src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author haftmann
Tue Oct 13 09:21:15 2015 +0200 (2015-10-13)
changeset 61424 c3658c18b7bc
parent 61306 9dd394c866fc
child 61426 d53db136e8fd
permissions -rw-r--r--
prod_case as canonical name for product type eliminator
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(*  title:      HOL/Library/Topology_Euclidian_Space.thy
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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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  Complex_Main
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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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lemma dist_0_norm:
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  fixes x :: "'a::real_normed_vector"
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  shows "dist 0 x = norm x"
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unfolding dist_norm by simp
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lemma dist_double: "dist x y < d / 2 \<Longrightarrow> dist x z < d / 2 \<Longrightarrow> dist y z < d"
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  using dist_triangle[of y z x] by (simp add: dist_commute)
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(* LEGACY *)
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lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s \<circ> r) ----> l"
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  by (rule LIMSEQ_subseq_LIMSEQ)
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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 Lim_within_open:
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  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
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  shows "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
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  by (fact tendsto_within_open)
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lemma continuous_on_union:
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  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
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  by (fact continuous_on_closed_Un)
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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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  def A \<equiv> "(\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n) ` N)) ` (Collect finite::nat set set)"
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  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>
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        (\<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)"
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  proof (safe intro!: exI[where x=A])
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    show "countable A"
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      unfolding A_def by (intro countable_image countable_Collect_finite)
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    fix a
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    assume "a \<in> A"
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    then show "x \<in> a" "open a"
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      using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
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  next
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    let ?int = "\<lambda>N. \<Inter>(from_nat_into A' ` N)"
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    fix a b
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    assume "a \<in> A" "b \<in> A"
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    then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)"
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      by (auto simp: A_def)
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    then show "a \<inter> b \<in> A"
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      by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])
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  next
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    fix S
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    assume "open S" "x \<in> S"
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    then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast
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    then show "\<exists>a\<in>A. a \<subseteq> S" using a A'
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      by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])
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  qed
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qed
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lemma (in topological_space) first_countableI:
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  assumes "countable A"
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    and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
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    and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
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  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))"
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proof (safe intro!: exI[of _ "from_nat_into A"])
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  fix i
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  have "A \<noteq> {}" using 2[of UNIV] by auto
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  show "x \<in> from_nat_into A i" "open (from_nat_into A i)"
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    using range_from_nat_into_subset[OF \<open>A \<noteq> {}\<close>] 1 by auto
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next
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  fix S
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  assume "open S" "x\<in>S" from 2[OF this]
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  show "\<exists>i. from_nat_into A i \<subseteq> S"
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    using subset_range_from_nat_into[OF \<open>countable A\<close>] by auto
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qed
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instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
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   283
proof
hoelzl@50883
   284
  fix x :: "'a \<times> 'b"
wenzelm@55522
   285
  obtain A where A:
wenzelm@55522
   286
      "countable A"
wenzelm@55522
   287
      "\<And>a. a \<in> A \<Longrightarrow> fst x \<in> a"
wenzelm@55522
   288
      "\<And>a. a \<in> A \<Longrightarrow> open a"
wenzelm@55522
   289
      "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
wenzelm@55522
   290
    by (rule first_countable_basisE[of "fst x"]) blast
wenzelm@55522
   291
  obtain B where B:
wenzelm@55522
   292
      "countable B"
wenzelm@55522
   293
      "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
wenzelm@55522
   294
      "\<And>a. a \<in> B \<Longrightarrow> open a"
wenzelm@55522
   295
      "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
wenzelm@55522
   296
    by (rule first_countable_basisE[of "snd x"]) blast
wenzelm@53282
   297
  show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set.
wenzelm@53282
   298
    (\<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
   299
  proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
wenzelm@53255
   300
    fix a b
wenzelm@53255
   301
    assume x: "a \<in> A" "b \<in> B"
wenzelm@53640
   302
    with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" and "open (a \<times> b)"
wenzelm@53640
   303
      unfolding mem_Times_iff
wenzelm@53640
   304
      by (auto intro: open_Times)
hoelzl@50883
   305
  next
wenzelm@53255
   306
    fix S
wenzelm@53255
   307
    assume "open S" "x \<in> S"
wenzelm@55522
   308
    then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
wenzelm@55522
   309
      by (rule open_prod_elim)
wenzelm@55522
   310
    moreover
wenzelm@55522
   311
    from a'b' A(4)[of a'] B(4)[of b']
wenzelm@55522
   312
    obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
wenzelm@55522
   313
      by auto
wenzelm@55522
   314
    ultimately
wenzelm@55522
   315
    show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"
hoelzl@50883
   316
      by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
hoelzl@50883
   317
  qed (simp add: A B)
hoelzl@50883
   318
qed
hoelzl@50883
   319
hoelzl@50881
   320
class second_countable_topology = topological_space +
wenzelm@53282
   321
  assumes ex_countable_subbasis:
wenzelm@53282
   322
    "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
hoelzl@51343
   323
begin
hoelzl@51343
   324
hoelzl@51343
   325
lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
hoelzl@51343
   326
proof -
wenzelm@53255
   327
  from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
wenzelm@53255
   328
    by blast
hoelzl@51343
   329
  let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
hoelzl@51343
   330
hoelzl@51343
   331
  show ?thesis
hoelzl@51343
   332
  proof (intro exI conjI)
hoelzl@51343
   333
    show "countable ?B"
hoelzl@51343
   334
      by (intro countable_image countable_Collect_finite_subset B)
wenzelm@53255
   335
    {
wenzelm@53255
   336
      fix S
wenzelm@53255
   337
      assume "open S"
hoelzl@51343
   338
      then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
hoelzl@51343
   339
        unfolding B
hoelzl@51343
   340
      proof induct
wenzelm@53255
   341
        case UNIV
wenzelm@53255
   342
        show ?case by (intro exI[of _ "{{}}"]) simp
hoelzl@51343
   343
      next
hoelzl@51343
   344
        case (Int a b)
hoelzl@51343
   345
        then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
hoelzl@51343
   346
          and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
hoelzl@51343
   347
          by blast
hoelzl@51343
   348
        show ?case
hoelzl@51343
   349
          unfolding x y Int_UN_distrib2
hoelzl@51343
   350
          by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
hoelzl@51343
   351
      next
hoelzl@51343
   352
        case (UN K)
hoelzl@51343
   353
        then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
wenzelm@55522
   354
        then obtain k where
wenzelm@55522
   355
            "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> UNION (k ka) Inter = ka"
wenzelm@55522
   356
          unfolding bchoice_iff ..
hoelzl@51343
   357
        then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
hoelzl@51343
   358
          by (intro exI[of _ "UNION K k"]) auto
hoelzl@51343
   359
      next
wenzelm@53255
   360
        case (Basis S)
wenzelm@53255
   361
        then show ?case
hoelzl@51343
   362
          by (intro exI[of _ "{{S}}"]) auto
hoelzl@51343
   363
      qed
hoelzl@51343
   364
      then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
hoelzl@51343
   365
        unfolding subset_image_iff by blast }
hoelzl@51343
   366
    then show "topological_basis ?B"
hoelzl@51343
   367
      unfolding topological_space_class.topological_basis_def
wenzelm@53282
   368
      by (safe intro!: topological_space_class.open_Inter)
hoelzl@51343
   369
         (simp_all add: B generate_topology.Basis subset_eq)
hoelzl@51343
   370
  qed
hoelzl@51343
   371
qed
hoelzl@51343
   372
hoelzl@51343
   373
end
hoelzl@51343
   374
hoelzl@51343
   375
sublocale second_countable_topology <
hoelzl@51343
   376
  countable_basis "SOME B. countable B \<and> topological_basis B"
hoelzl@51343
   377
  using someI_ex[OF ex_countable_basis]
hoelzl@51343
   378
  by unfold_locales safe
immler@50094
   379
hoelzl@50882
   380
instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
hoelzl@50882
   381
proof
hoelzl@50882
   382
  obtain A :: "'a set set" where "countable A" "topological_basis A"
hoelzl@50882
   383
    using ex_countable_basis by auto
hoelzl@50882
   384
  moreover
hoelzl@50882
   385
  obtain B :: "'b set set" where "countable B" "topological_basis B"
hoelzl@50882
   386
    using ex_countable_basis by auto
hoelzl@51343
   387
  ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
hoelzl@51343
   388
    by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
hoelzl@51343
   389
      topological_basis_imp_subbasis)
hoelzl@50882
   390
qed
hoelzl@50882
   391
hoelzl@50883
   392
instance second_countable_topology \<subseteq> first_countable_topology
hoelzl@50883
   393
proof
hoelzl@50883
   394
  fix x :: 'a
hoelzl@50883
   395
  def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"
hoelzl@50883
   396
  then have B: "countable B" "topological_basis B"
hoelzl@50883
   397
    using countable_basis is_basis
hoelzl@50883
   398
    by (auto simp: countable_basis is_basis)
wenzelm@53282
   399
  then show "\<exists>A::nat \<Rightarrow> 'a set.
wenzelm@53282
   400
    (\<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
   401
    by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
hoelzl@51473
   402
       (fastforce simp: topological_space_class.topological_basis_def)+
hoelzl@50883
   403
qed
hoelzl@50883
   404
wenzelm@53255
   405
wenzelm@60420
   406
subsection \<open>Polish spaces\<close>
wenzelm@60420
   407
wenzelm@60420
   408
text \<open>Textbooks define Polish spaces as completely metrizable.
wenzelm@60420
   409
  We assume the topology to be complete for a given metric.\<close>
immler@50087
   410
hoelzl@50881
   411
class polish_space = complete_space + second_countable_topology
immler@50087
   412
wenzelm@60420
   413
subsection \<open>General notion of a topology as a value\<close>
himmelma@33175
   414
wenzelm@53255
   415
definition "istopology L \<longleftrightarrow>
wenzelm@60585
   416
  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
   417
wenzelm@49834
   418
typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
himmelma@33175
   419
  morphisms "openin" "topology"
himmelma@33175
   420
  unfolding istopology_def by blast
himmelma@33175
   421
himmelma@33175
   422
lemma istopology_open_in[intro]: "istopology(openin U)"
himmelma@33175
   423
  using openin[of U] by blast
himmelma@33175
   424
himmelma@33175
   425
lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
huffman@44170
   426
  using topology_inverse[unfolded mem_Collect_eq] .
himmelma@33175
   427
himmelma@33175
   428
lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
himmelma@33175
   429
  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
himmelma@33175
   430
himmelma@33175
   431
lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
wenzelm@53255
   432
proof
wenzelm@53255
   433
  assume "T1 = T2"
wenzelm@53255
   434
  then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
wenzelm@53255
   435
next
wenzelm@53255
   436
  assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
wenzelm@53255
   437
  then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
wenzelm@53255
   438
  then have "topology (openin T1) = topology (openin T2)" by simp
wenzelm@53255
   439
  then show "T1 = T2" unfolding openin_inverse .
himmelma@33175
   440
qed
himmelma@33175
   441
wenzelm@60420
   442
text\<open>Infer the "universe" from union of all sets in the topology.\<close>
himmelma@33175
   443
wenzelm@53640
   444
definition "topspace T = \<Union>{S. openin T S}"
himmelma@33175
   445
wenzelm@60420
   446
subsubsection \<open>Main properties of open sets\<close>
himmelma@33175
   447
himmelma@33175
   448
lemma openin_clauses:
himmelma@33175
   449
  fixes U :: "'a topology"
wenzelm@53282
   450
  shows
wenzelm@53282
   451
    "openin U {}"
wenzelm@53282
   452
    "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
wenzelm@53282
   453
    "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
wenzelm@53282
   454
  using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
himmelma@33175
   455
himmelma@33175
   456
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
himmelma@33175
   457
  unfolding topspace_def by blast
wenzelm@53255
   458
wenzelm@53255
   459
lemma openin_empty[simp]: "openin U {}"
wenzelm@53255
   460
  by (simp add: openin_clauses)
himmelma@33175
   461
himmelma@33175
   462
lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
huffman@36362
   463
  using openin_clauses by simp
huffman@36362
   464
wenzelm@60585
   465
lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
huffman@36362
   466
  using openin_clauses by simp
himmelma@33175
   467
himmelma@33175
   468
lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
himmelma@33175
   469
  using openin_Union[of "{S,T}" U] by auto
himmelma@33175
   470
wenzelm@53255
   471
lemma openin_topspace[intro, simp]: "openin U (topspace U)"
wenzelm@53255
   472
  by (simp add: openin_Union topspace_def)
himmelma@33175
   473
wenzelm@49711
   474
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
   475
  (is "?lhs \<longleftrightarrow> ?rhs")
huffman@36584
   476
proof
wenzelm@49711
   477
  assume ?lhs
wenzelm@49711
   478
  then show ?rhs by auto
huffman@36584
   479
next
huffman@36584
   480
  assume H: ?rhs
huffman@36584
   481
  let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
huffman@36584
   482
  have "openin U ?t" by (simp add: openin_Union)
huffman@36584
   483
  also have "?t = S" using H by auto
huffman@36584
   484
  finally show "openin U S" .
himmelma@33175
   485
qed
himmelma@33175
   486
wenzelm@49711
   487
wenzelm@60420
   488
subsubsection \<open>Closed sets\<close>
himmelma@33175
   489
himmelma@33175
   490
definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
himmelma@33175
   491
wenzelm@53255
   492
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
wenzelm@53255
   493
  by (metis closedin_def)
wenzelm@53255
   494
wenzelm@53255
   495
lemma closedin_empty[simp]: "closedin U {}"
wenzelm@53255
   496
  by (simp add: closedin_def)
wenzelm@53255
   497
wenzelm@53255
   498
lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
wenzelm@53255
   499
  by (simp add: closedin_def)
wenzelm@53255
   500
himmelma@33175
   501
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
himmelma@33175
   502
  by (auto simp add: Diff_Un closedin_def)
himmelma@33175
   503
wenzelm@60585
   504
lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union>{A - s|s. s\<in>S}"
wenzelm@53255
   505
  by auto
wenzelm@53255
   506
wenzelm@53255
   507
lemma closedin_Inter[intro]:
wenzelm@53255
   508
  assumes Ke: "K \<noteq> {}"
wenzelm@53255
   509
    and Kc: "\<forall>S \<in>K. closedin U S"
wenzelm@60585
   510
  shows "closedin U (\<Inter>K)"
wenzelm@53255
   511
  using Ke Kc unfolding closedin_def Diff_Inter by auto
himmelma@33175
   512
himmelma@33175
   513
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
himmelma@33175
   514
  using closedin_Inter[of "{S,T}" U] by auto
himmelma@33175
   515
wenzelm@53255
   516
lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B"
wenzelm@53255
   517
  by blast
wenzelm@53255
   518
himmelma@33175
   519
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
himmelma@33175
   520
  apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
himmelma@33175
   521
  apply (metis openin_subset subset_eq)
himmelma@33175
   522
  done
himmelma@33175
   523
wenzelm@53255
   524
lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
himmelma@33175
   525
  by (simp add: openin_closedin_eq)
himmelma@33175
   526
wenzelm@53255
   527
lemma openin_diff[intro]:
wenzelm@53255
   528
  assumes oS: "openin U S"
wenzelm@53255
   529
    and cT: "closedin U T"
wenzelm@53255
   530
  shows "openin U (S - T)"
wenzelm@53255
   531
proof -
himmelma@33175
   532
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
himmelma@33175
   533
    by (auto simp add: topspace_def openin_subset)
wenzelm@53282
   534
  then show ?thesis using oS cT
wenzelm@53282
   535
    by (auto simp add: closedin_def)
himmelma@33175
   536
qed
himmelma@33175
   537
wenzelm@53255
   538
lemma closedin_diff[intro]:
wenzelm@53255
   539
  assumes oS: "closedin U S"
wenzelm@53255
   540
    and cT: "openin U T"
wenzelm@53255
   541
  shows "closedin U (S - T)"
wenzelm@53255
   542
proof -
wenzelm@53255
   543
  have "S - T = S \<inter> (topspace U - T)"
wenzelm@53282
   544
    using closedin_subset[of U S] oS cT by (auto simp add: topspace_def)
wenzelm@53255
   545
  then show ?thesis
wenzelm@53255
   546
    using oS cT by (auto simp add: openin_closedin_eq)
wenzelm@53255
   547
qed
wenzelm@53255
   548
himmelma@33175
   549
wenzelm@60420
   550
subsubsection \<open>Subspace topology\<close>
huffman@44170
   551
huffman@44170
   552
definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
huffman@44170
   553
huffman@44170
   554
lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
huffman@44170
   555
  (is "istopology ?L")
wenzelm@53255
   556
proof -
huffman@44170
   557
  have "?L {}" by blast
wenzelm@53255
   558
  {
wenzelm@53255
   559
    fix A B
wenzelm@53255
   560
    assume A: "?L A" and B: "?L B"
wenzelm@53255
   561
    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
   562
      by blast
wenzelm@53255
   563
    have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
wenzelm@53255
   564
      using Sa Sb by blast+
wenzelm@53255
   565
    then have "?L (A \<inter> B)" by blast
wenzelm@53255
   566
  }
himmelma@33175
   567
  moreover
wenzelm@53255
   568
  {
wenzelm@53282
   569
    fix K
wenzelm@53282
   570
    assume K: "K \<subseteq> Collect ?L"
huffman@44170
   571
    have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
lp15@55775
   572
      by blast
himmelma@33175
   573
    from K[unfolded th0 subset_image_iff]
wenzelm@53255
   574
    obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
wenzelm@53255
   575
      by blast
wenzelm@53255
   576
    have "\<Union>K = (\<Union>Sk) \<inter> V"
wenzelm@53255
   577
      using Sk by auto
wenzelm@60585
   578
    moreover have "openin U (\<Union>Sk)"
wenzelm@53255
   579
      using Sk by (auto simp add: subset_eq)
wenzelm@53255
   580
    ultimately have "?L (\<Union>K)" by blast
wenzelm@53255
   581
  }
huffman@44170
   582
  ultimately show ?thesis
huffman@44170
   583
    unfolding subset_eq mem_Collect_eq istopology_def by blast
himmelma@33175
   584
qed
himmelma@33175
   585
wenzelm@53255
   586
lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
himmelma@33175
   587
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
huffman@44170
   588
  by auto
himmelma@33175
   589
wenzelm@53255
   590
lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
himmelma@33175
   591
  by (auto simp add: topspace_def openin_subtopology)
himmelma@33175
   592
wenzelm@53255
   593
lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
himmelma@33175
   594
  unfolding closedin_def topspace_subtopology
lp15@55775
   595
  by (auto simp add: openin_subtopology)
himmelma@33175
   596
himmelma@33175
   597
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
himmelma@33175
   598
  unfolding openin_subtopology
lp15@55775
   599
  by auto (metis IntD1 in_mono openin_subset)
wenzelm@49711
   600
wenzelm@49711
   601
lemma subtopology_superset:
wenzelm@49711
   602
  assumes UV: "topspace U \<subseteq> V"
himmelma@33175
   603
  shows "subtopology U V = U"
wenzelm@53255
   604
proof -
wenzelm@53255
   605
  {
wenzelm@53255
   606
    fix S
wenzelm@53255
   607
    {
wenzelm@53255
   608
      fix T
wenzelm@53255
   609
      assume T: "openin U T" "S = T \<inter> V"
wenzelm@53255
   610
      from T openin_subset[OF T(1)] UV have eq: "S = T"
wenzelm@53255
   611
        by blast
wenzelm@53255
   612
      have "openin U S"
wenzelm@53255
   613
        unfolding eq using T by blast
wenzelm@53255
   614
    }
himmelma@33175
   615
    moreover
wenzelm@53255
   616
    {
wenzelm@53255
   617
      assume S: "openin U S"
wenzelm@53255
   618
      then have "\<exists>T. openin U T \<and> S = T \<inter> V"
wenzelm@53255
   619
        using openin_subset[OF S] UV by auto
wenzelm@53255
   620
    }
wenzelm@53255
   621
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
wenzelm@53255
   622
      by blast
wenzelm@53255
   623
  }
wenzelm@53255
   624
  then show ?thesis
wenzelm@53255
   625
    unfolding topology_eq openin_subtopology by blast
himmelma@33175
   626
qed
himmelma@33175
   627
himmelma@33175
   628
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
himmelma@33175
   629
  by (simp add: subtopology_superset)
himmelma@33175
   630
himmelma@33175
   631
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
himmelma@33175
   632
  by (simp add: subtopology_superset)
himmelma@33175
   633
wenzelm@53255
   634
wenzelm@60420
   635
subsubsection \<open>The standard Euclidean topology\<close>
himmelma@33175
   636
wenzelm@53255
   637
definition euclidean :: "'a::topological_space topology"
wenzelm@53255
   638
  where "euclidean = topology open"
himmelma@33175
   639
himmelma@33175
   640
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
himmelma@33175
   641
  unfolding euclidean_def
himmelma@33175
   642
  apply (rule cong[where x=S and y=S])
himmelma@33175
   643
  apply (rule topology_inverse[symmetric])
himmelma@33175
   644
  apply (auto simp add: istopology_def)
huffman@44170
   645
  done
himmelma@33175
   646
himmelma@33175
   647
lemma topspace_euclidean: "topspace euclidean = UNIV"
himmelma@33175
   648
  apply (simp add: topspace_def)
nipkow@39302
   649
  apply (rule set_eqI)
wenzelm@53255
   650
  apply (auto simp add: open_openin[symmetric])
wenzelm@53255
   651
  done
himmelma@33175
   652
himmelma@33175
   653
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
himmelma@33175
   654
  by (simp add: topspace_euclidean topspace_subtopology)
himmelma@33175
   655
himmelma@33175
   656
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
himmelma@33175
   657
  by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
himmelma@33175
   658
himmelma@33175
   659
lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
himmelma@33175
   660
  by (simp add: open_openin openin_subopen[symmetric])
himmelma@33175
   661
wenzelm@60420
   662
text \<open>Basic "localization" results are handy for connectedness.\<close>
huffman@44210
   663
huffman@44210
   664
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
huffman@44210
   665
  by (auto simp add: openin_subtopology open_openin[symmetric])
huffman@44210
   666
huffman@44210
   667
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
huffman@44210
   668
  by (auto simp add: openin_open)
huffman@44210
   669
huffman@44210
   670
lemma open_openin_trans[trans]:
wenzelm@53255
   671
  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
huffman@44210
   672
  by (metis Int_absorb1  openin_open_Int)
huffman@44210
   673
wenzelm@53255
   674
lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
huffman@44210
   675
  by (auto simp add: openin_open)
huffman@44210
   676
huffman@44210
   677
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
huffman@44210
   678
  by (simp add: closedin_subtopology closed_closedin Int_ac)
huffman@44210
   679
wenzelm@53291
   680
lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
huffman@44210
   681
  by (metis closedin_closed)
huffman@44210
   682
wenzelm@53282
   683
lemma closed_closedin_trans:
wenzelm@53282
   684
  "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
lp15@55775
   685
  by (metis closedin_closed inf.absorb2)
huffman@44210
   686
huffman@44210
   687
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
huffman@44210
   688
  by (auto simp add: closedin_closed)
huffman@44210
   689
huffman@44210
   690
lemma openin_euclidean_subtopology_iff:
huffman@44210
   691
  fixes S U :: "'a::metric_space set"
wenzelm@53255
   692
  shows "openin (subtopology euclidean U) S \<longleftrightarrow>
wenzelm@53255
   693
    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
   694
  (is "?lhs \<longleftrightarrow> ?rhs")
huffman@44210
   695
proof
wenzelm@53255
   696
  assume ?lhs
wenzelm@53282
   697
  then show ?rhs
wenzelm@53282
   698
    unfolding openin_open open_dist by blast
huffman@44210
   699
next
huffman@44210
   700
  def T \<equiv> "{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
   701
  have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
huffman@44210
   702
    unfolding T_def
huffman@44210
   703
    apply clarsimp
huffman@44210
   704
    apply (rule_tac x="d - dist x a" in exI)
huffman@44210
   705
    apply (clarsimp simp add: less_diff_eq)
lp15@55775
   706
    by (metis dist_commute dist_triangle_lt)
wenzelm@53282
   707
  assume ?rhs then have 2: "S = U \<inter> T"
lp15@60141
   708
    unfolding T_def
lp15@55775
   709
    by auto (metis dist_self)
huffman@44210
   710
  from 1 2 show ?lhs
huffman@44210
   711
    unfolding openin_open open_dist by fast
huffman@44210
   712
qed
lp15@61306
   713
 
lp15@61306
   714
lemma connected_open_in:
lp15@61306
   715
      "connected s \<longleftrightarrow>
lp15@61306
   716
       ~(\<exists>e1 e2. openin (subtopology euclidean s) e1 \<and>
lp15@61306
   717
                 openin (subtopology euclidean s) e2 \<and>
lp15@61306
   718
                 s \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and> e1 \<noteq> {} \<and> e2 \<noteq> {})"
lp15@61306
   719
  apply (simp add: connected_def openin_open, safe)
lp15@61306
   720
  apply (simp_all, blast+)
lp15@61306
   721
  done
lp15@61306
   722
lp15@61306
   723
lemma connected_open_in_eq:
lp15@61306
   724
      "connected s \<longleftrightarrow>
lp15@61306
   725
       ~(\<exists>e1 e2. openin (subtopology euclidean s) e1 \<and>
lp15@61306
   726
                 openin (subtopology euclidean s) e2 \<and>
lp15@61306
   727
                 e1 \<union> e2 = s \<and> e1 \<inter> e2 = {} \<and>
lp15@61306
   728
                 e1 \<noteq> {} \<and> e2 \<noteq> {})"
lp15@61306
   729
  apply (simp add: connected_open_in, safe)
lp15@61306
   730
  apply blast
lp15@61306
   731
  by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)
lp15@61306
   732
lp15@61306
   733
lemma connected_closed_in:
lp15@61306
   734
      "connected s \<longleftrightarrow>
lp15@61306
   735
       ~(\<exists>e1 e2.
lp15@61306
   736
             closedin (subtopology euclidean s) e1 \<and>
lp15@61306
   737
             closedin (subtopology euclidean s) e2 \<and>
lp15@61306
   738
             s \<subseteq> e1 \<union> e2 \<and> e1 \<inter> e2 = {} \<and>
lp15@61306
   739
             e1 \<noteq> {} \<and> e2 \<noteq> {})"
lp15@61306
   740
proof -
lp15@61306
   741
  { fix A B x x'
lp15@61306
   742
    assume s_sub: "s \<subseteq> A \<union> B"
lp15@61306
   743
       and disj: "A \<inter> B \<inter> s = {}"
lp15@61306
   744
       and x: "x \<in> s" "x \<in> B" and x': "x' \<in> s" "x' \<in> A"
lp15@61306
   745
       and cl: "closed A" "closed B"
lp15@61306
   746
    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
   747
    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
   748
      by (metis (no_types) Int_Un_distrib Int_assoc)
lp15@61306
   749
    moreover have "s \<inter> (A \<inter> B) = {}" "s \<inter> (A \<union> B) = s" "s \<inter> B \<noteq> {}"
lp15@61306
   750
      using disj s_sub x by blast+
lp15@61306
   751
    ultimately have "s \<inter> A = {}"
lp15@61306
   752
      using cl by (metis inf.left_commute inf_bot_right order_refl)
lp15@61306
   753
    then have False
lp15@61306
   754
      using x' by blast
lp15@61306
   755
  } note * = this
lp15@61306
   756
  show ?thesis
lp15@61306
   757
    apply (simp add: connected_closed closedin_closed)
lp15@61306
   758
    apply (safe; simp)
lp15@61306
   759
    apply blast
lp15@61306
   760
    apply (blast intro: *)
lp15@61306
   761
    done
lp15@61306
   762
qed
lp15@61306
   763
lp15@61306
   764
lemma connected_closed_in_eq:
lp15@61306
   765
      "connected s \<longleftrightarrow>
lp15@61306
   766
           ~(\<exists>e1 e2.
lp15@61306
   767
                 closedin (subtopology euclidean s) e1 \<and>
lp15@61306
   768
                 closedin (subtopology euclidean s) e2 \<and>
lp15@61306
   769
                 e1 \<union> e2 = s \<and> e1 \<inter> e2 = {} \<and>
lp15@61306
   770
                 e1 \<noteq> {} \<and> e2 \<noteq> {})"
lp15@61306
   771
  apply (simp add: connected_closed_in, safe)
lp15@61306
   772
  apply blast
lp15@61306
   773
  by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)
lp15@61306
   774
    
wenzelm@60420
   775
text \<open>These "transitivity" results are handy too\<close>
huffman@44210
   776
wenzelm@53255
   777
lemma openin_trans[trans]:
wenzelm@53255
   778
  "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
wenzelm@53255
   779
    openin (subtopology euclidean U) S"
huffman@44210
   780
  unfolding open_openin openin_open by blast
huffman@44210
   781
huffman@44210
   782
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
huffman@44210
   783
  by (auto simp add: openin_open intro: openin_trans)
huffman@44210
   784
huffman@44210
   785
lemma closedin_trans[trans]:
wenzelm@53255
   786
  "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
wenzelm@53255
   787
    closedin (subtopology euclidean U) S"
huffman@44210
   788
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
huffman@44210
   789
huffman@44210
   790
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
huffman@44210
   791
  by (auto simp add: closedin_closed intro: closedin_trans)
huffman@44210
   792
huffman@44210
   793
wenzelm@60420
   794
subsection \<open>Open and closed balls\<close>
himmelma@33175
   795
wenzelm@53255
   796
definition ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
wenzelm@53255
   797
  where "ball x e = {y. dist x y < e}"
wenzelm@53255
   798
wenzelm@53255
   799
definition cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
wenzelm@53255
   800
  where "cball x e = {y. dist x y \<le> e}"
himmelma@33175
   801
huffman@45776
   802
lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
huffman@45776
   803
  by (simp add: ball_def)
huffman@45776
   804
huffman@45776
   805
lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
huffman@45776
   806
  by (simp add: cball_def)
huffman@45776
   807
huffman@45776
   808
lemma mem_ball_0:
himmelma@33175
   809
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   810
  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
himmelma@33175
   811
  by (simp add: dist_norm)
himmelma@33175
   812
huffman@45776
   813
lemma mem_cball_0:
himmelma@33175
   814
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   815
  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
himmelma@33175
   816
  by (simp add: dist_norm)
himmelma@33175
   817
huffman@45776
   818
lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"
huffman@45776
   819
  by simp
huffman@45776
   820
huffman@45776
   821
lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
huffman@45776
   822
  by simp
huffman@45776
   823
wenzelm@53255
   824
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e"
wenzelm@53255
   825
  by (simp add: subset_eq)
wenzelm@53255
   826
wenzelm@53282
   827
lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
wenzelm@53255
   828
  by (simp add: subset_eq)
wenzelm@53255
   829
wenzelm@53282
   830
lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
wenzelm@53255
   831
  by (simp add: subset_eq)
wenzelm@53255
   832
himmelma@33175
   833
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
nipkow@39302
   834
  by (simp add: set_eq_iff) arith
himmelma@33175
   835
himmelma@33175
   836
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
nipkow@39302
   837
  by (simp add: set_eq_iff)
himmelma@33175
   838
wenzelm@53255
   839
lemma diff_less_iff:
wenzelm@53255
   840
  "(a::real) - b > 0 \<longleftrightarrow> a > b"
himmelma@33175
   841
  "(a::real) - b < 0 \<longleftrightarrow> a < b"
wenzelm@53255
   842
  "a - b < c \<longleftrightarrow> a < c + b" "a - b > c \<longleftrightarrow> a > c + b"
wenzelm@53255
   843
  by arith+
wenzelm@53255
   844
wenzelm@53255
   845
lemma diff_le_iff:
wenzelm@53255
   846
  "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b"
wenzelm@53255
   847
  "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
wenzelm@53255
   848
  "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
wenzelm@53255
   849
  "a - b \<ge> c \<longleftrightarrow> a \<ge> c + b"
wenzelm@53255
   850
  by arith+
himmelma@33175
   851
huffman@54070
   852
lemma open_ball [intro, simp]: "open (ball x e)"
huffman@54070
   853
proof -
huffman@54070
   854
  have "open (dist x -` {..<e})"
hoelzl@56371
   855
    by (intro open_vimage open_lessThan continuous_intros)
huffman@54070
   856
  also have "dist x -` {..<e} = ball x e"
huffman@54070
   857
    by auto
huffman@54070
   858
  finally show ?thesis .
huffman@54070
   859
qed
himmelma@33175
   860
himmelma@33175
   861
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   862
  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
himmelma@33175
   863
hoelzl@33714
   864
lemma openE[elim?]:
wenzelm@53282
   865
  assumes "open S" "x\<in>S"
hoelzl@33714
   866
  obtains e where "e>0" "ball x e \<subseteq> S"
hoelzl@33714
   867
  using assms unfolding open_contains_ball by auto
hoelzl@33714
   868
himmelma@33175
   869
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   870
  by (metis open_contains_ball subset_eq centre_in_ball)
himmelma@33175
   871
himmelma@33175
   872
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
nipkow@39302
   873
  unfolding mem_ball set_eq_iff
himmelma@33175
   874
  apply (simp add: not_less)
wenzelm@52624
   875
  apply (metis zero_le_dist order_trans dist_self)
wenzelm@52624
   876
  done
himmelma@33175
   877
wenzelm@53291
   878
lemma ball_empty[intro]: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
himmelma@33175
   879
hoelzl@50526
   880
lemma euclidean_dist_l2:
hoelzl@50526
   881
  fixes x y :: "'a :: euclidean_space"
hoelzl@50526
   882
  shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
hoelzl@50526
   883
  unfolding dist_norm norm_eq_sqrt_inner setL2_def
hoelzl@50526
   884
  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
hoelzl@50526
   885
immler@56189
   886
wenzelm@60420
   887
subsection \<open>Boxes\<close>
immler@56189
   888
hoelzl@57447
   889
abbreviation One :: "'a::euclidean_space"
hoelzl@57447
   890
  where "One \<equiv> \<Sum>Basis"
hoelzl@57447
   891
immler@54775
   892
definition (in euclidean_space) eucl_less (infix "<e" 50)
immler@54775
   893
  where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
immler@54775
   894
immler@54775
   895
definition box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"
immler@56188
   896
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
   897
immler@54775
   898
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
   899
  and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b"
immler@56188
   900
  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
   901
    "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
   902
  by (auto simp: box_eucl_less eucl_less_def cbox_def)
immler@56188
   903
lp15@60615
   904
lemma cbox_Pair_eq: "cbox (a, c) (b, d) = cbox a b \<times> cbox c d"
lp15@60615
   905
  by (force simp: cbox_def Basis_prod_def)
lp15@60615
   906
lp15@60615
   907
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
   908
  by (force simp: cbox_Pair_eq)
lp15@60615
   909
lp15@60615
   910
lemma cbox_Pair_eq_0: "cbox (a, c) (b, d) = {} \<longleftrightarrow> cbox a b = {} \<or> cbox c d = {}"
lp15@60615
   911
  by (force simp: cbox_Pair_eq)
lp15@60615
   912
lp15@60615
   913
lemma swap_cbox_Pair [simp]: "prod.swap ` cbox (c, a) (d, b) = cbox (a,c) (b,d)"
lp15@60615
   914
  by auto
lp15@60615
   915
immler@56188
   916
lemma mem_box_real[simp]:
immler@56188
   917
  "(x::real) \<in> box a b \<longleftrightarrow> a < x \<and> x < b"
immler@56188
   918
  "(x::real) \<in> cbox a b \<longleftrightarrow> a \<le> x \<and> x \<le> b"
immler@56188
   919
  by (auto simp: mem_box)
immler@56188
   920
immler@56188
   921
lemma box_real[simp]:
immler@56188
   922
  fixes a b:: real
immler@56188
   923
  shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
immler@56188
   924
  by auto
hoelzl@50526
   925
hoelzl@57447
   926
lemma box_Int_box:
hoelzl@57447
   927
  fixes a :: "'a::euclidean_space"
hoelzl@57447
   928
  shows "box a b \<inter> box c d =
hoelzl@57447
   929
    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
   930
  unfolding set_eq_iff and Int_iff and mem_box by auto
hoelzl@57447
   931
immler@50087
   932
lemma rational_boxes:
wenzelm@61076
   933
  fixes x :: "'a::euclidean_space"
wenzelm@53291
   934
  assumes "e > 0"
hoelzl@50526
   935
  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
   936
proof -
immler@50087
   937
  def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
wenzelm@53291
   938
  then have e: "e' > 0"
nipkow@56541
   939
    using assms by (auto simp: DIM_positive)
hoelzl@50526
   940
  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
   941
  proof
wenzelm@53255
   942
    fix i
wenzelm@53255
   943
    from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e
wenzelm@53255
   944
    show "?th i" by auto
immler@50087
   945
  qed
wenzelm@55522
   946
  from choice[OF this] obtain a where
wenzelm@55522
   947
    a: "\<forall>xa. a xa \<in> \<rat> \<and> a xa < x \<bullet> xa \<and> x \<bullet> xa - a xa < e'" ..
hoelzl@50526
   948
  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
   949
  proof
wenzelm@53255
   950
    fix i
wenzelm@53255
   951
    from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e
wenzelm@53255
   952
    show "?th i" by auto
immler@50087
   953
  qed
wenzelm@55522
   954
  from choice[OF this] obtain b where
wenzelm@55522
   955
    b: "\<forall>xa. b xa \<in> \<rat> \<and> x \<bullet> xa < b xa \<and> b xa - x \<bullet> xa < e'" ..
hoelzl@50526
   956
  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
hoelzl@50526
   957
  show ?thesis
hoelzl@50526
   958
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
wenzelm@53255
   959
    fix y :: 'a
wenzelm@53255
   960
    assume *: "y \<in> box ?a ?b"
wenzelm@53015
   961
    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
immler@50087
   962
      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
hoelzl@50526
   963
    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
immler@50087
   964
    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
wenzelm@53255
   965
      fix i :: "'a"
wenzelm@53255
   966
      assume i: "i \<in> Basis"
wenzelm@53255
   967
      have "a i < y\<bullet>i \<and> y\<bullet>i < b i"
wenzelm@53255
   968
        using * i by (auto simp: box_def)
wenzelm@53255
   969
      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"
wenzelm@53255
   970
        using a by auto
wenzelm@53255
   971
      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"
wenzelm@53255
   972
        using b by auto
wenzelm@53255
   973
      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'"
wenzelm@53255
   974
        by auto
hoelzl@50526
   975
      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
immler@50087
   976
        unfolding e'_def by (auto simp: dist_real_def)
wenzelm@53015
   977
      then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
immler@50087
   978
        by (rule power_strict_mono) auto
wenzelm@53015
   979
      then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
immler@50087
   980
        by (simp add: power_divide)
immler@50087
   981
    qed auto
wenzelm@53255
   982
    also have "\<dots> = e"
wenzelm@60420
   983
      using \<open>0 < e\<close> by (simp add: real_eq_of_nat)
wenzelm@53255
   984
    finally show "y \<in> ball x e"
wenzelm@53255
   985
      by (auto simp: ball_def)
hoelzl@50526
   986
  qed (insert a b, auto simp: box_def)
hoelzl@50526
   987
qed
immler@51103
   988
hoelzl@50526
   989
lemma open_UNION_box:
wenzelm@61076
   990
  fixes M :: "'a::euclidean_space set"
wenzelm@53282
   991
  assumes "open M"
hoelzl@50526
   992
  defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
hoelzl@50526
   993
  defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
wenzelm@53015
   994
  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
hoelzl@50526
   995
  shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
wenzelm@52624
   996
proof -
wenzelm@60462
   997
  have "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))" if "x \<in> M" for x
wenzelm@60462
   998
  proof -
wenzelm@52624
   999
    obtain e where e: "e > 0" "ball x e \<subseteq> M"
wenzelm@60420
  1000
      using openE[OF \<open>open M\<close> \<open>x \<in> M\<close>] by auto
wenzelm@53282
  1001
    moreover obtain a b where ab:
wenzelm@53282
  1002
      "x \<in> box a b"
wenzelm@53282
  1003
      "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
wenzelm@53282
  1004
      "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>"
wenzelm@53282
  1005
      "box a b \<subseteq> ball x e"
wenzelm@52624
  1006
      using rational_boxes[OF e(1)] by metis
wenzelm@60462
  1007
    ultimately show ?thesis
wenzelm@52624
  1008
       by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
wenzelm@52624
  1009
          (auto simp: euclidean_representation I_def a'_def b'_def)
wenzelm@60462
  1010
  qed
wenzelm@52624
  1011
  then show ?thesis by (auto simp: I_def)
wenzelm@52624
  1012
qed
wenzelm@52624
  1013
immler@56189
  1014
lemma box_eq_empty:
immler@56189
  1015
  fixes a :: "'a::euclidean_space"
immler@56189
  1016
  shows "(box a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1)
immler@56189
  1017
    and "(cbox a b = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
immler@56189
  1018
proof -
immler@56189
  1019
  {
immler@56189
  1020
    fix i x
immler@56189
  1021
    assume i: "i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>box a b"
immler@56189
  1022
    then have "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i"
immler@56189
  1023
      unfolding mem_box by (auto simp: box_def)
immler@56189
  1024
    then have "a\<bullet>i < b\<bullet>i" by auto
immler@56189
  1025
    then have False using as by auto
immler@56189
  1026
  }
immler@56189
  1027
  moreover
immler@56189
  1028
  {
immler@56189
  1029
    assume as: "\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
immler@56189
  1030
    let ?x = "(1/2) *\<^sub>R (a + b)"
immler@56189
  1031
    {
immler@56189
  1032
      fix i :: 'a
immler@56189
  1033
      assume i: "i \<in> Basis"
immler@56189
  1034
      have "a\<bullet>i < b\<bullet>i"
immler@56189
  1035
        using as[THEN bspec[where x=i]] i by auto
immler@56189
  1036
      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
  1037
        by (auto simp: inner_add_left)
immler@56189
  1038
    }
immler@56189
  1039
    then have "box a b \<noteq> {}"
immler@56189
  1040
      using mem_box(1)[of "?x" a b] by auto
immler@56189
  1041
  }
immler@56189
  1042
  ultimately show ?th1 by blast
immler@56189
  1043
immler@56189
  1044
  {
immler@56189
  1045
    fix i x
immler@56189
  1046
    assume i: "i \<in> Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>cbox a b"
immler@56189
  1047
    then have "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
immler@56189
  1048
      unfolding mem_box by auto
immler@56189
  1049
    then have "a\<bullet>i \<le> b\<bullet>i" by auto
immler@56189
  1050
    then have False using as by auto
immler@56189
  1051
  }
immler@56189
  1052
  moreover
immler@56189
  1053
  {
immler@56189
  1054
    assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
immler@56189
  1055
    let ?x = "(1/2) *\<^sub>R (a + b)"
immler@56189
  1056
    {
immler@56189
  1057
      fix i :: 'a
immler@56189
  1058
      assume i:"i \<in> Basis"
immler@56189
  1059
      have "a\<bullet>i \<le> b\<bullet>i"
immler@56189
  1060
        using as[THEN bspec[where x=i]] i by auto
immler@56189
  1061
      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
  1062
        by (auto simp: inner_add_left)
immler@56189
  1063
    }
immler@56189
  1064
    then have "cbox a b \<noteq> {}"
immler@56189
  1065
      using mem_box(2)[of "?x" a b] by auto
immler@56189
  1066
  }
immler@56189
  1067
  ultimately show ?th2 by blast
immler@56189
  1068
qed
immler@56189
  1069
immler@56189
  1070
lemma box_ne_empty:
immler@56189
  1071
  fixes a :: "'a::euclidean_space"
immler@56189
  1072
  shows "cbox a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)"
immler@56189
  1073
  and "box a b \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
immler@56189
  1074
  unfolding box_eq_empty[of a b] by fastforce+
immler@56189
  1075
immler@56189
  1076
lemma
immler@56189
  1077
  fixes a :: "'a::euclidean_space"
immler@56189
  1078
  shows cbox_sing: "cbox a a = {a}"
immler@56189
  1079
    and box_sing: "box a a = {}"
immler@56189
  1080
  unfolding set_eq_iff mem_box eq_iff [symmetric]
immler@56189
  1081
  by (auto intro!: euclidean_eqI[where 'a='a])
immler@56189
  1082
     (metis all_not_in_conv nonempty_Basis)
immler@56189
  1083
immler@56189
  1084
lemma subset_box_imp:
immler@56189
  1085
  fixes a :: "'a::euclidean_space"
immler@56189
  1086
  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
  1087
    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
  1088
    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
  1089
     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
  1090
  unfolding subset_eq[unfolded Ball_def] unfolding mem_box
wenzelm@58757
  1091
  by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+
immler@56189
  1092
immler@56189
  1093
lemma box_subset_cbox:
immler@56189
  1094
  fixes a :: "'a::euclidean_space"
immler@56189
  1095
  shows "box a b \<subseteq> cbox a b"
immler@56189
  1096
  unfolding subset_eq [unfolded Ball_def] mem_box
immler@56189
  1097
  by (fast intro: less_imp_le)
immler@56189
  1098
immler@56189
  1099
lemma subset_box:
immler@56189
  1100
  fixes a :: "'a::euclidean_space"
immler@56189
  1101
  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
  1102
    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
  1103
    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
  1104
    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
  1105
proof -
immler@56189
  1106
  show ?th1
immler@56189
  1107
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1108
    by (auto intro: order_trans)
immler@56189
  1109
  show ?th2
immler@56189
  1110
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1111
    by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
immler@56189
  1112
  {
immler@56189
  1113
    assume as: "box c d \<subseteq> cbox a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
immler@56189
  1114
    then have "box c d \<noteq> {}"
immler@56189
  1115
      unfolding box_eq_empty by auto
immler@56189
  1116
    fix i :: 'a
immler@56189
  1117
    assume i: "i \<in> Basis"
immler@56189
  1118
    (** TODO combine the following two parts as done in the HOL_light version. **)
immler@56189
  1119
    {
immler@56189
  1120
      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
  1121
      assume as2: "a\<bullet>i > c\<bullet>i"
immler@56189
  1122
      {
immler@56189
  1123
        fix j :: 'a
immler@56189
  1124
        assume j: "j \<in> Basis"
immler@56189
  1125
        then have "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
immler@56189
  1126
          apply (cases "j = i")
immler@56189
  1127
          using as(2)[THEN bspec[where x=j]] i
immler@56189
  1128
          apply (auto simp add: as2)
immler@56189
  1129
          done
immler@56189
  1130
      }
immler@56189
  1131
      then have "?x\<in>box c d"
immler@56189
  1132
        using i unfolding mem_box by auto
immler@56189
  1133
      moreover
immler@56189
  1134
      have "?x \<notin> cbox a b"
immler@56189
  1135
        unfolding mem_box
immler@56189
  1136
        apply auto
immler@56189
  1137
        apply (rule_tac x=i in bexI)
immler@56189
  1138
        using as(2)[THEN bspec[where x=i]] and as2 i
immler@56189
  1139
        apply auto
immler@56189
  1140
        done
immler@56189
  1141
      ultimately have False using as by auto
immler@56189
  1142
    }
immler@56189
  1143
    then have "a\<bullet>i \<le> c\<bullet>i" by (rule ccontr) auto
immler@56189
  1144
    moreover
immler@56189
  1145
    {
immler@56189
  1146
      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
  1147
      assume as2: "b\<bullet>i < d\<bullet>i"
immler@56189
  1148
      {
immler@56189
  1149
        fix j :: 'a
immler@56189
  1150
        assume "j\<in>Basis"
immler@56189
  1151
        then have "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"
immler@56189
  1152
          apply (cases "j = i")
immler@56189
  1153
          using as(2)[THEN bspec[where x=j]]
immler@56189
  1154
          apply (auto simp add: as2)
immler@56189
  1155
          done
immler@56189
  1156
      }
immler@56189
  1157
      then have "?x\<in>box c d"
immler@56189
  1158
        unfolding mem_box by auto
immler@56189
  1159
      moreover
immler@56189
  1160
      have "?x\<notin>cbox a b"
immler@56189
  1161
        unfolding mem_box
immler@56189
  1162
        apply auto
immler@56189
  1163
        apply (rule_tac x=i in bexI)
immler@56189
  1164
        using as(2)[THEN bspec[where x=i]] and as2 using i
immler@56189
  1165
        apply auto
immler@56189
  1166
        done
immler@56189
  1167
      ultimately have False using as by auto
immler@56189
  1168
    }
immler@56189
  1169
    then have "b\<bullet>i \<ge> d\<bullet>i" by (rule ccontr) auto
immler@56189
  1170
    ultimately
immler@56189
  1171
    have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
immler@56189
  1172
  } note part1 = this
immler@56189
  1173
  show ?th3
immler@56189
  1174
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1175
    apply (rule, rule, rule, rule)
immler@56189
  1176
    apply (rule part1)
immler@56189
  1177
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1178
    prefer 4
immler@56189
  1179
    apply auto
immler@56189
  1180
    apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+
immler@56189
  1181
    done
immler@56189
  1182
  {
immler@56189
  1183
    assume as: "box c d \<subseteq> box a b" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
immler@56189
  1184
    fix i :: 'a
immler@56189
  1185
    assume i:"i\<in>Basis"
immler@56189
  1186
    from as(1) have "box c d \<subseteq> cbox a b"
immler@56189
  1187
      using box_subset_cbox[of a b] by auto
immler@56189
  1188
    then have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i"
immler@56189
  1189
      using part1 and as(2) using i by auto
immler@56189
  1190
  } note * = this
immler@56189
  1191
  show ?th4
immler@56189
  1192
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1193
    apply (rule, rule, rule, rule)
immler@56189
  1194
    apply (rule *)
immler@56189
  1195
    unfolding subset_eq and Ball_def and mem_box
immler@56189
  1196
    prefer 4
immler@56189
  1197
    apply auto
immler@56189
  1198
    apply (erule_tac x=xa in allE, simp)+
immler@56189
  1199
    done
immler@56189
  1200
qed
immler@56189
  1201
immler@56189
  1202
lemma inter_interval:
immler@56189
  1203
  fixes a :: "'a::euclidean_space"
immler@56189
  1204
  shows "cbox a b \<inter> cbox c d =
immler@56189
  1205
    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
  1206
  unfolding set_eq_iff and Int_iff and mem_box
immler@56189
  1207
  by auto
immler@56189
  1208
immler@56189
  1209
lemma disjoint_interval:
immler@56189
  1210
  fixes a::"'a::euclidean_space"
immler@56189
  1211
  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
  1212
    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
  1213
    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
  1214
    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
  1215
proof -
immler@56189
  1216
  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
  1217
  have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
immler@56189
  1218
      (\<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
  1219
    by blast
immler@56189
  1220
  note * = set_eq_iff Int_iff empty_iff mem_box ball_conj_distrib[symmetric] eq_False ball_simps(10)
immler@56189
  1221
  show ?th1 unfolding * by (intro **) auto
immler@56189
  1222
  show ?th2 unfolding * by (intro **) auto
immler@56189
  1223
  show ?th3 unfolding * by (intro **) auto
immler@56189
  1224
  show ?th4 unfolding * by (intro **) auto
immler@56189
  1225
qed
immler@56189
  1226
hoelzl@57447
  1227
lemma UN_box_eq_UNIV: "(\<Union>i::nat. box (- (real i *\<^sub>R One)) (real i *\<^sub>R One)) = UNIV"
hoelzl@57447
  1228
proof -
wenzelm@60462
  1229
  have "\<bar>x \<bullet> b\<bar> < real (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)) + 1)"
wenzelm@60462
  1230
    if [simp]: "b \<in> Basis" for x b :: 'a
wenzelm@60462
  1231
  proof -
nipkow@59587
  1232
    have "\<bar>x \<bullet> b\<bar> \<le> real (ceiling \<bar>x \<bullet> b\<bar>)"
nipkow@59587
  1233
      by (rule real_of_int_ceiling_ge)
nipkow@59587
  1234
    also have "\<dots> \<le> real (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)))"
nipkow@59587
  1235
      by (auto intro!: ceiling_mono)
nipkow@59587
  1236
    also have "\<dots> < real (ceiling (Max ((\<lambda>b. \<bar>x \<bullet> b\<bar>)`Basis)) + 1)"
hoelzl@57447
  1237
      by simp
wenzelm@60462
  1238
    finally show ?thesis .
wenzelm@60462
  1239
  qed
wenzelm@60462
  1240
  then have "\<exists>n::nat. \<forall>b\<in>Basis. \<bar>x \<bullet> b\<bar> < real n" for x :: 'a
nipkow@59587
  1241
    by (metis order.strict_trans reals_Archimedean2)
hoelzl@57447
  1242
  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
  1243
    by auto
hoelzl@57447
  1244
  ultimately show ?thesis
hoelzl@57447
  1245
    by (auto simp: box_def inner_setsum_left inner_Basis setsum.If_cases)
hoelzl@57447
  1246
qed
hoelzl@57447
  1247
wenzelm@60420
  1248
text \<open>Intervals in general, including infinite and mixtures of open and closed.\<close>
immler@56189
  1249
immler@56189
  1250
definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
immler@56189
  1251
  (\<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
  1252
immler@56189
  1253
lemma is_interval_cbox: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
immler@56189
  1254
  and is_interval_box: "is_interval (box a b)" (is ?th2)
immler@56189
  1255
  unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
immler@56189
  1256
  by (meson order_trans le_less_trans less_le_trans less_trans)+
immler@56189
  1257
immler@56189
  1258
lemma is_interval_empty:
immler@56189
  1259
 "is_interval {}"
immler@56189
  1260
  unfolding is_interval_def
immler@56189
  1261
  by simp
immler@56189
  1262
immler@56189
  1263
lemma is_interval_univ:
immler@56189
  1264
 "is_interval UNIV"
immler@56189
  1265
  unfolding is_interval_def
immler@56189
  1266
  by simp
immler@56189
  1267
immler@56189
  1268
lemma mem_is_intervalI:
immler@56189
  1269
  assumes "is_interval s"
immler@56189
  1270
  assumes "a \<in> s" "b \<in> s"
immler@56189
  1271
  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
  1272
  shows "x \<in> s"
immler@56189
  1273
  by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])
immler@56189
  1274
immler@56189
  1275
lemma interval_subst:
immler@56189
  1276
  fixes S::"'a::euclidean_space set"
immler@56189
  1277
  assumes "is_interval S"
immler@56189
  1278
  assumes "x \<in> S" "y j \<in> S"
immler@56189
  1279
  assumes "j \<in> Basis"
immler@56189
  1280
  shows "(\<Sum>i\<in>Basis. (if i = j then y i \<bullet> i else x \<bullet> i) *\<^sub>R i) \<in> S"
immler@56189
  1281
  by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)
immler@56189
  1282
immler@56189
  1283
lemma mem_box_componentwiseI:
immler@56189
  1284
  fixes S::"'a::euclidean_space set"
immler@56189
  1285
  assumes "is_interval S"
immler@56189
  1286
  assumes "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<in> ((\<lambda>x. x \<bullet> i) ` S)"
immler@56189
  1287
  shows "x \<in> S"
immler@56189
  1288
proof -
immler@56189
  1289
  from assms have "\<forall>i \<in> Basis. \<exists>s \<in> S. x \<bullet> i = s \<bullet> i"
immler@56189
  1290
    by auto
immler@56189
  1291
  with finite_Basis obtain s and bs::"'a list" where
immler@56189
  1292
    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
  1293
    bs: "set bs = Basis" "distinct bs"
immler@56189
  1294
    by (metis finite_distinct_list)
immler@56189
  1295
  from nonempty_Basis s obtain j where j: "j \<in> Basis" "s j \<in> S" by blast
immler@56189
  1296
  def y \<equiv> "rec_list
immler@56189
  1297
    (s j)
immler@56189
  1298
    (\<lambda>j _ Y. (\<Sum>i\<in>Basis. (if i = j then s i \<bullet> i else Y \<bullet> i) *\<^sub>R i))"
immler@56189
  1299
  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
  1300
    using bs by (auto simp add: s(1)[symmetric] euclidean_representation)
immler@56189
  1301
  also have [symmetric]: "y bs = \<dots>"
immler@56189
  1302
    using bs(2) bs(1)[THEN equalityD1]
immler@56189
  1303
    by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
immler@56189
  1304
  also have "y bs \<in> S"
immler@56189
  1305
    using bs(1)[THEN equalityD1]
immler@56189
  1306
    apply (induct bs)
immler@56189
  1307
    apply (auto simp: y_def j)
immler@56189
  1308
    apply (rule interval_subst[OF assms(1)])
immler@56189
  1309
    apply (auto simp: s)
immler@56189
  1310
    done
immler@56189
  1311
  finally show ?thesis .
immler@56189
  1312
qed
immler@56189
  1313
himmelma@33175
  1314
wenzelm@60420
  1315
subsection\<open>Connectedness\<close>
himmelma@33175
  1316
himmelma@33175
  1317
lemma connected_local:
wenzelm@53255
  1318
 "connected S \<longleftrightarrow>
wenzelm@53255
  1319
  \<not> (\<exists>e1 e2.
wenzelm@53255
  1320
      openin (subtopology euclidean S) e1 \<and>
wenzelm@53255
  1321
      openin (subtopology euclidean S) e2 \<and>
wenzelm@53255
  1322
      S \<subseteq> e1 \<union> e2 \<and>
wenzelm@53255
  1323
      e1 \<inter> e2 = {} \<and>
wenzelm@53255
  1324
      e1 \<noteq> {} \<and>
wenzelm@53255
  1325
      e2 \<noteq> {})"
wenzelm@53282
  1326
  unfolding connected_def openin_open
lp15@59765
  1327
  by safe blast+
himmelma@33175
  1328
huffman@34105
  1329
lemma exists_diff:
huffman@34105
  1330
  fixes P :: "'a set \<Rightarrow> bool"
wenzelm@60462
  1331
  shows "(\<exists>S. P (- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53255
  1332
proof -
wenzelm@53255
  1333
  {
wenzelm@53255
  1334
    assume "?lhs"
wenzelm@53255
  1335
    then have ?rhs by blast
wenzelm@53255
  1336
  }
himmelma@33175
  1337
  moreover
wenzelm@53255
  1338
  {
wenzelm@53255
  1339
    fix S
wenzelm@53255
  1340
    assume H: "P S"
huffman@34105
  1341
    have "S = - (- S)" by auto
wenzelm@53255
  1342
    with H have "P (- (- S))" by metis
wenzelm@53255
  1343
  }
himmelma@33175
  1344
  ultimately show ?thesis by metis
himmelma@33175
  1345
qed
himmelma@33175
  1346
himmelma@33175
  1347
lemma connected_clopen: "connected S \<longleftrightarrow>
wenzelm@53255
  1348
  (\<forall>T. openin (subtopology euclidean S) T \<and>
wenzelm@53255
  1349
     closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53255
  1350
proof -
wenzelm@53255
  1351
  have "\<not> connected S \<longleftrightarrow>
wenzelm@53255
  1352
    (\<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
  1353
    unfolding connected_def openin_open closedin_closed
lp15@55775
  1354
    by (metis double_complement)
wenzelm@53282
  1355
  then have th0: "connected S \<longleftrightarrow>
wenzelm@53255
  1356
    \<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
  1357
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")
wenzelm@52624
  1358
    apply (simp add: closed_def)
wenzelm@52624
  1359
    apply metis
wenzelm@52624
  1360
    done
himmelma@33175
  1361
  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
  1362
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
himmelma@33175
  1363
    unfolding connected_def openin_open closedin_closed by auto
wenzelm@53255
  1364
  {
wenzelm@53255
  1365
    fix e2
wenzelm@53255
  1366
    {
wenzelm@53255
  1367
      fix e1
wenzelm@53282
  1368
      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
  1369
        by auto
wenzelm@53255
  1370
    }
wenzelm@53255
  1371
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
wenzelm@53255
  1372
      by metis
wenzelm@53255
  1373
  }
wenzelm@53255
  1374
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
wenzelm@53255
  1375
    by blast
wenzelm@53255
  1376
  then show ?thesis
wenzelm@53255
  1377
    unfolding th0 th1 by simp
himmelma@33175
  1378
qed
himmelma@33175
  1379
wenzelm@60420
  1380
subsection\<open>Limit points\<close>
himmelma@33175
  1381
wenzelm@53282
  1382
definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)
wenzelm@53255
  1383
  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
  1384
himmelma@33175
  1385
lemma islimptI:
himmelma@33175
  1386
  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
himmelma@33175
  1387
  shows "x islimpt S"
himmelma@33175
  1388
  using assms unfolding islimpt_def by auto
himmelma@33175
  1389
himmelma@33175
  1390
lemma islimptE:
himmelma@33175
  1391
  assumes "x islimpt S" and "x \<in> T" and "open T"
himmelma@33175
  1392
  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
himmelma@33175
  1393
  using assms unfolding islimpt_def by auto
himmelma@33175
  1394
huffman@44584
  1395
lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
huffman@44584
  1396
  unfolding islimpt_def eventually_at_topological by auto
huffman@44584
  1397
wenzelm@53255
  1398
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
huffman@44584
  1399
  unfolding islimpt_def by fast
himmelma@33175
  1400
himmelma@33175
  1401
lemma islimpt_approachable:
himmelma@33175
  1402
  fixes x :: "'a::metric_space"
himmelma@33175
  1403
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
huffman@44584
  1404
  unfolding islimpt_iff_eventually eventually_at by fast
himmelma@33175
  1405
himmelma@33175
  1406
lemma islimpt_approachable_le:
himmelma@33175
  1407
  fixes x :: "'a::metric_space"
wenzelm@53640
  1408
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
himmelma@33175
  1409
  unfolding islimpt_approachable
huffman@44584
  1410
  using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
huffman@44584
  1411
    THEN arg_cong [where f=Not]]
huffman@44584
  1412
  by (simp add: Bex_def conj_commute conj_left_commute)
himmelma@33175
  1413
huffman@44571
  1414
lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
huffman@44571
  1415
  unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
huffman@44571
  1416
hoelzl@51351
  1417
lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
hoelzl@51351
  1418
  unfolding islimpt_def by blast
hoelzl@51351
  1419
wenzelm@60420
  1420
text \<open>A perfect space has no isolated points.\<close>
huffman@44210
  1421
huffman@44571
  1422
lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
huffman@44571
  1423
  unfolding islimpt_UNIV_iff by (rule not_open_singleton)
himmelma@33175
  1424
himmelma@33175
  1425
lemma perfect_choose_dist:
huffman@44072
  1426
  fixes x :: "'a::{perfect_space, metric_space}"
himmelma@33175
  1427
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
wenzelm@53255
  1428
  using islimpt_UNIV [of x]
wenzelm@53255
  1429
  by (simp add: islimpt_approachable)
himmelma@33175
  1430
himmelma@33175
  1431
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
himmelma@33175
  1432
  unfolding closed_def
himmelma@33175
  1433
  apply (subst open_subopen)
huffman@34105
  1434
  apply (simp add: islimpt_def subset_eq)
wenzelm@52624
  1435
  apply (metis ComplE ComplI)
wenzelm@52624
  1436
  done
himmelma@33175
  1437
himmelma@33175
  1438
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
himmelma@33175
  1439
  unfolding islimpt_def by auto
himmelma@33175
  1440
himmelma@33175
  1441
lemma finite_set_avoid:
himmelma@33175
  1442
  fixes a :: "'a::metric_space"
wenzelm@53255
  1443
  assumes fS: "finite S"
wenzelm@53640
  1444
  shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
wenzelm@53255
  1445
proof (induct rule: finite_induct[OF fS])
wenzelm@53255
  1446
  case 1
wenzelm@53255
  1447
  then show ?case by (auto intro: zero_less_one)
himmelma@33175
  1448
next
himmelma@33175
  1449
  case (2 x F)
wenzelm@60462
  1450
  from 2 obtain d where d: "d > 0" "\<forall>x\<in>F. x \<noteq> a \<longrightarrow> d \<le> dist a x"
wenzelm@53255
  1451
    by blast
wenzelm@53255
  1452
  show ?case
wenzelm@53255
  1453
  proof (cases "x = a")
wenzelm@53255
  1454
    case True
wenzelm@53255
  1455
    then show ?thesis using d by auto
wenzelm@53255
  1456
  next
wenzelm@53255
  1457
    case False
himmelma@33175
  1458
    let ?d = "min d (dist a x)"
wenzelm@53255
  1459
    have dp: "?d > 0"
wenzelm@53255
  1460
      using False d(1) using dist_nz by auto
wenzelm@60462
  1461
    from d have d': "\<forall>x\<in>F. x \<noteq> a \<longrightarrow> ?d \<le> dist a x"
wenzelm@53255
  1462
      by auto
wenzelm@53255
  1463
    with dp False show ?thesis
wenzelm@53255
  1464
      by (auto intro!: exI[where x="?d"])
wenzelm@53255
  1465
  qed
himmelma@33175
  1466
qed
himmelma@33175
  1467
himmelma@33175
  1468
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
huffman@50897
  1469
  by (simp add: islimpt_iff_eventually eventually_conj_iff)
himmelma@33175
  1470
himmelma@33175
  1471
lemma discrete_imp_closed:
himmelma@33175
  1472
  fixes S :: "'a::metric_space set"
wenzelm@53255
  1473
  assumes e: "0 < e"
wenzelm@53255
  1474
    and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
himmelma@33175
  1475
  shows "closed S"
wenzelm@53255
  1476
proof -
wenzelm@53255
  1477
  {
wenzelm@53255
  1478
    fix x
wenzelm@53255
  1479
    assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
himmelma@33175
  1480
    from e have e2: "e/2 > 0" by arith
wenzelm@53282
  1481
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
wenzelm@53255
  1482
      by blast
himmelma@33175
  1483
    let ?m = "min (e/2) (dist x y) "
wenzelm@53255
  1484
    from e2 y(2) have mp: "?m > 0"
wenzelm@53291
  1485
      by (simp add: dist_nz[symmetric])
wenzelm@53282
  1486
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
wenzelm@53255
  1487
      by blast
himmelma@33175
  1488
    have th: "dist z y < e" using z y
himmelma@33175
  1489
      by (intro dist_triangle_lt [where z=x], simp)
himmelma@33175
  1490
    from d[rule_format, OF y(1) z(1) th] y z
himmelma@33175
  1491
    have False by (auto simp add: dist_commute)}
wenzelm@53255
  1492
  then show ?thesis
wenzelm@53255
  1493
    by (metis islimpt_approachable closed_limpt [where 'a='a])
himmelma@33175
  1494
qed
himmelma@33175
  1495
huffman@44210
  1496
wenzelm@60420
  1497
subsection \<open>Interior of a Set\<close>
huffman@44210
  1498
huffman@44519
  1499
definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
huffman@44519
  1500
huffman@44519
  1501
lemma interiorI [intro?]:
huffman@44519
  1502
  assumes "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
  1503
  shows "x \<in> interior S"
huffman@44519
  1504
  using assms unfolding interior_def by fast
huffman@44519
  1505
huffman@44519
  1506
lemma interiorE [elim?]:
huffman@44519
  1507
  assumes "x \<in> interior S"
huffman@44519
  1508
  obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
  1509
  using assms unfolding interior_def by fast
huffman@44519
  1510
huffman@44519
  1511
lemma open_interior [simp, intro]: "open (interior S)"
huffman@44519
  1512
  by (simp add: interior_def open_Union)
huffman@44519
  1513
huffman@44519
  1514
lemma interior_subset: "interior S \<subseteq> S"
huffman@44519
  1515
  by (auto simp add: interior_def)
huffman@44519
  1516
huffman@44519
  1517
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
huffman@44519
  1518
  by (auto simp add: interior_def)
huffman@44519
  1519
huffman@44519
  1520
lemma interior_open: "open S \<Longrightarrow> interior S = S"
huffman@44519
  1521
  by (intro equalityI interior_subset interior_maximal subset_refl)
himmelma@33175
  1522
himmelma@33175
  1523
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
huffman@44519
  1524
  by (metis open_interior interior_open)
huffman@44519
  1525
huffman@44519
  1526
lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
himmelma@33175
  1527
  by (metis interior_maximal interior_subset subset_trans)
himmelma@33175
  1528
huffman@44519
  1529
lemma interior_empty [simp]: "interior {} = {}"
huffman@44519
  1530
  using open_empty by (rule interior_open)
huffman@44519
  1531
huffman@44522
  1532
lemma interior_UNIV [simp]: "interior UNIV = UNIV"
huffman@44522
  1533
  using open_UNIV by (rule interior_open)
huffman@44522
  1534
huffman@44519
  1535
lemma interior_interior [simp]: "interior (interior S) = interior S"
huffman@44519
  1536
  using open_interior by (rule interior_open)
huffman@44519
  1537
huffman@44522
  1538
lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
huffman@44522
  1539
  by (auto simp add: interior_def)
huffman@44519
  1540
huffman@44519
  1541
lemma interior_unique:
huffman@44519
  1542
  assumes "T \<subseteq> S" and "open T"
huffman@44519
  1543
  assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
huffman@44519
  1544
  shows "interior S = T"
huffman@44519
  1545
  by (intro equalityI assms interior_subset open_interior interior_maximal)
huffman@44519
  1546
huffman@44519
  1547
lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
huffman@44522
  1548
  by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
huffman@44519
  1549
    Int_lower2 interior_maximal interior_subset open_Int open_interior)
huffman@44519
  1550
huffman@44519
  1551
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
huffman@44519
  1552
  using open_contains_ball_eq [where S="interior S"]
huffman@44519
  1553
  by (simp add: open_subset_interior)
himmelma@33175
  1554
himmelma@33175
  1555
lemma interior_limit_point [intro]:
himmelma@33175
  1556
  fixes x :: "'a::perfect_space"
wenzelm@53255
  1557
  assumes x: "x \<in> interior S"
wenzelm@53255
  1558
  shows "x islimpt S"
huffman@44072
  1559
  using x islimpt_UNIV [of x]
huffman@44072
  1560
  unfolding interior_def islimpt_def
huffman@44072
  1561
  apply (clarsimp, rename_tac T T')
huffman@44072
  1562
  apply (drule_tac x="T \<inter> T'" in spec)
huffman@44072
  1563
  apply (auto simp add: open_Int)
huffman@44072
  1564
  done
himmelma@33175
  1565
himmelma@33175
  1566
lemma interior_closed_Un_empty_interior:
wenzelm@53255
  1567
  assumes cS: "closed S"
wenzelm@53255
  1568
    and iT: "interior T = {}"
huffman@44519
  1569
  shows "interior (S \<union> T) = interior S"
himmelma@33175
  1570
proof
huffman@44519
  1571
  show "interior S \<subseteq> interior (S \<union> T)"
wenzelm@53255
  1572
    by (rule interior_mono) (rule Un_upper1)
himmelma@33175
  1573
  show "interior (S \<union> T) \<subseteq> interior S"
himmelma@33175
  1574
  proof
wenzelm@53255
  1575
    fix x
wenzelm@53255
  1576
    assume "x \<in> interior (S \<union> T)"
huffman@44519
  1577
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
himmelma@33175
  1578
    show "x \<in> interior S"
himmelma@33175
  1579
    proof (rule ccontr)
himmelma@33175
  1580
      assume "x \<notin> interior S"
wenzelm@60420
  1581
      with \<open>x \<in> R\<close> \<open>open R\<close> obtain y where "y \<in> R - S"
huffman@44519
  1582
        unfolding interior_def by fast
wenzelm@60420
  1583
      from \<open>open R\<close> \<open>closed S\<close> have "open (R - S)"
wenzelm@53282
  1584
        by (rule open_Diff)
wenzelm@60420
  1585
      from \<open>R \<subseteq> S \<union> T\<close> have "R - S \<subseteq> T"
wenzelm@53282
  1586
        by fast
wenzelm@60420
  1587
      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
  1588
        unfolding interior_def by fast
himmelma@33175
  1589
    qed
himmelma@33175
  1590
  qed
himmelma@33175
  1591
qed
himmelma@33175
  1592
huffman@44365
  1593
lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
huffman@44365
  1594
proof (rule interior_unique)
huffman@44365
  1595
  show "interior A \<times> interior B \<subseteq> A \<times> B"
huffman@44365
  1596
    by (intro Sigma_mono interior_subset)
huffman@44365
  1597
  show "open (interior A \<times> interior B)"
huffman@44365
  1598
    by (intro open_Times open_interior)
wenzelm@53255
  1599
  fix T
wenzelm@53255
  1600
  assume "T \<subseteq> A \<times> B" and "open T"
wenzelm@53255
  1601
  then show "T \<subseteq> interior A \<times> interior B"
wenzelm@53282
  1602
  proof safe
wenzelm@53255
  1603
    fix x y
wenzelm@53255
  1604
    assume "(x, y) \<in> T"
huffman@44519
  1605
    then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
wenzelm@60420
  1606
      using \<open>open T\<close> unfolding open_prod_def by fast
wenzelm@53255
  1607
    then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
wenzelm@60420
  1608
      using \<open>T \<subseteq> A \<times> B\<close> by auto
wenzelm@53255
  1609
    then show "x \<in> interior A" and "y \<in> interior B"
huffman@44519
  1610
      by (auto intro: interiorI)
huffman@44519
  1611
  qed
huffman@44365
  1612
qed
huffman@44365
  1613
hoelzl@61245
  1614
lemma interior_Ici:
hoelzl@61245
  1615
  fixes x :: "'a :: {dense_linorder, linorder_topology}"
hoelzl@61245
  1616
  assumes "b < x"
hoelzl@61245
  1617
  shows "interior { x ..} = { x <..}"
hoelzl@61245
  1618
proof (rule interior_unique)
hoelzl@61245
  1619
  fix T assume "T \<subseteq> {x ..}" "open T"
hoelzl@61245
  1620
  moreover have "x \<notin> T"
hoelzl@61245
  1621
  proof
hoelzl@61245
  1622
    assume "x \<in> T"
hoelzl@61245
  1623
    obtain y where "y < x" "{y <.. x} \<subseteq> T"
hoelzl@61245
  1624
      using open_left[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>b < x\<close>] by auto
hoelzl@61245
  1625
    with dense[OF \<open>y < x\<close>] obtain z where "z \<in> T" "z < x"
hoelzl@61245
  1626
      by (auto simp: subset_eq Ball_def)
hoelzl@61245
  1627
    with \<open>T \<subseteq> {x ..}\<close> show False by auto
hoelzl@61245
  1628
  qed
hoelzl@61245
  1629
  ultimately show "T \<subseteq> {x <..}"
hoelzl@61245
  1630
    by (auto simp: subset_eq less_le)
hoelzl@61245
  1631
qed auto
hoelzl@61245
  1632
hoelzl@61245
  1633
lemma interior_Iic:
hoelzl@61245
  1634
  fixes x :: "'a :: {dense_linorder, linorder_topology}"
hoelzl@61245
  1635
  assumes "x < b"
hoelzl@61245
  1636
  shows "interior {.. x} = {..< x}"
hoelzl@61245
  1637
proof (rule interior_unique)
hoelzl@61245
  1638
  fix T assume "T \<subseteq> {.. x}" "open T"
hoelzl@61245
  1639
  moreover have "x \<notin> T"
hoelzl@61245
  1640
  proof
hoelzl@61245
  1641
    assume "x \<in> T"
hoelzl@61245
  1642
    obtain y where "x < y" "{x ..< y} \<subseteq> T"
hoelzl@61245
  1643
      using open_right[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>x < b\<close>] by auto
hoelzl@61245
  1644
    with dense[OF \<open>x < y\<close>] obtain z where "z \<in> T" "x < z"
hoelzl@61245
  1645
      by (auto simp: subset_eq Ball_def less_le)
hoelzl@61245
  1646
    with \<open>T \<subseteq> {.. x}\<close> show False by auto
hoelzl@61245
  1647
  qed
hoelzl@61245
  1648
  ultimately show "T \<subseteq> {..< x}"
hoelzl@61245
  1649
    by (auto simp: subset_eq less_le)
hoelzl@61245
  1650
qed auto
himmelma@33175
  1651
wenzelm@60420
  1652
subsection \<open>Closure of a Set\<close>
himmelma@33175
  1653
himmelma@33175
  1654
definition "closure S = S \<union> {x | x. x islimpt S}"
himmelma@33175
  1655
huffman@44518
  1656
lemma interior_closure: "interior S = - (closure (- S))"
huffman@44518
  1657
  unfolding interior_def closure_def islimpt_def by auto
huffman@44518
  1658
huffman@34105
  1659
lemma closure_interior: "closure S = - interior (- S)"
huffman@44518
  1660
  unfolding interior_closure by simp
himmelma@33175
  1661
himmelma@33175
  1662
lemma closed_closure[simp, intro]: "closed (closure S)"
huffman@44518
  1663
  unfolding closure_interior by (simp add: closed_Compl)
huffman@44518
  1664
huffman@44518
  1665
lemma closure_subset: "S \<subseteq> closure S"
huffman@44518
  1666
  unfolding closure_def by simp
himmelma@33175
  1667
himmelma@33175
  1668
lemma closure_hull: "closure S = closed hull S"
huffman@44519
  1669
  unfolding hull_def closure_interior interior_def by auto
himmelma@33175
  1670
himmelma@33175
  1671
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
huffman@44519
  1672
  unfolding closure_hull using closed_Inter by (rule hull_eq)
huffman@44519
  1673
huffman@44519
  1674
lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
huffman@44519
  1675
  unfolding closure_eq .
huffman@44519
  1676
huffman@44519
  1677
lemma closure_closure [simp]: "closure (closure S) = closure S"
huffman@44518
  1678
  unfolding closure_hull by (rule hull_hull)
himmelma@33175
  1679
huffman@44522
  1680
lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
huffman@44518
  1681
  unfolding closure_hull by (rule hull_mono)
himmelma@33175
  1682
huffman@44519
  1683
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
huffman@44518
  1684
  unfolding closure_hull by (rule hull_minimal)
himmelma@33175
  1685
huffman@44519
  1686
lemma closure_unique:
wenzelm@53255
  1687
  assumes "S \<subseteq> T"
wenzelm@53255
  1688
    and "closed T"
wenzelm@53255
  1689
    and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
huffman@44519
  1690
  shows "closure S = T"
huffman@44519
  1691
  using assms unfolding closure_hull by (rule hull_unique)
huffman@44519
  1692
huffman@44519
  1693
lemma closure_empty [simp]: "closure {} = {}"
huffman@44518
  1694
  using closed_empty by (rule closure_closed)
himmelma@33175
  1695
huffman@44522
  1696
lemma closure_UNIV [simp]: "closure UNIV = UNIV"
huffman@44518
  1697
  using closed_UNIV by (rule closure_closed)
huffman@44518
  1698
huffman@44518
  1699
lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
huffman@44518
  1700
  unfolding closure_interior by simp
himmelma@33175
  1701
lp15@60974
  1702
lemma closure_eq_empty [iff]: "closure S = {} \<longleftrightarrow> S = {}"
himmelma@33175
  1703
  using closure_empty closure_subset[of S]
himmelma@33175
  1704
  by blast
himmelma@33175
  1705
himmelma@33175
  1706
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
  1707
  using closure_eq[of S] closure_subset[of S]
himmelma@33175
  1708
  by simp
himmelma@33175
  1709
himmelma@33175
  1710
lemma open_inter_closure_eq_empty:
himmelma@33175
  1711
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
huffman@34105
  1712
  using open_subset_interior[of S "- T"]
huffman@34105
  1713
  using interior_subset[of "- T"]
himmelma@33175
  1714
  unfolding closure_interior
himmelma@33175
  1715
  by auto
himmelma@33175
  1716
himmelma@33175
  1717
lemma open_inter_closure_subset:
himmelma@33175
  1718
  "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
himmelma@33175
  1719
proof
himmelma@33175
  1720
  fix x
himmelma@33175
  1721
  assume as: "open S" "x \<in> S \<inter> closure T"
wenzelm@53255
  1722
  {
wenzelm@53282
  1723
    assume *: "x islimpt T"
himmelma@33175
  1724
    have "x islimpt (S \<inter> T)"
himmelma@33175
  1725
    proof (rule islimptI)
himmelma@33175
  1726
      fix A
himmelma@33175
  1727
      assume "x \<in> A" "open A"
himmelma@33175
  1728
      with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
himmelma@33175
  1729
        by (simp_all add: open_Int)
himmelma@33175
  1730
      with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
himmelma@33175
  1731
        by (rule islimptE)
wenzelm@53255
  1732
      then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
himmelma@33175
  1733
        by simp_all
wenzelm@53255
  1734
      then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
himmelma@33175
  1735
    qed
himmelma@33175
  1736
  }
himmelma@33175
  1737
  then show "x \<in> closure (S \<inter> T)" using as
himmelma@33175
  1738
    unfolding closure_def
himmelma@33175
  1739
    by blast
himmelma@33175
  1740
qed
himmelma@33175
  1741
huffman@44519
  1742
lemma closure_complement: "closure (- S) = - interior S"
huffman@44518
  1743
  unfolding closure_interior by simp
himmelma@33175
  1744
huffman@44519
  1745
lemma interior_complement: "interior (- S) = - closure S"
huffman@44518
  1746
  unfolding closure_interior by simp
himmelma@33175
  1747
huffman@44365
  1748
lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
huffman@44519
  1749
proof (rule closure_unique)
huffman@44365
  1750
  show "A \<times> B \<subseteq> closure A \<times> closure B"
huffman@44365
  1751
    by (intro Sigma_mono closure_subset)
huffman@44365
  1752
  show "closed (closure A \<times> closure B)"
huffman@44365
  1753
    by (intro closed_Times closed_closure)
wenzelm@53282
  1754
  fix T
wenzelm@53282
  1755
  assume "A \<times> B \<subseteq> T" and "closed T"
wenzelm@53282
  1756
  then show "closure A \<times> closure B \<subseteq> T"
huffman@44365
  1757
    apply (simp add: closed_def open_prod_def, clarify)
huffman@44365
  1758
    apply (rule ccontr)
huffman@44365
  1759
    apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
huffman@44365
  1760
    apply (simp add: closure_interior interior_def)
huffman@44365
  1761
    apply (drule_tac x=C in spec)
huffman@44365
  1762
    apply (drule_tac x=D in spec)
huffman@44365
  1763
    apply auto
huffman@44365
  1764
    done
huffman@44365
  1765
qed
huffman@44365
  1766
hoelzl@51351
  1767
lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"
hoelzl@51351
  1768
  unfolding closure_def using islimpt_punctured by blast
hoelzl@51351
  1769
lp15@61306
  1770
lemma connected_imp_connected_closure: "connected s \<Longrightarrow> connected (closure s)"
lp15@61306
  1771
    by (rule connectedI) (meson closure_subset open_Int open_inter_closure_eq_empty subset_trans connectedD)
lp15@61306
  1772
lp15@61306
  1773
lemma limpt_of_limpts:
lp15@61306
  1774
      fixes x :: "'a::metric_space"
lp15@61306
  1775
      shows "x islimpt {y. y islimpt s} \<Longrightarrow> x islimpt s"
lp15@61306
  1776
  apply (clarsimp simp add: islimpt_approachable)
lp15@61306
  1777
  apply (drule_tac x="e/2" in spec)
lp15@61306
  1778
  apply (auto simp: simp del: less_divide_eq_numeral1)
lp15@61306
  1779
  apply (drule_tac x="dist x' x" in spec)
lp15@61306
  1780
  apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1)
lp15@61306
  1781
  apply (erule rev_bexI)
lp15@61306
  1782
  by (metis dist_commute dist_triangle_half_r less_trans less_irrefl)
lp15@61306
  1783
lp15@61306
  1784
lemma closed_limpts:  "closed {x::'a::metric_space. x islimpt s}"
lp15@61306
  1785
  using closed_limpt limpt_of_limpts by blast
lp15@61306
  1786
lp15@61306
  1787
lemma limpt_of_closure:
lp15@61306
  1788
      fixes x :: "'a::metric_space"
lp15@61306
  1789
      shows "x islimpt closure s \<longleftrightarrow> x islimpt s"
lp15@61306
  1790
  by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)
lp15@61306
  1791
lp15@61306
  1792
lemma closed_in_limpt:
lp15@61306
  1793
   "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
  1794
  apply (simp add: closedin_closed, safe)
lp15@61306
  1795
  apply (simp add: closed_limpt islimpt_subset)
lp15@61306
  1796
  apply (rule_tac x="closure s" in exI)
lp15@61306
  1797
  apply simp
lp15@61306
  1798
  apply (force simp: closure_def)
lp15@61306
  1799
  done
lp15@61306
  1800
lp15@61306
  1801
subsection\<open>Connected components, considered as a connectedness relation or a set\<close>
lp15@61306
  1802
lp15@61306
  1803
definition
lp15@61306
  1804
   "connected_component s x y \<equiv> \<exists>t. connected t \<and> t \<subseteq> s \<and> x \<in> t \<and> y \<in> t"
lp15@61306
  1805
lp15@61306
  1806
abbreviation
lp15@61306
  1807
   "connected_component_set s x \<equiv> Collect (connected_component s x)"
lp15@61306
  1808
lp15@61306
  1809
lemma connected_component_in: "connected_component s x y \<Longrightarrow> x \<in> s \<and> y \<in> s"
lp15@61306
  1810
  by (auto simp: connected_component_def)
lp15@61306
  1811
lp15@61306
  1812
lemma connected_component_refl: "x \<in> s \<Longrightarrow> connected_component s x x"
lp15@61306
  1813
  apply (auto simp: connected_component_def)
lp15@61306
  1814
  using connected_sing by blast
lp15@61306
  1815
lp15@61306
  1816
lemma connected_component_refl_eq [simp]: "connected_component s x x \<longleftrightarrow> x \<in> s"
lp15@61306
  1817
  by (auto simp: connected_component_refl) (auto simp: connected_component_def)
lp15@61306
  1818
lp15@61306
  1819
lemma connected_component_sym: "connected_component s x y \<Longrightarrow> connected_component s y x"
lp15@61306
  1820
  by (auto simp: connected_component_def)
lp15@61306
  1821
lp15@61306
  1822
lemma connected_component_trans:
lp15@61306
  1823
    "\<lbrakk>connected_component s x y; connected_component s y z\<rbrakk> \<Longrightarrow> connected_component s x z"
lp15@61306
  1824
  unfolding connected_component_def
lp15@61306
  1825
  by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)
lp15@61306
  1826
lp15@61306
  1827
lemma connected_component_of_subset: "\<lbrakk>connected_component s x y; s \<subseteq> t\<rbrakk> \<Longrightarrow> connected_component t x y"
lp15@61306
  1828
  by (auto simp: connected_component_def)
lp15@61306
  1829
lp15@61306
  1830
lemma connected_component_Union: "connected_component_set s x = Union {t. connected t \<and> x \<in> t \<and> t \<subseteq> s}"
lp15@61306
  1831
  by (auto simp: connected_component_def)
lp15@61306
  1832
lp15@61306
  1833
lemma connected_connected_component [iff]: "connected (connected_component_set s x)"
lp15@61306
  1834
  by (auto simp: connected_component_Union intro: connected_Union)
lp15@61306
  1835
lp15@61306
  1836
lemma connected_iff_eq_connected_component_set: "connected s \<longleftrightarrow> (\<forall>x \<in> s. connected_component_set s x = s)"
lp15@61306
  1837
proof (cases "s={}")
lp15@61306
  1838
  case True then show ?thesis by simp
lp15@61306
  1839
next
lp15@61306
  1840
  case False
lp15@61306
  1841
  then obtain x where "x \<in> s" by auto
lp15@61306
  1842
  show ?thesis
lp15@61306
  1843
  proof
lp15@61306
  1844
    assume "connected s"
lp15@61306
  1845
    then show "\<forall>x \<in> s. connected_component_set s x = s"
lp15@61306
  1846
      by (force simp: connected_component_def)
lp15@61306
  1847
  next
lp15@61306
  1848
    assume "\<forall>x \<in> s. connected_component_set s x = s"
lp15@61306
  1849
    then show "connected s"
lp15@61306
  1850
      by (metis `x \<in> s` connected_connected_component)
lp15@61306
  1851
  qed
lp15@61306
  1852
qed
lp15@61306
  1853
lp15@61306
  1854
lemma connected_component_subset: "connected_component_set s x \<subseteq> s"
lp15@61306
  1855
  using connected_component_in by blast
lp15@61306
  1856
lp15@61306
  1857
lemma connected_component_eq_self: "\<lbrakk>connected s; x \<in> s\<rbrakk> \<Longrightarrow> connected_component_set s x = s"
lp15@61306
  1858
  by (simp add: connected_iff_eq_connected_component_set)
lp15@61306
  1859
lp15@61306
  1860
lemma connected_iff_connected_component:
lp15@61306
  1861
    "connected s \<longleftrightarrow> (\<forall>x \<in> s. \<forall>y \<in> s. connected_component s x y)"
lp15@61306
  1862
  using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)
lp15@61306
  1863
lp15@61306
  1864
lemma connected_component_maximal:
lp15@61306
  1865
    "\<lbrakk>x \<in> t; connected t; t \<subseteq> s\<rbrakk> \<Longrightarrow> t \<subseteq> (connected_component_set s x)"
lp15@61306
  1866
  using connected_component_eq_self connected_component_of_subset by blast
lp15@61306
  1867
lp15@61306
  1868
lemma connected_component_mono:
lp15@61306
  1869
    "s \<subseteq> t \<Longrightarrow> (connected_component_set s x) \<subseteq> (connected_component_set t x)"
lp15@61306
  1870
  by (simp add: Collect_mono connected_component_of_subset)
lp15@61306
  1871
lp15@61306
  1872
lemma connected_component_eq_empty [simp]: "connected_component_set s x = {} \<longleftrightarrow> (x \<notin> s)"
lp15@61306
  1873
  using connected_component_refl by (fastforce simp: connected_component_in)
lp15@61306
  1874
lp15@61306
  1875
lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"
lp15@61306
  1876
  using connected_component_eq_empty by blast
lp15@61306
  1877
lp15@61306
  1878
lemma connected_component_eq:
lp15@61306
  1879
    "y \<in> connected_component_set s x
lp15@61306
  1880
     \<Longrightarrow> (connected_component_set s y = connected_component_set s x)"
lp15@61306
  1881
  by (metis (no_types, lifting) Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)
lp15@61306
  1882
lp15@61306
  1883
lemma closed_connected_component:
lp15@61306
  1884
  assumes s: "closed s" shows "closed (connected_component_set s x)"
lp15@61306
  1885
proof (cases "x \<in> s")
lp15@61306
  1886
  case False then show ?thesis
lp15@61306
  1887
    by (metis connected_component_eq_empty closed_empty)
lp15@61306
  1888
next
lp15@61306
  1889
  case True
lp15@61306
  1890
  show ?thesis
lp15@61306
  1891
    unfolding closure_eq [symmetric]
lp15@61306
  1892
    proof
lp15@61306
  1893
      show "closure (connected_component_set s x) \<subseteq> connected_component_set s x"
lp15@61306
  1894
        apply (rule connected_component_maximal)
lp15@61306
  1895
        apply (simp add: closure_def True)
lp15@61306
  1896
        apply (simp add: connected_imp_connected_closure)
lp15@61306
  1897
        apply (simp add: s closure_minimal connected_component_subset)
lp15@61306
  1898
        done
lp15@61306
  1899
    next
lp15@61306
  1900
      show "connected_component_set s x \<subseteq> closure (connected_component_set s x)"
lp15@61306
  1901
        by (simp add: closure_subset)
lp15@61306
  1902
  qed
lp15@61306
  1903
qed
lp15@61306
  1904
lp15@61306
  1905
lemma connected_component_disjoint:
lp15@61306
  1906
    "(connected_component_set s a) \<inter> (connected_component_set s b) = {} \<longleftrightarrow>
lp15@61306
  1907
     a \<notin> connected_component_set s b"
lp15@61306
  1908
apply (auto simp: connected_component_eq)
lp15@61306
  1909
using connected_component_eq connected_component_sym by blast
lp15@61306
  1910
lp15@61306
  1911
lemma connected_component_nonoverlap:
lp15@61306
  1912
    "(connected_component_set s a) \<inter> (connected_component_set s b) = {} \<longleftrightarrow>
lp15@61306
  1913
     (a \<notin> s \<or> b \<notin> s \<or> connected_component_set s a \<noteq> connected_component_set s b)"
lp15@61306
  1914
  apply (auto simp: connected_component_in)
lp15@61306
  1915
  using connected_component_refl_eq apply blast
lp15@61306
  1916
  apply (metis connected_component_eq mem_Collect_eq)
lp15@61306
  1917
  apply (metis connected_component_eq mem_Collect_eq)
lp15@61306
  1918
  done
lp15@61306
  1919
lp15@61306
  1920
lemma connected_component_overlap:
lp15@61306
  1921
    "(connected_component_set s a \<inter> connected_component_set s b \<noteq> {}) =
lp15@61306
  1922
     (a \<in> s \<and> b \<in> s \<and> connected_component_set s a = connected_component_set s b)"
lp15@61306
  1923
  by (auto simp: connected_component_nonoverlap)
lp15@61306
  1924
lp15@61306
  1925
lemma connected_component_sym_eq: "connected_component s x y \<longleftrightarrow> connected_component s y x"
lp15@61306
  1926
  using connected_component_sym by blast
lp15@61306
  1927
lp15@61306
  1928
lemma connected_component_eq_eq:
lp15@61306
  1929
    "connected_component_set s x = connected_component_set s y \<longleftrightarrow>
lp15@61306
  1930
     x \<notin> s \<and> y \<notin> s \<or> x \<in> s \<and> y \<in> s \<and> connected_component s x y"
lp15@61306
  1931
  apply (case_tac "y \<in> s")
lp15@61306
  1932
   apply (simp add:)
lp15@61306
  1933
   apply (metis connected_component_eq connected_component_eq_empty connected_component_refl_eq mem_Collect_eq)
lp15@61306
  1934
  apply (case_tac "x \<in> s")
lp15@61306
  1935
   apply (simp add:)
lp15@61306
  1936
   apply (metis connected_component_eq_empty)
lp15@61306
  1937
  using connected_component_eq_empty by blast
lp15@61306
  1938
lp15@61306
  1939
lemma connected_iff_connected_component_eq:
lp15@61306
  1940
    "connected s \<longleftrightarrow>
lp15@61306
  1941
       (\<forall>x \<in> s. \<forall>y \<in> s. connected_component_set s x = connected_component_set s y)"
lp15@61306
  1942
  by (simp add: connected_component_eq_eq connected_iff_connected_component)
lp15@61306
  1943
lp15@61306
  1944
lemma connected_component_idemp:
lp15@61306
  1945
    "connected_component_set (connected_component_set s x) x = connected_component_set s x"
lp15@61306
  1946
apply (rule subset_antisym)
lp15@61306
  1947
apply (simp add: connected_component_subset)
lp15@61306
  1948
by (metis connected_component_eq_empty connected_component_maximal connected_component_refl_eq connected_connected_component mem_Collect_eq set_eq_subset)
lp15@61306
  1949
lp15@61306
  1950
lemma connected_component_unique:
lp15@61306
  1951
  "\<lbrakk>x \<in> c; c \<subseteq> s; connected c;
lp15@61306
  1952
    \<And>c'. x \<in> c' \<and> c' \<subseteq> s \<and> connected c'
lp15@61306
  1953
              \<Longrightarrow> c' \<subseteq> c\<rbrakk>
lp15@61306
  1954
        \<Longrightarrow> connected_component_set s x = c"
lp15@61306
  1955
apply (rule subset_antisym)
lp15@61306
  1956
apply (meson connected_component_maximal connected_component_subset connected_connected_component contra_subsetD)
lp15@61306
  1957
by (simp add: connected_component_maximal)
lp15@61306
  1958
lp15@61306
  1959
lemma joinable_connected_component_eq:
lp15@61306
  1960
  "\<lbrakk>connected t; t \<subseteq> s;
lp15@61306
  1961
    connected_component_set s x \<inter> t \<noteq> {};
lp15@61306
  1962
    connected_component_set s y \<inter> t \<noteq> {}\<rbrakk>
lp15@61306
  1963
    \<Longrightarrow> connected_component_set s x = connected_component_set s y"
lp15@61306
  1964
apply (simp add: ex_in_conv [symmetric])
lp15@61306
  1965
apply (rule connected_component_eq)
lp15@61306
  1966
by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in connected_component_maximal subsetD mem_Collect_eq)
lp15@61306
  1967
lp15@61306
  1968
lp15@61306
  1969
lemma Union_connected_component: "Union (connected_component_set s ` s) = s"
lp15@61306
  1970
  apply (rule subset_antisym)
lp15@61306
  1971
  apply (simp add: SUP_least connected_component_subset)
lp15@61306
  1972
  using connected_component_refl_eq
lp15@61306
  1973
  by force
lp15@61306
  1974
lp15@61306
  1975
lp15@61306
  1976
lemma complement_connected_component_unions:
lp15@61306
  1977
    "s - connected_component_set s x =
lp15@61306
  1978
     Union (connected_component_set s ` s - {connected_component_set s x})"
lp15@61306
  1979
  apply (subst Union_connected_component [symmetric], auto)
lp15@61306
  1980
  apply (metis connected_component_eq_eq connected_component_in)
lp15@61306
  1981
  by (metis connected_component_eq mem_Collect_eq)
lp15@61306
  1982
lp15@61306
  1983
lemma connected_component_intermediate_subset:
lp15@61306
  1984
        "\<lbrakk>connected_component_set u a \<subseteq> t; t \<subseteq> u\<rbrakk>
lp15@61306
  1985
        \<Longrightarrow> connected_component_set t a = connected_component_set u a"
lp15@61306
  1986
  apply (case_tac "a \<in> u")
lp15@61306
  1987
  apply (simp add: connected_component_maximal connected_component_mono subset_antisym)
lp15@61306
  1988
  using connected_component_eq_empty by blast
lp15@61306
  1989
lp15@61306
  1990
subsection\<open>The set of connected components of a set\<close>
lp15@61306
  1991
lp15@61306
  1992
definition components:: "'a::topological_space set \<Rightarrow> 'a set set" where
lp15@61306
  1993
  "components s \<equiv> connected_component_set s ` s"
lp15@61306
  1994
lp15@61306
  1995
lemma components_iff: "s \<in> components u \<longleftrightarrow> (\<exists>x. x \<in> u \<and> s = connected_component_set u x)"
lp15@61306
  1996
  by (auto simp: components_def)
lp15@61306
  1997
lp15@61306
  1998
lemma Union_components: "u = Union (components u)"
lp15@61306
  1999
  apply (rule subset_antisym)
lp15@61306
  2000
  apply (metis Union_connected_component components_def set_eq_subset)
lp15@61306
  2001
  using Union_connected_component components_def by fastforce
lp15@61306
  2002
lp15@61306
  2003
lemma pairwise_disjoint_components: "pairwise (\<lambda>X Y. X \<inter> Y = {}) (components u)"
lp15@61306
  2004
  apply (simp add: pairwise_def)
lp15@61306
  2005
  apply (auto simp: components_iff)
lp15@61306
  2006
  apply (metis connected_component_eq_eq connected_component_in)+
lp15@61306
  2007
  done
lp15@61306
  2008
lp15@61306
  2009
lemma in_components_nonempty: "c \<in> components s \<Longrightarrow> c \<noteq> {}"
lp15@61306
  2010
    by (metis components_iff connected_component_eq_empty)
lp15@61306
  2011
lp15@61306
  2012
lemma in_components_subset: "c \<in> components s \<Longrightarrow> c \<subseteq> s"
lp15@61306
  2013
  using Union_components by blast
lp15@61306
  2014
lp15@61306
  2015
lemma in_components_connected: "c \<in> components s \<Longrightarrow> connected c"
lp15@61306
  2016
  by (metis components_iff connected_connected_component)
lp15@61306
  2017
lp15@61306
  2018
lemma in_components_maximal:
lp15@61306
  2019
     "c \<in> components s \<longleftrightarrow>
lp15@61306
  2020
      (c \<noteq> {} \<and> c \<subseteq> s \<and> connected c \<and> (\<forall>d. d \<noteq> {} \<and> c \<subseteq> d \<and> d \<subseteq> s \<and> connected d \<longrightarrow> d = c))"
lp15@61306
  2021
  apply (rule iffI)
lp15@61306
  2022
  apply (simp add: in_components_nonempty in_components_connected)
lp15@61306
  2023
  apply (metis (full_types) components_iff connected_component_eq_self connected_component_intermediate_subset connected_component_refl in_components_subset mem_Collect_eq rev_subsetD)
lp15@61306
  2024
  by (metis bot.extremum_uniqueI components_iff connected_component_eq_empty connected_component_maximal connected_component_subset connected_connected_component subset_emptyI)
lp15@61306
  2025
lp15@61306
  2026
lemma joinable_components_eq:
lp15@61306
  2027
    "connected t \<and> t \<subseteq> s \<and> c1 \<in> components s \<and> c2 \<in> components s \<and> c1 \<inter> t \<noteq> {} \<and> c2 \<inter> t \<noteq> {} \<Longrightarrow> c1 = c2"
lp15@61306
  2028
  by (metis (full_types) components_iff joinable_connected_component_eq)
lp15@61306
  2029
lp15@61306
  2030
lemma closed_components: "\<lbrakk>closed s; c \<in> components s\<rbrakk> \<Longrightarrow> closed c"
lp15@61306
  2031
  by (metis closed_connected_component components_iff)
lp15@61306
  2032
lp15@61306
  2033
lemma components_nonoverlap:
lp15@61306
  2034
    "\<lbrakk>c \<in> components s; c' \<in> components s\<rbrakk> \<Longrightarrow> (c \<inter> c' = {}) \<longleftrightarrow> (c \<noteq> c')"
lp15@61306
  2035
  apply (auto simp: in_components_nonempty components_iff)
lp15@61306
  2036
    using connected_component_refl apply blast
lp15@61306
  2037
   apply (metis connected_component_eq_eq connected_component_in)
lp15@61306
  2038
  by (metis connected_component_eq mem_Collect_eq)
lp15@61306
  2039
lp15@61306
  2040
lemma components_eq: "\<lbrakk>c \<in> components s; c' \<in> components s\<rbrakk> \<Longrightarrow> (c = c' \<longleftrightarrow> c \<inter> c' \<noteq> {})"
lp15@61306
  2041
  by (metis components_nonoverlap)
lp15@61306
  2042
lp15@61306
  2043
lemma components_eq_empty [simp]: "components s = {} \<longleftrightarrow> s = {}"
lp15@61306
  2044
  by (simp add: components_def)
lp15@61306
  2045
lp15@61306
  2046
lemma components_empty [simp]: "components {} = {}"
lp15@61306
  2047
  by simp
lp15@61306
  2048
lp15@61306
  2049
lemma connected_eq_connected_components_eq: "connected s \<longleftrightarrow> (\<forall>c \<in> components s. \<forall>c' \<in> components s. c = c')"
lp15@61306
  2050
  by (metis (no_types, hide_lams) components_iff connected_component_eq_eq connected_iff_connected_component)
lp15@61306
  2051
lp15@61306
  2052
lemma components_eq_sing_iff: "components s = {s} \<longleftrightarrow> connected s \<and> s \<noteq> {}"
lp15@61306
  2053
  apply (rule iffI)
lp15@61306
  2054
   using in_components_connected apply fastforce
lp15@61306
  2055
  apply safe
lp15@61306
  2056
    using Union_components apply fastforce
lp15@61306
  2057
   apply (metis components_iff connected_component_eq_self)
lp15@61306
  2058
  using in_components_maximal by auto
lp15@61306
  2059
lp15@61306
  2060
lemma components_eq_sing_exists: "(\<exists>a. components s = {a}) \<longleftrightarrow> connected s \<and> s \<noteq> {}"
lp15@61306
  2061
  apply (rule iffI)
lp15@61306
  2062
   using connected_eq_connected_components_eq apply fastforce
lp15@61306
  2063
  by (metis components_eq_sing_iff)
lp15@61306
  2064
lp15@61306
  2065
lemma connected_eq_components_subset_sing: "connected s \<longleftrightarrow> components s \<subseteq> {s}"
lp15@61306
  2066
  by (metis Union_components components_empty components_eq_sing_iff connected_empty insert_subset order_refl subset_singletonD)
lp15@61306
  2067
lp15@61306
  2068
lemma connected_eq_components_subset_sing_exists: "connected s \<longleftrightarrow> (\<exists>a. components s \<subseteq> {a})"
lp15@61306
  2069
  by (metis components_eq_sing_exists connected_eq_components_subset_sing empty_iff subset_iff subset_singletonD)
lp15@61306
  2070
lp15@61306
  2071
lemma in_components_self: "s \<in> components s \<longleftrightarrow> connected s \<and> s \<noteq> {}"
lp15@61306
  2072
  by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)
lp15@61306
  2073
lp15@61306
  2074
lemma components_maximal: "\<lbrakk>c \<in> components s; connected t; t \<subseteq> s; c \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> t \<subseteq> c"
lp15@61306
  2075
  apply (simp add: components_def ex_in_conv [symmetric], clarify)
lp15@61306
  2076
  by (meson connected_component_def connected_component_trans)
lp15@61306
  2077
lp15@61306
  2078
lemma exists_component_superset: "\<lbrakk>t \<subseteq> s; s \<noteq> {}; connected t\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> components s \<and> t \<subseteq> c"
lp15@61306
  2079
  apply (case_tac "t = {}")
lp15@61306
  2080
   apply force
lp15@61306
  2081
  by (metis components_def ex_in_conv connected_component_maximal contra_subsetD image_eqI)
lp15@61306
  2082
lp15@61306
  2083
lemma components_intermediate_subset: "\<lbrakk>s \<in> components u; s \<subseteq> t; t \<subseteq> u\<rbrakk> \<Longrightarrow> s \<in> components t"
lp15@61306
  2084
  apply (auto simp: components_iff)
lp15@61306
  2085
  by (metis connected_component_eq_empty connected_component_intermediate_subset)
lp15@61306
  2086
lp15@61306
  2087
lemma in_components_unions_complement: "c \<in> components s \<Longrightarrow> s - c = Union(components s - {c})"
lp15@61306
  2088
  by (metis complement_connected_component_unions components_def components_iff)
lp15@61306
  2089
lp15@61306
  2090
lemma connected_intermediate_closure:
lp15@61306
  2091
  assumes cs: "connected s" and st: "s \<subseteq> t" and ts: "t \<subseteq> closure s"
lp15@61306
  2092
    shows "connected t"
lp15@61306
  2093
proof (rule connectedI)
lp15@61306
  2094
  fix A B
lp15@61306
  2095
  assume A: "open A" and B: "open B" and Alap: "A \<inter> t \<noteq> {}" and Blap: "B \<inter> t \<noteq> {}"
lp15@61306
  2096
     and disj: "A \<inter> B \<inter> t = {}" and cover: "t \<subseteq> A \<union> B"
lp15@61306
  2097
  have disjs: "A \<inter> B \<inter> s = {}"
lp15@61306
  2098
    using disj st by auto
lp15@61306
  2099
  have "A \<inter> closure s \<noteq> {}"
lp15@61306
  2100
    using Alap Int_absorb1 ts by blast
lp15@61306
  2101
  then have Alaps: "A \<inter> s \<noteq> {}"
lp15@61306
  2102
    by (simp add: A open_inter_closure_eq_empty)
lp15@61306
  2103
  have "B \<inter> closure s \<noteq> {}"
lp15@61306
  2104
    using Blap Int_absorb1 ts by blast
lp15@61306
  2105
  then have Blaps: "B \<inter> s \<noteq> {}"
lp15@61306
  2106
    by (simp add: B open_inter_closure_eq_empty)
lp15@61306
  2107
  then show False
lp15@61306
  2108
    using cs [unfolded connected_def] A B disjs Alaps Blaps cover st
lp15@61306
  2109
    by blast
lp15@61306
  2110
qed
lp15@61306
  2111
lp15@61306
  2112
lemma closed_in_connected_component: "closedin (subtopology euclidean s) (connected_component_set s x)"
lp15@61306
  2113
proof (cases "connected_component_set s x = {}")
lp15@61306
  2114
  case True then show ?thesis 
lp15@61306
  2115
    by (metis closedin_empty)
lp15@61306
  2116
next
lp15@61306
  2117
  case False
lp15@61306
  2118
  then obtain y where y: "connected_component s x y"
lp15@61306
  2119
    by blast
lp15@61306
  2120
  have 1: "connected_component_set s x \<subseteq> s \<inter> closure (connected_component_set s x)"
lp15@61306
  2121
    by (auto simp: closure_def connected_component_in)
lp15@61306
  2122
  have 2: "connected_component s x y \<Longrightarrow> s \<inter> closure (connected_component_set s x) \<subseteq> connected_component_set s x"
lp15@61306
  2123
    apply (rule connected_component_maximal)
lp15@61306
  2124
    apply (simp add:)
lp15@61306
  2125
    using closure_subset connected_component_in apply fastforce
lp15@61306
  2126
    using "1" connected_intermediate_closure apply blast+
lp15@61306
  2127
    done
lp15@61306
  2128
  show ?thesis using y
lp15@61306
  2129
    apply (simp add: Topology_Euclidean_Space.closedin_closed)
lp15@61306
  2130
    using 1 2 by auto
lp15@61306
  2131
qed
hoelzl@51351
  2132
wenzelm@60420
  2133
subsection \<open>Frontier (aka boundary)\<close>
himmelma@33175
  2134
himmelma@33175
  2135
definition "frontier S = closure S - interior S"
himmelma@33175
  2136
wenzelm@53255
  2137
lemma frontier_closed: "closed (frontier S)"
himmelma@33175
  2138
  by (simp add: frontier_def closed_Diff)
himmelma@33175
  2139
huffman@34105
  2140
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
himmelma@33175
  2141
  by (auto simp add: frontier_def interior_closure)
himmelma@33175
  2142
himmelma@33175
  2143
lemma frontier_straddle:
himmelma@33175
  2144
  fixes a :: "'a::metric_space"
huffman@44909
  2145
  shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
huffman@44909
  2146
  unfolding frontier_def closure_interior
huffman@44909
  2147
  by (auto simp add: mem_interior subset_eq ball_def)
himmelma@33175
  2148
himmelma@33175
  2149
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
himmelma@33175
  2150
  by (metis frontier_def closure_closed Diff_subset)
himmelma@33175
  2151
hoelzl@34964
  2152
lemma frontier_empty[simp]: "frontier {} = {}"
huffman@36362
  2153
  by (simp add: frontier_def)
himmelma@33175
  2154
himmelma@33175
  2155
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
wenzelm@58757
  2156
proof -
wenzelm@53255
  2157
  {
wenzelm@53255
  2158
    assume "frontier S \<subseteq> S"
wenzelm@53255
  2159
    then have "closure S \<subseteq> S"
wenzelm@53255
  2160
      using interior_subset unfolding frontier_def by auto
wenzelm@53255
  2161
    then have "closed S"
wenzelm@53255
  2162
      using closure_subset_eq by auto
himmelma@33175
  2163
  }
wenzelm@53255
  2164
  then show ?thesis using frontier_subset_closed[of S] ..
himmelma@33175
  2165
qed
himmelma@33175
  2166
wenzelm@58757
  2167
lemma frontier_complement: "frontier (- S) = frontier S"
himmelma@33175
  2168
  by (auto simp add: frontier_def closure_complement interior_complement)
himmelma@33175
  2169
himmelma@33175
  2170
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
huffman@34105
  2171
  using frontier_complement frontier_subset_eq[of "- S"]
huffman@34105
  2172
  unfolding open_closed by auto
himmelma@33175
  2173
wenzelm@58757
  2174
wenzelm@60420
  2175
subsection \<open>Filters and the ``eventually true'' quantifier\<close>
huffman@44081
  2176
wenzelm@52624
  2177
definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
wenzelm@52624
  2178
    (infixr "indirection" 70)
wenzelm@52624
  2179
  where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
himmelma@33175
  2180
wenzelm@60420
  2181
text \<open>Identify Trivial limits, where we can't approach arbitrarily closely.\<close>
himmelma@33175
  2182
wenzelm@52624
  2183
lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
himmelma@33175
  2184
proof
himmelma@33175
  2185
  assume "trivial_limit (at a within S)"
wenzelm@53255
  2186
  then show "\<not> a islimpt S"
himmelma@33175
  2187
    unfolding trivial_limit_def
hoelzl@51641
  2188
    unfolding eventually_at_topological
himmelma@33175
  2189
    unfolding islimpt_def
nipkow@39302
  2190
    apply (clarsimp simp add: set_eq_iff)
himmelma@33175
  2191
    apply (rename_tac T, rule_tac x=T in exI)
huffman@36358
  2192
    apply (clarsimp, drule_tac x=y in bspec, simp_all)
himmelma@33175
  2193
    done
himmelma@33175
  2194
next
himmelma@33175
  2195
  assume "\<not> a islimpt S"
wenzelm@53255
  2196
  then show "trivial_limit (at a within S)"
lp15@55775
  2197
    unfolding trivial_limit_def eventually_at_topological islimpt_def
lp15@55775
  2198
    by metis
himmelma@33175
  2199
qed
himmelma@33175
  2200
himmelma@33175
  2201
lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
huffman@45031
  2202
  using trivial_limit_within [of a UNIV] by simp
himmelma@33175
  2203
himmelma@33175
  2204
lemma trivial_limit_at:
himmelma@33175
  2205
  fixes a :: "'a::perfect_space"
himmelma@33175
  2206
  shows "\<not> trivial_limit (at a)"
huffman@44571
  2207
  by (rule at_neq_bot)
himmelma@33175
  2208
himmelma@33175
  2209
lemma trivial_limit_at_infinity:
huffman@44081
  2210
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
huffman@36358
  2211
  unfolding trivial_limit_def eventually_at_infinity
huffman@36358
  2212
  apply clarsimp
huffman@44072
  2213
  apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
huffman@44072
  2214
   apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
huffman@44072
  2215
  apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
huffman@44072
  2216
  apply (drule_tac x=UNIV in spec, simp)
himmelma@33175
  2217
  done
himmelma@33175
  2218
wenzelm@53640
  2219
lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
wenzelm@53640
  2220
  using islimpt_in_closure
wenzelm@53640
  2221
  by (metis trivial_limit_within)
hoelzl@51351
  2222
wenzelm@60420
  2223
text \<open>Some property holds "sufficiently close" to the limit point.\<close>
himmelma@33175
  2224
himmelma@33175
  2225
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
huffman@45031
  2226
  by simp
himmelma@33175
  2227
himmelma@33175
  2228
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
huffman@44342
  2229
  by (simp add: filter_eq_iff)
himmelma@33175
  2230
wenzelm@60420
  2231
text\<open>Combining theorems for "eventually"\<close>
himmelma@33175
  2232
himmelma@33175
  2233
lemma eventually_rev_mono:
himmelma@33175
  2234
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
wenzelm@53255
  2235
  using eventually_mono [of P Q] by fast
himmelma@33175
  2236
wenzelm@53282
  2237
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"
himmelma@33175
  2238
  by (simp add: eventually_False)
himmelma@33175
  2239
huffman@44210
  2240
wenzelm@60420
  2241
subsection \<open>Limits\<close>