eq [symmetric]
lp15@61306
  2193
    proof
lp15@61306
  2194
      show "closure (connected_component_set s x) \<subseteq> connected_component_set s x"
lp15@61306
  2195
        apply (rule connected_component_maximal)
lp15@61306
  2196
        apply (simp add: closure_def True)
lp15@61306
  2197
        apply (simp add: connected_imp_connected_closure)
lp15@61306
  2198
        apply (simp add: s closure_minimal connected_component_subset)
lp15@61306
  2199
        done
lp15@61306
  2200
    next
lp15@61306
  2201
      show "connected_component_set s x \<subseteq> closure (connected_component_set s x)"
lp15@61306
  2202
        by (simp add: closure_subset)
lp15@61306
  2203
  qed
lp15@61306
  2204
qed
lp15@61306
  2205
lp15@61306
  2206
lemma connected_component_disjoint:
lp15@61306
  2207
    "(connected_component_set s a) \<inter> (connected_component_set s b) = {} \<longleftrightarrow>
lp15@61306
  2208
     a \<notin> connected_component_set s b"
lp15@61306
  2209
apply (auto simp: connected_component_eq)
lp15@61306
  2210
using connected_component_eq connected_component_sym by blast
lp15@61306
  2211
lp15@61306
  2212
lemma connected_component_nonoverlap:
lp15@61306
  2213
    "(connected_component_set s a) \<inter> (connected_component_set s b) = {} \<longleftrightarrow>
lp15@61306
  2214
     (a \<notin> s \<or> b \<notin> s \<or> connected_component_set s a \<noteq> connected_component_set s b)"
lp15@61306
  2215
  apply (auto simp: connected_component_in)
lp15@61306
  2216
  using connected_component_refl_eq apply blast
lp15@61306
  2217
  apply (metis connected_component_eq mem_Collect_eq)
lp15@61306
  2218
  apply (metis connected_component_eq mem_Collect_eq)
lp15@61306
  2219
  done
lp15@61306
  2220
lp15@61306
  2221
lemma connected_component_overlap:
lp15@61306
  2222
    "(connected_component_set s a \<inter> connected_component_set s b \<noteq> {}) =
lp15@61306
  2223
     (a \<in> s \<and> b \<in> s \<and> connected_component_set s a = connected_component_set s b)"
lp15@61306
  2224
  by (auto simp: connected_component_nonoverlap)
lp15@61306
  2225
lp15@61306
  2226
lemma connected_component_sym_eq: "connected_component s x y \<longleftrightarrow> connected_component s y x"
lp15@61306
  2227
  using connected_component_sym by blast
lp15@61306
  2228
lp15@61306
  2229
lemma connected_component_eq_eq:
lp15@61306
  2230
    "connected_component_set s x = connected_component_set s y \<longleftrightarrow>
lp15@61306
  2231
     x \<notin> s \<and> y \<notin> s \<or> x \<in> s \<and> y \<in> s \<and> connected_component s x y"
lp15@61306
  2232
  apply (case_tac "y \<in> s")
lp15@61306
  2233
   apply (simp add:)