src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
author hoelzl
Tue Mar 05 15:43:22 2013 +0100 (2013-03-05)
changeset 51351 dd1dd470690b
parent 51350 490f34774a9a
child 51361 21e5b6efb317
permissions -rw-r--r--
generalized lemmas in Extended_Real_Limits
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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/Glbs"
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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 o r) ----> l"
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  by (rule LIMSEQ_subseq_LIMSEQ)
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lemmas real_isGlb_unique = isGlb_unique[where 'a=real]
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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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subsection {* Topological Basis *}
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context topological_space
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begin
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definition "topological_basis B =
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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 = (\<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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  hence "(\<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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  thus "\<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" using assms unfolding topological_basis_def
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  proof safe
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    fix O'::"'a set" 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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    thus "\<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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  assumes "\<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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  assumes "open O'"
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  assumes "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'" 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'" 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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  assumes "X \<in> B"
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  shows "open X"
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  using assms
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  by (simp add: topological_basis_def)
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lemma topological_basis_imp_subbasis:
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  assumes B: "topological_basis B" shows "open = generate_topology B"
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proof (intro ext iffI)
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  fix S :: "'a set" 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" assume "generate_topology B S" 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" and f::"'a set \<Rightarrow> 'a"
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  assumes "topological_basis B"
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  assumes 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" assume "open X" "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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  thus "\<exists>B'\<in>B. f B' \<in> X" 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" 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" 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 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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  assumes 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 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 by (atomize_elim) simp
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lemma countable_dense_exists:
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  shows "\<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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class first_countable_topology = topological_space +
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  assumes first_countable_basis:
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    "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
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lemma (in first_countable_topology) countable_basis_at_decseq:
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  obtains A :: "nat \<Rightarrow> 'a set" where
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    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
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    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
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proof atomize_elim
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  from first_countable_basis[of x] obtain A
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    where "countable A"
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    and nhds: "\<And>a. a \<in> A \<Longrightarrow> open a" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a"
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    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"  by auto
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  then have "A \<noteq> {}" by auto
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  with `countable A` have r: "A = range (from_nat_into A)" by auto
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  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. from_nat_into A i"
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  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
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      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
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  proof (safe intro!: exI[of _ F])
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    fix i
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    show "open (F i)" using nhds(1) r by (auto simp: F_def intro!: open_INT)
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    show "x \<in> F i" using nhds(2) r by (auto simp: F_def)
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  next
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    fix S assume "open S" "x \<in> S"
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    from incl[OF this] obtain i where "F i \<subseteq> S"
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      by (subst (asm) r) (auto simp: F_def)
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    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
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      by (auto simp: F_def)
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    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
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      by (auto simp: eventually_sequentially)
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  qed
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qed
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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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  by atomize_elim auto
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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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  thus "\<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" unfolding A_def by (intro countable_image countable_Collect_finite)
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    fix a assume "a \<in> A"
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    thus "x \<in> a" "open a" 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 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)" by (auto simp: A_def)
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    thus "a \<inter> b \<in> A" by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])
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  next
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    fix S assume "open S" "x \<in> S" then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast
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    thus "\<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 countable_basis:
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  obtains A :: "nat \<Rightarrow> 'a::first_countable_topology set" where
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    "\<And>i. open (A i)" "\<And>i. x \<in> A i"
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    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
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proof atomize_elim
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  from countable_basis_at_decseq[of x] guess A . note A = this
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  { fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
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    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
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      by (auto elim: eventually_elim1 simp: subset_eq) }
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  with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F ----> x)"
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    by (intro exI[of _ A]) (auto simp: tendsto_def)
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qed
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lemma sequentially_imp_eventually_nhds_within:
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  fixes a :: "'a::first_countable_topology"
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  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
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  shows "eventually P (nhds a within s)"
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proof (rule ccontr)
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  from countable_basis[of a] guess A . note A = this
hoelzl@51350
   268
  assume "\<not> eventually P (nhds a within s)"
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   269
  with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
hoelzl@51350
   270
    unfolding Limits.eventually_within eventually_nhds by (intro choice) fastforce
hoelzl@51350
   271
  then guess F ..
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   272
  hence F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
hoelzl@51350
   273
    by fast+
hoelzl@51350
   274
  with A have "F ----> a" by auto
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   275
  hence "eventually (\<lambda>n. P (F n)) sequentially"
hoelzl@51350
   276
    using assms F0 by simp
hoelzl@51350
   277
  thus "False" by (simp add: F3)
hoelzl@51350
   278
qed
hoelzl@51350
   279
hoelzl@51350
   280
lemma eventually_nhds_within_iff_sequentially:
hoelzl@51350
   281
  fixes a :: "'a::first_countable_topology"
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   282
  shows "eventually P (nhds a within s) \<longleftrightarrow> 
hoelzl@51350
   283
    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51350
   284
proof (safe intro!: sequentially_imp_eventually_nhds_within)
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   285
  assume "eventually P (nhds a within s)" 
hoelzl@51350
   286
  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
hoelzl@51350
   287
    by (auto simp: Limits.eventually_within eventually_nhds)
hoelzl@51350
   288
  moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
hoelzl@51350
   289
  ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51350
   290
    by (auto dest!: topological_tendstoD elim: eventually_elim1)
hoelzl@51350
   291
qed
hoelzl@51350
   292
hoelzl@51350
   293
lemma eventually_nhds_iff_sequentially:
hoelzl@51350
   294
  fixes a :: "'a::first_countable_topology"
hoelzl@51350
   295
  shows "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51350
   296
  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
hoelzl@51350
   297
hoelzl@51350
   298
lemma not_eventually_sequentiallyD:
hoelzl@51350
   299
  assumes P: "\<not> eventually P sequentially"
hoelzl@51350
   300
  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
hoelzl@51350
   301
proof -
hoelzl@51350
   302
  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
hoelzl@51350
   303
    unfolding eventually_sequentially by (simp add: not_less)
hoelzl@51350
   304
  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
hoelzl@51350
   305
    by (auto simp: choice_iff)
hoelzl@51350
   306
  then show ?thesis
hoelzl@51350
   307
    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
hoelzl@51350
   308
             simp: less_eq_Suc_le subseq_Suc_iff)
hoelzl@51350
   309
qed
hoelzl@51350
   310
hoelzl@51350
   311
hoelzl@50883
   312
instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
hoelzl@50883
   313
proof
hoelzl@50883
   314
  fix x :: "'a \<times> 'b"
hoelzl@50883
   315
  from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this
hoelzl@50883
   316
  from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this
hoelzl@50883
   317
  show "\<exists>A::('a\<times>'b) set set. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
hoelzl@50883
   318
  proof (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
hoelzl@50883
   319
    fix a b assume x: "a \<in> A" "b \<in> B"
hoelzl@50883
   320
    with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"
hoelzl@50883
   321
      unfolding mem_Times_iff by (auto intro: open_Times)
hoelzl@50883
   322
  next
hoelzl@50883
   323
    fix S assume "open S" "x \<in> S"
hoelzl@50883
   324
    from open_prod_elim[OF this] guess a' b' .
hoelzl@50883
   325
    moreover with A(4)[of a'] B(4)[of b']
hoelzl@50883
   326
    obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'" by auto
hoelzl@50883
   327
    ultimately show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (A \<times> B). a \<subseteq> S"
hoelzl@50883
   328
      by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
hoelzl@50883
   329
  qed (simp add: A B)
hoelzl@50883
   330
qed
hoelzl@50883
   331
hoelzl@50883
   332
instance metric_space \<subseteq> first_countable_topology
hoelzl@50883
   333
proof
hoelzl@50883
   334
  fix x :: 'a
hoelzl@50883
   335
  show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
hoelzl@50883
   336
  proof (intro exI, safe)
hoelzl@50883
   337
    fix S assume "open S" "x \<in> S"
hoelzl@50883
   338
    then obtain r where "0 < r" "{y. dist x y < r} \<subseteq> S"
hoelzl@50883
   339
      by (auto simp: open_dist dist_commute subset_eq)
hoelzl@50883
   340
    moreover from reals_Archimedean[OF `0 < r`] guess n ..
hoelzl@50883
   341
    moreover
hoelzl@50883
   342
    then have "{y. dist x y < inverse (Suc n)} \<subseteq> {y. dist x y < r}"
hoelzl@50883
   343
      by (auto simp: inverse_eq_divide)
hoelzl@50883
   344
    ultimately show "\<exists>a\<in>range (\<lambda>n. {y. dist x y < inverse (Suc n)}). a \<subseteq> S"
hoelzl@50883
   345
      by auto
hoelzl@50883
   346
  qed (auto intro: open_ball)
hoelzl@50883
   347
qed
hoelzl@50883
   348
hoelzl@50881
   349
class second_countable_topology = topological_space +
hoelzl@51343
   350
  assumes ex_countable_subbasis: "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
hoelzl@51343
   351
begin
hoelzl@51343
   352
hoelzl@51343
   353
lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
hoelzl@51343
   354
proof -
hoelzl@51343
   355
  from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B" by blast
hoelzl@51343
   356
  let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
hoelzl@51343
   357
hoelzl@51343
   358
  show ?thesis
hoelzl@51343
   359
  proof (intro exI conjI)
hoelzl@51343
   360
    show "countable ?B"
hoelzl@51343
   361
      by (intro countable_image countable_Collect_finite_subset B)
hoelzl@51343
   362
    { fix S assume "open S"
hoelzl@51343
   363
      then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
hoelzl@51343
   364
        unfolding B
hoelzl@51343
   365
      proof induct
hoelzl@51343
   366
        case UNIV show ?case by (intro exI[of _ "{{}}"]) simp
hoelzl@51343
   367
      next
hoelzl@51343
   368
        case (Int a b)
hoelzl@51343
   369
        then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
hoelzl@51343
   370
          and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
hoelzl@51343
   371
          by blast
hoelzl@51343
   372
        show ?case
hoelzl@51343
   373
          unfolding x y Int_UN_distrib2
hoelzl@51343
   374
          by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
hoelzl@51343
   375
      next
hoelzl@51343
   376
        case (UN K)
hoelzl@51343
   377
        then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
hoelzl@51343
   378
        then guess k unfolding bchoice_iff ..
hoelzl@51343
   379
        then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
hoelzl@51343
   380
          by (intro exI[of _ "UNION K k"]) auto
hoelzl@51343
   381
      next
hoelzl@51343
   382
        case (Basis S) then show ?case
hoelzl@51343
   383
          by (intro exI[of _ "{{S}}"]) auto
hoelzl@51343
   384
      qed
hoelzl@51343
   385
      then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
hoelzl@51343
   386
        unfolding subset_image_iff by blast }
hoelzl@51343
   387
    then show "topological_basis ?B"
hoelzl@51343
   388
      unfolding topological_space_class.topological_basis_def
hoelzl@51343
   389
      by (safe intro!: topological_space_class.open_Inter) 
hoelzl@51343
   390
         (simp_all add: B generate_topology.Basis subset_eq)
hoelzl@51343
   391
  qed
hoelzl@51343
   392
qed
hoelzl@51343
   393
hoelzl@51343
   394
end
hoelzl@51343
   395
hoelzl@51343
   396
sublocale second_countable_topology <
hoelzl@51343
   397
  countable_basis "SOME B. countable B \<and> topological_basis B"
hoelzl@51343
   398
  using someI_ex[OF ex_countable_basis]
hoelzl@51343
   399
  by unfold_locales safe
immler@50094
   400
hoelzl@50882
   401
instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
hoelzl@50882
   402
proof
hoelzl@50882
   403
  obtain A :: "'a set set" where "countable A" "topological_basis A"
hoelzl@50882
   404
    using ex_countable_basis by auto
hoelzl@50882
   405
  moreover
hoelzl@50882
   406
  obtain B :: "'b set set" where "countable B" "topological_basis B"
hoelzl@50882
   407
    using ex_countable_basis by auto
hoelzl@51343
   408
  ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
hoelzl@51343
   409
    by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
hoelzl@51343
   410
      topological_basis_imp_subbasis)
hoelzl@50882
   411
qed
hoelzl@50882
   412
hoelzl@50883
   413
instance second_countable_topology \<subseteq> first_countable_topology
hoelzl@50883
   414
proof
hoelzl@50883
   415
  fix x :: 'a
hoelzl@50883
   416
  def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"
hoelzl@50883
   417
  then have B: "countable B" "topological_basis B"
hoelzl@50883
   418
    using countable_basis is_basis
hoelzl@50883
   419
    by (auto simp: countable_basis is_basis)
hoelzl@50883
   420
  then show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
hoelzl@50883
   421
    by (intro exI[of _ "{b\<in>B. x \<in> b}"])
hoelzl@50883
   422
       (fastforce simp: topological_space_class.topological_basis_def)
hoelzl@50883
   423
qed
hoelzl@50883
   424
immler@50087
   425
subsection {* Polish spaces *}
immler@50087
   426
immler@50087
   427
text {* Textbooks define Polish spaces as completely metrizable.
immler@50087
   428
  We assume the topology to be complete for a given metric. *}
immler@50087
   429
hoelzl@50881
   430
class polish_space = complete_space + second_countable_topology
immler@50087
   431
huffman@44517
   432
subsection {* General notion of a topology as a value *}
himmelma@33175
   433
huffman@44170
   434
definition "istopology L \<longleftrightarrow> 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@49834
   435
typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
himmelma@33175
   436
  morphisms "openin" "topology"
himmelma@33175
   437
  unfolding istopology_def by blast
himmelma@33175
   438
himmelma@33175
   439
lemma istopology_open_in[intro]: "istopology(openin U)"
himmelma@33175
   440
  using openin[of U] by blast
himmelma@33175
   441
himmelma@33175
   442
lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
huffman@44170
   443
  using topology_inverse[unfolded mem_Collect_eq] .
himmelma@33175
   444
himmelma@33175
   445
lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
himmelma@33175
   446
  using topology_inverse[of U] istopology_open_in[of "topology U"] by auto
himmelma@33175
   447
himmelma@33175
   448
lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
himmelma@33175
   449
proof-
wenzelm@49711
   450
  { assume "T1=T2"
wenzelm@49711
   451
    hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp }
himmelma@33175
   452
  moreover
wenzelm@49711
   453
  { assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
huffman@44170
   454
    hence "openin T1 = openin T2" by (simp add: fun_eq_iff)
himmelma@33175
   455
    hence "topology (openin T1) = topology (openin T2)" by simp
wenzelm@49711
   456
    hence "T1 = T2" unfolding openin_inverse .
wenzelm@49711
   457
  }
himmelma@33175
   458
  ultimately show ?thesis by blast
himmelma@33175
   459
qed
himmelma@33175
   460
himmelma@33175
   461
text{* Infer the "universe" from union of all sets in the topology. *}
himmelma@33175
   462
himmelma@33175
   463
definition "topspace T =  \<Union>{S. openin T S}"
himmelma@33175
   464
huffman@44210
   465
subsubsection {* Main properties of open sets *}
himmelma@33175
   466
himmelma@33175
   467
lemma openin_clauses:
himmelma@33175
   468
  fixes U :: "'a topology"
himmelma@33175
   469
  shows "openin U {}"
himmelma@33175
   470
  "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
himmelma@33175
   471
  "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
huffman@44170
   472
  using openin[of U] unfolding istopology_def mem_Collect_eq
huffman@44170
   473
  by fast+
himmelma@33175
   474
himmelma@33175
   475
lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
himmelma@33175
   476
  unfolding topspace_def by blast
himmelma@33175
   477
lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)
himmelma@33175
   478
himmelma@33175
   479
lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
huffman@36362
   480
  using openin_clauses by simp
huffman@36362
   481
huffman@36362
   482
lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
huffman@36362
   483
  using openin_clauses by simp
himmelma@33175
   484
himmelma@33175
   485
lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
himmelma@33175
   486
  using openin_Union[of "{S,T}" U] by auto
himmelma@33175
   487
himmelma@33175
   488
lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)
himmelma@33175
   489
wenzelm@49711
   490
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
   491
  (is "?lhs \<longleftrightarrow> ?rhs")
huffman@36584
   492
proof
wenzelm@49711
   493
  assume ?lhs
wenzelm@49711
   494
  then show ?rhs by auto
huffman@36584
   495
next
huffman@36584
   496
  assume H: ?rhs
huffman@36584
   497
  let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
huffman@36584
   498
  have "openin U ?t" by (simp add: openin_Union)
huffman@36584
   499
  also have "?t = S" using H by auto
huffman@36584
   500
  finally show "openin U S" .
himmelma@33175
   501
qed
himmelma@33175
   502
wenzelm@49711
   503
huffman@44210
   504
subsubsection {* Closed sets *}
himmelma@33175
   505
himmelma@33175
   506
definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
himmelma@33175
   507
himmelma@33175
   508
lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
himmelma@33175
   509
lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
himmelma@33175
   510
lemma closedin_topspace[intro,simp]:
himmelma@33175
   511
  "closedin U (topspace U)" by (simp add: closedin_def)
himmelma@33175
   512
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
himmelma@33175
   513
  by (auto simp add: Diff_Un closedin_def)
himmelma@33175
   514
himmelma@33175
   515
lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
himmelma@33175
   516
lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
himmelma@33175
   517
  shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto
himmelma@33175
   518
himmelma@33175
   519
lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
himmelma@33175
   520
  using closedin_Inter[of "{S,T}" U] by auto
himmelma@33175
   521
himmelma@33175
   522
lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
himmelma@33175
   523
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
himmelma@33175
   524
  apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
himmelma@33175
   525
  apply (metis openin_subset subset_eq)
himmelma@33175
   526
  done
himmelma@33175
   527
himmelma@33175
   528
lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
himmelma@33175
   529
  by (simp add: openin_closedin_eq)
himmelma@33175
   530
himmelma@33175
   531
lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
himmelma@33175
   532
proof-
himmelma@33175
   533
  have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
himmelma@33175
   534
    by (auto simp add: topspace_def openin_subset)
himmelma@33175
   535
  then show ?thesis using oS cT by (auto simp add: closedin_def)
himmelma@33175
   536
qed
himmelma@33175
   537
himmelma@33175
   538
lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
himmelma@33175
   539
proof-
himmelma@33175
   540
  have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
himmelma@33175
   541
    by (auto simp add: topspace_def )
himmelma@33175
   542
  then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
himmelma@33175
   543
qed
himmelma@33175
   544
huffman@44210
   545
subsubsection {* Subspace topology *}
huffman@44170
   546
huffman@44170
   547
definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
huffman@44170
   548
huffman@44170
   549
lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
huffman@44170
   550
  (is "istopology ?L")
himmelma@33175
   551
proof-
huffman@44170
   552
  have "?L {}" by blast
huffman@44170
   553
  {fix A B assume A: "?L A" and B: "?L B"
himmelma@33175
   554
    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" by blast
himmelma@33175
   555
    have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
huffman@44170
   556
    then have "?L (A \<inter> B)" by blast}
himmelma@33175
   557
  moreover
huffman@44170
   558
  {fix K assume K: "K \<subseteq> Collect ?L"
huffman@44170
   559
    have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
nipkow@39302
   560
      apply (rule set_eqI)
himmelma@33175
   561
      apply (simp add: Ball_def image_iff)
huffman@44170
   562
      by metis
himmelma@33175
   563
    from K[unfolded th0 subset_image_iff]
huffman@44170
   564
    obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk" by blast
himmelma@33175
   565
    have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
huffman@44170
   566
    moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)
huffman@44170
   567
    ultimately have "?L (\<Union>K)" by blast}
huffman@44170
   568
  ultimately show ?thesis
huffman@44170
   569
    unfolding subset_eq mem_Collect_eq istopology_def by blast
himmelma@33175
   570
qed
himmelma@33175
   571
himmelma@33175
   572
lemma openin_subtopology:
himmelma@33175
   573
  "openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
himmelma@33175
   574
  unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
huffman@44170
   575
  by auto
himmelma@33175
   576
himmelma@33175
   577
lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
himmelma@33175
   578
  by (auto simp add: topspace_def openin_subtopology)
himmelma@33175
   579
himmelma@33175
   580
lemma closedin_subtopology:
himmelma@33175
   581
  "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
himmelma@33175
   582
  unfolding closedin_def topspace_subtopology
himmelma@33175
   583
  apply (simp add: openin_subtopology)
himmelma@33175
   584
  apply (rule iffI)
himmelma@33175
   585
  apply clarify
himmelma@33175
   586
  apply (rule_tac x="topspace U - T" in exI)
himmelma@33175
   587
  by auto
himmelma@33175
   588
himmelma@33175
   589
lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
himmelma@33175
   590
  unfolding openin_subtopology
himmelma@33175
   591
  apply (rule iffI, clarify)
himmelma@33175
   592
  apply (frule openin_subset[of U])  apply blast
himmelma@33175
   593
  apply (rule exI[where x="topspace U"])
wenzelm@49711
   594
  apply auto
wenzelm@49711
   595
  done
wenzelm@49711
   596
wenzelm@49711
   597
lemma subtopology_superset:
wenzelm@49711
   598
  assumes UV: "topspace U \<subseteq> V"
himmelma@33175
   599
  shows "subtopology U V = U"
himmelma@33175
   600
proof-
himmelma@33175
   601
  {fix S
himmelma@33175
   602
    {fix T assume T: "openin U T" "S = T \<inter> V"
himmelma@33175
   603
      from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
himmelma@33175
   604
      have "openin U S" unfolding eq using T by blast}
himmelma@33175
   605
    moreover
himmelma@33175
   606
    {assume S: "openin U S"
himmelma@33175
   607
      hence "\<exists>T. openin U T \<and> S = T \<inter> V"
himmelma@33175
   608
        using openin_subset[OF S] UV by auto}
himmelma@33175
   609
    ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
himmelma@33175
   610
  then show ?thesis unfolding topology_eq openin_subtopology by blast
himmelma@33175
   611
qed
himmelma@33175
   612
himmelma@33175
   613
lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
himmelma@33175
   614
  by (simp add: subtopology_superset)
himmelma@33175
   615
himmelma@33175
   616
lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
himmelma@33175
   617
  by (simp add: subtopology_superset)
himmelma@33175
   618
huffman@44210
   619
subsubsection {* The standard Euclidean topology *}
himmelma@33175
   620
himmelma@33175
   621
definition
himmelma@33175
   622
  euclidean :: "'a::topological_space topology" where
himmelma@33175
   623
  "euclidean = topology open"
himmelma@33175
   624
himmelma@33175
   625
lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
himmelma@33175
   626
  unfolding euclidean_def
himmelma@33175
   627
  apply (rule cong[where x=S and y=S])
himmelma@33175
   628
  apply (rule topology_inverse[symmetric])
himmelma@33175
   629
  apply (auto simp add: istopology_def)
huffman@44170
   630
  done
himmelma@33175
   631
himmelma@33175
   632
lemma topspace_euclidean: "topspace euclidean = UNIV"
himmelma@33175
   633
  apply (simp add: topspace_def)
nipkow@39302
   634
  apply (rule set_eqI)
himmelma@33175
   635
  by (auto simp add: open_openin[symmetric])
himmelma@33175
   636
himmelma@33175
   637
lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
himmelma@33175
   638
  by (simp add: topspace_euclidean topspace_subtopology)
himmelma@33175
   639
himmelma@33175
   640
lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
himmelma@33175
   641
  by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)
himmelma@33175
   642
himmelma@33175
   643
lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
himmelma@33175
   644
  by (simp add: open_openin openin_subopen[symmetric])
himmelma@33175
   645
huffman@44210
   646
text {* Basic "localization" results are handy for connectedness. *}
huffman@44210
   647
huffman@44210
   648
lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
huffman@44210
   649
  by (auto simp add: openin_subtopology open_openin[symmetric])
huffman@44210
   650
huffman@44210
   651
lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
huffman@44210
   652
  by (auto simp add: openin_open)
huffman@44210
   653
huffman@44210
   654
lemma open_openin_trans[trans]:
huffman@44210
   655
 "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
huffman@44210
   656
  by (metis Int_absorb1  openin_open_Int)
huffman@44210
   657
huffman@44210
   658
lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
huffman@44210
   659
  by (auto simp add: openin_open)
huffman@44210
   660
huffman@44210
   661
lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
huffman@44210
   662
  by (simp add: closedin_subtopology closed_closedin Int_ac)
huffman@44210
   663
huffman@44210
   664
lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
huffman@44210
   665
  by (metis closedin_closed)
huffman@44210
   666
huffman@44210
   667
lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
huffman@44210
   668
  apply (subgoal_tac "S \<inter> T = T" )
huffman@44210
   669
  apply auto
huffman@44210
   670
  apply (frule closedin_closed_Int[of T S])
huffman@44210
   671
  by simp
huffman@44210
   672
huffman@44210
   673
lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
huffman@44210
   674
  by (auto simp add: closedin_closed)
huffman@44210
   675
huffman@44210
   676
lemma openin_euclidean_subtopology_iff:
huffman@44210
   677
  fixes S U :: "'a::metric_space set"
huffman@44210
   678
  shows "openin (subtopology euclidean U) S
huffman@44210
   679
  \<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
huffman@44210
   680
proof
huffman@44210
   681
  assume ?lhs thus ?rhs unfolding openin_open open_dist by blast
huffman@44210
   682
next
huffman@44210
   683
  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
   684
  have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
huffman@44210
   685
    unfolding T_def
huffman@44210
   686
    apply clarsimp
huffman@44210
   687
    apply (rule_tac x="d - dist x a" in exI)
huffman@44210
   688
    apply (clarsimp simp add: less_diff_eq)
huffman@44210
   689
    apply (erule rev_bexI)
huffman@44210
   690
    apply (rule_tac x=d in exI, clarify)
huffman@44210
   691
    apply (erule le_less_trans [OF dist_triangle])
huffman@44210
   692
    done
huffman@44210
   693
  assume ?rhs hence 2: "S = U \<inter> T"
huffman@44210
   694
    unfolding T_def
huffman@44210
   695
    apply auto
huffman@44210
   696
    apply (drule (1) bspec, erule rev_bexI)
huffman@44210
   697
    apply auto
huffman@44210
   698
    done
huffman@44210
   699
  from 1 2 show ?lhs
huffman@44210
   700
    unfolding openin_open open_dist by fast
huffman@44210
   701
qed
huffman@44210
   702
huffman@44210
   703
text {* These "transitivity" results are handy too *}
huffman@44210
   704
huffman@44210
   705
lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
huffman@44210
   706
  \<Longrightarrow> openin (subtopology euclidean U) S"
huffman@44210
   707
  unfolding open_openin openin_open by blast
huffman@44210
   708
huffman@44210
   709
lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
huffman@44210
   710
  by (auto simp add: openin_open intro: openin_trans)
huffman@44210
   711
huffman@44210
   712
lemma closedin_trans[trans]:
huffman@44210
   713
 "closedin (subtopology euclidean T) S \<Longrightarrow>
huffman@44210
   714
           closedin (subtopology euclidean U) T
huffman@44210
   715
           ==> closedin (subtopology euclidean U) S"
huffman@44210
   716
  by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)
huffman@44210
   717
huffman@44210
   718
lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
huffman@44210
   719
  by (auto simp add: closedin_closed intro: closedin_trans)
huffman@44210
   720
huffman@44210
   721
huffman@44210
   722
subsection {* Open and closed balls *}
himmelma@33175
   723
himmelma@33175
   724
definition
himmelma@33175
   725
  ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
himmelma@33175
   726
  "ball x e = {y. dist x y < e}"
himmelma@33175
   727
himmelma@33175
   728
definition
himmelma@33175
   729
  cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
himmelma@33175
   730
  "cball x e = {y. dist x y \<le> e}"
himmelma@33175
   731
huffman@45776
   732
lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
huffman@45776
   733
  by (simp add: ball_def)
huffman@45776
   734
huffman@45776
   735
lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
huffman@45776
   736
  by (simp add: cball_def)
huffman@45776
   737
huffman@45776
   738
lemma mem_ball_0:
himmelma@33175
   739
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   740
  shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
himmelma@33175
   741
  by (simp add: dist_norm)
himmelma@33175
   742
huffman@45776
   743
lemma mem_cball_0:
himmelma@33175
   744
  fixes x :: "'a::real_normed_vector"
himmelma@33175
   745
  shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
himmelma@33175
   746
  by (simp add: dist_norm)
himmelma@33175
   747
huffman@45776
   748
lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"
huffman@45776
   749
  by simp
huffman@45776
   750
huffman@45776
   751
lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
huffman@45776
   752
  by simp
huffman@45776
   753
himmelma@33175
   754
lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
himmelma@33175
   755
lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
himmelma@33175
   756
lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
himmelma@33175
   757
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
nipkow@39302
   758
  by (simp add: set_eq_iff) arith
himmelma@33175
   759
himmelma@33175
   760
lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
nipkow@39302
   761
  by (simp add: set_eq_iff)
himmelma@33175
   762
himmelma@33175
   763
lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
himmelma@33175
   764
  "(a::real) - b < 0 \<longleftrightarrow> a < b"
himmelma@33175
   765
  "a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
himmelma@33175
   766
lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
himmelma@33175
   767
  "a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+
himmelma@33175
   768
himmelma@33175
   769
lemma open_ball[intro, simp]: "open (ball x e)"
huffman@44170
   770
  unfolding open_dist ball_def mem_Collect_eq Ball_def
himmelma@33175
   771
  unfolding dist_commute
himmelma@33175
   772
  apply clarify
himmelma@33175
   773
  apply (rule_tac x="e - dist xa x" in exI)
himmelma@33175
   774
  using dist_triangle_alt[where z=x]
himmelma@33175
   775
  apply (clarsimp simp add: diff_less_iff)
himmelma@33175
   776
  apply atomize
himmelma@33175
   777
  apply (erule_tac x="y" in allE)
himmelma@33175
   778
  apply (erule_tac x="xa" in allE)
himmelma@33175
   779
  by arith
himmelma@33175
   780
himmelma@33175
   781
lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   782
  unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..
himmelma@33175
   783
hoelzl@33714
   784
lemma openE[elim?]:
hoelzl@33714
   785
  assumes "open S" "x\<in>S" 
hoelzl@33714
   786
  obtains e where "e>0" "ball x e \<subseteq> S"
hoelzl@33714
   787
  using assms unfolding open_contains_ball by auto
hoelzl@33714
   788
himmelma@33175
   789
lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
himmelma@33175
   790
  by (metis open_contains_ball subset_eq centre_in_ball)
himmelma@33175
   791
himmelma@33175
   792
lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
nipkow@39302
   793
  unfolding mem_ball set_eq_iff
himmelma@33175
   794
  apply (simp add: not_less)
himmelma@33175
   795
  by (metis zero_le_dist order_trans dist_self)
himmelma@33175
   796
himmelma@33175
   797
lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp
himmelma@33175
   798
hoelzl@50526
   799
lemma euclidean_dist_l2:
hoelzl@50526
   800
  fixes x y :: "'a :: euclidean_space"
hoelzl@50526
   801
  shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
hoelzl@50526
   802
  unfolding dist_norm norm_eq_sqrt_inner setL2_def
hoelzl@50526
   803
  by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)
hoelzl@50526
   804
hoelzl@50526
   805
definition "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"
hoelzl@50526
   806
immler@50087
   807
lemma rational_boxes:
hoelzl@50526
   808
  fixes x :: "'a\<Colon>euclidean_space"
immler@50087
   809
  assumes "0 < e"
hoelzl@50526
   810
  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
   811
proof -
immler@50087
   812
  def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
hoelzl@50526
   813
  then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive)
hoelzl@50526
   814
  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
   815
  proof
hoelzl@50526
   816
    fix i from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e show "?th i" by auto
immler@50087
   817
  qed
immler@50087
   818
  from choice[OF this] guess a .. note a = this
hoelzl@50526
   819
  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
   820
  proof
hoelzl@50526
   821
    fix i from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e show "?th i" by auto
immler@50087
   822
  qed
immler@50087
   823
  from choice[OF this] guess b .. note b = this
hoelzl@50526
   824
  let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
hoelzl@50526
   825
  show ?thesis
hoelzl@50526
   826
  proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
hoelzl@50526
   827
    fix y :: 'a assume *: "y \<in> box ?a ?b"
hoelzl@50526
   828
    have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<twosuperior>)"
immler@50087
   829
      unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
hoelzl@50526
   830
    also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
immler@50087
   831
    proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
hoelzl@50526
   832
      fix i :: "'a" assume i: "i \<in> Basis"
hoelzl@50526
   833
      have "a i < y\<bullet>i \<and> y\<bullet>i < b i" using * i by (auto simp: box_def)
hoelzl@50526
   834
      moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'" using a by auto
hoelzl@50526
   835
      moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'" using b by auto
hoelzl@50526
   836
      ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'" by auto
hoelzl@50526
   837
      then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
immler@50087
   838
        unfolding e'_def by (auto simp: dist_real_def)
hoelzl@50526
   839
      then have "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
immler@50087
   840
        by (rule power_strict_mono) auto
hoelzl@50526
   841
      then show "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
immler@50087
   842
        by (simp add: power_divide)
immler@50087
   843
    qed auto
hoelzl@50526
   844
    also have "\<dots> = e" using `0 < e` by (simp add: real_eq_of_nat)
hoelzl@50526
   845
    finally show "y \<in> ball x e" by (auto simp: ball_def)
hoelzl@50526
   846
  qed (insert a b, auto simp: box_def)
hoelzl@50526
   847
qed
immler@51103
   848
hoelzl@50526
   849
lemma open_UNION_box:
hoelzl@50526
   850
  fixes M :: "'a\<Colon>euclidean_space set"
hoelzl@50526
   851
  assumes "open M" 
hoelzl@50526
   852
  defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
hoelzl@50526
   853
  defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
hoelzl@50526
   854
  defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^isub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
hoelzl@50526
   855
  shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
immler@50087
   856
proof safe
immler@50087
   857
  fix x assume "x \<in> M"
immler@50087
   858
  obtain e where e: "e > 0" "ball x e \<subseteq> M"
hoelzl@50526
   859
    using openE[OF `open M` `x \<in> M`] by auto
hoelzl@50526
   860
  moreover then obtain a b where ab: "x \<in> box a b"
hoelzl@50526
   861
    "\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"
hoelzl@50526
   862
    using rational_boxes[OF e(1)] by metis
hoelzl@50526
   863
  ultimately show "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"
hoelzl@50526
   864
     by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
hoelzl@50526
   865
        (auto simp: euclidean_representation I_def a'_def b'_def)
hoelzl@50526
   866
qed (auto simp: I_def)
himmelma@33175
   867
himmelma@33175
   868
subsection{* Connectedness *}
himmelma@33175
   869
himmelma@33175
   870
definition "connected S \<longleftrightarrow>
himmelma@33175
   871
  ~(\<exists>e1 e2. open e1 \<and> open e2 \<and> S \<subseteq> (e1 \<union> e2) \<and> (e1 \<inter> e2 \<inter> S = {})
himmelma@33175
   872
  \<and> ~(e1 \<inter> S = {}) \<and> ~(e2 \<inter> S = {}))"
himmelma@33175
   873
himmelma@33175
   874
lemma connected_local:
himmelma@33175
   875
 "connected S \<longleftrightarrow> ~(\<exists>e1 e2.
himmelma@33175
   876
                 openin (subtopology euclidean S) e1 \<and>
himmelma@33175
   877
                 openin (subtopology euclidean S) e2 \<and>
himmelma@33175
   878
                 S \<subseteq> e1 \<union> e2 \<and>
himmelma@33175
   879
                 e1 \<inter> e2 = {} \<and>
himmelma@33175
   880
                 ~(e1 = {}) \<and>
himmelma@33175
   881
                 ~(e2 = {}))"
himmelma@33175
   882
unfolding connected_def openin_open by (safe, blast+)
himmelma@33175
   883
huffman@34105
   884
lemma exists_diff:
huffman@34105
   885
  fixes P :: "'a set \<Rightarrow> bool"
huffman@34105
   886
  shows "(\<exists>S. P(- S)) \<longleftrightarrow> (\<exists>S. P S)" (is "?lhs \<longleftrightarrow> ?rhs")
himmelma@33175
   887
proof-
himmelma@33175
   888
  {assume "?lhs" hence ?rhs by blast }
himmelma@33175
   889
  moreover
himmelma@33175
   890
  {fix S assume H: "P S"
huffman@34105
   891
    have "S = - (- S)" by auto
huffman@34105
   892
    with H have "P (- (- S))" by metis }
himmelma@33175
   893
  ultimately show ?thesis by metis
himmelma@33175
   894
qed
himmelma@33175
   895
himmelma@33175
   896
lemma connected_clopen: "connected S \<longleftrightarrow>
himmelma@33175
   897
        (\<forall>T. openin (subtopology euclidean S) T \<and>
himmelma@33175
   898
            closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
himmelma@33175
   899
proof-
huffman@34105
   900
  have " \<not> connected S \<longleftrightarrow> (\<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
   901
    unfolding connected_def openin_open closedin_closed
himmelma@33175
   902
    apply (subst exists_diff) by blast
huffman@34105
   903
  hence th0: "connected S \<longleftrightarrow> \<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> {})"
huffman@34105
   904
    (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)") apply (simp add: closed_def) by metis
himmelma@33175
   905
himmelma@33175
   906
  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
   907
    (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
himmelma@33175
   908
    unfolding connected_def openin_open closedin_closed by auto
himmelma@33175
   909
  {fix e2
himmelma@33175
   910
    {fix e1 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)"
himmelma@33175
   911
        by auto}
himmelma@33175
   912
    then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
himmelma@33175
   913
  then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
himmelma@33175
   914
  then show ?thesis unfolding th0 th1 by simp
himmelma@33175
   915
qed
himmelma@33175
   916
himmelma@33175
   917
lemma connected_empty[simp, intro]: "connected {}"
himmelma@33175
   918
  by (simp add: connected_def)
himmelma@33175
   919
huffman@44210
   920
himmelma@33175
   921
subsection{* Limit points *}
himmelma@33175
   922
huffman@44207
   923
definition (in topological_space)
huffman@44207
   924
  islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
himmelma@33175
   925
  "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
   926
himmelma@33175
   927
lemma islimptI:
himmelma@33175
   928
  assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
himmelma@33175
   929
  shows "x islimpt S"
himmelma@33175
   930
  using assms unfolding islimpt_def by auto
himmelma@33175
   931
himmelma@33175
   932
lemma islimptE:
himmelma@33175
   933
  assumes "x islimpt S" and "x \<in> T" and "open T"
himmelma@33175
   934
  obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
himmelma@33175
   935
  using assms unfolding islimpt_def by auto
himmelma@33175
   936
huffman@44584
   937
lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
huffman@44584
   938
  unfolding islimpt_def eventually_at_topological by auto
huffman@44584
   939
huffman@44584
   940
lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"
huffman@44584
   941
  unfolding islimpt_def by fast
himmelma@33175
   942
himmelma@33175
   943
lemma islimpt_approachable:
himmelma@33175
   944
  fixes x :: "'a::metric_space"
himmelma@33175
   945
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
huffman@44584
   946
  unfolding islimpt_iff_eventually eventually_at by fast
himmelma@33175
   947
himmelma@33175
   948
lemma islimpt_approachable_le:
himmelma@33175
   949
  fixes x :: "'a::metric_space"
himmelma@33175
   950
  shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
himmelma@33175
   951
  unfolding islimpt_approachable
huffman@44584
   952
  using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
huffman@44584
   953
    THEN arg_cong [where f=Not]]
huffman@44584
   954
  by (simp add: Bex_def conj_commute conj_left_commute)
himmelma@33175
   955
huffman@44571
   956
lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
huffman@44571
   957
  unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
huffman@44571
   958
hoelzl@51351
   959
lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
hoelzl@51351
   960
  unfolding islimpt_def by blast
hoelzl@51351
   961
huffman@44210
   962
text {* A perfect space has no isolated points. *}
huffman@44210
   963
huffman@44571
   964
lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
huffman@44571
   965
  unfolding islimpt_UNIV_iff by (rule not_open_singleton)
himmelma@33175
   966
himmelma@33175
   967
lemma perfect_choose_dist:
huffman@44072
   968
  fixes x :: "'a::{perfect_space, metric_space}"
himmelma@33175
   969
  shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
himmelma@33175
   970
using islimpt_UNIV [of x]
himmelma@33175
   971
by (simp add: islimpt_approachable)
himmelma@33175
   972
himmelma@33175
   973
lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
himmelma@33175
   974
  unfolding closed_def
himmelma@33175
   975
  apply (subst open_subopen)
huffman@34105
   976
  apply (simp add: islimpt_def subset_eq)
huffman@44170
   977
  by (metis ComplE ComplI)
himmelma@33175
   978
himmelma@33175
   979
lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
himmelma@33175
   980
  unfolding islimpt_def by auto
himmelma@33175
   981
himmelma@33175
   982
lemma finite_set_avoid:
himmelma@33175
   983
  fixes a :: "'a::metric_space"
himmelma@33175
   984
  assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
himmelma@33175
   985
proof(induct rule: finite_induct[OF fS])
boehmes@41863
   986
  case 1 thus ?case by (auto intro: zero_less_one)
himmelma@33175
   987
next
himmelma@33175
   988
  case (2 x F)
himmelma@33175
   989
  from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
himmelma@33175
   990
  {assume "x = a" hence ?case using d by auto  }
himmelma@33175
   991
  moreover
himmelma@33175
   992
  {assume xa: "x\<noteq>a"
himmelma@33175
   993
    let ?d = "min d (dist a x)"
himmelma@33175
   994
    have dp: "?d > 0" using xa d(1) using dist_nz by auto
himmelma@33175
   995
    from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
himmelma@33175
   996
    with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
himmelma@33175
   997
  ultimately show ?case by blast
himmelma@33175
   998
qed
himmelma@33175
   999
himmelma@33175
  1000
lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
huffman@50897
  1001
  by (simp add: islimpt_iff_eventually eventually_conj_iff)
himmelma@33175
  1002
himmelma@33175
  1003
lemma discrete_imp_closed:
himmelma@33175
  1004
  fixes S :: "'a::metric_space set"
himmelma@33175
  1005
  assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
himmelma@33175
  1006
  shows "closed S"
himmelma@33175
  1007
proof-
himmelma@33175
  1008
  {fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
himmelma@33175
  1009
    from e have e2: "e/2 > 0" by arith
himmelma@33175
  1010
    from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
himmelma@33175
  1011
    let ?m = "min (e/2) (dist x y) "
himmelma@33175
  1012
    from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
himmelma@33175
  1013
    from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
himmelma@33175
  1014
    have th: "dist z y < e" using z y
himmelma@33175
  1015
      by (intro dist_triangle_lt [where z=x], simp)
himmelma@33175
  1016
    from d[rule_format, OF y(1) z(1) th] y z
himmelma@33175
  1017
    have False by (auto simp add: dist_commute)}
himmelma@33175
  1018
  then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
himmelma@33175
  1019
qed
himmelma@33175
  1020
huffman@44210
  1021
huffman@44210
  1022
subsection {* Interior of a Set *}
huffman@44210
  1023
huffman@44519
  1024
definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
huffman@44519
  1025
huffman@44519
  1026
lemma interiorI [intro?]:
huffman@44519
  1027
  assumes "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
  1028
  shows "x \<in> interior S"
huffman@44519
  1029
  using assms unfolding interior_def by fast
huffman@44519
  1030
huffman@44519
  1031
lemma interiorE [elim?]:
huffman@44519
  1032
  assumes "x \<in> interior S"
huffman@44519
  1033
  obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
huffman@44519
  1034
  using assms unfolding interior_def by fast
huffman@44519
  1035
huffman@44519
  1036
lemma open_interior [simp, intro]: "open (interior S)"
huffman@44519
  1037
  by (simp add: interior_def open_Union)
huffman@44519
  1038
huffman@44519
  1039
lemma interior_subset: "interior S \<subseteq> S"
huffman@44519
  1040
  by (auto simp add: interior_def)
huffman@44519
  1041
huffman@44519
  1042
lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
huffman@44519
  1043
  by (auto simp add: interior_def)
huffman@44519
  1044
huffman@44519
  1045
lemma interior_open: "open S \<Longrightarrow> interior S = S"
huffman@44519
  1046
  by (intro equalityI interior_subset interior_maximal subset_refl)
himmelma@33175
  1047
himmelma@33175
  1048
lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
huffman@44519
  1049
  by (metis open_interior interior_open)
huffman@44519
  1050
huffman@44519
  1051
lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
himmelma@33175
  1052
  by (metis interior_maximal interior_subset subset_trans)
himmelma@33175
  1053
huffman@44519
  1054
lemma interior_empty [simp]: "interior {} = {}"
huffman@44519
  1055
  using open_empty by (rule interior_open)
huffman@44519
  1056
huffman@44522
  1057
lemma interior_UNIV [simp]: "interior UNIV = UNIV"
huffman@44522
  1058
  using open_UNIV by (rule interior_open)
huffman@44522
  1059
huffman@44519
  1060
lemma interior_interior [simp]: "interior (interior S) = interior S"
huffman@44519
  1061
  using open_interior by (rule interior_open)
huffman@44519
  1062
huffman@44522
  1063
lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
huffman@44522
  1064
  by (auto simp add: interior_def)
huffman@44519
  1065
huffman@44519
  1066
lemma interior_unique:
huffman@44519
  1067
  assumes "T \<subseteq> S" and "open T"
huffman@44519
  1068
  assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
huffman@44519
  1069
  shows "interior S = T"
huffman@44519
  1070
  by (intro equalityI assms interior_subset open_interior interior_maximal)
huffman@44519
  1071
huffman@44519
  1072
lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
huffman@44522
  1073
  by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
huffman@44519
  1074
    Int_lower2 interior_maximal interior_subset open_Int open_interior)
huffman@44519
  1075
huffman@44519
  1076
lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
huffman@44519
  1077
  using open_contains_ball_eq [where S="interior S"]
huffman@44519
  1078
  by (simp add: open_subset_interior)
himmelma@33175
  1079
himmelma@33175
  1080
lemma interior_limit_point [intro]:
himmelma@33175
  1081
  fixes x :: "'a::perfect_space"
himmelma@33175
  1082
  assumes x: "x \<in> interior S" shows "x islimpt S"
huffman@44072
  1083
  using x islimpt_UNIV [of x]
huffman@44072
  1084
  unfolding interior_def islimpt_def
huffman@44072
  1085
  apply (clarsimp, rename_tac T T')
huffman@44072
  1086
  apply (drule_tac x="T \<inter> T'" in spec)
huffman@44072
  1087
  apply (auto simp add: open_Int)
huffman@44072
  1088
  done
himmelma@33175
  1089
himmelma@33175
  1090
lemma interior_closed_Un_empty_interior:
himmelma@33175
  1091
  assumes cS: "closed S" and iT: "interior T = {}"
huffman@44519
  1092
  shows "interior (S \<union> T) = interior S"
himmelma@33175
  1093
proof
huffman@44519
  1094
  show "interior S \<subseteq> interior (S \<union> T)"
huffman@44522
  1095
    by (rule interior_mono, rule Un_upper1)
himmelma@33175
  1096
next
himmelma@33175
  1097
  show "interior (S \<union> T) \<subseteq> interior S"
himmelma@33175
  1098
  proof
himmelma@33175
  1099
    fix x assume "x \<in> interior (S \<union> T)"
huffman@44519
  1100
    then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
himmelma@33175
  1101
    show "x \<in> interior S"
himmelma@33175
  1102
    proof (rule ccontr)
himmelma@33175
  1103
      assume "x \<notin> interior S"
himmelma@33175
  1104
      with `x \<in> R` `open R` obtain y where "y \<in> R - S"
huffman@44519
  1105
        unfolding interior_def by fast
himmelma@33175
  1106
      from `open R` `closed S` have "open (R - S)" by (rule open_Diff)
himmelma@33175
  1107
      from `R \<subseteq> S \<union> T` have "R - S \<subseteq> T" by fast
himmelma@33175
  1108
      from `y \<in> R - S` `open (R - S)` `R - S \<subseteq> T` `interior T = {}`
himmelma@33175
  1109
      show "False" unfolding interior_def by fast
himmelma@33175
  1110
    qed
himmelma@33175
  1111
  qed
himmelma@33175
  1112
qed
himmelma@33175
  1113
huffman@44365
  1114
lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
huffman@44365
  1115
proof (rule interior_unique)
huffman@44365
  1116
  show "interior A \<times> interior B \<subseteq> A \<times> B"
huffman@44365
  1117
    by (intro Sigma_mono interior_subset)
huffman@44365
  1118
  show "open (interior A \<times> interior B)"
huffman@44365
  1119
    by (intro open_Times open_interior)
huffman@44519
  1120
  fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"
huffman@44519
  1121
  proof (safe)
huffman@44519
  1122
    fix x y assume "(x, y) \<in> T"
huffman@44519
  1123
    then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
huffman@44519
  1124
      using `open T` unfolding open_prod_def by fast
huffman@44519
  1125
    hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
huffman@44519
  1126
      using `T \<subseteq> A \<times> B` by auto
huffman@44519
  1127
    thus "x \<in> interior A" and "y \<in> interior B"
huffman@44519
  1128
      by (auto intro: interiorI)
huffman@44519
  1129
  qed
huffman@44365
  1130
qed
huffman@44365
  1131
himmelma@33175
  1132
huffman@44210
  1133
subsection {* Closure of a Set *}
himmelma@33175
  1134
himmelma@33175
  1135
definition "closure S = S \<union> {x | x. x islimpt S}"
himmelma@33175
  1136
huffman@44518
  1137
lemma interior_closure: "interior S = - (closure (- S))"
huffman@44518
  1138
  unfolding interior_def closure_def islimpt_def by auto
huffman@44518
  1139
huffman@34105
  1140
lemma closure_interior: "closure S = - interior (- S)"
huffman@44518
  1141
  unfolding interior_closure by simp
himmelma@33175
  1142
himmelma@33175
  1143
lemma closed_closure[simp, intro]: "closed (closure S)"
huffman@44518
  1144
  unfolding closure_interior by (simp add: closed_Compl)
huffman@44518
  1145
huffman@44518
  1146
lemma closure_subset: "S \<subseteq> closure S"
huffman@44518
  1147
  unfolding closure_def by simp
himmelma@33175
  1148
himmelma@33175
  1149
lemma closure_hull: "closure S = closed hull S"
huffman@44519
  1150
  unfolding hull_def closure_interior interior_def by auto
himmelma@33175
  1151
himmelma@33175
  1152
lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
huffman@44519
  1153
  unfolding closure_hull using closed_Inter by (rule hull_eq)
huffman@44519
  1154
huffman@44519
  1155
lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
huffman@44519
  1156
  unfolding closure_eq .
huffman@44519
  1157
huffman@44519
  1158
lemma closure_closure [simp]: "closure (closure S) = closure S"
huffman@44518
  1159
  unfolding closure_hull by (rule hull_hull)
himmelma@33175
  1160
huffman@44522
  1161
lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
huffman@44518
  1162
  unfolding closure_hull by (rule hull_mono)
himmelma@33175
  1163
huffman@44519
  1164
lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
huffman@44518
  1165
  unfolding closure_hull by (rule hull_minimal)
himmelma@33175
  1166
huffman@44519
  1167
lemma closure_unique:
huffman@44519
  1168
  assumes "S \<subseteq> T" and "closed T"
huffman@44519
  1169
  assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
huffman@44519
  1170
  shows "closure S = T"
huffman@44519
  1171
  using assms unfolding closure_hull by (rule hull_unique)
huffman@44519
  1172
huffman@44519
  1173
lemma closure_empty [simp]: "closure {} = {}"
huffman@44518
  1174
  using closed_empty by (rule closure_closed)
himmelma@33175
  1175
huffman@44522
  1176
lemma closure_UNIV [simp]: "closure UNIV = UNIV"
huffman@44518
  1177
  using closed_UNIV by (rule closure_closed)
huffman@44518
  1178
huffman@44518
  1179
lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
huffman@44518
  1180
  unfolding closure_interior by simp
himmelma@33175
  1181
himmelma@33175
  1182
lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
himmelma@33175
  1183
  using closure_empty closure_subset[of S]
himmelma@33175
  1184
  by blast
himmelma@33175
  1185
himmelma@33175
  1186
lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
  1187
  using closure_eq[of S] closure_subset[of S]
himmelma@33175
  1188
  by simp
himmelma@33175
  1189
himmelma@33175
  1190
lemma open_inter_closure_eq_empty:
himmelma@33175
  1191
  "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
huffman@34105
  1192
  using open_subset_interior[of S "- T"]
huffman@34105
  1193
  using interior_subset[of "- T"]
himmelma@33175
  1194
  unfolding closure_interior
himmelma@33175
  1195
  by auto
himmelma@33175
  1196
himmelma@33175
  1197
lemma open_inter_closure_subset:
himmelma@33175
  1198
  "open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
himmelma@33175
  1199
proof
himmelma@33175
  1200
  fix x
himmelma@33175
  1201
  assume as: "open S" "x \<in> S \<inter> closure T"
himmelma@33175
  1202
  { assume *:"x islimpt T"
himmelma@33175
  1203
    have "x islimpt (S \<inter> T)"
himmelma@33175
  1204
    proof (rule islimptI)
himmelma@33175
  1205
      fix A
himmelma@33175
  1206
      assume "x \<in> A" "open A"
himmelma@33175
  1207
      with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
himmelma@33175
  1208
        by (simp_all add: open_Int)
himmelma@33175
  1209
      with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
himmelma@33175
  1210
        by (rule islimptE)
himmelma@33175
  1211
      hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
himmelma@33175
  1212
        by simp_all
himmelma@33175
  1213
      thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
himmelma@33175
  1214
    qed
himmelma@33175
  1215
  }
himmelma@33175
  1216
  then show "x \<in> closure (S \<inter> T)" using as
himmelma@33175
  1217
    unfolding closure_def
himmelma@33175
  1218
    by blast
himmelma@33175
  1219
qed
himmelma@33175
  1220
huffman@44519
  1221
lemma closure_complement: "closure (- S) = - interior S"
huffman@44518
  1222
  unfolding closure_interior by simp
himmelma@33175
  1223
huffman@44519
  1224
lemma interior_complement: "interior (- S) = - closure S"
huffman@44518
  1225
  unfolding closure_interior by simp
himmelma@33175
  1226
huffman@44365
  1227
lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
huffman@44519
  1228
proof (rule closure_unique)
huffman@44365
  1229
  show "A \<times> B \<subseteq> closure A \<times> closure B"
huffman@44365
  1230
    by (intro Sigma_mono closure_subset)
huffman@44365
  1231
  show "closed (closure A \<times> closure B)"
huffman@44365
  1232
    by (intro closed_Times closed_closure)
huffman@44519
  1233
  fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"
huffman@44365
  1234
    apply (simp add: closed_def open_prod_def, clarify)
huffman@44365
  1235
    apply (rule ccontr)
huffman@44365
  1236
    apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
huffman@44365
  1237
    apply (simp add: closure_interior interior_def)
huffman@44365
  1238
    apply (drule_tac x=C in spec)
huffman@44365
  1239
    apply (drule_tac x=D in spec)
huffman@44365
  1240
    apply auto
huffman@44365
  1241
    done
huffman@44365
  1242
qed
huffman@44365
  1243
huffman@44210
  1244
hoelzl@51351
  1245
lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"
hoelzl@51351
  1246
  unfolding closure_def using islimpt_punctured by blast
hoelzl@51351
  1247
hoelzl@51351
  1248
huffman@44210
  1249
subsection {* Frontier (aka boundary) *}
himmelma@33175
  1250
himmelma@33175
  1251
definition "frontier S = closure S - interior S"
himmelma@33175
  1252
himmelma@33175
  1253
lemma frontier_closed: "closed(frontier S)"
himmelma@33175
  1254
  by (simp add: frontier_def closed_Diff)
himmelma@33175
  1255
huffman@34105
  1256
lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
himmelma@33175
  1257
  by (auto simp add: frontier_def interior_closure)
himmelma@33175
  1258
himmelma@33175
  1259
lemma frontier_straddle:
himmelma@33175
  1260
  fixes a :: "'a::metric_space"
huffman@44909
  1261
  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
  1262
  unfolding frontier_def closure_interior
huffman@44909
  1263
  by (auto simp add: mem_interior subset_eq ball_def)
himmelma@33175
  1264
himmelma@33175
  1265
lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
himmelma@33175
  1266
  by (metis frontier_def closure_closed Diff_subset)
himmelma@33175
  1267
hoelzl@34964
  1268
lemma frontier_empty[simp]: "frontier {} = {}"
huffman@36362
  1269
  by (simp add: frontier_def)
himmelma@33175
  1270
himmelma@33175
  1271
lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
himmelma@33175
  1272
proof-
himmelma@33175
  1273
  { assume "frontier S \<subseteq> S"
himmelma@33175
  1274
    hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
himmelma@33175
  1275
    hence "closed S" using closure_subset_eq by auto
himmelma@33175
  1276
  }
huffman@36362
  1277
  thus ?thesis using frontier_subset_closed[of S] ..
himmelma@33175
  1278
qed
himmelma@33175
  1279
huffman@34105
  1280
lemma frontier_complement: "frontier(- S) = frontier S"
himmelma@33175
  1281
  by (auto simp add: frontier_def closure_complement interior_complement)
himmelma@33175
  1282
himmelma@33175
  1283
lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
huffman@34105
  1284
  using frontier_complement frontier_subset_eq[of "- S"]
huffman@34105
  1285
  unfolding open_closed by auto
himmelma@33175
  1286
huffman@44081
  1287
subsection {* Filters and the ``eventually true'' quantifier *}
huffman@44081
  1288
himmelma@33175
  1289
definition
huffman@44081
  1290
  indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
huffman@44081
  1291
    (infixr "indirection" 70) where
himmelma@33175
  1292
  "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
himmelma@33175
  1293
huffman@36437
  1294
text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}
himmelma@33175
  1295
himmelma@33175
  1296
lemma trivial_limit_within:
himmelma@33175
  1297
  shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
himmelma@33175
  1298
proof
himmelma@33175
  1299
  assume "trivial_limit (at a within S)"
himmelma@33175
  1300
  thus "\<not> a islimpt S"
himmelma@33175
  1301
    unfolding trivial_limit_def
huffman@36358
  1302
    unfolding eventually_within eventually_at_topological
himmelma@33175
  1303
    unfolding islimpt_def
nipkow@39302
  1304
    apply (clarsimp simp add: set_eq_iff)
himmelma@33175
  1305
    apply (rename_tac T, rule_tac x=T in exI)
huffman@36358
  1306
    apply (clarsimp, drule_tac x=y in bspec, simp_all)
himmelma@33175
  1307
    done
himmelma@33175
  1308
next
himmelma@33175
  1309
  assume "\<not> a islimpt S"
himmelma@33175
  1310
  thus "trivial_limit (at a within S)"
himmelma@33175
  1311
    unfolding trivial_limit_def
huffman@36358
  1312
    unfolding eventually_within eventually_at_topological
himmelma@33175
  1313
    unfolding islimpt_def
huffman@36358
  1314
    apply clarsimp
huffman@36358
  1315
    apply (rule_tac x=T in exI)
huffman@36358
  1316
    apply auto
himmelma@33175
  1317
    done
himmelma@33175
  1318
qed
himmelma@33175
  1319
himmelma@33175
  1320
lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
huffman@45031
  1321
  using trivial_limit_within [of a UNIV] by simp
himmelma@33175
  1322
himmelma@33175
  1323
lemma trivial_limit_at:
himmelma@33175
  1324
  fixes a :: "'a::perfect_space"
himmelma@33175
  1325
  shows "\<not> trivial_limit (at a)"
huffman@44571
  1326
  by (rule at_neq_bot)
himmelma@33175
  1327
himmelma@33175
  1328
lemma trivial_limit_at_infinity:
huffman@44081
  1329
  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
huffman@36358
  1330
  unfolding trivial_limit_def eventually_at_infinity
huffman@36358
  1331
  apply clarsimp
huffman@44072
  1332
  apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
huffman@44072
  1333
   apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
huffman@44072
  1334
  apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
huffman@44072
  1335
  apply (drule_tac x=UNIV in spec, simp)
himmelma@33175
  1336
  done
himmelma@33175
  1337
hoelzl@51351
  1338
lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"
hoelzl@51351
  1339
  using islimpt_in_closure by (metis trivial_limit_within)
hoelzl@51351
  1340
huffman@36437
  1341
text {* Some property holds "sufficiently close" to the limit point. *}
himmelma@33175
  1342
himmelma@33175
  1343
lemma eventually_at: (* FIXME: this replaces Limits.eventually_at *)
himmelma@33175
  1344
  "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
himmelma@33175
  1345
unfolding eventually_at dist_nz by auto
himmelma@33175
  1346
hoelzl@50526
  1347
lemma eventually_within: (* FIXME: this replaces Limits.eventually_within *)
hoelzl@50526
  1348
  "eventually P (at a within S) \<longleftrightarrow>
himmelma@33175
  1349
        (\<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
hoelzl@50526
  1350
  by (rule eventually_within_less)
himmelma@33175
  1351
himmelma@33175
  1352
lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
huffman@36358
  1353
  unfolding trivial_limit_def
huffman@36358
  1354
  by (auto elim: eventually_rev_mp)
himmelma@33175
  1355
himmelma@33175
  1356
lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
huffman@45031
  1357
  by simp
himmelma@33175
  1358
himmelma@33175
  1359
lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
huffman@44342
  1360
  by (simp add: filter_eq_iff)
himmelma@33175
  1361
himmelma@33175
  1362
text{* Combining theorems for "eventually" *}
himmelma@33175
  1363
himmelma@33175
  1364
lemma eventually_rev_mono:
himmelma@33175
  1365
  "eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
himmelma@33175
  1366
using eventually_mono [of P Q] by fast
himmelma@33175
  1367
himmelma@33175
  1368
lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"
himmelma@33175
  1369
  by (simp add: eventually_False)
himmelma@33175
  1370
huffman@44210
  1371
huffman@36437
  1372
subsection {* Limits *}
himmelma@33175
  1373
huffman@44081
  1374
text{* Notation Lim to avoid collition with lim defined in analysis *}
huffman@44081
  1375
huffman@44081
  1376
definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
huffman@44081
  1377
  where "Lim A f = (THE l. (f ---> l) A)"
himmelma@33175
  1378
himmelma@33175
  1379
lemma Lim:
himmelma@33175
  1380
 "(f ---> l) net \<longleftrightarrow>
himmelma@33175
  1381
        trivial_limit net \<or>
himmelma@33175
  1382
        (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
himmelma@33175
  1383
  unfolding tendsto_iff trivial_limit_eq by auto
himmelma@33175
  1384
himmelma@33175
  1385
text{* Show that they yield usual definitions in the various cases. *}
himmelma@33175
  1386
himmelma@33175
  1387
lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
himmelma@33175
  1388
           (\<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)"
himmelma@33175
  1389
  by (auto simp add: tendsto_iff eventually_within_le)
himmelma@33175
  1390
himmelma@33175
  1391
lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
himmelma@33175
  1392
        (\<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)"
himmelma@33175
  1393
  by (auto simp add: tendsto_iff eventually_within)
himmelma@33175
  1394
himmelma@33175
  1395
lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
himmelma@33175
  1396
        (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
himmelma@33175
  1397
  by (auto simp add: tendsto_iff eventually_at)
himmelma@33175
  1398
himmelma@33175
  1399
lemma Lim_at_infinity:
himmelma@33175
  1400
  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
himmelma@33175
  1401
  by (auto simp add: tendsto_iff eventually_at_infinity)
himmelma@33175
  1402
himmelma@33175
  1403
lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
himmelma@33175
  1404
  by (rule topological_tendstoI, auto elim: eventually_rev_mono)
himmelma@33175
  1405
himmelma@33175
  1406
text{* The expected monotonicity property. *}
himmelma@33175
  1407
himmelma@33175
  1408
lemma Lim_within_empty: "(f ---> l) (net within {})"
himmelma@33175
  1409
  unfolding tendsto_def Limits.eventually_within by simp
himmelma@33175
  1410
himmelma@33175
  1411
lemma Lim_within_subset: "(f ---> l) (net within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (net within T)"
himmelma@33175
  1412
  unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1413
  by (auto elim!: eventually_elim1)
himmelma@33175
  1414
himmelma@33175
  1415
lemma Lim_Un: assumes "(f ---> l) (net within S)" "(f ---> l) (net within T)"
himmelma@33175
  1416
  shows "(f ---> l) (net within (S \<union> T))"
himmelma@33175
  1417
  using assms unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1418
  apply clarify
himmelma@33175
  1419
  apply (drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1420
  apply (drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1421
  apply (auto elim: eventually_elim2)
himmelma@33175
  1422
  done
himmelma@33175
  1423
himmelma@33175
  1424
lemma Lim_Un_univ:
himmelma@33175
  1425
 "(f ---> l) (net within S) \<Longrightarrow> (f ---> l) (net within T) \<Longrightarrow>  S \<union> T = UNIV
himmelma@33175
  1426
        ==> (f ---> l) net"
himmelma@33175
  1427
  by (metis Lim_Un within_UNIV)
himmelma@33175
  1428
himmelma@33175
  1429
text{* Interrelations between restricted and unrestricted limits. *}
himmelma@33175
  1430
himmelma@33175
  1431
lemma Lim_at_within: "(f ---> l) net ==> (f ---> l)(net within S)"
himmelma@33175
  1432
  (* FIXME: rename *)
himmelma@33175
  1433
  unfolding tendsto_def Limits.eventually_within
himmelma@33175
  1434
  apply (clarify, drule spec, drule (1) mp, drule (1) mp)
himmelma@33175
  1435
  by (auto elim!: eventually_elim1)
himmelma@33175
  1436
huffman@44210
  1437
lemma eventually_within_interior:
huffman@44210
  1438
  assumes "x \<in> interior S"
huffman@44210
  1439
  shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
huffman@44210
  1440
proof-
huffman@44519
  1441
  from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
huffman@44210
  1442
  { assume "?lhs"
huffman@44210
  1443
    then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
huffman@44210
  1444
      unfolding Limits.eventually_within Limits.eventually_at_topological
huffman@44210
  1445
      by auto
huffman@44210
  1446
    with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
huffman@44210
  1447
      by auto
huffman@44210
  1448
    then have "?rhs"
huffman@44210
  1449
      unfolding Limits.eventually_at_topological by auto
huffman@44210
  1450
  } moreover
huffman@44210
  1451
  { assume "?rhs" hence "?lhs"
huffman@44210
  1452
      unfolding Limits.eventually_within
huffman@44210
  1453
      by (auto elim: eventually_elim1)
huffman@44210
  1454
  } ultimately
huffman@44210
  1455
  show "?thesis" ..
huffman@44210
  1456
qed
huffman@44210
  1457
huffman@44210
  1458
lemma at_within_interior:
huffman@44210
  1459
  "x \<in> interior S \<Longrightarrow> at x within S = at x"
huffman@44210
  1460
  by (simp add: filter_eq_iff eventually_within_interior)
huffman@44210
  1461
huffman@44210
  1462
lemma at_within_open:
huffman@44210
  1463
  "\<lbrakk>x \<in> S; open S\<rbrakk> \<Longrightarrow> at x within S = at x"
huffman@44210
  1464
  by (simp only: at_within_interior interior_open)
huffman@44210
  1465
himmelma@33175
  1466
lemma Lim_within_open:
himmelma@33175
  1467
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
himmelma@33175
  1468
  assumes"a \<in> S" "open S"
huffman@44210
  1469
  shows "(f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
huffman@44210
  1470
  using assms by (simp only: at_within_open)
himmelma@33175
  1471
hoelzl@43338
  1472
lemma Lim_within_LIMSEQ:
huffman@44584
  1473
  fixes a :: "'a::metric_space"
hoelzl@43338
  1474
  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
  1475
  shows "(X ---> L) (at a within T)"
huffman@44584
  1476
  using assms unfolding tendsto_def [where l=L]
huffman@44584
  1477
  by (simp add: sequentially_imp_eventually_within)
hoelzl@43338
  1478
hoelzl@43338
  1479
lemma Lim_right_bound:
hoelzl@43338
  1480
  fixes f :: "real \<Rightarrow> real"
hoelzl@43338
  1481
  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"
hoelzl@43338
  1482
  assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
hoelzl@43338
  1483
  shows "(f ---> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
hoelzl@43338
  1484
proof cases
hoelzl@43338
  1485
  assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
hoelzl@43338
  1486
next
hoelzl@43338
  1487
  assume [simp]: "{x<..} \<inter> I \<noteq> {}"
hoelzl@43338
  1488
  show ?thesis
hoelzl@43338
  1489
  proof (rule Lim_within_LIMSEQ, safe)
hoelzl@43338
  1490
    fix S assume S: "\<forall>n. S n \<noteq> x \<and> S n \<in> {x <..} \<inter> I" "S ----> x"
hoelzl@43338
  1491
    
hoelzl@43338
  1492
    show "(\<lambda>n. f (S n)) ----> Inf (f ` ({x<..} \<inter> I))"
hoelzl@43338
  1493
    proof (rule LIMSEQ_I, rule ccontr)
hoelzl@43338
  1494
      fix r :: real assume "0 < r"
hoelzl@43338
  1495
      with Inf_close[of "f ` ({x<..} \<inter> I)" r]
hoelzl@43338
  1496
      obtain y where y: "x < y" "y \<in> I" "f y < Inf (f ` ({x <..} \<inter> I)) + r" by auto
hoelzl@43338
  1497
      from `x < y` have "0 < y - x" by auto
hoelzl@43338
  1498
      from S(2)[THEN LIMSEQ_D, OF this]
hoelzl@43338
  1499
      obtain N where N: "\<And>n. N \<le> n \<Longrightarrow> \<bar>S n - x\<bar> < y - x" by auto
hoelzl@43338
  1500
      
hoelzl@43338
  1501
      assume "\<not> (\<exists>N. \<forall>n\<ge>N. norm (f (S n) - Inf (f ` ({x<..} \<inter> I))) < r)"
hoelzl@43338
  1502
      moreover have "\<And>n. Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
hoelzl@43338
  1503
        using S bnd by (intro Inf_lower[where z=K]) auto
hoelzl@43338
  1504
      ultimately obtain n where n: "N \<le> n" "r + Inf (f ` ({x<..} \<inter> I)) \<le> f (S n)"
hoelzl@43338
  1505
        by (auto simp: not_less field_simps)
hoelzl@43338
  1506
      with N[OF n(1)] mono[OF _ `y \<in> I`, of "S n"] S(1)[THEN spec, of n] y
hoelzl@43338
  1507
      show False by auto
hoelzl@43338
  1508
    qed
hoelzl@43338
  1509
  qed
hoelzl@43338
  1510
qed
hoelzl@43338
  1511
himmelma@33175
  1512
text{* Another limit point characterization. *}
himmelma@33175
  1513
himmelma@33175
  1514
lemma islimpt_sequential:
hoelzl@50883
  1515
  fixes x :: "'a::first_countable_topology"
hoelzl@50883
  1516
  shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"
himmelma@33175
  1517
    (is "?lhs = ?rhs")
himmelma@33175
  1518
proof
himmelma@33175
  1519
  assume ?lhs
hoelzl@50883
  1520
  from countable_basis_at_decseq[of x] guess A . note A = this
hoelzl@50883
  1521
  def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
hoelzl@50883
  1522
  { fix n
hoelzl@50883
  1523
    from `?lhs` have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
hoelzl@50883
  1524
      unfolding islimpt_def using A(1,2)[of n] by auto
hoelzl@50883
  1525
    then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
hoelzl@50883
  1526
      unfolding f_def by (rule someI_ex)
hoelzl@50883
  1527
    then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto }
hoelzl@50883
  1528
  then have "\<forall>n. f n \<in> S - {x}" by auto
hoelzl@50883
  1529
  moreover have "(\<lambda>n. f n) ----> x"
hoelzl@50883
  1530
  proof (rule topological_tendstoI)
hoelzl@50883
  1531
    fix S assume "open S" "x \<in> S"
hoelzl@50883
  1532
    from A(3)[OF this] `\<And>n. f n \<in> A n`
hoelzl@50883
  1533
    show "eventually (\<lambda>x. f x \<in> S) sequentially" by (auto elim!: eventually_elim1)
huffman@44584
  1534
  qed
huffman@44584
  1535
  ultimately show ?rhs by fast
himmelma@33175
  1536
next
himmelma@33175
  1537
  assume ?rhs
hoelzl@50883
  1538
  then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x" by auto
hoelzl@50883
  1539
  show ?lhs
hoelzl@50883
  1540
    unfolding islimpt_def
hoelzl@50883
  1541
  proof safe
hoelzl@50883
  1542
    fix T assume "open T" "x \<in> T"
hoelzl@50883
  1543
    from lim[THEN topological_tendstoD, OF this] f
hoelzl@50883
  1544
    show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
hoelzl@50883
  1545
      unfolding eventually_sequentially by auto
hoelzl@50883
  1546
  qed
himmelma@33175
  1547
qed
himmelma@33175
  1548
huffman@44125
  1549
lemma Lim_inv: (* TODO: delete *)
huffman@44081
  1550
  fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
huffman@44081
  1551
  assumes "(f ---> l) A" and "l \<noteq> 0"
huffman@44081
  1552
  shows "((inverse o f) ---> inverse l) A"
huffman@36437
  1553
  unfolding o_def using assms by (rule tendsto_inverse)
huffman@36437
  1554
himmelma@33175
  1555
lemma Lim_null:
himmelma@33175
  1556
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
huffman@44125
  1557
  shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"
himmelma@33175
  1558
  by (simp add: Lim dist_norm)
himmelma@33175
  1559
himmelma@33175
  1560
lemma Lim_null_comparison:
himmelma@33175
  1561
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1562
  assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
himmelma@33175
  1563
  shows "(f ---> 0) net"
huffman@44252
  1564
proof (rule metric_tendsto_imp_tendsto)
huffman@44252
  1565
  show "(g ---> 0) net" by fact
huffman@44252
  1566
  show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
huffman@44252
  1567
    using assms(1) by (rule eventually_elim1, simp add: dist_norm)
himmelma@33175
  1568
qed
himmelma@33175
  1569
himmelma@33175
  1570
lemma Lim_transform_bound:
himmelma@33175
  1571
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1572
  fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
himmelma@33175
  1573
  assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
himmelma@33175
  1574
  shows "(f ---> 0) net"
huffman@44252
  1575
  using assms(1) tendsto_norm_zero [OF assms(2)]
huffman@44252
  1576
  by (rule Lim_null_comparison)
himmelma@33175
  1577
himmelma@33175
  1578
text{* Deducing things about the limit from the elements. *}
himmelma@33175
  1579
himmelma@33175
  1580
lemma Lim_in_closed_set:
himmelma@33175
  1581
  assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
himmelma@33175
  1582
  shows "l \<in> S"
himmelma@33175
  1583
proof (rule ccontr)
himmelma@33175
  1584
  assume "l \<notin> S"
himmelma@33175
  1585
  with `closed S` have "open (- S)" "l \<in> - S"
himmelma@33175
  1586
    by (simp_all add: open_Compl)
himmelma@33175
  1587
  with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
himmelma@33175
  1588
    by (rule topological_tendstoD)
himmelma@33175
  1589
  with assms(2) have "eventually (\<lambda>x. False) net"
himmelma@33175
  1590
    by (rule eventually_elim2) simp
himmelma@33175
  1591
  with assms(3) show "False"
himmelma@33175
  1592
    by (simp add: eventually_False)
himmelma@33175
  1593
qed
himmelma@33175
  1594
himmelma@33175
  1595
text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}
himmelma@33175
  1596
himmelma@33175
  1597
lemma Lim_dist_ubound:
himmelma@33175
  1598
  assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
himmelma@33175
  1599
  shows "dist a l <= e"
huffman@44252
  1600
proof-
huffman@44252
  1601
  have "dist a l \<in> {..e}"
huffman@44252
  1602
  proof (rule Lim_in_closed_set)
huffman@44252
  1603
    show "closed {..e}" by simp
huffman@44252
  1604
    show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
huffman@44252
  1605
    show "\<not> trivial_limit net" by fact
huffman@44252
  1606
    show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
huffman@44252
  1607
  qed
huffman@44252
  1608
  thus ?thesis by simp
himmelma@33175
  1609
qed
himmelma@33175
  1610
himmelma@33175
  1611
lemma Lim_norm_ubound:
himmelma@33175
  1612
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1613
  assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
himmelma@33175
  1614
  shows "norm(l) <= e"
huffman@44252
  1615
proof-
huffman@44252
  1616
  have "norm l \<in> {..e}"
huffman@44252
  1617
  proof (rule Lim_in_closed_set)
huffman@44252
  1618
    show "closed {..e}" by simp
huffman@44252
  1619
    show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
huffman@44252
  1620
    show "\<not> trivial_limit net" by fact
huffman@44252
  1621
    show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
huffman@44252
  1622
  qed
huffman@44252
  1623
  thus ?thesis by simp
himmelma@33175
  1624
qed
himmelma@33175
  1625
himmelma@33175
  1626
lemma Lim_norm_lbound:
himmelma@33175
  1627
  fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1628
  assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
himmelma@33175
  1629
  shows "e \<le> norm l"
huffman@44252
  1630
proof-
huffman@44252
  1631
  have "norm l \<in> {e..}"
huffman@44252
  1632
  proof (rule Lim_in_closed_set)
huffman@44252
  1633
    show "closed {e..}" by simp
huffman@44252
  1634
    show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
huffman@44252
  1635
    show "\<not> trivial_limit net" by fact
huffman@44252
  1636
    show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
huffman@44252
  1637
  qed
huffman@44252
  1638
  thus ?thesis by simp
himmelma@33175
  1639
qed
himmelma@33175
  1640
himmelma@33175
  1641
text{* Uniqueness of the limit, when nontrivial. *}
himmelma@33175
  1642
himmelma@33175
  1643
lemma tendsto_Lim:
himmelma@33175
  1644
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
himmelma@33175
  1645
  shows "~(trivial_limit net) \<Longrightarrow> (f ---> l) net ==> Lim net f = l"
hoelzl@41970
  1646
  unfolding Lim_def using tendsto_unique[of net f] by auto
himmelma@33175
  1647
himmelma@33175
  1648
text{* Limit under bilinear function *}
himmelma@33175
  1649
himmelma@33175
  1650
lemma Lim_bilinear:
himmelma@33175
  1651
  assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
himmelma@33175
  1652
  shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
himmelma@33175
  1653
using `bounded_bilinear h` `(f ---> l) net` `(g ---> m) net`
himmelma@33175
  1654
by (rule bounded_bilinear.tendsto)
himmelma@33175
  1655
himmelma@33175
  1656
text{* These are special for limits out of the same vector space. *}
himmelma@33175
  1657
himmelma@33175
  1658
lemma Lim_within_id: "(id ---> a) (at a within s)"
huffman@45031
  1659
  unfolding id_def by (rule tendsto_ident_at_within)
himmelma@33175
  1660
himmelma@33175
  1661
lemma Lim_at_id: "(id ---> a) (at a)"
huffman@45031
  1662
  unfolding id_def by (rule tendsto_ident_at)
himmelma@33175
  1663
himmelma@33175
  1664
lemma Lim_at_zero:
himmelma@33175
  1665
  fixes a :: "'a::real_normed_vector"
himmelma@33175
  1666
  fixes l :: "'b::topological_space"
himmelma@33175
  1667
  shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
huffman@44252
  1668
  using LIM_offset_zero LIM_offset_zero_cancel ..
himmelma@33175
  1669
huffman@44081
  1670
text{* It's also sometimes useful to extract the limit point from the filter. *}
himmelma@33175
  1671
himmelma@33175
  1672
definition
huffman@44081
  1673
  netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
himmelma@33175
  1674
  "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
himmelma@33175
  1675
himmelma@33175
  1676
lemma netlimit_within:
himmelma@33175
  1677
  assumes "\<not> trivial_limit (at a within S)"
himmelma@33175
  1678
  shows "netlimit (at a within S) = a"
himmelma@33175
  1679
unfolding netlimit_def
himmelma@33175
  1680
apply (rule some_equality)
himmelma@33175
  1681
apply (rule Lim_at_within)
huffman@44568
  1682
apply (rule tendsto_ident_at)
hoelzl@41970
  1683
apply (erule tendsto_unique [OF assms])
himmelma@33175
  1684
apply (rule Lim_at_within)
huffman@44568
  1685
apply (rule tendsto_ident_at)
himmelma@33175
  1686
done
himmelma@33175
  1687
himmelma@33175
  1688
lemma netlimit_at:
huffman@44072
  1689
  fixes a :: "'a::{perfect_space,t2_space}"
himmelma@33175
  1690
  shows "netlimit (at a) = a"
huffman@45031
  1691
  using netlimit_within [of a UNIV] by simp
himmelma@33175
  1692
huffman@44210
  1693
lemma lim_within_interior:
huffman@44210
  1694
  "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
huffman@44210
  1695
  by (simp add: at_within_interior)
huffman@44210
  1696
huffman@44210
  1697
lemma netlimit_within_interior:
huffman@44210
  1698
  fixes x :: "'a::{t2_space,perfect_space}"
huffman@44210
  1699
  assumes "x \<in> interior S"
huffman@44210
  1700
  shows "netlimit (at x within S) = x"
huffman@44210
  1701
using assms by (simp add: at_within_interior netlimit_at)
huffman@44210
  1702
himmelma@33175
  1703
text{* Transformation of limit. *}
himmelma@33175
  1704
himmelma@33175
  1705
lemma Lim_transform:
himmelma@33175
  1706
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
himmelma@33175
  1707
  assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
himmelma@33175
  1708
  shows "(g ---> l) net"
huffman@44252
  1709
  using tendsto_diff [OF assms(2) assms(1)] by simp
himmelma@33175
  1710
himmelma@33175
  1711
lemma Lim_transform_eventually:
huffman@36667
  1712
  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
himmelma@33175
  1713
  apply (rule topological_tendstoI)
himmelma@33175
  1714
  apply (drule (2) topological_tendstoD)
himmelma@33175
  1715
  apply (erule (1) eventually_elim2, simp)
himmelma@33175
  1716
  done
himmelma@33175
  1717
himmelma@33175
  1718
lemma Lim_transform_within:
huffman@36667
  1719
  assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
huffman@36667
  1720
  and "(f ---> l) (at x within S)"
huffman@36667
  1721
  shows "(g ---> l) (at x within S)"
huffman@36667
  1722
proof (rule Lim_transform_eventually)
huffman@36667
  1723
  show "eventually (\<lambda>x. f x = g x) (at x within S)"
huffman@36667
  1724
    unfolding eventually_within
huffman@36667
  1725
    using assms(1,2) by auto
huffman@36667
  1726
  show "(f ---> l) (at x within S)" by fact
huffman@36667
  1727
qed
himmelma@33175
  1728
himmelma@33175
  1729
lemma Lim_transform_at:
huffman@36667
  1730
  assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
huffman@36667
  1731
  and "(f ---> l) (at x)"
huffman@36667
  1732
  shows "(g ---> l) (at x)"
huffman@36667
  1733
proof (rule Lim_transform_eventually)
huffman@36667
  1734
  show "eventually (\<lambda>x. f x = g x) (at x)"
huffman@36667
  1735
    unfolding eventually_at
huffman@36667
  1736
    using assms(1,2) by auto
huffman@36667
  1737
  show "(f ---> l) (at x)" by fact
huffman@36667
  1738
qed
himmelma@33175
  1739
himmelma@33175
  1740
text{* Common case assuming being away from some crucial point like 0. *}
himmelma@33175
  1741
himmelma@33175
  1742
lemma Lim_transform_away_within:
huffman@36669
  1743
  fixes a b :: "'a::t1_space"
huffman@36667
  1744
  assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
himmelma@33175
  1745
  and "(f ---> l) (at a within S)"
himmelma@33175
  1746
  shows "(g ---> l) (at a within S)"
huffman@36669
  1747
proof (rule Lim_transform_eventually)
huffman@36669
  1748
  show "(f ---> l) (at a within S)" by fact
huffman@36669
  1749
  show "eventually (\<lambda>x. f x = g x) (at a within S)"
huffman@36669
  1750
    unfolding Limits.eventually_within eventually_at_topological
huffman@36669
  1751
    by (rule exI [where x="- {b}"], simp add: open_Compl assms)
himmelma@33175
  1752
qed
himmelma@33175
  1753
himmelma@33175
  1754
lemma Lim_transform_away_at:
huffman@36669
  1755
  fixes a b :: "'a::t1_space"
himmelma@33175
  1756
  assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
himmelma@33175
  1757
  and fl: "(f ---> l) (at a)"
himmelma@33175
  1758
  shows "(g ---> l) (at a)"
himmelma@33175
  1759
  using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
huffman@45031
  1760
  by simp
himmelma@33175
  1761
himmelma@33175
  1762
text{* Alternatively, within an open set. *}
himmelma@33175
  1763
himmelma@33175
  1764
lemma Lim_transform_within_open:
huffman@36667
  1765
  assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
huffman@36667
  1766
  and "(f ---> l) (at a)"
himmelma@33175
  1767
  shows "(g ---> l) (at a)"
huffman@36667
  1768
proof (rule Lim_transform_eventually)
huffman@36667
  1769
  show "eventually (\<lambda>x. f x = g x) (at a)"
huffman@36667
  1770
    unfolding eventually_at_topological
huffman@36667
  1771
    using assms(1,2,3) by auto
huffman@36667
  1772
  show "(f ---> l) (at a)" by fact
himmelma@33175
  1773
qed
himmelma@33175
  1774
himmelma@33175
  1775
text{* A congruence rule allowing us to transform limits assuming not at point. *}
himmelma@33175
  1776
himmelma@33175
  1777
(* FIXME: Only one congruence rule for tendsto can be used at a time! *)
himmelma@33175
  1778
huffman@36362
  1779
lemma Lim_cong_within(*[cong add]*):
hoelzl@43338
  1780
  assumes "a = b" "x = y" "S = T"
hoelzl@43338
  1781
  assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
hoelzl@43338
  1782
  shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
huffman@36667
  1783
  unfolding tendsto_def Limits.eventually_within eventually_at_topological
huffman@36667
  1784
  using assms by simp
huffman@36667
  1785
huffman@36667
  1786
lemma Lim_cong_at(*[cong add]*):
hoelzl@43338
  1787
  assumes "a = b" "x = y"
huffman@36667
  1788
  assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
hoelzl@43338
  1789
  shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
huffman@36667
  1790
  unfolding tendsto_def eventually_at_topological
huffman@36667
  1791
  using assms by simp
himmelma@33175
  1792
himmelma@33175
  1793
text{* Useful lemmas on closure and set of possible sequential limits.*}
himmelma@33175
  1794
himmelma@33175
  1795
lemma closure_sequential:
hoelzl@50883
  1796
  fixes l :: "'a::first_countable_topology"
himmelma@33175
  1797
  shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
himmelma@33175
  1798
proof
himmelma@33175
  1799
  assume "?lhs" moreover
himmelma@33175
  1800
  { assume "l \<in> S"
huffman@44125
  1801
    hence "?rhs" using tendsto_const[of l sequentially] by auto
himmelma@33175
  1802
  } moreover
himmelma@33175
  1803
  { assume "l islimpt S"
himmelma@33175
  1804
    hence "?rhs" unfolding islimpt_sequential by auto
himmelma@33175
  1805
  } ultimately
himmelma@33175
  1806
  show "?rhs" unfolding closure_def by auto
himmelma@33175
  1807
next
himmelma@33175
  1808
  assume "?rhs"
himmelma@33175
  1809
  thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
himmelma@33175
  1810
qed
himmelma@33175
  1811
himmelma@33175
  1812
lemma closed_sequential_limits:
hoelzl@50883
  1813
  fixes S :: "'a::first_countable_topology set"
himmelma@33175
  1814
  shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
himmelma@33175
  1815
  unfolding closed_limpt
himmelma@33175
  1816
  using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]
himmelma@33175
  1817
  by metis
himmelma@33175
  1818
himmelma@33175
  1819
lemma closure_approachable:
himmelma@33175
  1820
  fixes S :: "'a::metric_space set"
himmelma@33175
  1821
  shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
himmelma@33175
  1822
  apply (auto simp add: closure_def islimpt_approachable)
himmelma@33175
  1823
  by (metis dist_self)
himmelma@33175
  1824
himmelma@33175
  1825
lemma closed_approachable:
himmelma@33175
  1826
  fixes S :: "'a::metric_space set"
himmelma@33175
  1827
  shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
himmelma@33175
  1828
  by (metis closure_closed closure_approachable)
himmelma@33175
  1829
hoelzl@51351
  1830
lemma closure_contains_Inf:
hoelzl@51351
  1831
  fixes S :: "real set"
hoelzl@51351
  1832
  assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"
hoelzl@51351
  1833
  shows "Inf S \<in> closure S"
hoelzl@51351
  1834
  unfolding closure_approachable
hoelzl@51351
  1835
proof safe
hoelzl@51351
  1836
  have *: "\<forall>x\<in>S. Inf S \<le> x"
hoelzl@51351
  1837
    using Inf_lower_EX[of _ S] assms by metis
hoelzl@51351
  1838
hoelzl@51351
  1839
  fix e :: real assume "0 < e"
hoelzl@51351
  1840
  then obtain x where x: "x \<in> S" "x < Inf S + e"
hoelzl@51351
  1841
    using Inf_close `S \<noteq> {}` by auto
hoelzl@51351
  1842
  moreover then have "x > Inf S - e" using * by auto
hoelzl@51351
  1843
  ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)
hoelzl@51351
  1844
  then show "\<exists>x\<in>S. dist x (Inf S) < e"
hoelzl@51351
  1845
    using x by (auto simp: dist_norm)
hoelzl@51351
  1846
qed
hoelzl@51351
  1847
hoelzl@51351
  1848
lemma closed_contains_Inf:
hoelzl@51351
  1849
  fixes S :: "real set"
hoelzl@51351
  1850
  assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"
hoelzl@51351
  1851
    and "closed S"
hoelzl@51351
  1852
  shows "Inf S \<in> S"
hoelzl@51351
  1853
  by (metis closure_contains_Inf closure_closed assms)
hoelzl@51351
  1854
hoelzl@51351
  1855
hoelzl@51351
  1856
lemma not_trivial_limit_within_ball:
hoelzl@51351
  1857
  "(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
hoelzl@51351
  1858
  (is "?lhs = ?rhs")
hoelzl@51351
  1859
proof -
hoelzl@51351
  1860
  { assume "?lhs"
hoelzl@51351
  1861
    { fix e :: real
hoelzl@51351
  1862
      assume "e>0"
hoelzl@51351
  1863
      then obtain y where "y:(S-{x}) & dist y x < e"
hoelzl@51351
  1864
        using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
hoelzl@51351
  1865
        by auto
hoelzl@51351
  1866
      then have "y : (S Int ball x e - {x})"
hoelzl@51351
  1867
        unfolding ball_def by (simp add: dist_commute)
hoelzl@51351
  1868
      then have "S Int ball x e - {x} ~= {}" by blast
hoelzl@51351
  1869
    } then have "?rhs" by auto
hoelzl@51351
  1870
  }
hoelzl@51351
  1871
  moreover
hoelzl@51351
  1872
  { assume "?rhs"
hoelzl@51351
  1873
    { fix e :: real
hoelzl@51351
  1874
      assume "e>0"
hoelzl@51351
  1875
      then obtain y where "y : (S Int ball x e - {x})" using `?rhs` by blast
hoelzl@51351
  1876
      then have "y:(S-{x}) & dist y x < e"
hoelzl@51351
  1877
        unfolding ball_def by (simp add: dist_commute)
hoelzl@51351
  1878
      then have "EX y:(S-{x}). dist y x < e" by auto
hoelzl@51351
  1879
    }
hoelzl@51351
  1880
    then have "?lhs"
hoelzl@51351
  1881
      using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
hoelzl@51351
  1882
  }
hoelzl@51351
  1883
  ultimately show ?thesis by auto
hoelzl@51351
  1884
qed
hoelzl@51351
  1885
immler@50087
  1886
subsection {* Infimum Distance *}
immler@50087
  1887
immler@50087
  1888
definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"
immler@50087
  1889
immler@50087
  1890
lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"
immler@50087
  1891
  by (simp add: infdist_def)
immler@50087
  1892
immler@50087
  1893
lemma infdist_nonneg:
immler@50087
  1894
  shows "0 \<le> infdist x A"
immler@50087
  1895
  using assms by (auto simp add: infdist_def)
immler@50087
  1896
immler@50087
  1897
lemma infdist_le:
immler@50087
  1898
  assumes "a \<in> A"
immler@50087
  1899
  assumes "d = dist x a"
immler@50087
  1900
  shows "infdist x A \<le> d"
immler@50087
  1901
  using assms by (auto intro!: SupInf.Inf_lower[where z=0] simp add: infdist_def)
immler@50087
  1902
immler@50087
  1903
lemma infdist_zero[simp]:
immler@50087
  1904
  assumes "a \<in> A" shows "infdist a A = 0"
immler@50087
  1905
proof -
immler@50087
  1906
  from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto
immler@50087
  1907
  with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto
immler@50087
  1908
qed
immler@50087
  1909
immler@50087
  1910
lemma infdist_triangle:
immler@50087
  1911
  shows "infdist x A \<le> infdist y A + dist x y"
immler@50087
  1912
proof cases
immler@50087
  1913
  assume "A = {}" thus ?thesis by (simp add: infdist_def)
immler@50087
  1914
next
immler@50087
  1915
  assume "A \<noteq> {}" then obtain a where "a \<in> A" by auto
immler@50087
  1916
  have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"
immler@50087
  1917
  proof
immler@50087
  1918
    from `A \<noteq> {}` show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}" by simp
immler@50087
  1919
    fix d assume "d \<in> {dist x y + dist y a |a. a \<in> A}"
immler@50087
  1920
    then obtain a where d: "d = dist x y + dist y a" "a \<in> A" by auto
immler@50087
  1921
    show "infdist x A \<le> d"
immler@50087
  1922
      unfolding infdist_notempty[OF `A \<noteq> {}`]
immler@50087
  1923
    proof (rule Inf_lower2)
immler@50087
  1924
      show "dist x a \<in> {dist x a |a. a \<in> A}" using `a \<in> A` by auto
immler@50087
  1925
      show "dist x a \<le> d" unfolding d by (rule dist_triangle)
immler@50087
  1926
      fix d assume "d \<in> {dist x a |a. a \<in> A}"
immler@50087
  1927
      then obtain a where "a \<in> A" "d = dist x a" by auto
immler@50087
  1928
      thus "infdist x A \<le> d" by (rule infdist_le)
immler@50087
  1929
    qed
immler@50087
  1930
  qed
immler@50087
  1931
  also have "\<dots> = dist x y + infdist y A"
immler@50087
  1932
  proof (rule Inf_eq, safe)
immler@50087
  1933
    fix a assume "a \<in> A"
immler@50087
  1934
    thus "dist x y + infdist y A \<le> dist x y + dist y a" by (auto intro: infdist_le)
immler@50087
  1935
  next
immler@50087
  1936
    fix i assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
immler@50087
  1937
    hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF `A \<noteq> {}`] using `a \<in> A`
immler@50087
  1938
      by (intro Inf_greatest) (auto simp: field_simps)
immler@50087
  1939
    thus "i \<le> dist x y + infdist y A" by simp
immler@50087
  1940
  qed
immler@50087
  1941
  finally show ?thesis by simp
immler@50087
  1942
qed
immler@50087
  1943
immler@50087
  1944
lemma
immler@50087
  1945
  in_closure_iff_infdist_zero:
immler@50087
  1946
  assumes "A \<noteq> {}"
immler@50087
  1947
  shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
immler@50087
  1948
proof
immler@50087
  1949
  assume "x \<in> closure A"
immler@50087
  1950
  show "infdist x A = 0"
immler@50087
  1951
  proof (rule ccontr)
immler@50087
  1952
    assume "infdist x A \<noteq> 0"
immler@50087
  1953
    with infdist_nonneg[of x A] have "infdist x A > 0" by auto
immler@50087
  1954
    hence "ball x (infdist x A) \<inter> closure A = {}" apply auto
immler@50087
  1955
      by (metis `0 < infdist x A` `x \<in> closure A` closure_approachable dist_commute
immler@50087
  1956
        eucl_less_not_refl euclidean_trans(2) infdist_le)
immler@50087
  1957
    hence "x \<notin> closure A" by (metis `0 < infdist x A` centre_in_ball disjoint_iff_not_equal)
immler@50087
  1958
    thus False using `x \<in> closure A` by simp
immler@50087
  1959
  qed
immler@50087
  1960
next
immler@50087
  1961
  assume x: "infdist x A = 0"
immler@50087
  1962
  then obtain a where "a \<in> A" by atomize_elim (metis all_not_in_conv assms)
immler@50087
  1963
  show "x \<in> closure A" unfolding closure_approachable
immler@50087
  1964
  proof (safe, rule ccontr)
immler@50087
  1965
    fix e::real assume "0 < e"
immler@50087
  1966
    assume "\<not> (\<exists>y\<in>A. dist y x < e)"
immler@50087
  1967
    hence "infdist x A \<ge> e" using `a \<in> A`
immler@50087
  1968
      unfolding infdist_def
hoelzl@50526
  1969
      by (force simp: dist_commute)
immler@50087
  1970
    with x `0 < e` show False by auto
immler@50087
  1971
  qed
immler@50087
  1972
qed
immler@50087
  1973
immler@50087
  1974
lemma
immler@50087
  1975
  in_closed_iff_infdist_zero:
immler@50087
  1976
  assumes "closed A" "A \<noteq> {}"
immler@50087
  1977
  shows "x \<in> A \<longleftrightarrow> infdist x A = 0"
immler@50087
  1978
proof -
immler@50087
  1979
  have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
immler@50087
  1980
    by (rule in_closure_iff_infdist_zero) fact
immler@50087
  1981
  with assms show ?thesis by simp
immler@50087
  1982
qed
immler@50087
  1983
immler@50087
  1984
lemma tendsto_infdist [tendsto_intros]:
immler@50087
  1985
  assumes f: "(f ---> l) F"
immler@50087
  1986
  shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"
immler@50087
  1987
proof (rule tendstoI)
immler@50087
  1988
  fix e ::real assume "0 < e"
immler@50087
  1989
  from tendstoD[OF f this]
immler@50087
  1990
  show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"
immler@50087
  1991
  proof (eventually_elim)
immler@50087
  1992
    fix x
immler@50087
  1993
    from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
immler@50087
  1994
    have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"
immler@50087
  1995
      by (simp add: dist_commute dist_real_def)
immler@50087
  1996
    also assume "dist (f x) l < e"
immler@50087
  1997
    finally show "dist (infdist (f x) A) (infdist l A) < e" .
immler@50087
  1998
  qed
immler@50087
  1999
qed
immler@50087
  2000
himmelma@33175
  2001
text{* Some other lemmas about sequences. *}
himmelma@33175
  2002
huffman@36441
  2003
lemma sequentially_offset:
huffman@36441
  2004
  assumes "eventually (\<lambda>i. P i) sequentially"
huffman@36441
  2005
  shows "eventually (\<lambda>i. P (i + k)) sequentially"
huffman@36441
  2006
  using assms unfolding eventually_sequentially by (metis trans_le_add1)
huffman@36441
  2007
himmelma@33175
  2008
lemma seq_offset:
huffman@36441
  2009
  assumes "(f ---> l) sequentially"
huffman@36441
  2010
  shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
huffman@44584
  2011
  using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)
himmelma@33175
  2012
himmelma@33175
  2013
lemma seq_offset_neg:
himmelma@33175
  2014
  "(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
himmelma@33175
  2015
  apply (rule topological_tendstoI)
himmelma@33175
  2016
  apply (drule (2) topological_tendstoD)
himmelma@33175
  2017
  apply (simp only: eventually_sequentially)
himmelma@33175
  2018
  apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
himmelma@33175
  2019
  apply metis
himmelma@33175
  2020
  by arith
himmelma@33175
  2021
himmelma@33175
  2022
lemma seq_offset_rev:
himmelma@33175
  2023
  "((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
huffman@44584
  2024
  by (rule LIMSEQ_offset) (* FIXME: redundant *)
himmelma@33175
  2025
himmelma@33175
  2026
lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
huffman@44584
  2027
  using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)
himmelma@33175
  2028
huffman@44210
  2029
subsection {* More properties of closed balls *}
himmelma@33175
  2030
himmelma@33175
  2031
lemma closed_cball: "closed (cball x e)"
himmelma@33175
  2032
unfolding cball_def closed_def
himmelma@33175
  2033
unfolding Collect_neg_eq [symmetric] not_le
himmelma@33175
  2034
apply (clarsimp simp add: open_dist, rename_tac y)
himmelma@33175
  2035
apply (rule_tac x="dist x y - e" in exI, clarsimp)
himmelma@33175
  2036
apply (rename_tac x')
himmelma@33175
  2037
apply (cut_tac x=x and y=x' and z=y in dist_triangle)
himmelma@33175
  2038
apply simp
himmelma@33175
  2039
done
himmelma@33175
  2040
himmelma@33175
  2041
lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
himmelma@33175
  2042
proof-
himmelma@33175
  2043
  { fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
himmelma@33175
  2044
    hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
himmelma@33175
  2045
  } moreover
himmelma@33175
  2046
  { fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
himmelma@33175
  2047
    hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
himmelma@33175
  2048
  } ultimately
himmelma@33175
  2049
  show ?thesis unfolding open_contains_ball by auto
himmelma@33175
  2050
qed
himmelma@33175
  2051
himmelma@33175
  2052
lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
huffman@44170
  2053
  by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
himmelma@33175
  2054
himmelma@33175
  2055
lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
himmelma@33175
  2056
  apply (simp add: interior_def, safe)
himmelma@33175
  2057
  apply (force simp add: open_contains_cball)
himmelma@33175
  2058
  apply (rule_tac x="ball x e" in exI)
huffman@36362
  2059
  apply (simp add: subset_trans [OF ball_subset_cball])
himmelma@33175
  2060
  done
himmelma@33175
  2061
himmelma@33175
  2062
lemma islimpt_ball:
himmelma@33175
  2063
  fixes x y :: "'a::{real_normed_vector,perfect_space}"
himmelma@33175
  2064
  shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e" (is "?lhs = ?rhs")
himmelma@33175
  2065
proof
himmelma@33175
  2066
  assume "?lhs"
himmelma@33175
  2067
  { assume "e \<le> 0"
himmelma@33175
  2068
    hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
himmelma@33175
  2069
    have False using `?lhs` unfolding * using islimpt_EMPTY[of y] by auto
himmelma@33175
  2070
  }
himmelma@33175
  2071
  hence "e > 0" by (metis not_less)
himmelma@33175
  2072
  moreover
himmelma@33175
  2073
  have "y \<in> cball x e" using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"] ball_subset_cball[of x e] `?lhs` unfolding closed_limpt by auto
himmelma@33175
  2074
  ultimately show "?rhs" by auto
himmelma@33175
  2075
next
himmelma@33175
  2076
  assume "?rhs" hence "e>0"  by auto
himmelma@33175
  2077
  { fix d::real assume "d>0"
himmelma@33175
  2078
    have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  2079
    proof(cases "d \<le> dist x y")
himmelma@33175
  2080
      case True thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  2081
      proof(cases "x=y")
himmelma@33175
  2082
        case True hence False using `d \<le> dist x y` `d>0` by auto
himmelma@33175
  2083
        thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by auto
himmelma@33175
  2084
      next
himmelma@33175
  2085
        case False
himmelma@33175
  2086
himmelma@33175
  2087
        have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x))
himmelma@33175
  2088
              = norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
himmelma@33175
  2089
          unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[THEN sym] by auto
himmelma@33175
  2090
        also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
himmelma@33175
  2091
          using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", THEN sym, of "y - x"]
himmelma@33175
  2092
          unfolding scaleR_minus_left scaleR_one
himmelma@33175
  2093
          by (auto simp add: norm_minus_commute)
himmelma@33175
  2094
        also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
himmelma@33175
  2095
          unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
webertj@49962
  2096
          unfolding distrib_right using `x\<noteq>y`[unfolded dist_nz, unfolded dist_norm] by auto
himmelma@33175
  2097
        also have "\<dots> \<le> e - d/2" using `d \<le> dist x y` and `d>0` and `?rhs` by(auto simp add: dist_norm)
himmelma@33175
  2098
        finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using `d>0` by auto
himmelma@33175
  2099
himmelma@33175
  2100
        moreover
himmelma@33175
  2101
himmelma@33175
  2102
        have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
himmelma@33175
  2103
          using `x\<noteq>y`[unfolded dist_nz] `d>0` unfolding scaleR_eq_0_iff by (auto simp add: dist_commute)
himmelma@33175
  2104
        moreover
himmelma@33175
  2105
        have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d" unfolding dist_norm apply simp unfolding norm_minus_cancel
himmelma@33175
  2106
          using `d>0` `x\<noteq>y`[unfolded dist_nz] dist_commute[of x y]
himmelma@33175
  2107
          unfolding dist_norm by auto
himmelma@33175
  2108
        ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac  x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
himmelma@33175
  2109
      qed
himmelma@33175
  2110
    next
himmelma@33175
  2111
      case False hence "d > dist x y" by auto
himmelma@33175
  2112
      show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  2113
      proof(cases "x=y")
himmelma@33175
  2114
        case True
himmelma@33175
  2115
        obtain z where **: "z \<noteq> y" "dist z y < min e d"
himmelma@33175
  2116
          using perfect_choose_dist[of "min e d" y]
himmelma@33175
  2117
          using `d > 0` `e>0` by auto
himmelma@33175
  2118
        show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  2119
          unfolding `x = y`
himmelma@33175
  2120
          using `z \<noteq> y` **
himmelma@33175
  2121
          by (rule_tac x=z in bexI, auto simp add: dist_commute)
himmelma@33175
  2122
      next
himmelma@33175
  2123
        case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
himmelma@33175
  2124
          using `d>0` `d > dist x y` `?rhs` by(rule_tac x=x in bexI, auto)
himmelma@33175
  2125
      qed
himmelma@33175
  2126
    qed  }
himmelma@33175
  2127
  thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
himmelma@33175
  2128
qed
himmelma@33175
  2129
himmelma@33175
  2130
lemma closure_ball_lemma:
himmelma@33175
  2131
  fixes x y :: "'a::real_normed_vector"
himmelma@33175
  2132
  assumes "x \<noteq> y" shows "y islimpt ball x (dist x y)"
himmelma@33175
  2133
proof (rule islimptI)
himmelma@33175
  2134
  fix T assume "y \<in> T" "open T"
himmelma@33175
  2135
  then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
himmelma@33175
  2136
    unfolding open_dist by fast
himmelma@33175
  2137
  (* choose point between x and y, within distance r of y. *)
himmelma@33175
  2138
  def k \<equiv> "min 1 (r / (2 * dist x y))"
himmelma@33175
  2139
  def z \<equiv> "y + scaleR k (x - y)"
himmelma@33175
  2140
  have z_def2: "z = x + scaleR (1 - k) (y - x)"
himmelma@33175
  2141
    unfolding z_def by (simp add: algebra_simps)
himmelma@33175
  2142
  have "dist z y < r"
himmelma@33175
  2143
    unfolding z_def k_def using `0 < r`
himmelma@33175
  2144
    by (simp add: dist_norm min_def)