src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author hoelzl
Wed Dec 18 11:53:40 2013 +0100 (2013-12-18)
changeset 54797 be020ec8560c
parent 54780 6fae499e0827
child 54863 82acc20ded73
permissions -rw-r--r--
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
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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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header {* Elementary topology in Euclidean space. *}
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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 {* Topological Basis *}
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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 `topological_basis B` `open X` choosefrom_basis[OF `X \<noteq> {}`]]
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  guess B' . note B' = this
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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 S` 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 from topological_basisE[OF A a] guess A0 .
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    moreover from topological_basisE[OF B b] guess B0 .
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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 {* Countable Basis *}
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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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  from first_countable_basisE[of x] guess A' . note A' = this
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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 `A \<noteq> {}`] 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 `countable A`] 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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proof
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  fix x :: "'a \<times> 'b"
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  from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this
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  from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this
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  show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set.
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    (\<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 (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
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    fix a b
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   282
    assume x: "a \<in> A" "b \<in> B"
wenzelm@53640
   283
    with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" and "open (a \<times> b)"
wenzelm@53640
   284
      unfolding mem_Times_iff
wenzelm@53640
   285
      by (auto intro: open_Times)
hoelzl@50883
   286
  next
wenzelm@53255
   287
    fix S
wenzelm@53255
   288
    assume "open S" "x \<in> S"
wenzelm@53374
   289
    from open_prod_elim[OF this] guess a' b' . note a'b' = this
wenzelm@53374
   290
    moreover from a'b' A(4)[of a'] B(4)[of b']
hoelzl@50883
   291
    obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'" by auto
hoelzl@50883
   292
    ultimately show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"
hoelzl@50883
   293
      by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
hoelzl@50883
   294
  qed (simp add: A B)
hoelzl@50883
   295
qed
hoelzl@50883
   296
hoelzl@50881
   297
class second_countable_topology = topological_space +
wenzelm@53282
   298
  assumes ex_countable_subbasis:
wenzelm@53282
   299
    "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
hoelzl@51343
   300
begin
hoelzl@51343
   301
hoelzl@51343
   302
lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
hoelzl@51343
   303
proof -
wenzelm@53255
   304
  from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
wenzelm@53255
   305
    by blast
hoelzl@51343
   306
  let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
hoelzl@51343
   307
hoelzl@51343
   308
  show ?thesis
hoelzl@51343
   309
  proof (intro exI conjI)
hoelzl@51343
   310
    show "countable ?B"
hoelzl@51343
   311
      by (intro countable_image countable_Collect_finite_subset B)
wenzelm@53255
   312
    {
wenzelm@53255
   313
      fix S
wenzelm@53255
   314
      assume "open S"
hoelzl@51343
   315
      then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
hoelzl@51343
   316
        unfolding B
hoelzl@51343
   317
      proof induct
wenzelm@53255
   318
        case UNIV
wenzelm@53255
   319
        show ?case by (intro exI[of _ "{{}}"]) simp
hoelzl@51343
   320
      next
hoelzl@51343
   321
        case (Int a b)
hoelzl@51343
   322
        then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
hoelzl@51343
   323
          and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
hoelzl@51343
   324
          by blast
hoelzl@51343
   325
        show ?case
hoelzl@51343
   326
          unfolding x y Int_UN_distrib2
hoelzl@51343
   327
          by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
hoelzl@51343
   328
      next
hoelzl@51343
   329
        case (UN K)
hoelzl@51343
   330
        then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
hoelzl@51343
   331
        then guess k unfolding bchoice_iff ..
hoelzl@51343
   332
        then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
hoelzl@51343
   333
          by (intro exI[of _ "UNION K k"]) auto
hoelzl@51343
   334
      next
wenzelm@53255
   335
        case (Basis S)
wenzelm@53255
   336
        then show ?case
hoelzl@51343
   337
          by (intro exI[of _ "{{S}}"]) auto
hoelzl@51343
   338
      qed
hoelzl@51343
   339
      then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
hoelzl@51343
   340
        unfolding subset_image_iff by blast }
hoelzl@51343
   341
    then show "topological_basis ?B"
hoelzl@51343
   342
      unfolding topological_space_class.topological_basis_def
wenzelm@53282
   343
      by (safe intro!: topological_space_class.open_Inter)
hoelzl@51343
   344
         (simp_all add: B generate_topology.Basis subset_eq)
hoelzl@51343
   345
  qed
hoelzl@51343
   346
qed
hoelzl@51343
   347
hoelzl@51343
   348
end
hoelzl@51343
   349
hoelzl@51343
   350
sublocale second_countable_topology <
hoelzl@51343
   351
  countable_basis "SOME B. countable B \<and> topological_basis B"
hoelzl@51343
   352
  using someI_ex[OF ex_countable_basis]
hoelzl@51343
   353
  by unfold_locales safe
immler@50094
   354
hoelzl@50882
   355
instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
hoelzl@50882
   356
proof
hoelzl@50882
   357
  obtain A :: "'a set set" where "countable A" "topological_basis A"
hoelzl@50882
   358
    using ex_countable_basis by auto
hoelzl@50882
   359
  moreover
hoelzl@50882
   360
  obtain B :: "'b set set" where "countable B" "topological_basis B"
hoelzl@50882
   361
    using ex_countable_basis by auto
hoelzl@51343
   362
  ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
hoelzl@51343
   363
    by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
hoelzl@51343
   364
      topological_basis_imp_subbasis)
hoelzl@50882
   365
qed
hoelzl@50882
   366
hoelzl@50883
   367
instance second_countable_topology \<subseteq> first_countable_topology
hoelzl@50883
   368
proof
hoelzl@50883
   369
  fix x :: 'a
hoelzl@50883
   370
  def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"
hoelzl@50883
   371
  then have B: "countable B" "topological_basis B"
hoelzl@50883
   372
    using countable_basis is_basis
hoelzl@50883
   373
    by (auto simp: countable_basis is_basis)
wenzelm@53282
   374
  then show "\<exists>A::nat \<Rightarrow> 'a set.
wenzelm@53282
   375
    (\<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
   376
    by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
hoelzl@51473
   377
       (fastforce simp: topological_space_class.topological_basis_def)+
hoelzl@50883
   378
qed
hoelzl@50883
   379
wenzelm@53255
   380
immler@50087
   381
subsection {* Polish spaces *}
immler@50087
   382
immler@50087
   383
text {* Textbooks define Polish spaces as completely metrizable.
immler@50087
   384
  We assume the topology to be complete for a given metric. *}
immler@50087
   385
hoelzl@50881
   386
class polish_space = complete_space + second_countable_topology
immler@50087
   387
huffman@44517
   388
subsection {* General notion of a topology as a value *}
himmelma@33175
   389
wenzelm@53255
   390
definition "istopology L \<longleftrightarrow>
wenzelm@53255
   391
  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
   392
wenzelm@49834
   393
typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
himmelma@33175
   394
  morphisms "openin" "topology"
himmelma@33175
   395
  unfolding istopology_def by blast
himmelma@33175
   396
himmelma@33175
   397
lemma istopology_open_in[intro]: "istopology(openin U)"
himmelma@33175
   398
  using openin[of U] by blast
himmelma@33175
   399
himmelma@33175
   400
lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
huffman@44170
   401
  using topology_inverse[unfolded mem_Collect_eq] .
himmelma@33175
   402
himmelma@33175
   403
lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
himmelma@33175
   404
  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
himmelma@33175
   405
himmelma@33175
   406
lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
wenzelm@53255
   407
proof
wenzelm@53255
   408
  assume "T1 = T2"
wenzelm@53255
   409
  then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
wenzelm@53255
   410
next
wenzelm@53255
   411
  assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
wenzelm@53255
   412
  then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
wenzelm@53255
   413
  then have "topology (openin T1) = topology (openin T2)" by simp
wenzelm@53255
   414
  then show "T1 = T2" unfolding openin_inverse .
himmelma@33175
   415
qed
himmelma@33175
   416
himmelma@33175
   417
text{* Infer the "universe" from union of all sets in the topology. *}
himmelma@33175
   418
wenzelm@53640
   419
definition "topspace T = \<Union>{S. openin T S}"
himmelma@33175
   420
huffman@44210
   421
subsubsection {* Main properties of open sets *}
himmelma@33175
   422
himmelma@33175
   423
lemma openin_clauses:
himmelma@33175
   424
  fixes U :: "'a topology"
wenzelm@53282
   425
  shows
wenzelm@53282
   426
    "openin U {}"
wenzelm@53282
   427
    "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
wenzelm@53282
   428
    "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
wenzelm@53282
   429
  using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
himmelma@33175
   430
himmelma@33175
   431
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
himmelma@33175
   432
  unfolding topspace_def by blast
wenzelm@53255
   433
wenzelm@53255
   434
lemma openin_empty[simp]: "openin U {}"
wenzelm@53255
   435
  by (simp add: openin_clauses)
himmelma@33175
   436
himmelma@33175
   437
lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
huffman@36362
   438
  using openin_clauses by simp
huffman@36362
   439
huffman@36362
   440
lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
huffman@36362
   441
  using openin_clauses by simp
himmelma@33175
   442
himmelma@33175
   443
lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
himmelma@33175
   444
  using openin_Union[of "{S,T}" U] by auto
himmelma@33175
   445
wenzelm@53255
   446
lemma openin_topspace[intro, simp]: "openin U (topspace U)"
wenzelm@53255
   447
  by (simp add: openin_Union topspace_def)
himmelma@33175
   448
wenzelm@49711
   449
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
   450
  (is "?lhs \<longleftrightarrow> ?rhs")
huffman@36584
   451
proof
wenzelm@49711
   452
  assume ?lhs
wenzelm@49711
   453
  then show ?rhs by auto
huffman@36584
   454
next
huffman@36584
   455
  assume H: ?rhs
huffman@36584
   456
  let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
huffman@36584
   457
  have "openin U ?t" by (simp add: openin_Union)
huffman@36584
   458
  also have "?t = S" using H by auto
huffman@36584
   459
  finally show "openin U S" .
himmelma@33175
   460
qed
himmelma@33175
   461
wenzelm@49711
   462
huffman@44210
   463
subsubsection {* Closed sets *}
himmelma@33175
   464
himmelma@33175
   465
definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
himmelma@33175
   466
wenzelm@53255
   467
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
wenzelm@53255
   468
  by (metis closedin_def)
wenzelm@53255
   469
wenzelm@53255
   470
lemma closedin_empty[simp]: "closedin U {}"
wenzelm@53255
   471
  by (simp add: closedin_def)
wenzelm@53255
   472
wenzelm@53255
   473
lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
wenzelm@53255
   474
  by (simp add: closedin_def)
wenzelm@53255
   475
himmelma@33175
   476
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
himmelma@33175
   477
  by (auto simp add: Diff_Un closedin_def)
himmelma@33175
   478
wenzelm@53255
   479
lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}"
wenzelm@53255
   480
  by auto
wenzelm@53255
   481
wenzelm@53255
   482
lemma closedin_Inter[intro]:
wenzelm@53255
   483
  assumes Ke: "K \<noteq> {}"
wenzelm@53255
   484
    and Kc: "\<forall>S \<in>K. closedin U S"
wenzelm@53255
   485
  shows "closedin U (\<Inter> K)"
wenzelm@53255
   486
  using Ke Kc unfolding closedin_def Diff_Inter by auto
himmelma@33175
   487
himmelma@33175
   488
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
himmelma@33175
   489
  using closedin_Inter[of "{S,T}" U] by auto
himmelma@33175
   490
wenzelm@53255
   491
lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B"
wenzelm@53255
   492
  by blast
wenzelm@53255
   493
himmelma@33175
   494
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
himmelma@33175
   495
  apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
himmelma@33175
   496
  apply (metis openin_subset subset_eq)
himmelma@33175
   497
  done
himmelma@33175
   498
wenzelm@53255
   499
lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
himmelma@33175
   500
  by (simp add: openin_closedin_eq)
himmelma@33175
   501
wenzelm@53255
   502
lemma openin_diff[intro]:
wenzelm@53255
   503
  assumes oS: "openin U S"
wenzelm@53255
   504
    and cT: "closedin U T"
wenzelm@53255
   505
  shows "openin U (S - T)"
wenzelm@53255
   506
proof -
himmelma@33175
   507
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
himmelma@33175
   508
    by (auto simp add: topspace_def openin_subset)
wenzelm@53282
   509
  then show ?thesis using oS cT
wenzelm@53282
   510
    by (auto simp add: closedin_def)
himmelma@33175
   511
qed
himmelma@33175
   512
wenzelm@53255
   513
lemma closedin_diff[intro]:
wenzelm@53255
   514
  assumes oS: "closedin U S"
wenzelm@53255
   515
    and cT: "openin U T"
wenzelm@53255
   516
  shows "closedin U (S - T)"
wenzelm@53255
   517
proof -
wenzelm@53255
   518
  have "S - T = S \<inter> (topspace U - T)"
wenzelm@53282
   519
    using closedin_subset[of U S] oS cT by (auto simp add: topspace_def)
wenzelm@53255
   520
  then show ?thesis
wenzelm@53255
   521
    using oS cT by (auto simp add: openin_closedin_eq)
wenzelm@53255
   522
qed
wenzelm@53255
   523
himmelma@33175
   524
huffman@44210
   525
subsubsection {* Subspace topology *}
huffman@44170
   526
huffman@44170
   527
definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
huffman@44170
   528
huffman@44170
   529
lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
huffman@44170
   530
  (is "istopology ?L")
wenzelm@53255
   531
proof -
huffman@44170
   532
  have "?L {}" by blast
wenzelm@53255
   533
  {
wenzelm@53255
   534
    fix A B
wenzelm@53255
   535
    assume A: "?L A" and B: "?L B"
wenzelm@53255
   536
    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
   537
      by blast
wenzelm@53255
   538
    have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
wenzelm@53255
   539
      using Sa Sb by blast+
wenzelm@53255
   540
    then have "?L (A \<inter> B)" by blast
wenzelm@53255
   541
  }
himmelma@33175
   542
  moreover
wenzelm@53255
   543
  {
wenzelm@53282
   544
    fix K
wenzelm@53282
   545
    assume K: "K \<subseteq> Collect ?L"
huffman@44170
   546
    have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
nipkow@39302
   547
      apply (rule set_eqI)
himmelma@33175
   548
      apply (simp add: Ball_def image_iff)
wenzelm@53255
   549
      apply metis
wenzelm@53255
   550
      done
himmelma@33175
   551
    from K[unfolded th0 subset_image_iff]
wenzelm@53255
   552
    obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
wenzelm@53255
   553
      by blast
wenzelm@53255
   554
    have "\<Union>K = (\<Union>Sk) \<inter> V"
wenzelm@53255
   555
      using Sk by auto
wenzelm@53255
   556
    moreover have "openin U (\<Union> Sk)"
wenzelm@53255
   557
      using Sk by (auto simp add: subset_eq)
wenzelm@53255
   558
    ultimately have "?L (\<Union>K)" by blast
wenzelm@53255
   559
  }
huffman@44170
   560
  ultimately show ?thesis
huffman@44170
   561
    unfolding subset_eq mem_Collect_eq istopology_def by blast
himmelma@33175
   562
qed
himmelma@33175
   563
wenzelm@53255
   564
lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
himmelma@33175
   565
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
huffman@44170
   566
  by auto
himmelma@33175
   567
wenzelm@53255
   568
lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
himmelma@33175
   569
  by (auto simp add: topspace_def openin_subtopology)
himmelma@33175
   570
wenzelm@53255
   571
lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
himmelma@33175
   572
  unfolding closedin_def topspace_subtopology
himmelma@33175
   573
  apply (simp add: openin_subtopology)
himmelma@33175
   574
  apply (rule iffI)
himmelma@33175
   575
  apply clarify
himmelma@33175
   576
  apply (rule_tac x="topspace U - T" in exI)
wenzelm@53255
   577
  apply auto
wenzelm@53255
   578
  done
himmelma@33175
   579
himmelma@33175
   580
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
himmelma@33175
   581
  unfolding openin_subtopology
himmelma@33175
   582
  apply (rule iffI, clarify)
wenzelm@53255
   583
  apply (frule openin_subset[of U])
wenzelm@53255
   584
  apply blast
himmelma@33175
   585
  apply (rule exI[where x="topspace U"])
wenzelm@49711
   586
  apply auto
wenzelm@49711
   587
  done
wenzelm@49711
   588
wenzelm@49711
   589
lemma subtopology_superset:
wenzelm@49711
   590
  assumes UV: "topspace U \<subseteq> V"
himmelma@33175
   591
  shows "subtopology U V = U"
wenzelm@53255
   592
proof -
wenzelm@53255
   593
  {
wenzelm@53255
   594
    fix S
wenzelm@53255
   595
    {
wenzelm@53255
   596
      fix T
wenzelm@53255
   597
      assume T: "openin U T" "S = T \<inter> V"
wenzelm@53255
   598
      from T openin_subset[OF T(1)] UV have eq: "S = T"
wenzelm@53255
   599
        by blast
wenzelm@53255
   600
      have "openin U S"
wenzelm@53255
   601
        unfolding eq using T by blast
wenzelm@53255
   602
    }
himmelma@33175
   603
    moreover
wenzelm@53255
   604
    {
wenzelm@53255
   605
      assume S: "openin U S"
wenzelm@53255
   606
      then have "\<exists>T. openin U T \<and> S = T \<inter> V"
wenzelm@53255
   607
        using openin_subset[OF S] UV by auto
wenzelm@53255
   608
    }
wenzelm@53255
   609
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
wenzelm@53255
   610
      by blast
wenzelm@53255
   611
  }
wenzelm@53255
   612
  then show ?thesis
wenzelm@53255
   613
    unfolding topology_eq openin_subtopology by blast
himmelma@33175
   614
qed
himmelma@33175
   615
himmelma@33175
   616
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
himmelma@33175
   617
  by (simp add: subtopology_superset)
himmelma@33175
   618
himmelma@33175
   619
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
himmelma@33175
   620
  by (simp add: subtopology_superset)
himmelma@33175
   621
wenzelm@53255
   622
huffman@44210
   623
subsubsection {* The standard Euclidean topology *}
himmelma@33175
   624
wenzelm@53255
   625
definition euclidean :: "'a::topological_space topology"
wenzelm@53255
   626
  where "euclidean = topology open"
himmelma@33175
   627
himmelma@33175
   628
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
himmelma@33175
   629
  unfolding euclidean_def
himmelma@33175
   630
  apply (rule cong[where x=S and y=S])
himmelma@33175
   631
  apply (rule topology_inverse[symmetric])
himmelma@33175
   632
  apply (auto simp add: istopology_def)
huffman@44170
   633
  done
himmelma@33175
   634
himmelma@33175
   635
lemma topspace_euclidean: "topspace euclidean = UNIV"
himmelma@33175
   636
  apply (simp add: topspace_def)
nipkow@39302
   637
  apply (rule set_eqI)
wenzelm@53255
   638
  apply (auto simp add: open_openin[symmetric])
wenzelm@53255
   639
  done
himmelma@33175
   640
himmelma@33175
   641
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
himmelma@33175
   642
  by (simp add: topspace_euclidean topspace_subtopology)
himmelma@33175
   643
himmelma@33175
   644
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
himmelma@33175
   645
  by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
himmelma@33175
   646
himmelma@33175
   647
lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
himmelma@33175
   648
  by (simp add: open_openin openin_subopen[symmetric])
himmelma@33175
   649
huffman@44210
   650
text {* Basic "localization" results are handy for connectedness. *}
huffman@44210
   651
huffman@44210
   652
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
huffman@44210
   653
  by (auto simp add: openin_subtopology open_openin[symmetric])
huffman@44210
   654
huffman@44210
   655
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
huffman@44210
   656
  by (auto simp add: openin_open)
huffman@44210
   657
huffman@44210
   658
lemma open_openin_trans[trans]:
wenzelm@53255
   659
  "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
huffman@44210
   660
  by (metis Int_absorb1  openin_open_Int)
huffman@44210
   661
wenzelm@53255
   662
lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
huffman@44210
   663
  by (auto simp add: openin_open)
huffman@44210
   664
huffman@44210
   665
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
huffman@44210
   666
  by (simp add: closedin_subtopology closed_closedin Int_ac)
huffman@44210
   667
wenzelm@53291
   668
lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
huffman@44210
   669
  by (metis closedin_closed)
huffman@44210
   670
wenzelm@53282
   671
lemma closed_closedin_trans:
wenzelm@53282
   672
  "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
huffman@44210
   673
  apply (subgoal_tac "S \<inter> T = T" )
huffman@44210
   674
  apply auto
huffman@44210
   675
  apply (frule closedin_closed_Int[of T S])
wenzelm@52624
   676
  apply simp
wenzelm@52624
   677
  done
huffman@44210
   678
huffman@44210
   679
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
huffman@44210
   680
  by (auto simp add: closedin_closed)
huffman@44210
   681
huffman@44210
   682
lemma openin_euclidean_subtopology_iff:
huffman@44210
   683
  fixes S U :: "'a::metric_space set"
wenzelm@53255
   684
  shows "openin (subtopology euclidean U) S \<longleftrightarrow>
wenzelm@53255
   685
    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
   686
  (is "?lhs \<longleftrightarrow> ?rhs")
huffman@44210
   687
proof
wenzelm@53255
   688
  assume ?lhs
wenzelm@53282
   689
  then show ?rhs
wenzelm@53282
   690
    unfolding openin_open open_dist by blast
huffman@44210
   691
next
huffman@44210
   692
  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
   693
  have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
huffman@44210
   694
    unfolding T_def
huffman@44210
   695
    apply clarsimp
huffman@44210
   696
    apply (rule_tac x="d - dist x a" in exI)
huffman@44210
   697
    apply (clarsimp simp add: less_diff_eq)
huffman@44210
   698
    apply (erule rev_bexI)
huffman@44210
   699
    apply (rule_tac x=d in exI, clarify)
huffman@44210
   700
    apply (erule le_less_trans [OF dist_triangle])
huffman@44210
   701
    done
wenzelm@53282
   702
  assume ?rhs then have 2: "S = U \<inter> T"
huffman@44210
   703
    unfolding T_def
huffman@44210
   704
    apply auto
huffman@44210
   705
    apply (drule (1) bspec, erule rev_bexI)
huffman@44210
   706
    apply auto
huffman@44210
   707
    done
huffman@44210
   708
  from 1 2 show ?lhs
huffman@44210
   709
    unfolding openin_open open_dist by fast
huffman@44210
   710
qed
huffman@44210
   711
huffman@44210
   712
text {* These "transitivity" results are handy too *}
huffman@44210
   713
wenzelm@53255
   714
lemma openin_trans[trans]:
wenzelm@53255
   715
  "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
wenzelm@53255
   716
    openin (subtopology euclidean U) S"
huffman@44210
   717
  unfolding open_openin openin_open by blast
huffman@44210
   718
huffman@44210
   719
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
huffman@44210
   720
  by (auto simp add: openin_open intro: openin_trans)
huffman@44210
   721
huffman@44210
   722
lemma closedin_trans[trans]:
wenzelm@53255
   723
  "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
wenzelm@53255
   724
    closedin (subtopology euclidean U) S"
huffman@44210
   725
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
huffman@44210
   726
huffman@44210
   727
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
huffman@44210
   728
  by (auto simp add: closedin_closed intro: closedin_trans)
huffman@44210
   729
huffman@44210
   730
huffman@44210
   731
subsection {* Open and closed balls *}
himmelma@33175
   732
wenzelm@53255
   733
definition ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
wenzelm@53255
   734
  where "ball x e = {y. dist x y < e}"
wenzelm@53255
   735
wenzelm@53255
   736
definition cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
wenzelm@53255
   737
  where "cball x e = {y. dist x y \<le> e}"
himmelma@33175
   738
huffman@45776
   739
lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
huffman@45776
   740
  by (simp add: ball_def)
huffman@45776
   741
huffman@45776
   742
lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
huffman@45776
   743
  by (simp add: cball_def)
huffman@45776
   744
huffman@45776
   745
lemma mem_ball_0:
himmelma@33175
   746
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   747
  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
himmelma@33175
   748
  by (simp add: dist_norm)
himmelma@33175
   749
huffman@45776
   750
lemma mem_cball_0:
himmelma@33175
   751
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   752
  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
himmelma@33175
   753
  by (simp add: dist_norm)
himmelma@33175
   754
huffman@45776
   755
lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"
huffman@45776
   756
  by simp
huffman@45776
   757
huffman@45776
   758
lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
huffman@45776
   759
  by simp
huffman@45776
   760
wenzelm@53255
   761
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e"
wenzelm@53255
   762
  by (simp add: subset_eq)
wenzelm@53255
   763
wenzelm@53282
   764
lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
wenzelm@53255
   765
  by (simp add: subset_eq)
wenzelm@53255
   766
wenzelm@53282
   767
lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
wenzelm@53255
   768
  by (simp add: subset_eq)
wenzelm@53255
   769
himmelma@33175
   770
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
nipkow@39302
   771
  by (simp add: set_eq_iff) arith
himmelma@33175
   772
himmelma@33175
   773
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
nipkow@39302
   774
  by (simp add: set_eq_iff)
himmelma@33175
   775
wenzelm@53255
   776
lemma diff_less_iff:
wenzelm@53255
   777
  "(a::real) - b > 0 \<longleftrightarrow> a > b"
himmelma@33175
   778
  "(a::real) - b < 0 \<longleftrightarrow> a < b"
wenzelm@53255
   779
  "a - b < c \<longleftrightarrow> a < c + b" "a - b > c \<longleftrightarrow> a > c + b"
wenzelm@53255
   780
  by arith+
wenzelm@53255
   781
wenzelm@53255
   782
lemma diff_le_iff:
wenzelm@53255
   783
  "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b"
wenzelm@53255
   784
  "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
wenzelm@53255
   785
  "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
wenzelm@53255
   786
  "a - b \<ge> c \<longleftrightarrow> a \<ge> c + b"
wenzelm@53255
   787
  by arith+
himmelma@33175
   788
huffman@54070
   789
lemma open_vimage: (* TODO: move to Topological_Spaces.thy *)
huffman@54070
   790
  assumes "open s" and "continuous_on UNIV f"
huffman@54070
   791
  shows "open (vimage f s)"
huffman@54070
   792
  using assms unfolding continuous_on_open_vimage [OF open_UNIV]
huffman@54070
   793
  by simp
huffman@54070
   794
huffman@54070
   795
lemma open_ball [intro, simp]: "open (ball x e)"
huffman@54070
   796
proof -
huffman@54070
   797
  have "open (dist x -` {..<e})"
huffman@54070
   798
    by (intro open_vimage open_lessThan continuous_on_intros)
huffman@54070
   799
  also have "dist x -` {..<e} = ball x e"
huffman@54070
   800
    by auto
huffman@54070
   801
  finally show ?thesis .
huffman@54070
   802
qed
himmelma@33175
   803
himmelma@33175
   804
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   805
  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
himmelma@33175
   806
hoelzl@33714
   807
lemma openE[elim?]:
wenzelm@53282
   808
  assumes "open S" "x\<in>S"
hoelzl@33714
   809
  obtains e where "e>0" "ball x e \<subseteq> S"
hoelzl@33714
   810
  using assms unfolding open_contains_ball by auto
hoelzl@33714
   811
himmelma@33175
   812
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   813
  by (metis open_contains_ball subset_eq centre_in_ball)
himmelma@33175
   814
himmelma@33175
   815
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
nipkow@39302
   816
  unfolding mem_ball set_eq_iff
himmelma@33175
   817
  apply (simp add: not_less)
wenzelm@52624
   818
  apply (metis zero_le_dist order_trans dist_self)
wenzelm@52624
   819
  done
himmelma@33175
   820
wenzelm@53291
   821
lemma ball_empty[intro]: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
himmelma@33175
   822
hoelzl@50526
   823
lemma euclidean_dist_l2:
hoelzl@50526
   824
  fixes x y :: "'a :: euclidean_space"
hoelzl@50526
   825
  shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
hoelzl@50526
   826
  unfolding dist_norm norm_eq_sqrt_inner setL2_def
hoelzl@50526
   827
  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
hoelzl@50526
   828
immler@54775
   829
definition (in euclidean_space) eucl_less (infix "<e" 50)
immler@54775
   830
  where "eucl_less a b \<longleftrightarrow> (\<forall>i\<in>Basis. a \<bullet> i < b \<bullet> i)"
immler@54775
   831
immler@54775
   832
definition box_eucl_less: "box a b = {x. a <e x \<and> x <e b}"
immler@54775
   833
immler@54775
   834
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
   835
  and in_box_eucl_less: "x \<in> box a b \<longleftrightarrow> a <e x \<and> x <e b"
immler@54775
   836
  by (auto simp: box_eucl_less eucl_less_def)
hoelzl@50526
   837
immler@50087
   838
lemma rational_boxes:
hoelzl@50526
   839
  fixes x :: "'a\<Colon>euclidean_space"
wenzelm@53291
   840
  assumes "e > 0"
hoelzl@50526
   841
  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
   842
proof -
immler@50087
   843
  def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
wenzelm@53291
   844
  then have e: "e' > 0"
wenzelm@53255
   845
    using assms by (auto intro!: divide_pos_pos simp: DIM_positive)
hoelzl@50526
   846
  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
   847
  proof
wenzelm@53255
   848
    fix i
wenzelm@53255
   849
    from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e
wenzelm@53255
   850
    show "?th i" by auto
immler@50087
   851
  qed
immler@50087
   852
  from choice[OF this] guess a .. note a = this
hoelzl@50526
   853
  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
   854
  proof
wenzelm@53255
   855
    fix i
wenzelm@53255
   856
    from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e
wenzelm@53255
   857
    show "?th i" by auto
immler@50087
   858
  qed
immler@50087
   859
  from choice[OF this] guess b .. note b = this
hoelzl@50526
   860
  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
hoelzl@50526
   861
  show ?thesis
hoelzl@50526
   862
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
wenzelm@53255
   863
    fix y :: 'a
wenzelm@53255
   864
    assume *: "y \<in> box ?a ?b"
wenzelm@53015
   865
    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<^sup>2)"
immler@50087
   866
      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
hoelzl@50526
   867
    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
immler@50087
   868
    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
wenzelm@53255
   869
      fix i :: "'a"
wenzelm@53255
   870
      assume i: "i \<in> Basis"
wenzelm@53255
   871
      have "a i < y\<bullet>i \<and> y\<bullet>i < b i"
wenzelm@53255
   872
        using * i by (auto simp: box_def)
wenzelm@53255
   873
      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'"
wenzelm@53255
   874
        using a by auto
wenzelm@53255
   875
      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'"
wenzelm@53255
   876
        using b by auto
wenzelm@53255
   877
      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'"
wenzelm@53255
   878
        by auto
hoelzl@50526
   879
      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
immler@50087
   880
        unfolding e'_def by (auto simp: dist_real_def)
wenzelm@53015
   881
      then have "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < (e/sqrt (real (DIM('a))))\<^sup>2"
immler@50087
   882
        by (rule power_strict_mono) auto
wenzelm@53015
   883
      then show "(dist (x \<bullet> i) (y \<bullet> i))\<^sup>2 < e\<^sup>2 / real DIM('a)"
immler@50087
   884
        by (simp add: power_divide)
immler@50087
   885
    qed auto
wenzelm@53255
   886
    also have "\<dots> = e"
wenzelm@53255
   887
      using `0 < e` by (simp add: real_eq_of_nat)
wenzelm@53255
   888
    finally show "y \<in> ball x e"
wenzelm@53255
   889
      by (auto simp: ball_def)
hoelzl@50526
   890
  qed (insert a b, auto simp: box_def)
hoelzl@50526
   891
qed
immler@51103
   892
hoelzl@50526
   893
lemma open_UNION_box:
hoelzl@50526
   894
  fixes M :: "'a\<Colon>euclidean_space set"
wenzelm@53282
   895
  assumes "open M"
hoelzl@50526
   896
  defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
hoelzl@50526
   897
  defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
wenzelm@53015
   898
  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^sub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
hoelzl@50526
   899
  shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
wenzelm@52624
   900
proof -
wenzelm@52624
   901
  {
wenzelm@52624
   902
    fix x assume "x \<in> M"
wenzelm@52624
   903
    obtain e where e: "e > 0" "ball x e \<subseteq> M"
wenzelm@52624
   904
      using openE[OF `open M` `x \<in> M`] by auto
wenzelm@53282
   905
    moreover obtain a b where ab:
wenzelm@53282
   906
      "x \<in> box a b"
wenzelm@53282
   907
      "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>"
wenzelm@53282
   908
      "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>"
wenzelm@53282
   909
      "box a b \<subseteq> ball x e"
wenzelm@52624
   910
      using rational_boxes[OF e(1)] by metis
wenzelm@52624
   911
    ultimately have "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"
wenzelm@52624
   912
       by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
wenzelm@52624
   913
          (auto simp: euclidean_representation I_def a'_def b'_def)
wenzelm@52624
   914
  }
wenzelm@52624
   915
  then show ?thesis by (auto simp: I_def)
wenzelm@52624
   916
qed
wenzelm@52624
   917
himmelma@33175
   918
himmelma@33175
   919
subsection{* Connectedness *}
himmelma@33175
   920
himmelma@33175
   921
lemma connected_local:
wenzelm@53255
   922
 "connected S \<longleftrightarrow>
wenzelm@53255
   923
  \<not> (\<exists>e1 e2.
wenzelm@53255
   924
      openin (subtopology euclidean S) e1 \<and>
wenzelm@53255
   925
      openin (subtopology euclidean S) e2 \<and>
wenzelm@53255
   926
      S \<subseteq> e1 \<union> e2 \<and>
wenzelm@53255
   927
      e1 \<inter> e2 = {} \<and>
wenzelm@53255
   928
      e1 \<noteq> {} \<and>
wenzelm@53255
   929
      e2 \<noteq> {})"
wenzelm@53282
   930
  unfolding connected_def openin_open
wenzelm@53282
   931
  apply safe
wenzelm@53282
   932
  apply blast+
wenzelm@53282
   933
  done
himmelma@33175
   934
huffman@34105
   935
lemma exists_diff:
huffman@34105
   936
  fixes P :: "'a set \<Rightarrow> bool"
huffman@34105
   937
  shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53255
   938
proof -
wenzelm@53255
   939
  {
wenzelm@53255
   940
    assume "?lhs"
wenzelm@53255
   941
    then have ?rhs by blast
wenzelm@53255
   942
  }
himmelma@33175
   943
  moreover
wenzelm@53255
   944
  {
wenzelm@53255
   945
    fix S
wenzelm@53255
   946
    assume H: "P S"
huffman@34105
   947
    have "S = - (- S)" by auto
wenzelm@53255
   948
    with H have "P (- (- S))" by metis
wenzelm@53255
   949
  }
himmelma@33175
   950
  ultimately show ?thesis by metis
himmelma@33175
   951
qed
himmelma@33175
   952
himmelma@33175
   953
lemma connected_clopen: "connected S \<longleftrightarrow>
wenzelm@53255
   954
  (\<forall>T. openin (subtopology euclidean S) T \<and>
wenzelm@53255
   955
     closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
wenzelm@53255
   956
proof -
wenzelm@53255
   957
  have "\<not> connected S \<longleftrightarrow>
wenzelm@53255
   958
    (\<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
   959
    unfolding connected_def openin_open closedin_closed
wenzelm@52624
   960
    apply (subst exists_diff)
wenzelm@52624
   961
    apply blast
wenzelm@52624
   962
    done
wenzelm@53282
   963
  then have th0: "connected S \<longleftrightarrow>
wenzelm@53255
   964
    \<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
   965
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")
wenzelm@52624
   966
    apply (simp add: closed_def)
wenzelm@52624
   967
    apply metis
wenzelm@52624
   968
    done
himmelma@33175
   969
  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
   970
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
himmelma@33175
   971
    unfolding connected_def openin_open closedin_closed by auto
wenzelm@53255
   972
  {
wenzelm@53255
   973
    fix e2
wenzelm@53255
   974
    {
wenzelm@53255
   975
      fix e1
wenzelm@53282
   976
      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
   977
        by auto
wenzelm@53255
   978
    }
wenzelm@53255
   979
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
wenzelm@53255
   980
      by metis
wenzelm@53255
   981
  }
wenzelm@53255
   982
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
wenzelm@53255
   983
    by blast
wenzelm@53255
   984
  then show ?thesis
wenzelm@53255
   985
    unfolding th0 th1 by simp
himmelma@33175
   986
qed
himmelma@33175
   987
huffman@44210
   988
himmelma@33175
   989
subsection{* Limit points *}
himmelma@33175
   990
wenzelm@53282
   991
definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool"  (infixr "islimpt" 60)
wenzelm@53255
   992
  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
   993
himmelma@33175
   994
lemma islimptI:
himmelma@33175
   995
  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
himmelma@33175
   996
  shows "x islimpt S"
himmelma@33175
   997
  using assms unfolding islimpt_def by auto
himmelma@33175
   998
himmelma@33175
   999
lemma islimptE:
himmelma@33175
  1000
  assumes "x islimpt S" and "x \<in> T" and "open T"
himmelma@33175
  1001
  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
himmelma@33175
  1002
  using assms unfolding islimpt_def by auto
himmelma@33175
  1003
huffman@44584
  1004
lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
huffman@44584
  1005
  unfolding islimpt_def eventually_at_topological by auto
huffman@44584
  1006
wenzelm@53255
  1007
lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
huffman@44584
  1008
  unfolding islimpt_def by fast
himmelma@33175
  1009
himmelma@33175
  1010
lemma islimpt_approachable:
himmelma@33175
  1011
  fixes x :: "'a::metric_space"
himmelma@33175
  1012
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
huffman@44584
  1013
  unfolding islimpt_iff_eventually eventually_at by fast
himmelma@33175
  1014
himmelma@33175
  1015
lemma islimpt_approachable_le:
himmelma@33175
  1016
  fixes x :: "'a::metric_space"
wenzelm@53640
  1017
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
himmelma@33175
  1018
  unfolding islimpt_approachable
huffman@44584
  1019
  using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
huffman@44584
  1020
    THEN arg_cong [where f=Not]]
huffman@44584
  1021
  by (simp add: Bex_def conj_commute conj_left_commute)
himmelma@33175
  1022
huffman@44571
  1023
lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
huffman@44571
  1024
  unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
huffman@44571
  1025
hoelzl@51351
  1026
lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
hoelzl@51351
  1027
  unfolding islimpt_def by blast
hoelzl@51351
  1028
huffman@44210
  1029
text {* A perfect space has no isolated points. *}
huffman@44210
  1030
huffman@44571
  1031
lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
huffman@44571
  1032
  unfolding islimpt_UNIV_iff by (rule not_open_singleton)
himmelma@33175
  1033
himmelma@33175
  1034
lemma perfect_choose_dist:
huffman@44072
  1035
  fixes x :: "'a::{perfect_space, metric_space}"
himmelma@33175
  1036
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
wenzelm@53255
  1037
  using islimpt_UNIV [of x]
wenzelm@53255
  1038
  by (simp add: islimpt_approachable)
himmelma@33175
  1039
himmelma@33175
  1040
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
himmelma@33175
  1041
  unfolding closed_def
himmelma@33175
  1042
  apply (subst open_subopen)
huffman@34105
  1043
  apply (simp add: islimpt_def subset_eq)
wenzelm@52624
  1044
  apply (metis ComplE ComplI)
wenzelm@52624
  1045
  done
himmelma@33175
  1046
himmelma@33175
  1047
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
himmelma@33175
  1048
  unfolding islimpt_def by auto
himmelma@33175
  1049
himmelma@33175
  1050
lemma finite_set_avoid:
himmelma@33175
  1051
  fixes a :: "'a::metric_space"
wenzelm@53255
  1052
  assumes fS: "finite S"
wenzelm@53640
  1053
  shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
wenzelm@53255
  1054
proof (induct rule: finite_induct[OF fS])
wenzelm@53255
  1055
  case 1
wenzelm@53255
  1056
  then show ?case by (auto intro: zero_less_one)
himmelma@33175
  1057
next
himmelma@33175
  1058
  case (2 x F)
wenzelm@53255
  1059
  from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x"
wenzelm@53255
  1060
    by blast
wenzelm@53255
  1061
  show ?case
wenzelm@53255
  1062
  proof (cases "x = a")
wenzelm@53255
  1063
    case True
wenzelm@53255
  1064
    then show ?thesis using d by auto
wenzelm@53255
  1065
  next
wenzelm@53255
  1066
    case False
himmelma@33175
  1067
    let ?d = "min d (dist a x)"
wenzelm@53255
  1068
    have dp: "?d > 0"
wenzelm@53255
  1069
      using False d(1) using dist_nz by auto
wenzelm@53255
  1070
    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x"
wenzelm@53255
  1071
      by auto
wenzelm@53255
  1072
    with dp False show ?thesis
wenzelm@53255
  1073
      by (auto intro!: exI[where x="?d"])
wenzelm@53255
  1074
  qed
himmelma@33175
  1075
qed
himmelma@33175
  1076
himmelma@33175
  1077
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
huffman@50897
  1078
  by (simp add: islimpt_iff_eventually eventually_conj_iff)
himmelma@33175
  1079
himmelma@33175
  1080
lemma discrete_imp_closed:
himmelma@33175
  1081
  fixes S :: "'a::metric_space set"
wenzelm@53255
  1082
  assumes e: "0 < e"
wenzelm@53255
  1083
    and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
himmelma@33175
  1084
  shows "closed S"
wenzelm@53255
  1085
proof -
wenzelm@53255
  1086
  {
wenzelm@53255
  1087
    fix x
wenzelm@53255
  1088
    assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
himmelma@33175
  1089
    from e have e2: "e/2 > 0" by arith
wenzelm@53282
  1090
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
wenzelm@53255
  1091
      by blast
himmelma@33175
  1092
    let ?m = "min (e/2) (dist x y) "
wenzelm@53255
  1093
    from e2 y(2) have mp: "?m > 0"
wenzelm@53291
  1094
      by (simp add: dist_nz[symmetric])
wenzelm@53282
  1095
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
wenzelm@53255
  1096
      by blast
himmelma@33175
  1097
    have th: "dist z y < e" using z y
himmelma@33175
  1098
      by (intro dist_triangle_lt [where z=x], simp)
himmelma@33175
  1099
    from d[rule_format, OF y(1) z(1) th] y z
himmelma@33175
  1100
    have False by (auto simp add: dist_commute)}
wenzelm@53255
  1101
  then show ?thesis
wenzelm@53255
  1102
    by (metis islimpt_approachable closed_limpt [where 'a='a])
himmelma@33175
  1103
qed
himmelma@33175
  1104
huffman@44210
  1105
huffman@44210
  1106
subsection {* Interior of a Set *}
huffman@44210
  1107
huffman@44519
  1108
definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
huffman@44519
  1109
huffman@44519
  1110
lemma interiorI [intro?]:
huffman@44519
  1111
  assumes "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
  1112
  shows "x \<in> interior S"
huffman@44519
  1113
  using assms unfolding interior_def by fast
huffman@44519
  1114
huffman@44519
  1115
lemma interiorE [elim?]:
huffman@44519
  1116
  assumes "x \<in> interior S"
huffman@44519
  1117
  obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
  1118
  using assms unfolding interior_def by fast
huffman@44519
  1119
huffman@44519
  1120
lemma open_interior [simp, intro]: "open (interior S)"
huffman@44519
  1121
  by (simp add: interior_def open_Union)
huffman@44519
  1122
huffman@44519
  1123
lemma interior_subset: "interior S \<subseteq> S"
huffman@44519
  1124
  by (auto simp add: interior_def)
huffman@44519
  1125
huffman@44519
  1126
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
huffman@44519
  1127
  by (auto simp add: interior_def)
huffman@44519
  1128
huffman@44519
  1129
lemma interior_open: "open S \<Longrightarrow> interior S = S"
huffman@44519
  1130
  by (intro equalityI interior_subset interior_maximal subset_refl)
himmelma@33175
  1131
himmelma@33175
  1132
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
huffman@44519
  1133
  by (metis open_interior interior_open)
huffman@44519
  1134
huffman@44519
  1135
lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
himmelma@33175
  1136
  by (metis interior_maximal interior_subset subset_trans)
himmelma@33175
  1137
huffman@44519
  1138
lemma interior_empty [simp]: "interior {} = {}"
huffman@44519
  1139
  using open_empty by (rule interior_open)
huffman@44519
  1140
huffman@44522
  1141
lemma interior_UNIV [simp]: "interior UNIV = UNIV"
huffman@44522
  1142
  using open_UNIV by (rule interior_open)
huffman@44522
  1143
huffman@44519
  1144
lemma interior_interior [simp]: "interior (interior S) = interior S"
huffman@44519
  1145
  using open_interior by (rule interior_open)
huffman@44519
  1146
huffman@44522
  1147
lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
huffman@44522
  1148
  by (auto simp add: interior_def)
huffman@44519
  1149
huffman@44519
  1150
lemma interior_unique:
huffman@44519
  1151
  assumes "T \<subseteq> S" and "open T"
huffman@44519
  1152
  assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
huffman@44519
  1153
  shows "interior S = T"
huffman@44519
  1154
  by (intro equalityI assms interior_subset open_interior interior_maximal)
huffman@44519
  1155
huffman@44519
  1156
lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
huffman@44522
  1157
  by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
huffman@44519
  1158
    Int_lower2 interior_maximal interior_subset open_Int open_interior)
huffman@44519
  1159
huffman@44519
  1160
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
huffman@44519
  1161
  using open_contains_ball_eq [where S="interior S"]
huffman@44519
  1162
  by (simp add: open_subset_interior)
himmelma@33175
  1163
himmelma@33175
  1164
lemma interior_limit_point [intro]:
himmelma@33175
  1165
  fixes x :: "'a::perfect_space"
wenzelm@53255
  1166
  assumes x: "x \<in> interior S"
wenzelm@53255
  1167
  shows "x islimpt S"
huffman@44072
  1168
  using x islimpt_UNIV [of x]
huffman@44072
  1169
  unfolding interior_def islimpt_def
huffman@44072
  1170
  apply (clarsimp, rename_tac T T')
huffman@44072
  1171
  apply (drule_tac x="T \<inter> T'" in spec)
huffman@44072
  1172
  apply (auto simp add: open_Int)
huffman@44072
  1173
  done
himmelma@33175
  1174
himmelma@33175
  1175
lemma interior_closed_Un_empty_interior:
wenzelm@53255
  1176
  assumes cS: "closed S"
wenzelm@53255
  1177
    and iT: "interior T = {}"
huffman@44519
  1178
  shows "interior (S \<union> T) = interior S"
himmelma@33175
  1179
proof
huffman@44519
  1180
  show "interior S \<subseteq> interior (S \<union> T)"
wenzelm@53255
  1181
    by (rule interior_mono) (rule Un_upper1)
himmelma@33175
  1182
  show "interior (S \<union> T) \<subseteq> interior S"
himmelma@33175
  1183
  proof
wenzelm@53255
  1184
    fix x
wenzelm@53255
  1185
    assume "x \<in> interior (S \<union> T)"
huffman@44519
  1186
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
himmelma@33175
  1187
    show "x \<in> interior S"
himmelma@33175
  1188
    proof (rule ccontr)
himmelma@33175
  1189
      assume "x \<notin> interior S"
himmelma@33175
  1190
      with `x \<in> R` `open R` obtain y where "y \<in> R - S"
huffman@44519
  1191
        unfolding interior_def by fast
wenzelm@53282
  1192
      from `open R` `closed S` have "open (R - S)"
wenzelm@53282
  1193
        by (rule open_Diff)
wenzelm@53282
  1194
      from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T"
wenzelm@53282
  1195
        by fast
wenzelm@53282
  1196
      from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}` show False
wenzelm@53282
  1197
        unfolding interior_def by fast
himmelma@33175
  1198
    qed
himmelma@33175
  1199
  qed
himmelma@33175
  1200
qed
himmelma@33175
  1201
huffman@44365
  1202
lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
huffman@44365
  1203
proof (rule interior_unique)
huffman@44365
  1204
  show "interior A \<times> interior B \<subseteq> A \<times> B"
huffman@44365
  1205
    by (intro Sigma_mono interior_subset)
huffman@44365
  1206
  show "open (interior A \<times> interior B)"
huffman@44365
  1207
    by (intro open_Times open_interior)
wenzelm@53255
  1208
  fix T
wenzelm@53255
  1209
  assume "T \<subseteq> A \<times> B" and "open T"
wenzelm@53255
  1210
  then show "T \<subseteq> interior A \<times> interior B"
wenzelm@53282
  1211
  proof safe
wenzelm@53255
  1212
    fix x y
wenzelm@53255
  1213
    assume "(x, y) \<in> T"
huffman@44519
  1214
    then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
huffman@44519
  1215
      using `open T` unfolding open_prod_def by fast
wenzelm@53255
  1216
    then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
huffman@44519
  1217
      using `T \<subseteq> A \<times> B` by auto
wenzelm@53255
  1218
    then show "x \<in> interior A" and "y \<in> interior B"
huffman@44519
  1219
      by (auto intro: interiorI)
huffman@44519
  1220
  qed
huffman@44365
  1221
qed
huffman@44365
  1222
himmelma@33175
  1223
huffman@44210
  1224
subsection {* Closure of a Set *}
himmelma@33175
  1225
himmelma@33175
  1226
definition "closure S = S \<union> {x | x. x islimpt S}"
himmelma@33175
  1227
huffman@44518
  1228
lemma interior_closure: "interior S = - (closure (- S))"
huffman@44518
  1229
  unfolding interior_def closure_def islimpt_def by auto
huffman@44518
  1230
huffman@34105
  1231
lemma closure_interior: "closure S = - interior (- S)"
huffman@44518
  1232
  unfolding interior_closure by simp
himmelma@33175
  1233
himmelma@33175
  1234
lemma closed_closure[simp, intro]: "closed (closure S)"
huffman@44518
  1235
  unfolding closure_interior by (simp add: closed_Compl)
huffman@44518
  1236
huffman@44518
  1237
lemma closure_subset: "S \<subseteq> closure S"
huffman@44518
  1238
  unfolding closure_def by simp
himmelma@33175
  1239
himmelma@33175
  1240
lemma closure_hull: "closure S = closed hull S"
huffman@44519
  1241
  unfolding hull_def closure_interior interior_def by auto
himmelma@33175
  1242
himmelma@33175
  1243
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
huffman@44519
  1244
  unfolding closure_hull using closed_Inter by (rule hull_eq)
huffman@44519
  1245
huffman@44519
  1246
lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
huffman@44519
  1247
  unfolding closure_eq .
huffman@44519
  1248
huffman@44519
  1249
lemma closure_closure [simp]: "closure (closure S) = closure S"
huffman@44518
  1250
  unfolding closure_hull by (rule hull_hull)
himmelma@33175
  1251
huffman@44522
  1252
lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
huffman@44518
  1253
  unfolding closure_hull by (rule hull_mono)
himmelma@33175
  1254
huffman@44519
  1255
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
huffman@44518
  1256
  unfolding closure_hull by (rule hull_minimal)
himmelma@33175
  1257
huffman@44519
  1258
lemma closure_unique:
wenzelm@53255
  1259
  assumes "S \<subseteq> T"
wenzelm@53255
  1260
    and "closed T"
wenzelm@53255
  1261
    and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
huffman@44519
  1262
  shows "closure S = T"
huffman@44519
  1263
  using assms unfolding closure_hull by (rule hull_unique)
huffman@44519
  1264
huffman@44519
  1265
lemma closure_empty [simp]: "closure {} = {}"
huffman@44518
  1266
  using closed_empty by (rule closure_closed)
himmelma@33175
  1267
huffman@44522
  1268
lemma closure_UNIV [simp]: "closure UNIV = UNIV"
huffman@44518
  1269
  using closed_UNIV by (rule closure_closed)
huffman@44518
  1270
huffman@44518
  1271
lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
huffman@44518
  1272
  unfolding closure_interior by simp
himmelma@33175
  1273
himmelma@33175
  1274
lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
himmelma@33175
  1275
  using closure_empty closure_subset[of S]
himmelma@33175
  1276
  by blast
himmelma@33175
  1277
himmelma@33175
  1278
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
  1279
  using closure_eq[of S] closure_subset[of S]
himmelma@33175
  1280
  by simp
himmelma@33175
  1281
himmelma@33175
  1282
lemma open_inter_closure_eq_empty:
himmelma@33175
  1283
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
huffman@34105
  1284
  using open_subset_interior[of S "- T"]
huffman@34105
  1285
  using interior_subset[of "- T"]
himmelma@33175
  1286
  unfolding closure_interior
himmelma@33175
  1287
  by auto
himmelma@33175
  1288
himmelma@33175
  1289
lemma open_inter_closure_subset:
himmelma@33175
  1290
  "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
himmelma@33175
  1291
proof
himmelma@33175
  1292
  fix x
himmelma@33175
  1293
  assume as: "open S" "x \<in> S \<inter> closure T"
wenzelm@53255
  1294
  {
wenzelm@53282
  1295
    assume *: "x islimpt T"
himmelma@33175
  1296
    have "x islimpt (S \<inter> T)"
himmelma@33175
  1297
    proof (rule islimptI)
himmelma@33175
  1298
      fix A
himmelma@33175
  1299
      assume "x \<in> A" "open A"
himmelma@33175
  1300
      with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
himmelma@33175
  1301
        by (simp_all add: open_Int)
himmelma@33175
  1302
      with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
himmelma@33175
  1303
        by (rule islimptE)
wenzelm@53255
  1304
      then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
himmelma@33175
  1305
        by simp_all
wenzelm@53255
  1306
      then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
himmelma@33175
  1307
    qed
himmelma@33175
  1308
  }
himmelma@33175
  1309
  then show "x \<in> closure (S \<inter> T)" using as
himmelma@33175
  1310
    unfolding closure_def
himmelma@33175
  1311
    by blast
himmelma@33175
  1312
qed
himmelma@33175
  1313
huffman@44519
  1314
lemma closure_complement: "closure (- S) = - interior S"
huffman@44518
  1315
  unfolding closure_interior by simp
himmelma@33175
  1316
huffman@44519
  1317
lemma interior_complement: "interior (- S) = - closure S"
huffman@44518
  1318
  unfolding closure_interior by simp
himmelma@33175
  1319
huffman@44365
  1320
lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
huffman@44519
  1321
proof (rule closure_unique)
huffman@44365
  1322
  show "A \<times> B \<subseteq> closure A \<times> closure B"
huffman@44365
  1323
    by (intro Sigma_mono closure_subset)
huffman@44365
  1324
  show "closed (closure A \<times> closure B)"
huffman@44365
  1325
    by (intro closed_Times closed_closure)
wenzelm@53282
  1326
  fix T
wenzelm@53282
  1327
  assume "A \<times> B \<subseteq> T" and "closed T"
wenzelm@53282
  1328
  then show "closure A \<times> closure B \<subseteq> T"
huffman@44365
  1329
    apply (simp add: closed_def open_prod_def, clarify)
huffman@44365
  1330
    apply (rule ccontr)
huffman@44365
  1331
    apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
huffman@44365
  1332
    apply (simp add: closure_interior interior_def)
huffman@44365
  1333
    apply (drule_tac x=C in spec)
huffman@44365
  1334
    apply (drule_tac x=D in spec)
huffman@44365
  1335
    apply auto
huffman@44365
  1336
    done
huffman@44365
  1337
qed
huffman@44365
  1338
hoelzl@51351
  1339
lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"
hoelzl@51351
  1340
  unfolding closure_def using islimpt_punctured by blast
hoelzl@51351
  1341
hoelzl@51351
  1342
huffman@44210
  1343
subsection {* Frontier (aka boundary) *}
himmelma@33175
  1344
himmelma@33175
  1345
definition "frontier S = closure S - interior S"
himmelma@33175
  1346
wenzelm@53255
  1347
lemma frontier_closed: "closed (frontier S)"
himmelma@33175
  1348
  by (simp add: frontier_def closed_Diff)
himmelma@33175
  1349
huffman@34105
  1350
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
himmelma@33175
  1351
  by (auto simp add: frontier_def interior_closure)
himmelma@33175
  1352
himmelma@33175
  1353
lemma frontier_straddle:
himmelma@33175
  1354
  fixes a :: "'a::metric_space"
huffman@44909
  1355
  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
  1356
  unfolding frontier_def closure_interior
huffman@44909
  1357
  by (auto simp add: mem_interior subset_eq ball_def)
himmelma@33175
  1358
himmelma@33175
  1359
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
himmelma@33175
  1360
  by (metis frontier_def closure_closed Diff_subset)
himmelma@33175
  1361
hoelzl@34964
  1362
lemma frontier_empty[simp]: "frontier {} = {}"
huffman@36362
  1363
  by (simp add: frontier_def)
himmelma@33175
  1364
himmelma@33175
  1365
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
  1366
proof-
wenzelm@53255
  1367
  {
wenzelm@53255
  1368
    assume "frontier S \<subseteq> S"
wenzelm@53255
  1369
    then have "closure S \<subseteq> S"
wenzelm@53255
  1370
      using interior_subset unfolding frontier_def by auto
wenzelm@53255
  1371
    then have "closed S"
wenzelm@53255
  1372
      using closure_subset_eq by auto
himmelma@33175
  1373
  }
wenzelm@53255
  1374
  then show ?thesis using frontier_subset_closed[of S] ..
himmelma@33175
  1375
qed
himmelma@33175
  1376
huffman@34105
  1377
lemma frontier_complement: "frontier(- S) = frontier S"
himmelma@33175
  1378
  by (auto simp add: frontier_def closure_complement interior_complement)
himmelma@33175
  1379
himmelma@33175
  1380
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
huffman@34105
  1381
  using frontier_complement frontier_subset_eq[of "- S"]
huffman@34105
  1382
  unfolding open_closed by auto
himmelma@33175
  1383
huffman@44081
  1384
subsection {* Filters and the ``eventually true'' quantifier *}
huffman@44081
  1385
wenzelm@52624
  1386
definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
wenzelm@52624
  1387
    (infixr "indirection" 70)
wenzelm@52624
  1388
  where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
himmelma@33175
  1389
huffman@36437
  1390
text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
himmelma@33175
  1391
wenzelm@52624
  1392
lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
himmelma@33175
  1393
proof
himmelma@33175
  1394
  assume "trivial_limit (at a within S)"
wenzelm@53255
  1395
  then show "\<not> a islimpt S"
himmelma@33175
  1396
    unfolding trivial_limit_def
hoelzl@51641
  1397
    unfolding eventually_at_topological
himmelma@33175
  1398
    unfolding islimpt_def
nipkow@39302
  1399
    apply (clarsimp simp add: set_eq_iff)
himmelma@33175
  1400
    apply (rename_tac T, rule_tac x=T in exI)
huffman@36358
  1401
    apply (clarsimp, drule_tac x=y in bspec, simp_all)
himmelma@33175
  1402
    done
himmelma@33175
  1403
next
himmelma@33175
  1404
  assume "\<not> a islimpt S"
wenzelm@53255
  1405
  then show "trivial_limit (at a within S)"
himmelma@33175
  1406
    unfolding trivial_limit_def
hoelzl@51641
  1407
    unfolding eventually_at_topological
himmelma@33175
  1408
    unfolding islimpt_def
huffman@36358
  1409
    apply clarsimp
huffman@36358
  1410
    apply (rule_tac x=T in exI)
huffman@36358
  1411
    apply auto
himmelma@33175
  1412
    done
himmelma@33175
  1413
qed
himmelma@33175
  1414
himmelma@33175
  1415
lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
huffman@45031
  1416
  using trivial_limit_within [of a UNIV] by simp
himmelma@33175
  1417
himmelma@33175
  1418
lemma trivial_limit_at:
himmelma@33175
  1419
  fixes a :: "'a::perfect_space"
himmelma@33175
  1420
  shows "\<not> trivial_limit (at a)"
huffman@44571
  1421
  by (rule at_neq_bot)
himmelma@33175
  1422
himmelma@33175
  1423
lemma trivial_limit_at_infinity:
huffman@44081
  1424
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
huffman@36358
  1425
  unfolding trivial_limit_def eventually_at_infinity
huffman@36358
  1426
  apply clarsimp
huffman@44072
  1427
  apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
huffman@44072
  1428
   apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
huffman@44072
  1429
  apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
huffman@44072
  1430
  apply (drule_tac x=UNIV in spec, simp)
himmelma@33175
  1431
  done
himmelma@33175
  1432
wenzelm@53640
  1433
lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
wenzelm@53640
  1434
  using islimpt_in_closure
wenzelm@53640
  1435
  by (metis trivial_limit_within)
hoelzl@51351
  1436
huffman@36437
  1437
text {* Some property holds "sufficiently close" to the limit point. *}
himmelma@33175
  1438
hoelzl@51530
  1439
lemma eventually_at2:
himmelma@33175
  1440
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
wenzelm@53255
  1441
  unfolding eventually_at dist_nz by auto
wenzelm@53255
  1442
wenzelm@53255
  1443
lemma eventually_happens: "eventually P net \<Longrightarrow> trivial_limit net \<or> (\<exists>x. P x)"
huffman@36358
  1444
  unfolding trivial_limit_def
huffman@36358
  1445
  by (auto elim: eventually_rev_mp)
himmelma@33175
  1446
himmelma@33175
  1447
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
huffman@45031
  1448
  by simp
himmelma@33175
  1449
himmelma@33175
  1450
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
huffman@44342
  1451
  by (simp add: filter_eq_iff)
himmelma@33175
  1452
himmelma@33175
  1453
text{* Combining theorems for "eventually" *}
himmelma@33175
  1454
himmelma@33175
  1455
lemma eventually_rev_mono:
himmelma@33175
  1456
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
wenzelm@53255
  1457
  using eventually_mono [of P Q] by fast
himmelma@33175
  1458
wenzelm@53282
  1459
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> \<not> trivial_limit net \<Longrightarrow> \<not> eventually (\<lambda>x. P x) net"
himmelma@33175
  1460
  by (simp add: eventually_False)
himmelma@33175
  1461
huffman@44210
  1462
huffman@36437
  1463
subsection {* Limits *}
himmelma@33175
  1464
himmelma@33175
  1465
lemma Lim:
wenzelm@53255
  1466
  "(f ---> l) net \<longleftrightarrow>
himmelma@33175
  1467
        trivial_limit net \<or>
himmelma@33175
  1468
        (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
himmelma@33175
  1469
  unfolding tendsto_iff trivial_limit_eq by auto
himmelma@33175
  1470
himmelma@33175
  1471
text{* Show that they yield usual definitions in the various cases. *}
himmelma@33175
  1472
himmelma@33175
  1473
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
wenzelm@53640
  1474
    (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
hoelzl@51641
  1475
  by (auto simp add: tendsto_iff eventually_at_le dist_nz)
himmelma@33175
  1476
himmelma@33175
  1477
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
wenzelm@53640
  1478
    (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a  < d \<longrightarrow> dist (f x) l < e)"
hoelzl@51641
  1479
  by (auto simp add: tendsto_iff eventually_at dist_nz)
himmelma@33175
  1480
himmelma@33175
  1481
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
wenzelm@53640
  1482
    (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d  \<longrightarrow> dist (f x) l < e)"
hoelzl@51530
  1483
  by (auto simp add: tendsto_iff eventually_at2)
himmelma@33175
  1484
himmelma@33175
  1485
lemma Lim_at_infinity:
wenzelm@53640
  1486
  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
himmelma@33175
  1487
  by (auto simp add: tendsto_iff eventually_at_infinity)
himmelma@33175
  1488
himmelma@33175
  1489
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
himmelma@33175
  1490
  by (rule topological_tendstoI, auto elim: eventually_rev_mono)
himmelma@33175
  1491
himmelma@33175
  1492
text{* The expected monotonicity property. *}
himmelma@33175
  1493
wenzelm@53255
  1494
lemma Lim_Un:
wenzelm@53255
  1495
  assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"
hoelzl@51641
  1496
  shows "(f ---> l) (at x within (S \<union> T))"
huffman@53860
  1497
  using assms unfolding at_within_union by (rule filterlim_sup)
himmelma@33175
  1498
himmelma@33175
  1499
lemma Lim_Un_univ:
wenzelm@53282
  1500
  "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>
wenzelm@53255
  1501
    S \<union> T = UNIV \<Longrightarrow> (f ---> l) (at x)"
hoelzl@51641
  1502
  by (metis Lim_Un)
himmelma@33175
  1503
himmelma@33175
  1504
text{* Interrelations between restricted and unrestricted limits. *}
himmelma@33175
  1505
hoelzl@51641
  1506
lemma Lim_at_within: (* FIXME: rename *)
hoelzl@51641
  1507
  "(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"
hoelzl@51641
  1508
  by (metis order_refl filterlim_mono subset_UNIV at_le)
himmelma@33175
  1509
huffman@44210
  1510
lemma eventually_within_interior:
huffman@44210
  1511
  assumes "x \<in> interior S"
wenzelm@53255
  1512
  shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"
wenzelm@53255
  1513
  (is "?lhs = ?rhs")
wenzelm@53255
  1514
proof
huffman@44519
  1515
  from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
wenzelm@53255
  1516
  {
wenzelm@53255
  1517
    assume "?lhs"
wenzelm@53640
  1518
    then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
hoelzl@51641
  1519
      unfolding eventually_at_topological
huffman@44210
  1520
      by auto
wenzelm@53640
  1521
    with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
huffman@44210
  1522
      by auto
wenzelm@53255
  1523
    then show "?rhs"
hoelzl@51471
  1524
      unfolding eventually_at_topological by auto
wenzelm@53255
  1525
  next
wenzelm@53255
  1526
    assume "?rhs"
wenzelm@53255
  1527
    then show "?lhs"
hoelzl@51641
  1528
      by (auto elim: eventually_elim1 simp: eventually_at_filter)
wenzelm@52624
  1529
  }
huffman@44210
  1530
qed
huffman@44210
  1531
huffman@44210
  1532
lemma at_within_interior:
huffman@44210
  1533
  "x \<in> interior S \<Longrightarrow> at x within S = at x"
hoelzl@51641
  1534
  unfolding filter_eq_iff by (intro allI eventually_within_interior)
huffman@44210
  1535
hoelzl@43338
  1536
lemma Lim_within_LIMSEQ:
huffman@53862
  1537
  fixes a :: "'a::first_countable_topology"
hoelzl@43338
  1538
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
hoelzl@43338
  1539
  shows "(X ---> L) (at a within T)"
huffman@44584
  1540
  using assms unfolding tendsto_def [where l=L]
huffman@44584
  1541
  by (simp add: sequentially_imp_eventually_within)
hoelzl@43338
  1542
hoelzl@43338
  1543
lemma Lim_right_bound:
hoelzl@51773
  1544
  fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>
hoelzl@51773
  1545
    'b::{linorder_topology, conditionally_complete_linorder}"
hoelzl@43338
  1546
  assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
wenzelm@53255
  1547
    and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
hoelzl@43338
  1548
  shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
wenzelm@53640
  1549
proof (cases "{x<..} \<inter> I = {}")
wenzelm@53640
  1550
  case True
huffman@53859
  1551
  then show ?thesis by simp
hoelzl@43338
  1552
next
wenzelm@53640
  1553
  case False
hoelzl@43338
  1554
  show ?thesis
hoelzl@51518
  1555
  proof (rule order_tendstoI)
wenzelm@53282
  1556
    fix a
wenzelm@53282
  1557
    assume a: "a < Inf (f ` ({x<..} \<inter> I))"
wenzelm@53255
  1558
    {
wenzelm@53255
  1559
      fix y
wenzelm@53255
  1560
      assume "y \<in> {x<..} \<inter> I"
wenzelm@53640
  1561
      with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
hoelzl@54263
  1562
        by (auto intro!: cInf_lower bdd_belowI2)
wenzelm@53255
  1563
      with a have "a < f y"
wenzelm@53255
  1564
        by (blast intro: less_le_trans)
wenzelm@53255
  1565
    }
hoelzl@51518
  1566
    then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"
hoelzl@51641
  1567
      by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
hoelzl@51518
  1568
  next
wenzelm@53255
  1569
    fix a
wenzelm@53255
  1570
    assume "Inf (f ` ({x<..} \<inter> I)) < a"
wenzelm@53640
  1571
    from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
wenzelm@53255
  1572
      by auto
hoelzl@51641
  1573
    then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
hoelzl@51641
  1574
      unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)
hoelzl@51641
  1575
    then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"
hoelzl@51641
  1576
      unfolding eventually_at_filter by eventually_elim simp
hoelzl@43338
  1577
  qed
hoelzl@43338
  1578
qed
hoelzl@43338
  1579
himmelma@33175
  1580
text{* Another limit point characterization. *}
himmelma@33175
  1581
himmelma@33175
  1582
lemma islimpt_sequential:
hoelzl@50883
  1583
  fixes x :: "'a::first_countable_topology"
hoelzl@50883
  1584
  shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"
himmelma@33175
  1585
    (is "?lhs = ?rhs")
himmelma@33175
  1586
proof
himmelma@33175
  1587
  assume ?lhs
hoelzl@50883
  1588
  from countable_basis_at_decseq[of x] guess A . note A = this
hoelzl@50883
  1589
  def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
wenzelm@53255
  1590
  {
wenzelm@53255
  1591
    fix n
hoelzl@50883
  1592
    from `?lhs` have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
hoelzl@50883
  1593
      unfolding islimpt_def using A(1,2)[of n] by auto
hoelzl@50883
  1594
    then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
hoelzl@50883
  1595
      unfolding f_def by (rule someI_ex)
wenzelm@53255
  1596
    then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto
wenzelm@53255
  1597
  }
hoelzl@50883
  1598
  then have "\<forall>n. f n \<in> S - {x}" by auto
hoelzl@50883
  1599
  moreover have "(\<lambda>n. f n) ----> x"
hoelzl@50883
  1600
  proof (rule topological_tendstoI)
wenzelm@53255
  1601
    fix S
wenzelm@53255
  1602
    assume "open S" "x \<in> S"
hoelzl@50883
  1603
    from A(3)[OF this] `\<And>n. f n \<in> A n`
wenzelm@53255
  1604
    show "eventually (\<lambda>x. f x \<in> S) sequentially"
wenzelm@53255
  1605
      by (auto elim!: eventually_elim1)
huffman@44584
  1606
  qed
huffman@44584
  1607
  ultimately show ?rhs by fast
himmelma@33175
  1608
next
himmelma@33175
  1609
  assume ?rhs
wenzelm@53255
  1610
  then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x"
wenzelm@53255
  1611
    by auto
hoelzl@50883
  1612
  show ?lhs
hoelzl@50883
  1613
    unfolding islimpt_def
hoelzl@50883
  1614
  proof safe
wenzelm@53255
  1615
    fix T
wenzelm@53255
  1616
    assume "open T" "x \<in> T"
hoelzl@50883
  1617
    from lim[THEN topological_tendstoD, OF this] f
hoelzl@50883
  1618
    show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
hoelzl@50883
  1619
      unfolding eventually_sequentially by auto
hoelzl@50883
  1620
  qed
himmelma@33175
  1621
qed
himmelma@33175
  1622
himmelma@33175
  1623
lemma Lim_null:
himmelma@33175
  1624
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
huffman@44125
  1625
  shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
himmelma@33175
  1626
  by (simp add: Lim dist_norm)
himmelma@33175
  1627
himmelma@33175
  1628
lemma Lim_null_comparison:
himmelma@33175
  1629
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1630
  assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
himmelma@33175
  1631
  shows "(f ---> 0) net"
wenzelm@53282
  1632
  using assms(2)
huffman@44252
  1633
proof (rule metric_tendsto_imp_tendsto)
huffman@44252
  1634
  show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
wenzelm@53255
  1635
    using assms(1) by (rule eventually_elim1) (simp add: dist_norm)
himmelma@33175
  1636
qed
himmelma@33175
  1637
himmelma@33175
  1638
lemma Lim_transform_bound:
himmelma@33175
  1639
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
wenzelm@53255
  1640
    and g :: "'a \<Rightarrow> 'c::real_normed_vector"
wenzelm@53640
  1641
  assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
wenzelm@53255
  1642
    and "(g ---> 0) net"
himmelma@33175
  1643
  shows "(f ---> 0) net"
huffman@44252
  1644
  using assms(1) tendsto_norm_zero [OF assms(2)]
huffman@44252
  1645
  by (rule Lim_null_comparison)
himmelma@33175
  1646
himmelma@33175
  1647
text{* Deducing things about the limit from the elements. *}
himmelma@33175
  1648
himmelma@33175
  1649
lemma Lim_in_closed_set:
wenzelm@53255
  1650
  assumes "closed S"
wenzelm@53255
  1651
    and "eventually (\<lambda>x. f(x) \<in> S) net"
wenzelm@53640
  1652
    and "\<not> trivial_limit net" "(f ---> l) net"
himmelma@33175
  1653
  shows "l \<in> S"
himmelma@33175
  1654
proof (rule ccontr)
himmelma@33175
  1655
  assume "l \<notin> S"
himmelma@33175
  1656
  with `closed S` have "open (- S)" "l \<in> - S"
himmelma@33175
  1657
    by (simp_all add: open_Compl)
himmelma@33175
  1658
  with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
himmelma@33175
  1659
    by (rule topological_tendstoD)
himmelma@33175
  1660
  with assms(2) have "eventually (\<lambda>x. False) net"
himmelma@33175
  1661
    by (rule eventually_elim2) simp
himmelma@33175
  1662
  with assms(3) show "False"
himmelma@33175
  1663
    by (simp add: eventually_False)
himmelma@33175
  1664
qed
himmelma@33175
  1665
himmelma@33175
  1666
text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
himmelma@33175
  1667
himmelma@33175
  1668
lemma Lim_dist_ubound:
wenzelm@53255
  1669
  assumes "\<not>(trivial_limit net)"
wenzelm@53255
  1670
    and "(f ---> l) net"
wenzelm@53640
  1671
    and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
wenzelm@53640
  1672
  shows "dist a l \<le> e"
wenzelm@52624
  1673
proof -
huffman@44252
  1674
  have "dist a l \<in> {..e}"
huffman@44252
  1675
  proof (rule Lim_in_closed_set)
wenzelm@53255
  1676
    show "closed {..e}"
wenzelm@53255
  1677
      by simp
wenzelm@53255
  1678
    show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net"
wenzelm@53255
  1679
      by (simp add: assms)
wenzelm@53255
  1680
    show "\<not> trivial_limit net"
wenzelm@53255
  1681
      by fact
wenzelm@53255
  1682
    show "((\<lambda>x. dist a (f x)) ---> dist a l) net"
wenzelm@53255
  1683
      by (intro tendsto_intros assms)
huffman@44252
  1684
  qed
wenzelm@53255
  1685
  then show ?thesis by simp
himmelma@33175
  1686
qed
himmelma@33175
  1687
himmelma@33175
  1688
lemma Lim_norm_ubound:
himmelma@33175
  1689
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
wenzelm@53255
  1690
  assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
wenzelm@53255
  1691
  shows "norm(l) \<le> e"
wenzelm@52624
  1692
proof -
huffman@44252
  1693
  have "norm l \<in> {..e}"
huffman@44252
  1694
  proof (rule Lim_in_closed_set)
wenzelm@53255
  1695
    show "closed {..e}"
wenzelm@53255
  1696
      by simp
wenzelm@53255
  1697
    show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net"
wenzelm@53255
  1698
      by (simp add: assms)
wenzelm@53255
  1699
    show "\<not> trivial_limit net"
wenzelm@53255
  1700
      by fact
wenzelm@53255
  1701
    show "((\<lambda>x. norm (f x)) ---> norm l) net"
wenzelm@53255
  1702
      by (intro tendsto_intros assms)
huffman@44252
  1703
  qed
wenzelm@53255
  1704
  then show ?thesis by simp
himmelma@33175
  1705
qed
himmelma@33175
  1706
himmelma@33175
  1707
lemma Lim_norm_lbound:
himmelma@33175
  1708
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
wenzelm@53640
  1709
  assumes "\<not> trivial_limit net"
wenzelm@53640
  1710
    and "(f ---> l) net"
wenzelm@53640
  1711
    and "eventually (\<lambda>x. e \<le> norm (f x)) net"
himmelma@33175
  1712
  shows "e \<le> norm l"
wenzelm@52624
  1713
proof -
huffman@44252
  1714
  have "norm l \<in> {e..}"
huffman@44252
  1715
  proof (rule Lim_in_closed_set)
wenzelm@53255
  1716
    show "closed {e..}"
wenzelm@53255
  1717
      by simp
wenzelm@53255
  1718
    show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net"
wenzelm@53255
  1719
      by (simp add: assms)
wenzelm@53255
  1720
    show "\<not> trivial_limit net"
wenzelm@53255
  1721
      by fact
wenzelm@53255
  1722
    show "((\<lambda>x. norm (f x)) ---> norm l) net"
wenzelm@53255
  1723
      by (intro tendsto_intros assms)
huffman@44252
  1724
  qed
wenzelm@53255
  1725
  then show ?thesis by simp
himmelma@33175
  1726
qed
himmelma@33175
  1727
himmelma@33175
  1728
text{* Limit under bilinear function *}
himmelma@33175
  1729
himmelma@33175
  1730
lemma Lim_bilinear:
wenzelm@53282
  1731
  assumes "(f ---> l) net"
wenzelm@53282
  1732
    and "(g ---> m) net"
wenzelm@53282
  1733
    and "bounded_bilinear h"
himmelma@33175
  1734
  shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
wenzelm@52624
  1735
  using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
wenzelm@52624
  1736
  by (rule bounded_bilinear.tendsto)
himmelma@33175
  1737
himmelma@33175
  1738
text{* These are special for limits out of the same vector space. *}
himmelma@33175
  1739
himmelma@33175
  1740
lemma Lim_within_id: "(id ---> a) (at a within s)"
hoelzl@51641
  1741
  unfolding id_def by (rule tendsto_ident_at)
himmelma@33175
  1742
himmelma@33175
  1743
lemma Lim_at_id: "(id ---> a) (at a)"
huffman@45031
  1744
  unfolding id_def by (rule tendsto_ident_at)
himmelma@33175
  1745
himmelma@33175
  1746
lemma Lim_at_zero:
himmelma@33175
  1747
  fixes a :: "'a::real_normed_vector"
wenzelm@53291
  1748
    and l :: "'b::topological_space"
wenzelm@53282
  1749
  shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)"
huffman@44252
  1750
  using LIM_offset_zero LIM_offset_zero_cancel ..
himmelma@33175
  1751
huffman@44081
  1752
text{* It's also sometimes useful to extract the limit point from the filter. *}
himmelma@33175
  1753
wenzelm@52624
  1754
abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
wenzelm@52624
  1755
  where "netlimit F \<equiv> Lim F (\<lambda>x. x)"
himmelma@33175
  1756
wenzelm@53282
  1757
lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
hoelzl@51365
  1758
  by (rule tendsto_Lim) (auto intro: tendsto_intros)
himmelma@33175
  1759
himmelma@33175
  1760
lemma netlimit_at:
huffman@44072
  1761
  fixes a :: "'a::{perfect_space,t2_space}"
himmelma@33175
  1762
  shows "netlimit (at a) = a"
huffman@45031
  1763
  using netlimit_within [of a UNIV] by simp
himmelma@33175
  1764
huffman@44210
  1765
lemma lim_within_interior:
huffman@44210
  1766
  "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
hoelzl@51641
  1767
  by (metis at_within_interior)
huffman@44210
  1768
huffman@44210
  1769
lemma netlimit_within_interior:
huffman@44210
  1770
  fixes x :: "'a::{t2_space,perfect_space}"
huffman@44210
  1771
  assumes "x \<in> interior S"
huffman@44210
  1772
  shows "netlimit (at x within S) = x"
wenzelm@52624
  1773
  using assms by (metis at_within_interior netlimit_at)
huffman@44210
  1774
himmelma@33175
  1775
text{* Transformation of limit. *}
himmelma@33175
  1776
himmelma@33175
  1777
lemma Lim_transform:
himmelma@33175
  1778
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1779
  assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
himmelma@33175
  1780
  shows "(g ---> l) net"
huffman@44252
  1781
  using tendsto_diff [OF assms(2) assms(1)] by simp
himmelma@33175
  1782
himmelma@33175
  1783
lemma Lim_transform_eventually:
huffman@36667
  1784
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
himmelma@33175
  1785
  apply (rule topological_tendstoI)
himmelma@33175
  1786
  apply (drule (2) topological_tendstoD)
himmelma@33175
  1787
  apply (erule (1) eventually_elim2, simp)
himmelma@33175
  1788
  done
himmelma@33175
  1789
himmelma@33175
  1790
lemma Lim_transform_within:
wenzelm@53282
  1791
  assumes "0 < d"
wenzelm@53282
  1792
    and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
wenzelm@53282
  1793
    and "(f ---> l) (at x within S)"
huffman@36667
  1794
  shows "(g ---> l) (at x within S)"
huffman@36667
  1795
proof (rule Lim_transform_eventually)
huffman@36667
  1796
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
hoelzl@51641
  1797
    using assms(1,2) by (auto simp: dist_nz eventually_at)
huffman@36667
  1798
  show "(f ---> l) (at x within S)" by fact
huffman@36667
  1799
qed
himmelma@33175
  1800
himmelma@33175
  1801
lemma Lim_transform_at:
wenzelm@53282
  1802
  assumes "0 < d"
wenzelm@53282
  1803
    and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
wenzelm@53282
  1804
    and "(f ---> l) (at x)"
huffman@36667
  1805
  shows "(g ---> l) (at x)"
wenzelm@53282
  1806
  using _ assms(3)
huffman@36667
  1807
proof (rule Lim_transform_eventually)
huffman@36667
  1808
  show "eventually (\<lambda>x. f x = g x) (at x)"
hoelzl@51530
  1809
    unfolding eventually_at2
huffman@36667
  1810
    using assms(1,2) by auto
huffman@36667
  1811
qed
himmelma@33175
  1812
himmelma@33175
  1813
text{* Common case assuming being away from some crucial point like 0. *}
himmelma@33175
  1814
himmelma@33175
  1815
lemma Lim_transform_away_within:
huffman@36669
  1816
  fixes a b :: "'a::t1_space"
wenzelm@53282
  1817
  assumes "a \<noteq> b"
wenzelm@53282
  1818
    and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
wenzelm@53282
  1819
    and "(f ---> l) (at a within S)"
himmelma@33175
  1820
  shows "(g ---> l) (at a within S)"
huffman@36669
  1821
proof (rule Lim_transform_eventually)
huffman@36669
  1822
  show "(f ---> l) (at a within S)" by fact
huffman@36669
  1823
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
hoelzl@51641
  1824
    unfolding eventually_at_topological
huffman@36669
  1825
    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
himmelma@33175
  1826
qed
himmelma@33175
  1827
himmelma@33175
  1828
lemma Lim_transform_away_at:
huffman@36669
  1829
  fixes a b :: "'a::t1_space"
wenzelm@52624
  1830
  assumes ab: "a\<noteq>b"
wenzelm@52624
  1831
    and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
wenzelm@52624
  1832
    and fl: "(f ---> l) (at a)"
himmelma@33175
  1833
  shows "(g ---> l) (at a)"
wenzelm@52624
  1834
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl by simp
himmelma@33175
  1835
himmelma@33175
  1836
text{* Alternatively, within an open set. *}
himmelma@33175
  1837
himmelma@33175
  1838
lemma Lim_transform_within_open:
wenzelm@53282
  1839
  assumes "open S" and "a \<in> S"
wenzelm@53282
  1840
    and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
wenzelm@53282
  1841
    and "(f ---> l) (at a)"
himmelma@33175
  1842
  shows "(g ---> l) (at a)"
huffman@36667
  1843
proof (rule Lim_transform_eventually)
huffman@36667
  1844
  show "eventually (\<lambda>x. f x = g x) (at a)"
huffman@36667
  1845
    unfolding eventually_at_topological
huffman@36667
  1846
    using assms(1,2,3) by auto
huffman@36667
  1847
  show "(f ---> l) (at a)" by fact
himmelma@33175
  1848
qed
himmelma@33175
  1849
himmelma@33175
  1850
text{* A congruence rule allowing us to transform limits assuming not at point. *}
himmelma@33175
  1851
himmelma@33175
  1852
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
himmelma@33175
  1853
huffman@36362
  1854
lemma Lim_cong_within(*[cong add]*):
wenzelm@53282
  1855
  assumes "a = b"
wenzelm@53282
  1856
    and "x = y"
wenzelm@53282
  1857
    and "S = T"
wenzelm@53282
  1858
    and "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
hoelzl@43338
  1859
  shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
hoelzl@51641
  1860
  unfolding tendsto_def eventually_at_topological
huffman@36667
  1861
  using assms by simp
huffman@36667
  1862
huffman@36667
  1863
lemma Lim_cong_at(*[cong add]*):
hoelzl@43338
  1864
  assumes "a = b" "x = y"
wenzelm@53282
  1865
    and "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
hoelzl@43338
  1866
  shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
huffman@36667
  1867
  unfolding tendsto_def eventually_at_topological
huffman@36667
  1868
  using assms by simp
himmelma@33175
  1869
himmelma@33175
  1870
text{* Useful lemmas on closure and set of possible sequential limits.*}
himmelma@33175
  1871
himmelma@33175
  1872
lemma closure_sequential:
hoelzl@50883
  1873
  fixes l :: "'a::first_countable_topology"
wenzelm@53291
  1874
  shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)"
wenzelm@53291
  1875
  (is "?lhs = ?rhs")
himmelma@33175
  1876
proof
wenzelm@53282
  1877
  assume "?lhs"
wenzelm@53282
  1878
  moreover
wenzelm@53282
  1879
  {
wenzelm@53282
  1880
    assume "l \<in> S"
wenzelm@53282
  1881
    then have "?rhs" using tendsto_const[of l sequentially] by auto
wenzelm@52624
  1882
  }
wenzelm@52624
  1883
  moreover
wenzelm@53282
  1884
  {
wenzelm@53282
  1885
    assume "l islimpt S"
wenzelm@53282
  1886
    then have "?rhs" unfolding islimpt_sequential by auto
wenzelm@52624
  1887
  }
wenzelm@52624
  1888
  ultimately show "?rhs"
wenzelm@52624
  1889
    unfolding closure_def by auto
himmelma@33175
  1890
next
himmelma@33175
  1891
  assume "?rhs"
wenzelm@53282
  1892
  then show "?lhs" unfolding closure_def islimpt_sequential by auto
himmelma@33175
  1893
qed
himmelma@33175
  1894
himmelma@33175
  1895
lemma closed_sequential_limits:
hoelzl@50883
  1896
  fixes S :: "'a::first_countable_topology set"
himmelma@33175
  1897
  shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
wenzelm@52624
  1898
  using closure_sequential [where 'a='a] closure_closed [where 'a='a]
wenzelm@52624
  1899
    closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]
himmelma@33175
  1900
  by metis
himmelma@33175
  1901
himmelma@33175
  1902
lemma closure_approachable:
himmelma@33175
  1903
  fixes S :: "'a::metric_space set"
himmelma@33175
  1904
  shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
himmelma@33175
  1905
  apply (auto simp add: closure_def islimpt_approachable)
wenzelm@52624
  1906
  apply (metis dist_self)
wenzelm@52624
  1907
  done
himmelma@33175
  1908
himmelma@33175
  1909
lemma closed_approachable:
himmelma@33175
  1910
  fixes S :: "'a::metric_space set"
wenzelm@53291
  1911
  shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
himmelma@33175
  1912
  by (metis closure_closed closure_approachable)
himmelma@33175
  1913
hoelzl@51351
  1914
lemma closure_contains_Inf:
hoelzl@51351
  1915
  fixes S :: "real set"
hoelzl@54258
  1916
  assumes "S \<noteq> {}" "bdd_below S"
hoelzl@51351
  1917
  shows "Inf S \<in> closure S"
wenzelm@52624
  1918
proof -
hoelzl@51351
  1919
  have *: "\<forall>x\<in>S. Inf S \<le> x"
hoelzl@54258
  1920
    using cInf_lower[of _ S] assms by metis
wenzelm@52624
  1921
  {
wenzelm@53282
  1922
    fix e :: real
wenzelm@53282
  1923
    assume "e > 0"
wenzelm@52624
  1924
    then have "Inf S < Inf S + e" by simp
wenzelm@52624
  1925
    with assms obtain x where "x \<in> S" "x < Inf S + e"
hoelzl@54258
  1926
      by (subst (asm) cInf_less_iff) auto
wenzelm@52624
  1927
    with * have "\<exists>x\<in>S. dist x (Inf S) < e"
wenzelm@52624
  1928
      by (intro bexI[of _ x]) (auto simp add: dist_real_def)
wenzelm@52624
  1929
  }
wenzelm@52624
  1930
  then show ?thesis unfolding closure_approachable by auto
hoelzl@51351
  1931
qed
hoelzl@51351
  1932
hoelzl@51351
  1933
lemma closed_contains_Inf:
hoelzl@51351
  1934
  fixes S :: "real set"
hoelzl@54258
  1935
  shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"
hoelzl@51351
  1936
  by (metis closure_contains_Inf closure_closed assms)
hoelzl@51351
  1937
hoelzl@51351
  1938
lemma not_trivial_limit_within_ball:
wenzelm@53640
  1939
  "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
hoelzl@51351
  1940
  (is "?lhs = ?rhs")
hoelzl@51351
  1941
proof -
wenzelm@53282
  1942
  {
wenzelm@53282
  1943
    assume "?lhs"
wenzelm@53282
  1944
    {
wenzelm@53282
  1945
      fix e :: real
wenzelm@53282
  1946
      assume "e > 0"
wenzelm@53640
  1947
      then obtain y where "y \<in> S - {x}" and "dist y x < e"
hoelzl@51351
  1948
        using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
hoelzl@51351
  1949
        by auto
wenzelm@53640
  1950
      then have "y \<in> S \<inter> ball x e - {x}"
hoelzl@51351
  1951
        unfolding ball_def by (simp add: dist_commute)
wenzelm@53640
  1952
      then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
wenzelm@52624
  1953
    }
wenzelm@52624
  1954
    then have "?rhs" by auto
hoelzl@51351
  1955
  }
hoelzl@51351
  1956
  moreover
wenzelm@53282
  1957
  {
wenzelm@53282
  1958
    assume "?rhs"
wenzelm@53282
  1959
    {
wenzelm@53282
  1960
      fix e :: real
wenzelm@53282
  1961
      assume "e > 0"
wenzelm@53640
  1962
      then obtain y where "y \<in> S \<inter> ball x e - {x}"
wenzelm@53282
  1963
        using `?rhs` by blast
wenzelm@53640
  1964
      then have "y \<in> S - {x}" and "dist y x < e"
wenzelm@53640
  1965
        unfolding ball_def by (simp_all add: dist_commute)
wenzelm@53640
  1966
      then have "\<exists>y \<in> S - {x}. dist y x < e"
wenzelm@53282
  1967
        by auto
hoelzl@51351
  1968
    }
hoelzl@51351
  1969
    then have "?lhs"
wenzelm@53282
  1970
      using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
wenzelm@53282
  1971
      by auto
hoelzl@51351
  1972
  }
hoelzl@51351
  1973
  ultimately show ?thesis by auto
hoelzl@51351
  1974
qed
hoelzl@51351
  1975
wenzelm@52624
  1976
immler@50087
  1977
subsection {* Infimum Distance *}
immler@50087
  1978
hoelzl@54260
  1979
definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"
hoelzl@54260
  1980
hoelzl@54260
  1981
lemma bdd_below_infdist[intro, simp]: "bdd_below (dist x`A)"
hoelzl@54258
  1982
  by (auto intro!: zero_le_dist)
hoelzl@54258
  1983
hoelzl@54260
  1984
lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = (INF a:A. dist x a)"
immler@50087
  1985
  by (simp add: infdist_def)
immler@50087
  1986
wenzelm@52624
  1987
lemma infdist_nonneg: "0 \<le> infdist x A"
hoelzl@54260
  1988
  by (auto simp add: infdist_def intro: cINF_greatest)
hoelzl@54260
  1989
hoelzl@54260
  1990
lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"
hoelzl@54260
  1991
  by (auto intro: cINF_lower simp add: infdist_def)
hoelzl@54260
  1992
hoelzl@54260
  1993
lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"
hoelzl@54260
  1994
  by (auto intro!: cINF_lower2 simp add: infdist_def)
hoelzl@54258
  1995
hoelzl@54258
  1996
lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"
hoelzl@54260
  1997
  by (auto intro!: antisym infdist_nonneg infdist_le2)
immler@50087
  1998
wenzelm@52624
  1999
lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
wenzelm@53640
  2000
proof (cases "A = {}")
wenzelm@53640
  2001
  case True
wenzelm@53282
  2002
  then show ?thesis by (simp add: infdist_def)
immler@50087
  2003
next
wenzelm@53640
  2004
  case False
wenzelm@52624
  2005
  then obtain a where "a \<in> A" by auto
immler@50087
  2006
  have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"
hoelzl@51475
  2007
  proof (rule cInf_greatest)
wenzelm@53282
  2008
    from `A \<noteq> {}` show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"
wenzelm@53282
  2009
      by simp
wenzelm@53282
  2010
    fix d
wenzelm@53282
  2011
    assume "d \<in> {dist x y + dist y a |a. a \<in> A}"
wenzelm@53282
  2012
    then obtain a where d: "d = dist x y + dist y a" "a \<in> A"
wenzelm@53282
  2013
      by auto
immler@50087
  2014
    show "infdist x A \<le> d"
immler@50087
  2015
      unfolding infdist_notempty[OF `A \<noteq> {}`]
hoelzl@54260
  2016
    proof (rule cINF_lower2)
hoelzl@54260
  2017
      show "a \<in> A" by fact
wenzelm@53282
  2018
      show "dist x a \<le> d"
wenzelm@53282
  2019
        unfolding d by (rule dist_triangle)
hoelzl@54258
  2020
    qed simp
immler@50087
  2021
  qed
immler@50087
  2022
  also have "\<dots> = dist x y + infdist y A"
hoelzl@51475
  2023
  proof (rule cInf_eq, safe)
wenzelm@53282
  2024
    fix a
wenzelm@53282
  2025
    assume "a \<in> A"
wenzelm@53282
  2026
    then show "dist x y + infdist y A \<le> dist x y + dist y a"
wenzelm@53282
  2027
      by (auto intro: infdist_le)
immler@50087
  2028
  next
wenzelm@53282
  2029
    fix i
wenzelm@53282
  2030
    assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
wenzelm@53282
  2031
    then have "i - dist x y \<le> infdist y A"
wenzelm@53282
  2032
      unfolding infdist_notempty[OF `A \<noteq> {}`] using `a \<in> A`
hoelzl@54260
  2033
      by (intro cINF_greatest) (auto simp: field_simps)
wenzelm@53282
  2034
    then show "i \<le> dist x y + infdist y A"
wenzelm@53282
  2035
      by simp
immler@50087
  2036
  qed
immler@50087
  2037
  finally show ?thesis by simp
immler@50087
  2038
qed
immler@50087
  2039
hoelzl@51475
  2040
lemma in_closure_iff_infdist_zero:
immler@50087
  2041
  assumes "A \<noteq> {}"
immler@50087
  2042
  shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
immler@50087
  2043
proof
immler@50087
  2044
  assume "x \<in> closure A"
immler@50087
  2045
  show "infdist x A = 0"
immler@50087
  2046
  proof (rule ccontr)
immler@50087
  2047
    assume "infdist x A \<noteq> 0"
wenzelm@53282
  2048
    with infdist_nonneg[of x A] have "infdist x A > 0"
wenzelm@53282
  2049
      by auto
wenzelm@53282
  2050
    then have "ball x (infdist x A) \<inter> closure A = {}"
wenzelm@52624
  2051
      apply auto
immler@54775
  2052
      apply (metis `x \<in> closure A` closure_approachable dist_commute infdist_le not_less)
wenzelm@52624
  2053
      done
wenzelm@53282
  2054
    then have "x \<notin> closure A"
wenzelm@52624
  2055
      by (metis `0 < infdist x A` centre_in_ball disjoint_iff_not_equal)
wenzelm@53282
  2056
    then show False using `x \<in> closure A` by simp
immler@50087
  2057
  qed
immler@50087
  2058
next
immler@50087
  2059
  assume x: "infdist x A = 0"
wenzelm@53282
  2060
  then obtain a where "a \<in> A"
wenzelm@53282
  2061
    by atomize_elim (metis all_not_in_conv assms)
wenzelm@53282
  2062
  show "x \<in> closure A"
wenzelm@53282
  2063
    unfolding closure_approachable
wenzelm@53282
  2064
    apply safe
wenzelm@53282
  2065
  proof (rule ccontr)
wenzelm@53282
  2066
    fix e :: real
wenzelm@53282
  2067
    assume "e > 0"
immler@50087
  2068
    assume "\<not> (\<exists>y\<in>A. dist y x < e)"
wenzelm@53282
  2069
    then have "infdist x A \<ge> e" using `a \<in> A`
immler@50087
  2070
      unfolding infdist_def
hoelzl@54260
  2071
      by (force simp: dist_commute intro: cINF_greatest)
wenzelm@53282
  2072
    with x `e > 0` show False by auto
immler@50087
  2073
  qed
immler@50087
  2074
qed
immler@50087
  2075
hoelzl@51475
  2076
lemma in_closed_iff_infdist_zero:
immler@50087
  2077
  assumes "closed A" "A \<noteq> {}"
immler@50087
  2078
  shows "x \<in> A \<longleftrightarrow> infdist x A = 0"
immler@50087
  2079
proof -
immler@50087
  2080
  have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
immler@50087
  2081
    by (rule in_closure_iff_infdist_zero) fact
immler@50087
  2082
  with assms show ?thesis by simp
immler@50087
  2083
qed
immler@50087
  2084
immler@50087
  2085
lemma tendsto_infdist [tendsto_intros]:
immler@50087
  2086
  assumes f: "(f ---> l) F"
immler@50087
  2087
  shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"
immler@50087
  2088
proof (rule tendstoI)
wenzelm@53282
  2089
  fix e ::real
wenzelm@53282
  2090
  assume "e > 0"
immler@50087
  2091
  from tendstoD[OF f this]
immler@50087
  2092
  show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"
immler@50087
  2093
  proof (eventually_elim)
immler@50087
  2094
    fix x
immler@50087
  2095
    from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
immler@50087
  2096
    have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"
immler@50087
  2097
      by (simp add: dist_commute dist_real_def)
immler@50087
  2098
    also assume "dist (f x) l < e"
immler@50087
  2099
    finally show "dist (infdist (f x) A) (infdist l A) < e" .
immler@50087
  2100
  qed
immler@50087
  2101
qed
immler@50087
  2102
himmelma@33175
  2103
text{* Some other lemmas about sequences. *}
himmelma@33175
  2104
huffman@53597
  2105
lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)
huffman@36441
  2106
  assumes "eventually (\<lambda>i. P i) sequentially"
huffman@36441
  2107
  shows "eventually (\<lambda>i. P (i + k)) sequentially"
huffman@53597
  2108
  using assms by (rule eventually_sequentially_seg [THEN iffD2])
huffman@53597
  2109
huffman@53597
  2110
lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)
wenzelm@53291
  2111
  "(f ---> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) ---> l) sequentially"
huffman@53597
  2112
  apply (erule filterlim_compose)
huffman@53597
  2113
  apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)
wenzelm@52624
  2114
  apply arith
wenzelm@52624
  2115
  done
himmelma@33175
  2116
himmelma@33175
  2117
lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
huffman@53597
  2118
  using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)
himmelma@33175
  2119
huffman@44210
  2120
subsection {* More properties of closed balls *}
himmelma@33175
  2121
huffman@54070
  2122
lemma closed_vimage: (* TODO: move to Topological_Spaces.thy *)
huffman@54070
  2123
  assumes "closed s" and "continuous_on UNIV f"
huffman@54070
  2124
  shows "closed (vimage f s)"
huffman@54070
  2125
  using assms unfolding continuous_on_closed_vimage [OF closed_UNIV]
huffman@54070
  2126
  by simp
huffman@54070
  2127
himmelma@33175
  2128
lemma closed_cball: "closed (cball x e)"
huffman@54070
  2129
proof -
huffman@54070
  2130
  have "closed (dist x -` {..e})"
huffman@54070
  2131
    by (intro closed_vimage closed_atMost continuous_on_intros)
huffman@54070
  2132
  also have "dist x -` {..e} = cball x e"
huffman@54070
  2133
    by auto
huffman@54070
  2134
  finally show ?thesis .
huffman@54070
  2135
qed
himmelma@33175
  2136
himmelma@33175
  2137
lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
wenzelm@52624
  2138
proof -
wenzelm@52624
  2139
  {
wenzelm@52624
  2140
    fix x and e::real
wenzelm@52624
  2141
    assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
wenzelm@53282
  2142
    then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
wenzelm@52624
  2143
  }
wenzelm@52624
  2144
  moreover
wenzelm@52624
  2145
  {
wenzelm@52624
  2146
    fix x and e::real
wenzelm@52624
  2147
    assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
wenzelm@53282
  2148
    then have "\<exists>d>0. ball x d \<subseteq> S"
wenzelm@52624
  2149
      unfolding subset_eq
wenzelm@52624
  2150
      apply(rule_tac x="e/2" in exI)
wenzelm@52624
  2151
      apply auto
wenzelm@52624
  2152
      done
wenzelm@52624
  2153
  }
wenzelm@52624
  2154
  ultimately show ?thesis
wenzelm@52624
  2155
    unfolding open_contains_ball by auto
himmelma@33175
  2156
qed
himmelma@33175
  2157
wenzelm@53291
  2158
lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
huffman@44170
  2159
  by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
himmelma@33175
  2160
himmelma@33175
  2161
lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
himmelma@33175
  2162
  apply (simp add: interior_def, safe)
himmelma@33175
  2163
  apply (force simp add: open_contains_cball)
himmelma@33175
  2164
  apply (rule_tac x="ball x e" in exI)
huffman@36362
  2165
  apply (simp add: subset_trans [OF ball_subset_cball])
himmelma@33175
  2166
  done
himmelma@33175
  2167
himmelma@33175
  2168
lemma islimpt_ball:
himmelma@33175
  2169
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
wenzelm@53291
  2170
  shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"
wenzelm@53291
  2171
  (is "?lhs = ?rhs")
himmelma@33175
  2172
proof
himmelma@33175
  2173
  assume "?lhs"
wenzelm@53282
  2174
  {
wenzelm@53282
  2175
    assume "e \<le> 0"
wenzelm@53282
  2176
    then have *:"ball x e = {}"
wenzelm@53282
  2177
      using ball_eq_empty[of x e] by auto
wenzelm@53282
  2178
    have False using `?lhs`
wenzelm@53282
  2179
      unfolding * using islimpt_EMPTY[of y] by auto
himmelma@33175
  2180
  }
wenzelm@53282
  2181
  then have "e > 0" by (metis not_less)
himmelma@33175
  2182
  moreover
wenzelm@52624
  2183
  have "y \<in> cball x e"
wenzelm@52624
  2184
    using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
wenzelm@52624
  2185
      ball_subset_cball[of x e] `?lhs`
wenzelm@52624
  2186
    unfolding closed_limpt by auto
himmelma@33175
  2187
  ultimately show "?rhs" by auto
himmelma@33175
  2188
next
wenzelm@53282
  2189
  assume "?rhs"
wenzelm@53640
  2190
  then have "e > 0" by auto
wenzelm@53282
  2191
  {
wenzelm@53282
  2192
    fix d :: real
wenzelm@53282
  2193
    assume "d > 0"
himmelma@33175
  2194
    have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
wenzelm@53282
  2195
    proof (cases "d \<le> dist x y")
wenzelm@53282
  2196
      case True
wenzelm@53282
  2197
      then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
wenzelm@53282
  2198
      proof (cases "x = y")
wenzelm@53282
  2199
        case True
wenzelm@53282
  2200
        then have False
wenzelm@53282
  2201
          using `d \<le> dist x y` `d>0` by auto
wenzelm@53282
  2202
        then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
wenzelm@53282
  2203
          by auto
himmelma@33175
  2204
      next
himmelma@33175
  2205
        case False
wenzelm@53282
  2206
        have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =
wenzelm@53282
  2207
          norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
wenzelm@53291
  2208
          unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
wenzelm@53282
  2209
          by auto
himmelma@33175
  2210
        also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
wenzelm@53291
  2211
          using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
himmelma@33175
  2212
          unfolding scaleR_minus_left scaleR_one
himmelma@33175
  2213
          by (auto simp add: norm_minus_commute)
himmelma@33175
  2214
        also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
himmelma@33175
  2215
          unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
wenzelm@53282
  2216
          unfolding distrib_right using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm]
wenzelm@53282
  2217
          by auto
wenzelm@53282
  2218
        also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs`
wenzelm@53282
  2219
          by (auto simp add: dist_norm)
wenzelm@53282
  2220
        finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0`
wenzelm@53282
  2221
          by auto
himmelma@33175
  2222
        moreover
himmelma@33175
  2223
        have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
wenzelm@53282
  2224
          using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff
wenzelm@53282
  2225
          by (auto simp add: dist_commute)
himmelma@33175
  2226
        moreover
wenzelm@53282
  2227
        have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"
wenzelm@53282
  2228
          unfolding dist_norm
wenzelm@53282
  2229
          apply simp
wenzelm@53282
  2230
          unfolding norm_minus_cancel
wenzelm@53282
  2231
          using `d > 0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
wenzelm@53282
  2232
          unfolding dist_norm
wenzelm@53282
  2233
          apply auto
wenzelm@53282
  2234
          done
wenzelm@53282
  2235
        ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
wenzelm@53282
  2236
          apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)
wenzelm@53282
  2237
          apply auto
wenzelm@53282
  2238
          done
himmelma@33175
  2239
      qed
himmelma@33175
  2240
    next
wenzelm@53282
  2241
      case False
wenzelm@53282
  2242
      then have "d > dist x y" by auto
wenzelm@53282
  2243
      show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"
wenzelm@53282
  2244
      proof (cases "x = y")
himmelma@33175
  2245
        case True
himmelma@33175
  2246
        obtain z where **: "z \<noteq> y" "dist z y < min e d"
himmelma@33175
  2247
          using perfect_choose_dist[of "min e d" y]
himmelma@33175
  2248
          using `d > 0` `e>0` by auto
himmelma@33175
  2249
        show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  2250
          unfolding `x = y`
himmelma@33175
  2251
          using `z \<noteq> y` **
wenzelm@53282
  2252
          apply (rule_tac x=z in bexI)
wenzelm@53282
  2253
          apply (auto simp add: dist_commute)
wenzelm@53282
  2254
          done
himmelma@33175
  2255
      next
wenzelm@53282
  2256
        case False
wenzelm@53282
  2257
        then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
wenzelm@53282
  2258
          using `d>0` `d > dist x y` `?rhs`
wenzelm@53282
  2259
          apply (rule_tac x=x in bexI)
wenzelm@53282
  2260
          apply auto
wenzelm@53282
  2261
          done
himmelma@33175
  2262
      qed
wenzelm@53282
  2263
    qed
wenzelm@53282
  2264
  }
wenzelm@53282
  2265
  then show "?lhs"
wenzelm@53282
  2266
    unfolding mem_cball islimpt_approachable mem_ball by auto
himmelma@33175
  2267
qed
himmelma@33175
  2268
himmelma@33175
  2269
lemma closure_ball_lemma:
himmelma@33175
  2270
  fixes x y :: "'a::real_normed_vector"
wenzelm@53282
  2271
  assumes "x \<noteq> y"