src/HOL/Topological_Spaces.thy
author haftmann
Sun Jun 23 21:16:07 2013 +0200 (2013-06-23)
changeset 52435 6646bb548c6b
parent 52265 bb907eba5902
child 52729 412c9e0381a1
permissions -rw-r--r--
migration from code_(const|type|class|instance) to code_printing and from code_module to code_identifier
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(*  Title:      HOL/Topological_Spaces.thy
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    Author:     Brian Huffman
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    Author:     Johannes Hölzl
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*)
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header {* Topological Spaces *}
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theory Topological_Spaces
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imports Main Conditionally_Complete_Lattices
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begin
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subsection {* Topological space *}
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class "open" =
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  fixes "open" :: "'a set \<Rightarrow> bool"
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class topological_space = "open" +
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  assumes open_UNIV [simp, intro]: "open UNIV"
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  assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
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  assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
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begin
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definition
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  closed :: "'a set \<Rightarrow> bool" where
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  "closed S \<longleftrightarrow> open (- S)"
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lemma open_empty [intro, simp]: "open {}"
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  using open_Union [of "{}"] by simp
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lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
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  using open_Union [of "{S, T}"] by simp
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lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
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  unfolding SUP_def by (rule open_Union) auto
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lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
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  by (induct set: finite) auto
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lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
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  unfolding INF_def by (rule open_Inter) auto
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lemma openI:
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  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
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  shows "open S"
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proof -
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  have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
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  moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
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  ultimately show "open S" by simp
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qed
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lemma closed_empty [intro, simp]:  "closed {}"
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  unfolding closed_def by simp
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lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
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  unfolding closed_def by auto
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lemma closed_UNIV [intro, simp]: "closed UNIV"
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  unfolding closed_def by simp
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lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
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  unfolding closed_def by auto
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lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
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  unfolding closed_def by auto
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lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
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  unfolding closed_def uminus_Inf by auto
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lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
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  by (induct set: finite) auto
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lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
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  unfolding SUP_def by (rule closed_Union) auto
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lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
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  unfolding closed_def by simp
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lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
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  unfolding closed_def by simp
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lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
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  unfolding closed_open Diff_eq by (rule open_Int)
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lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
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  unfolding open_closed Diff_eq by (rule closed_Int)
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lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
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  unfolding closed_open .
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lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
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  unfolding open_closed .
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end
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subsection{* Hausdorff and other separation properties *}
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class t0_space = topological_space +
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  assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
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class t1_space = topological_space +
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  assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
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instance t1_space \<subseteq> t0_space
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proof qed (fast dest: t1_space)
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lemma separation_t1:
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  fixes x y :: "'a::t1_space"
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  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
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  using t1_space[of x y] by blast
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lemma closed_singleton:
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  fixes a :: "'a::t1_space"
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  shows "closed {a}"
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proof -
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  let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
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  have "open ?T" by (simp add: open_Union)
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  also have "?T = - {a}"
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    by (simp add: set_eq_iff separation_t1, auto)
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  finally show "closed {a}" unfolding closed_def .
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qed
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lemma closed_insert [simp]:
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  fixes a :: "'a::t1_space"
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  assumes "closed S" shows "closed (insert a S)"
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proof -
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  from closed_singleton assms
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  have "closed ({a} \<union> S)" by (rule closed_Un)
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  thus "closed (insert a S)" by simp
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qed
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lemma finite_imp_closed:
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  fixes S :: "'a::t1_space set"
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  shows "finite S \<Longrightarrow> closed S"
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by (induct set: finite, simp_all)
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text {* T2 spaces are also known as Hausdorff spaces. *}
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class t2_space = topological_space +
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  assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
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instance t2_space \<subseteq> t1_space
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proof qed (fast dest: hausdorff)
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lemma separation_t2:
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  fixes x y :: "'a::t2_space"
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  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
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  using hausdorff[of x y] by blast
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lemma separation_t0:
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  fixes x y :: "'a::t0_space"
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  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
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  using t0_space[of x y] by blast
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text {* A perfect space is a topological space with no isolated points. *}
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class perfect_space = topological_space +
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  assumes not_open_singleton: "\<not> open {x}"
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subsection {* Generators for toplogies *}
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inductive generate_topology for S where
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  UNIV: "generate_topology S UNIV"
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| Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)"
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| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)"
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| Basis: "s \<in> S \<Longrightarrow> generate_topology S s"
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hide_fact (open) UNIV Int UN Basis 
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lemma generate_topology_Union: 
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  "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"
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  unfolding SUP_def by (intro generate_topology.UN) auto
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lemma topological_space_generate_topology:
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  "class.topological_space (generate_topology S)"
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  by default (auto intro: generate_topology.intros)
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subsection {* Order topologies *}
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class order_topology = order + "open" +
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  assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
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begin
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subclass topological_space
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  unfolding open_generated_order
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  by (rule topological_space_generate_topology)
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lemma open_greaterThan [simp]: "open {a <..}"
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  unfolding open_generated_order by (auto intro: generate_topology.Basis)
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lemma open_lessThan [simp]: "open {..< a}"
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  unfolding open_generated_order by (auto intro: generate_topology.Basis)
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lemma open_greaterThanLessThan [simp]: "open {a <..< b}"
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   unfolding greaterThanLessThan_eq by (simp add: open_Int)
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end
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class linorder_topology = linorder + order_topology
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lemma closed_atMost [simp]: "closed {.. a::'a::linorder_topology}"
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  by (simp add: closed_open)
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lemma closed_atLeast [simp]: "closed {a::'a::linorder_topology ..}"
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  by (simp add: closed_open)
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lemma closed_atLeastAtMost [simp]: "closed {a::'a::linorder_topology .. b}"
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proof -
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  have "{a .. b} = {a ..} \<inter> {.. b}"
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    by auto
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  then show ?thesis
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    by (simp add: closed_Int)
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qed
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lemma (in linorder) less_separate:
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  assumes "x < y"
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  shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}"
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proof cases
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  assume "\<exists>z. x < z \<and> z < y"
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  then guess z ..
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  then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}"
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    by auto
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  then show ?thesis by blast
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next
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  assume "\<not> (\<exists>z. x < z \<and> z < y)"
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  with `x < y` have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}"
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    by auto
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  then show ?thesis by blast
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qed
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instance linorder_topology \<subseteq> t2_space
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proof
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  fix x y :: 'a
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  from less_separate[of x y] less_separate[of y x]
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  show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
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    by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+
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qed
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lemma (in linorder_topology) open_right:
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  assumes "open S" "x \<in> S" and gt_ex: "x < y" shows "\<exists>b>x. {x ..< b} \<subseteq> S"
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  using assms unfolding open_generated_order
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proof induction
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  case (Int A B)
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  then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B" by auto
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  then show ?case by (auto intro!: exI[of _ "min a b"])
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next
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  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] gt_ex)
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qed blast+
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lemma (in linorder_topology) open_left:
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  assumes "open S" "x \<in> S" and lt_ex: "y < x" shows "\<exists>b<x. {b <.. x} \<subseteq> S"
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  using assms unfolding open_generated_order
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proof induction
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  case (Int A B)
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  then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B" by auto
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  then show ?case by (auto intro!: exI[of _ "max a b"])
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next
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  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex)
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qed blast+
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subsection {* Filters *}
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text {*
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  This definition also allows non-proper filters.
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*}
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locale is_filter =
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  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
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  assumes True: "F (\<lambda>x. True)"
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  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
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  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
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typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
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proof
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  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
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qed
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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
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  using Rep_filter [of F] by simp
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lemma Abs_filter_inverse':
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  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
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  using assms by (simp add: Abs_filter_inverse)
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subsubsection {* Eventually *}
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
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  where "eventually P F \<longleftrightarrow> Rep_filter F P"
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lemma eventually_Abs_filter:
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  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
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  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
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lemma filter_eq_iff:
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  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
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  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
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  unfolding eventually_def
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  by (rule is_filter.True [OF is_filter_Rep_filter])
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
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   304
proof -
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   305
  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
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   306
  thus "eventually P F" by simp
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   307
qed
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   308
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   309
lemma eventually_mono:
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   310
  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
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   311
  unfolding eventually_def
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   312
  by (rule is_filter.mono [OF is_filter_Rep_filter])
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   313
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   314
lemma eventually_conj:
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   315
  assumes P: "eventually (\<lambda>x. P x) F"
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   316
  assumes Q: "eventually (\<lambda>x. Q x) F"
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   317
  shows "eventually (\<lambda>x. P x \<and> Q x) F"
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   318
  using assms unfolding eventually_def
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   319
  by (rule is_filter.conj [OF is_filter_Rep_filter])
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   320
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   321
lemma eventually_Ball_finite:
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   322
  assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
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   323
  shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
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   324
using assms by (induct set: finite, simp, simp add: eventually_conj)
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   325
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   326
lemma eventually_all_finite:
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   327
  fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
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   328
  assumes "\<And>y. eventually (\<lambda>x. P x y) net"
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   329
  shows "eventually (\<lambda>x. \<forall>y. P x y) net"
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   330
using eventually_Ball_finite [of UNIV P] assms by simp
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   331
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   332
lemma eventually_mp:
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   333
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
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   334
  assumes "eventually (\<lambda>x. P x) F"
hoelzl@51471
   335
  shows "eventually (\<lambda>x. Q x) F"
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   336
proof (rule eventually_mono)
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   337
  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
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   338
  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
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   339
    using assms by (rule eventually_conj)
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   340
qed
hoelzl@51471
   341
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   342
lemma eventually_rev_mp:
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   343
  assumes "eventually (\<lambda>x. P x) F"
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   344
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
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   345
  shows "eventually (\<lambda>x. Q x) F"
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   346
using assms(2) assms(1) by (rule eventually_mp)
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   347
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   348
lemma eventually_conj_iff:
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   349
  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
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   350
  by (auto intro: eventually_conj elim: eventually_rev_mp)
hoelzl@51471
   351
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   352
lemma eventually_elim1:
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   353
  assumes "eventually (\<lambda>i. P i) F"
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   354
  assumes "\<And>i. P i \<Longrightarrow> Q i"
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   355
  shows "eventually (\<lambda>i. Q i) F"
hoelzl@51471
   356
  using assms by (auto elim!: eventually_rev_mp)
hoelzl@51471
   357
hoelzl@51471
   358
lemma eventually_elim2:
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   359
  assumes "eventually (\<lambda>i. P i) F"
hoelzl@51471
   360
  assumes "eventually (\<lambda>i. Q i) F"
hoelzl@51471
   361
  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
hoelzl@51471
   362
  shows "eventually (\<lambda>i. R i) F"
hoelzl@51471
   363
  using assms by (auto elim!: eventually_rev_mp)
hoelzl@51471
   364
hoelzl@51471
   365
lemma eventually_subst:
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   366
  assumes "eventually (\<lambda>n. P n = Q n) F"
hoelzl@51471
   367
  shows "eventually P F = eventually Q F" (is "?L = ?R")
hoelzl@51471
   368
proof -
hoelzl@51471
   369
  from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@51471
   370
      and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
hoelzl@51471
   371
    by (auto elim: eventually_elim1)
hoelzl@51471
   372
  then show ?thesis by (auto elim: eventually_elim2)
hoelzl@51471
   373
qed
hoelzl@51471
   374
hoelzl@51471
   375
ML {*
hoelzl@51471
   376
  fun eventually_elim_tac ctxt thms thm =
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   377
    let
hoelzl@51471
   378
      val thy = Proof_Context.theory_of ctxt
hoelzl@51471
   379
      val mp_thms = thms RL [@{thm eventually_rev_mp}]
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   380
      val raw_elim_thm =
hoelzl@51471
   381
        (@{thm allI} RS @{thm always_eventually})
hoelzl@51471
   382
        |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
hoelzl@51471
   383
        |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
hoelzl@51471
   384
      val cases_prop = prop_of (raw_elim_thm RS thm)
hoelzl@51471
   385
      val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
hoelzl@51471
   386
    in
hoelzl@51471
   387
      CASES cases (rtac raw_elim_thm 1) thm
hoelzl@51471
   388
    end
hoelzl@51471
   389
*}
hoelzl@51471
   390
hoelzl@51471
   391
method_setup eventually_elim = {*
hoelzl@51471
   392
  Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
hoelzl@51471
   393
*} "elimination of eventually quantifiers"
hoelzl@51471
   394
hoelzl@51471
   395
hoelzl@51471
   396
subsubsection {* Finer-than relation *}
hoelzl@51471
   397
hoelzl@51471
   398
text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
hoelzl@51471
   399
filter @{term F'}. *}
hoelzl@51471
   400
hoelzl@51471
   401
instantiation filter :: (type) complete_lattice
hoelzl@51471
   402
begin
hoelzl@51471
   403
hoelzl@51471
   404
definition le_filter_def:
hoelzl@51471
   405
  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
hoelzl@51471
   406
hoelzl@51471
   407
definition
hoelzl@51471
   408
  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@51471
   409
hoelzl@51471
   410
definition
hoelzl@51471
   411
  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
hoelzl@51471
   412
hoelzl@51471
   413
definition
hoelzl@51471
   414
  "bot = Abs_filter (\<lambda>P. True)"
hoelzl@51471
   415
hoelzl@51471
   416
definition
hoelzl@51471
   417
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
hoelzl@51471
   418
hoelzl@51471
   419
definition
hoelzl@51471
   420
  "inf F F' = Abs_filter
hoelzl@51471
   421
      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@51471
   422
hoelzl@51471
   423
definition
hoelzl@51471
   424
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
hoelzl@51471
   425
hoelzl@51471
   426
definition
hoelzl@51471
   427
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
hoelzl@51471
   428
hoelzl@51471
   429
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
hoelzl@51471
   430
  unfolding top_filter_def
hoelzl@51471
   431
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@51471
   432
hoelzl@51471
   433
lemma eventually_bot [simp]: "eventually P bot"
hoelzl@51471
   434
  unfolding bot_filter_def
hoelzl@51471
   435
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@51471
   436
hoelzl@51471
   437
lemma eventually_sup:
hoelzl@51471
   438
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
hoelzl@51471
   439
  unfolding sup_filter_def
hoelzl@51471
   440
  by (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   441
     (auto elim!: eventually_rev_mp)
hoelzl@51471
   442
hoelzl@51471
   443
lemma eventually_inf:
hoelzl@51471
   444
  "eventually P (inf F F') \<longleftrightarrow>
hoelzl@51471
   445
   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@51471
   446
  unfolding inf_filter_def
hoelzl@51471
   447
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   448
  apply (fast intro: eventually_True)
hoelzl@51471
   449
  apply clarify
hoelzl@51471
   450
  apply (intro exI conjI)
hoelzl@51471
   451
  apply (erule (1) eventually_conj)
hoelzl@51471
   452
  apply (erule (1) eventually_conj)
hoelzl@51471
   453
  apply simp
hoelzl@51471
   454
  apply auto
hoelzl@51471
   455
  done
hoelzl@51471
   456
hoelzl@51471
   457
lemma eventually_Sup:
hoelzl@51471
   458
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
hoelzl@51471
   459
  unfolding Sup_filter_def
hoelzl@51471
   460
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   461
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
hoelzl@51471
   462
  done
hoelzl@51471
   463
hoelzl@51471
   464
instance proof
hoelzl@51471
   465
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
hoelzl@51471
   466
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@51471
   467
    by (rule less_filter_def) }
hoelzl@51471
   468
  { show "F \<le> F"
hoelzl@51471
   469
    unfolding le_filter_def by simp }
hoelzl@51471
   470
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
hoelzl@51471
   471
    unfolding le_filter_def by simp }
hoelzl@51471
   472
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
hoelzl@51471
   473
    unfolding le_filter_def filter_eq_iff by fast }
hoelzl@51471
   474
  { show "F \<le> top"
hoelzl@51471
   475
    unfolding le_filter_def eventually_top by (simp add: always_eventually) }
hoelzl@51471
   476
  { show "bot \<le> F"
hoelzl@51471
   477
    unfolding le_filter_def by simp }
hoelzl@51471
   478
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
hoelzl@51471
   479
    unfolding le_filter_def eventually_sup by simp_all }
hoelzl@51471
   480
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
hoelzl@51471
   481
    unfolding le_filter_def eventually_sup by simp }
hoelzl@51471
   482
  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
hoelzl@51471
   483
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
hoelzl@51471
   484
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
hoelzl@51471
   485
    unfolding le_filter_def eventually_inf
hoelzl@51471
   486
    by (auto elim!: eventually_mono intro: eventually_conj) }
hoelzl@51471
   487
  { assume "F \<in> S" thus "F \<le> Sup S"
hoelzl@51471
   488
    unfolding le_filter_def eventually_Sup by simp }
hoelzl@51471
   489
  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
hoelzl@51471
   490
    unfolding le_filter_def eventually_Sup by simp }
hoelzl@51471
   491
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
hoelzl@51471
   492
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
hoelzl@51471
   493
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
hoelzl@51471
   494
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
hoelzl@51471
   495
qed
hoelzl@51471
   496
hoelzl@51471
   497
end
hoelzl@51471
   498
hoelzl@51471
   499
lemma filter_leD:
hoelzl@51471
   500
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
hoelzl@51471
   501
  unfolding le_filter_def by simp
hoelzl@51471
   502
hoelzl@51471
   503
lemma filter_leI:
hoelzl@51471
   504
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
hoelzl@51471
   505
  unfolding le_filter_def by simp
hoelzl@51471
   506
hoelzl@51471
   507
lemma eventually_False:
hoelzl@51471
   508
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
hoelzl@51471
   509
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
hoelzl@51471
   510
hoelzl@51471
   511
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
hoelzl@51471
   512
  where "trivial_limit F \<equiv> F = bot"
hoelzl@51471
   513
hoelzl@51471
   514
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
hoelzl@51471
   515
  by (rule eventually_False [symmetric])
hoelzl@51471
   516
hoelzl@51471
   517
lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"
hoelzl@51471
   518
  by (cases P) (simp_all add: eventually_False)
hoelzl@51471
   519
hoelzl@51471
   520
hoelzl@51471
   521
subsubsection {* Map function for filters *}
hoelzl@51471
   522
hoelzl@51471
   523
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
hoelzl@51471
   524
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
hoelzl@51471
   525
hoelzl@51471
   526
lemma eventually_filtermap:
hoelzl@51471
   527
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
hoelzl@51471
   528
  unfolding filtermap_def
hoelzl@51471
   529
  apply (rule eventually_Abs_filter)
hoelzl@51471
   530
  apply (rule is_filter.intro)
hoelzl@51471
   531
  apply (auto elim!: eventually_rev_mp)
hoelzl@51471
   532
  done
hoelzl@51471
   533
hoelzl@51471
   534
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
hoelzl@51471
   535
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   536
hoelzl@51471
   537
lemma filtermap_filtermap:
hoelzl@51471
   538
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
hoelzl@51471
   539
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   540
hoelzl@51471
   541
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
hoelzl@51471
   542
  unfolding le_filter_def eventually_filtermap by simp
hoelzl@51471
   543
hoelzl@51471
   544
lemma filtermap_bot [simp]: "filtermap f bot = bot"
hoelzl@51471
   545
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   546
hoelzl@51471
   547
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
hoelzl@51471
   548
  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
hoelzl@51471
   549
hoelzl@51471
   550
subsubsection {* Order filters *}
hoelzl@51471
   551
hoelzl@51471
   552
definition at_top :: "('a::order) filter"
hoelzl@51471
   553
  where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
hoelzl@51471
   554
hoelzl@51471
   555
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
hoelzl@51471
   556
  unfolding at_top_def
hoelzl@51471
   557
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   558
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@51471
   559
  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
hoelzl@51471
   560
  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
hoelzl@51471
   561
  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
hoelzl@51471
   562
  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
hoelzl@51471
   563
qed auto
hoelzl@51471
   564
hoelzl@51471
   565
lemma eventually_ge_at_top:
hoelzl@51471
   566
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
hoelzl@51471
   567
  unfolding eventually_at_top_linorder by auto
hoelzl@51471
   568
hoelzl@51471
   569
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::dense_linorder. \<forall>n>N. P n)"
hoelzl@51471
   570
  unfolding eventually_at_top_linorder
hoelzl@51471
   571
proof safe
hoelzl@51471
   572
  fix N assume "\<forall>n\<ge>N. P n" then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])
hoelzl@51471
   573
next
hoelzl@51471
   574
  fix N assume "\<forall>n>N. P n"
hoelzl@51471
   575
  moreover from gt_ex[of N] guess y ..
hoelzl@51471
   576
  ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])
hoelzl@51471
   577
qed
hoelzl@51471
   578
hoelzl@51471
   579
lemma eventually_gt_at_top:
hoelzl@51471
   580
  "eventually (\<lambda>x. (c::_::dense_linorder) < x) at_top"
hoelzl@51471
   581
  unfolding eventually_at_top_dense by auto
hoelzl@51471
   582
hoelzl@51471
   583
definition at_bot :: "('a::order) filter"
hoelzl@51471
   584
  where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
hoelzl@51471
   585
hoelzl@51471
   586
lemma eventually_at_bot_linorder:
hoelzl@51471
   587
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@51471
   588
  unfolding at_bot_def
hoelzl@51471
   589
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   590
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@51471
   591
  assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
hoelzl@51471
   592
  then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
hoelzl@51471
   593
  then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
hoelzl@51471
   594
  then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
hoelzl@51471
   595
qed auto
hoelzl@51471
   596
hoelzl@51471
   597
lemma eventually_le_at_bot:
hoelzl@51471
   598
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
hoelzl@51471
   599
  unfolding eventually_at_bot_linorder by auto
hoelzl@51471
   600
hoelzl@51471
   601
lemma eventually_at_bot_dense:
hoelzl@51471
   602
  fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
hoelzl@51471
   603
  unfolding eventually_at_bot_linorder
hoelzl@51471
   604
proof safe
hoelzl@51471
   605
  fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N])
hoelzl@51471
   606
next
hoelzl@51471
   607
  fix N assume "\<forall>n<N. P n" 
hoelzl@51471
   608
  moreover from lt_ex[of N] guess y ..
hoelzl@51471
   609
  ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
hoelzl@51471
   610
qed
hoelzl@51471
   611
hoelzl@51471
   612
lemma eventually_gt_at_bot:
hoelzl@51471
   613
  "eventually (\<lambda>x. x < (c::_::dense_linorder)) at_bot"
hoelzl@51471
   614
  unfolding eventually_at_bot_dense by auto
hoelzl@51471
   615
hoelzl@51471
   616
subsection {* Sequentially *}
hoelzl@51471
   617
hoelzl@51471
   618
abbreviation sequentially :: "nat filter"
hoelzl@51471
   619
  where "sequentially == at_top"
hoelzl@51471
   620
hoelzl@51471
   621
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
hoelzl@51471
   622
  unfolding at_top_def by simp
hoelzl@51471
   623
hoelzl@51471
   624
lemma eventually_sequentially:
hoelzl@51471
   625
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@51471
   626
  by (rule eventually_at_top_linorder)
hoelzl@51471
   627
hoelzl@51471
   628
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
hoelzl@51471
   629
  unfolding filter_eq_iff eventually_sequentially by auto
hoelzl@51471
   630
hoelzl@51471
   631
lemmas trivial_limit_sequentially = sequentially_bot
hoelzl@51471
   632
hoelzl@51471
   633
lemma eventually_False_sequentially [simp]:
hoelzl@51471
   634
  "\<not> eventually (\<lambda>n. False) sequentially"
hoelzl@51471
   635
  by (simp add: eventually_False)
hoelzl@51471
   636
hoelzl@51471
   637
lemma le_sequentially:
hoelzl@51471
   638
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
hoelzl@51471
   639
  unfolding le_filter_def eventually_sequentially
hoelzl@51471
   640
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
hoelzl@51471
   641
hoelzl@51471
   642
lemma eventually_sequentiallyI:
hoelzl@51471
   643
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
hoelzl@51471
   644
  shows "eventually P sequentially"
hoelzl@51471
   645
using assms by (auto simp: eventually_sequentially)
hoelzl@51471
   646
hoelzl@51474
   647
lemma eventually_sequentially_seg:
hoelzl@51474
   648
  "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@51474
   649
  unfolding eventually_sequentially
hoelzl@51474
   650
  apply safe
hoelzl@51474
   651
   apply (rule_tac x="N + k" in exI)
hoelzl@51474
   652
   apply rule
hoelzl@51474
   653
   apply (erule_tac x="n - k" in allE)
hoelzl@51474
   654
   apply auto []
hoelzl@51474
   655
  apply (rule_tac x=N in exI)
hoelzl@51474
   656
  apply auto []
hoelzl@51474
   657
  done
hoelzl@51471
   658
hoelzl@51471
   659
subsubsection {* Standard filters *}
hoelzl@51471
   660
hoelzl@51641
   661
definition principal :: "'a set \<Rightarrow> 'a filter" where
hoelzl@51641
   662
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
hoelzl@51641
   663
hoelzl@51641
   664
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
hoelzl@51641
   665
  unfolding principal_def
hoelzl@51641
   666
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
hoelzl@51471
   667
hoelzl@51641
   668
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
hoelzl@51641
   669
  unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
hoelzl@51641
   670
hoelzl@51641
   671
lemma principal_UNIV[simp]: "principal UNIV = top"
hoelzl@51641
   672
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@51471
   673
hoelzl@51641
   674
lemma principal_empty[simp]: "principal {} = bot"
hoelzl@51641
   675
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@51641
   676
hoelzl@51641
   677
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
hoelzl@51641
   678
  by (auto simp: le_filter_def eventually_principal)
hoelzl@51471
   679
hoelzl@51641
   680
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
hoelzl@51641
   681
  unfolding le_filter_def eventually_principal
hoelzl@51641
   682
  apply safe
hoelzl@51641
   683
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
hoelzl@51641
   684
  apply (auto elim: eventually_elim1)
hoelzl@51641
   685
  done
hoelzl@51471
   686
hoelzl@51641
   687
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
hoelzl@51641
   688
  unfolding eq_iff by simp
hoelzl@51471
   689
hoelzl@51641
   690
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
hoelzl@51641
   691
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
hoelzl@51471
   692
hoelzl@51641
   693
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
hoelzl@51641
   694
  unfolding filter_eq_iff eventually_inf eventually_principal
hoelzl@51641
   695
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
hoelzl@51471
   696
hoelzl@51641
   697
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
hoelzl@51641
   698
  unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
hoelzl@51641
   699
hoelzl@51641
   700
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
hoelzl@51641
   701
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
hoelzl@51471
   702
hoelzl@51471
   703
subsubsection {* Topological filters *}
hoelzl@51471
   704
hoelzl@51471
   705
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
hoelzl@51471
   706
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
hoelzl@51471
   707
hoelzl@51641
   708
definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60)
hoelzl@51641
   709
  where "at a within s = inf (nhds a) (principal (s - {a}))"
hoelzl@51641
   710
hoelzl@51641
   711
abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where
hoelzl@51641
   712
  "at x \<equiv> at x within (CONST UNIV)"
hoelzl@51471
   713
hoelzl@51473
   714
abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   715
  "at_right x \<equiv> at x within {x <..}"
hoelzl@51471
   716
hoelzl@51473
   717
abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   718
  "at_left x \<equiv> at x within {..< x}"
hoelzl@51471
   719
hoelzl@51473
   720
lemma (in topological_space) eventually_nhds:
hoelzl@51471
   721
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
hoelzl@51471
   722
  unfolding nhds_def
hoelzl@51471
   723
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51473
   724
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
hoelzl@51471
   725
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
hoelzl@51471
   726
next
hoelzl@51471
   727
  fix P Q
hoelzl@51471
   728
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
hoelzl@51471
   729
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
hoelzl@51471
   730
  then obtain S T where
hoelzl@51471
   731
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
hoelzl@51471
   732
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
hoelzl@51471
   733
  hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
hoelzl@51471
   734
    by (simp add: open_Int)
hoelzl@51471
   735
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" ..
hoelzl@51471
   736
qed auto
hoelzl@51471
   737
hoelzl@51471
   738
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
hoelzl@51471
   739
  unfolding trivial_limit_def eventually_nhds by simp
hoelzl@51471
   740
hoelzl@51641
   741
lemma eventually_at_filter:
hoelzl@51641
   742
  "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
hoelzl@51641
   743
  unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)
hoelzl@51641
   744
hoelzl@51641
   745
lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
hoelzl@51641
   746
  unfolding at_within_def by (intro inf_mono) auto
hoelzl@51641
   747
hoelzl@51471
   748
lemma eventually_at_topological:
hoelzl@51641
   749
  "eventually P (at a within s) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
hoelzl@51641
   750
  unfolding eventually_nhds eventually_at_filter by simp
hoelzl@51471
   751
hoelzl@51481
   752
lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
hoelzl@51641
   753
  unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
hoelzl@51481
   754
hoelzl@51471
   755
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
hoelzl@51471
   756
  unfolding trivial_limit_def eventually_at_topological
hoelzl@51471
   757
  by (safe, case_tac "S = {a}", simp, fast, fast)
hoelzl@51471
   758
hoelzl@51471
   759
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
hoelzl@51471
   760
  by (simp add: at_eq_bot_iff not_open_singleton)
hoelzl@51471
   761
hoelzl@51471
   762
lemma eventually_at_right:
hoelzl@51471
   763
  fixes x :: "'a :: {no_top, linorder_topology}"
hoelzl@51471
   764
  shows "eventually P (at_right x) \<longleftrightarrow> (\<exists>b. x < b \<and> (\<forall>z. x < z \<and> z < b \<longrightarrow> P z))"
hoelzl@51641
   765
  unfolding eventually_at_topological
hoelzl@51471
   766
proof safe
hoelzl@51480
   767
  from gt_ex[of x] guess y ..
hoelzl@51480
   768
  moreover fix S assume "open S" "x \<in> S" note open_right[OF this, of y]
hoelzl@51480
   769
  moreover note gt_ex[of x]
hoelzl@51641
   770
  moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {x<..} \<longrightarrow> P s"
hoelzl@51471
   771
  ultimately show "\<exists>b>x. \<forall>z. x < z \<and> z < b \<longrightarrow> P z"
hoelzl@51471
   772
    by (auto simp: subset_eq Ball_def)
hoelzl@51471
   773
next
hoelzl@51471
   774
  fix b assume "x < b" "\<forall>z. x < z \<and> z < b \<longrightarrow> P z"
hoelzl@51641
   775
  then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>xa\<in>S. xa \<noteq> x \<longrightarrow> xa \<in> {x<..} \<longrightarrow> P xa)"
hoelzl@51471
   776
    by (intro exI[of _ "{..< b}"]) auto
hoelzl@51471
   777
qed
hoelzl@51471
   778
hoelzl@51471
   779
lemma eventually_at_left:
hoelzl@51471
   780
  fixes x :: "'a :: {no_bot, linorder_topology}"
hoelzl@51471
   781
  shows "eventually P (at_left x) \<longleftrightarrow> (\<exists>b. x > b \<and> (\<forall>z. b < z \<and> z < x \<longrightarrow> P z))"
hoelzl@51641
   782
  unfolding eventually_at_topological
hoelzl@51471
   783
proof safe
hoelzl@51480
   784
  from lt_ex[of x] guess y ..
hoelzl@51480
   785
  moreover fix S assume "open S" "x \<in> S" note open_left[OF this, of y]
hoelzl@51641
   786
  moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {..<x} \<longrightarrow> P s"
hoelzl@51471
   787
  ultimately show "\<exists>b<x. \<forall>z. b < z \<and> z < x \<longrightarrow> P z"
hoelzl@51471
   788
    by (auto simp: subset_eq Ball_def)
hoelzl@51471
   789
next
hoelzl@51471
   790
  fix b assume "b < x" "\<forall>z. b < z \<and> z < x \<longrightarrow> P z"
hoelzl@51641
   791
  then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {..<x} \<longrightarrow> P s)"
hoelzl@51471
   792
    by (intro exI[of _ "{b <..}"]) auto
hoelzl@51471
   793
qed
hoelzl@51471
   794
hoelzl@51471
   795
lemma trivial_limit_at_left_real [simp]:
hoelzl@51471
   796
  "\<not> trivial_limit (at_left (x::'a::{no_bot, dense_linorder, linorder_topology}))"
hoelzl@51471
   797
  unfolding trivial_limit_def eventually_at_left by (auto dest: dense)
hoelzl@51471
   798
hoelzl@51471
   799
lemma trivial_limit_at_right_real [simp]:
hoelzl@51471
   800
  "\<not> trivial_limit (at_right (x::'a::{no_top, dense_linorder, linorder_topology}))"
hoelzl@51471
   801
  unfolding trivial_limit_def eventually_at_right by (auto dest: dense)
hoelzl@51471
   802
hoelzl@51471
   803
lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"
hoelzl@51641
   804
  by (auto simp: eventually_at_filter filter_eq_iff eventually_sup 
hoelzl@51471
   805
           elim: eventually_elim2 eventually_elim1)
hoelzl@51471
   806
hoelzl@51471
   807
lemma eventually_at_split:
hoelzl@51471
   808
  "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
hoelzl@51471
   809
  by (subst at_eq_sup_left_right) (simp add: eventually_sup)
hoelzl@51471
   810
hoelzl@51471
   811
subsection {* Limits *}
hoelzl@51471
   812
hoelzl@51471
   813
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@51471
   814
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@51471
   815
hoelzl@51471
   816
syntax
hoelzl@51471
   817
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@51471
   818
hoelzl@51471
   819
translations
hoelzl@51471
   820
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@51471
   821
hoelzl@51471
   822
lemma filterlim_iff:
hoelzl@51471
   823
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
hoelzl@51471
   824
  unfolding filterlim_def le_filter_def eventually_filtermap ..
hoelzl@51471
   825
hoelzl@51471
   826
lemma filterlim_compose:
hoelzl@51471
   827
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@51471
   828
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@51471
   829
hoelzl@51471
   830
lemma filterlim_mono:
hoelzl@51471
   831
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@51471
   832
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@51471
   833
hoelzl@51471
   834
lemma filterlim_ident: "LIM x F. x :> F"
hoelzl@51471
   835
  by (simp add: filterlim_def filtermap_ident)
hoelzl@51471
   836
hoelzl@51471
   837
lemma filterlim_cong:
hoelzl@51471
   838
  "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
hoelzl@51471
   839
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
hoelzl@51471
   840
hoelzl@51641
   841
lemma filterlim_principal:
hoelzl@51641
   842
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@51641
   843
  unfolding filterlim_def eventually_filtermap le_principal ..
hoelzl@51641
   844
hoelzl@51641
   845
lemma filterlim_inf:
hoelzl@51641
   846
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
hoelzl@51641
   847
  unfolding filterlim_def by simp
hoelzl@51471
   848
hoelzl@51471
   849
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
hoelzl@51471
   850
  unfolding filterlim_def filtermap_filtermap ..
hoelzl@51471
   851
hoelzl@51471
   852
lemma filterlim_sup:
hoelzl@51471
   853
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
hoelzl@51471
   854
  unfolding filterlim_def filtermap_sup by auto
hoelzl@51471
   855
hoelzl@51471
   856
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
hoelzl@51471
   857
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
hoelzl@51471
   858
hoelzl@51471
   859
subsubsection {* Tendsto *}
hoelzl@51471
   860
hoelzl@51471
   861
abbreviation (in topological_space)
hoelzl@51471
   862
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
hoelzl@51471
   863
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
hoelzl@51471
   864
hoelzl@51478
   865
definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
   866
  "Lim A f = (THE l. (f ---> l) A)"
hoelzl@51478
   867
hoelzl@51471
   868
lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F"
hoelzl@51471
   869
  by simp
hoelzl@51471
   870
hoelzl@51471
   871
ML {*
hoelzl@51471
   872
hoelzl@51471
   873
structure Tendsto_Intros = Named_Thms
hoelzl@51471
   874
(
hoelzl@51471
   875
  val name = @{binding tendsto_intros}
hoelzl@51471
   876
  val description = "introduction rules for tendsto"
hoelzl@51471
   877
)
hoelzl@51471
   878
hoelzl@51471
   879
*}
hoelzl@51471
   880
hoelzl@51471
   881
setup {*
hoelzl@51471
   882
  Tendsto_Intros.setup #>
hoelzl@51471
   883
  Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},
hoelzl@51641
   884
    map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])) o Tendsto_Intros.get o Context.proof_of);
hoelzl@51471
   885
*}
hoelzl@51471
   886
hoelzl@51473
   887
lemma (in topological_space) tendsto_def:
hoelzl@51473
   888
   "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@51471
   889
  unfolding filterlim_def
hoelzl@51471
   890
proof safe
hoelzl@51471
   891
  fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
hoelzl@51471
   892
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   893
    unfolding eventually_nhds eventually_filtermap le_filter_def
hoelzl@51471
   894
    by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
hoelzl@51471
   895
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
hoelzl@51471
   896
hoelzl@51471
   897
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
hoelzl@51471
   898
  unfolding tendsto_def le_filter_def by fast
hoelzl@51471
   899
hoelzl@51641
   900
lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)"
hoelzl@51641
   901
  by (blast intro: tendsto_mono at_le)
hoelzl@51641
   902
hoelzl@51641
   903
lemma filterlim_at:
hoelzl@51641
   904
  "(LIM x F. f x :> at b within s) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f ---> b) F)"
hoelzl@51641
   905
  by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
hoelzl@51641
   906
hoelzl@51473
   907
lemma (in topological_space) topological_tendstoI:
hoelzl@51641
   908
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f ---> l) F"
hoelzl@51471
   909
  unfolding tendsto_def by auto
hoelzl@51471
   910
hoelzl@51473
   911
lemma (in topological_space) topological_tendstoD:
hoelzl@51471
   912
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   913
  unfolding tendsto_def by auto
hoelzl@51471
   914
hoelzl@51471
   915
lemma order_tendstoI:
hoelzl@51471
   916
  fixes y :: "_ :: order_topology"
hoelzl@51471
   917
  assumes "\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
hoelzl@51471
   918
  assumes "\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
hoelzl@51471
   919
  shows "(f ---> y) F"
hoelzl@51471
   920
proof (rule topological_tendstoI)
hoelzl@51471
   921
  fix S assume "open S" "y \<in> S"
hoelzl@51471
   922
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   923
    unfolding open_generated_order
hoelzl@51471
   924
  proof induct
hoelzl@51471
   925
    case (UN K)
hoelzl@51471
   926
    then obtain k where "y \<in> k" "k \<in> K" by auto
hoelzl@51471
   927
    with UN(2)[of k] show ?case
hoelzl@51471
   928
      by (auto elim: eventually_elim1)
hoelzl@51471
   929
  qed (insert assms, auto elim: eventually_elim2)
hoelzl@51471
   930
qed
hoelzl@51471
   931
hoelzl@51471
   932
lemma order_tendstoD:
hoelzl@51471
   933
  fixes y :: "_ :: order_topology"
hoelzl@51471
   934
  assumes y: "(f ---> y) F"
hoelzl@51471
   935
  shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
hoelzl@51471
   936
    and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
hoelzl@51471
   937
  using topological_tendstoD[OF y, of "{..< a}"] topological_tendstoD[OF y, of "{a <..}"] by auto
hoelzl@51471
   938
hoelzl@51471
   939
lemma order_tendsto_iff: 
hoelzl@51471
   940
  fixes f :: "_ \<Rightarrow> 'a :: order_topology"
hoelzl@51471
   941
  shows "(f ---> x) F \<longleftrightarrow>(\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)"
hoelzl@51471
   942
  by (metis order_tendstoI order_tendstoD)
hoelzl@51471
   943
hoelzl@51471
   944
lemma tendsto_bot [simp]: "(f ---> a) bot"
hoelzl@51471
   945
  unfolding tendsto_def by simp
hoelzl@51471
   946
hoelzl@51641
   947
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a within s)"
hoelzl@51471
   948
  unfolding tendsto_def eventually_at_topological by auto
hoelzl@51471
   949
hoelzl@51478
   950
lemma (in topological_space) tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
hoelzl@51471
   951
  by (simp add: tendsto_def)
hoelzl@51471
   952
hoelzl@51478
   953
lemma (in t2_space) tendsto_unique:
hoelzl@51471
   954
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
hoelzl@51471
   955
  shows "a = b"
hoelzl@51471
   956
proof (rule ccontr)
hoelzl@51471
   957
  assume "a \<noteq> b"
hoelzl@51471
   958
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
hoelzl@51471
   959
    using hausdorff [OF `a \<noteq> b`] by fast
hoelzl@51471
   960
  have "eventually (\<lambda>x. f x \<in> U) F"
hoelzl@51471
   961
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
hoelzl@51471
   962
  moreover
hoelzl@51471
   963
  have "eventually (\<lambda>x. f x \<in> V) F"
hoelzl@51471
   964
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
hoelzl@51471
   965
  ultimately
hoelzl@51471
   966
  have "eventually (\<lambda>x. False) F"
hoelzl@51471
   967
  proof eventually_elim
hoelzl@51471
   968
    case (elim x)
hoelzl@51471
   969
    hence "f x \<in> U \<inter> V" by simp
hoelzl@51471
   970
    with `U \<inter> V = {}` show ?case by simp
hoelzl@51471
   971
  qed
hoelzl@51471
   972
  with `\<not> trivial_limit F` show "False"
hoelzl@51471
   973
    by (simp add: trivial_limit_def)
hoelzl@51471
   974
qed
hoelzl@51471
   975
hoelzl@51478
   976
lemma (in t2_space) tendsto_const_iff:
hoelzl@51478
   977
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"
hoelzl@51471
   978
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
hoelzl@51471
   979
hoelzl@51471
   980
lemma increasing_tendsto:
hoelzl@51471
   981
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
   982
  assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
hoelzl@51471
   983
      and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
hoelzl@51471
   984
  shows "(f ---> l) F"
hoelzl@51471
   985
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
hoelzl@51471
   986
hoelzl@51471
   987
lemma decreasing_tendsto:
hoelzl@51471
   988
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
   989
  assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
hoelzl@51471
   990
      and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
hoelzl@51471
   991
  shows "(f ---> l) F"
hoelzl@51471
   992
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
hoelzl@51471
   993
hoelzl@51471
   994
lemma tendsto_sandwich:
hoelzl@51471
   995
  fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
hoelzl@51471
   996
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
hoelzl@51471
   997
  assumes lim: "(f ---> c) net" "(h ---> c) net"
hoelzl@51471
   998
  shows "(g ---> c) net"
hoelzl@51471
   999
proof (rule order_tendstoI)
hoelzl@51471
  1000
  fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
hoelzl@51471
  1001
    using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
  1002
next
hoelzl@51471
  1003
  fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
hoelzl@51471
  1004
    using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
  1005
qed
hoelzl@51471
  1006
hoelzl@51471
  1007
lemma tendsto_le:
hoelzl@51471
  1008
  fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
  1009
  assumes F: "\<not> trivial_limit F"
hoelzl@51471
  1010
  assumes x: "(f ---> x) F" and y: "(g ---> y) F"
hoelzl@51471
  1011
  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
hoelzl@51471
  1012
  shows "y \<le> x"
hoelzl@51471
  1013
proof (rule ccontr)
hoelzl@51471
  1014
  assume "\<not> y \<le> x"
hoelzl@51471
  1015
  with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
hoelzl@51471
  1016
    by (auto simp: not_le)
hoelzl@51471
  1017
  then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
hoelzl@51471
  1018
    using x y by (auto intro: order_tendstoD)
hoelzl@51471
  1019
  with ev have "eventually (\<lambda>x. False) F"
hoelzl@51471
  1020
    by eventually_elim (insert xy, fastforce)
hoelzl@51471
  1021
  with F show False
hoelzl@51471
  1022
    by (simp add: eventually_False)
hoelzl@51471
  1023
qed
hoelzl@51471
  1024
hoelzl@51471
  1025
lemma tendsto_le_const:
hoelzl@51471
  1026
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
  1027
  assumes F: "\<not> trivial_limit F"
hoelzl@51471
  1028
  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>x. a \<le> f x) F"
hoelzl@51471
  1029
  shows "a \<le> x"
hoelzl@51471
  1030
  using F x tendsto_const a by (rule tendsto_le)
hoelzl@51471
  1031
hoelzl@51478
  1032
subsubsection {* Rules about @{const Lim} *}
hoelzl@51478
  1033
hoelzl@51478
  1034
lemma (in t2_space) tendsto_Lim:
hoelzl@51478
  1035
  "\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l"
hoelzl@51478
  1036
  unfolding Lim_def using tendsto_unique[of net f] by auto
hoelzl@51478
  1037
hoelzl@51641
  1038
lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
hoelzl@51478
  1039
  by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
hoelzl@51478
  1040
hoelzl@51471
  1041
subsection {* Limits to @{const at_top} and @{const at_bot} *}
hoelzl@51471
  1042
hoelzl@51471
  1043
lemma filterlim_at_top:
hoelzl@51471
  1044
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@51471
  1045
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1046
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
hoelzl@51471
  1047
hoelzl@51471
  1048
lemma filterlim_at_top_dense:
hoelzl@51471
  1049
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
hoelzl@51471
  1050
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@51471
  1051
  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
hoelzl@51471
  1052
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
hoelzl@51471
  1053
hoelzl@51471
  1054
lemma filterlim_at_top_ge:
hoelzl@51471
  1055
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@51471
  1056
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1057
  unfolding filterlim_at_top
hoelzl@51471
  1058
proof safe
hoelzl@51471
  1059
  fix Z assume *: "\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F"
hoelzl@51471
  1060
  with *[THEN spec, of "max Z c"] show "eventually (\<lambda>x. Z \<le> f x) F"
hoelzl@51471
  1061
    by (auto elim!: eventually_elim1)
hoelzl@51471
  1062
qed simp
hoelzl@51471
  1063
hoelzl@51471
  1064
lemma filterlim_at_top_at_top:
hoelzl@51471
  1065
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
hoelzl@51471
  1066
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1067
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1068
  assumes Q: "eventually Q at_top"
hoelzl@51471
  1069
  assumes P: "eventually P at_top"
hoelzl@51471
  1070
  shows "filterlim f at_top at_top"
hoelzl@51471
  1071
proof -
hoelzl@51471
  1072
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
  1073
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
  1074
  show ?thesis
hoelzl@51471
  1075
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
  1076
    fix z assume "x \<le> z"
hoelzl@51471
  1077
    with x have "P z" by auto
hoelzl@51471
  1078
    have "eventually (\<lambda>x. g z \<le> x) at_top"
hoelzl@51471
  1079
      by (rule eventually_ge_at_top)
hoelzl@51471
  1080
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
hoelzl@51471
  1081
      by eventually_elim (metis mono bij `P z`)
hoelzl@51471
  1082
  qed
hoelzl@51471
  1083
qed
hoelzl@51471
  1084
hoelzl@51471
  1085
lemma filterlim_at_top_gt:
hoelzl@51471
  1086
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
hoelzl@51471
  1087
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1088
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
hoelzl@51471
  1089
hoelzl@51471
  1090
lemma filterlim_at_bot: 
hoelzl@51471
  1091
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@51471
  1092
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
hoelzl@51471
  1093
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
hoelzl@51471
  1094
hoelzl@51471
  1095
lemma filterlim_at_bot_le:
hoelzl@51471
  1096
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@51471
  1097
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@51471
  1098
  unfolding filterlim_at_bot
hoelzl@51471
  1099
proof safe
hoelzl@51471
  1100
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@51471
  1101
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@51471
  1102
    by (auto elim!: eventually_elim1)
hoelzl@51471
  1103
qed simp
hoelzl@51471
  1104
hoelzl@51471
  1105
lemma filterlim_at_bot_lt:
hoelzl@51471
  1106
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
hoelzl@51471
  1107
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@51471
  1108
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
hoelzl@51471
  1109
hoelzl@51471
  1110
lemma filterlim_at_bot_at_right:
hoelzl@51471
  1111
  fixes f :: "'a::{no_top, linorder_topology} \<Rightarrow> 'b::linorder"
hoelzl@51471
  1112
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1113
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1114
  assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
hoelzl@51471
  1115
  assumes P: "eventually P at_bot"
hoelzl@51471
  1116
  shows "filterlim f at_bot (at_right a)"
hoelzl@51471
  1117
proof -
hoelzl@51471
  1118
  from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
hoelzl@51471
  1119
    unfolding eventually_at_bot_linorder by auto
hoelzl@51471
  1120
  show ?thesis
hoelzl@51471
  1121
  proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
hoelzl@51471
  1122
    fix z assume "z \<le> x"
hoelzl@51471
  1123
    with x have "P z" by auto
hoelzl@51471
  1124
    have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
hoelzl@51471
  1125
      using bound[OF bij(2)[OF `P z`]]
hoelzl@51471
  1126
      unfolding eventually_at_right by (auto intro!: exI[of _ "g z"])
hoelzl@51471
  1127
    with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
hoelzl@51471
  1128
      by eventually_elim (metis bij `P z` mono)
hoelzl@51471
  1129
  qed
hoelzl@51471
  1130
qed
hoelzl@51471
  1131
hoelzl@51471
  1132
lemma filterlim_at_top_at_left:
hoelzl@51471
  1133
  fixes f :: "'a::{no_bot, linorder_topology} \<Rightarrow> 'b::linorder"
hoelzl@51471
  1134
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1135
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1136
  assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
hoelzl@51471
  1137
  assumes P: "eventually P at_top"
hoelzl@51471
  1138
  shows "filterlim f at_top (at_left a)"
hoelzl@51471
  1139
proof -
hoelzl@51471
  1140
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
  1141
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
  1142
  show ?thesis
hoelzl@51471
  1143
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
  1144
    fix z assume "x \<le> z"
hoelzl@51471
  1145
    with x have "P z" by auto
hoelzl@51471
  1146
    have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
hoelzl@51471
  1147
      using bound[OF bij(2)[OF `P z`]]
hoelzl@51471
  1148
      unfolding eventually_at_left by (auto intro!: exI[of _ "g z"])
hoelzl@51471
  1149
    with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
hoelzl@51471
  1150
      by eventually_elim (metis bij `P z` mono)
hoelzl@51471
  1151
  qed
hoelzl@51471
  1152
qed
hoelzl@51471
  1153
hoelzl@51471
  1154
lemma filterlim_split_at:
hoelzl@51471
  1155
  "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"
hoelzl@51471
  1156
  by (subst at_eq_sup_left_right) (rule filterlim_sup)
hoelzl@51471
  1157
hoelzl@51471
  1158
lemma filterlim_at_split:
hoelzl@51471
  1159
  "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
hoelzl@51471
  1160
  by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
hoelzl@51471
  1161
hoelzl@51471
  1162
hoelzl@51471
  1163
subsection {* Limits on sequences *}
hoelzl@51471
  1164
hoelzl@51471
  1165
abbreviation (in topological_space)
hoelzl@51471
  1166
  LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
hoelzl@51471
  1167
    ("((_)/ ----> (_))" [60, 60] 60) where
hoelzl@51471
  1168
  "X ----> L \<equiv> (X ---> L) sequentially"
hoelzl@51471
  1169
hoelzl@51478
  1170
abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
  1171
  "lim X \<equiv> Lim sequentially X"
hoelzl@51471
  1172
hoelzl@51471
  1173
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
hoelzl@51471
  1174
  "convergent X = (\<exists>L. X ----> L)"
hoelzl@51471
  1175
hoelzl@51478
  1176
lemma lim_def: "lim X = (THE L. X ----> L)"
hoelzl@51478
  1177
  unfolding Lim_def ..
hoelzl@51478
  1178
hoelzl@51471
  1179
subsubsection {* Monotone sequences and subsequences *}
hoelzl@51471
  1180
hoelzl@51471
  1181
definition
hoelzl@51471
  1182
  monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@51471
  1183
    --{*Definition of monotonicity.
hoelzl@51471
  1184
        The use of disjunction here complicates proofs considerably.
hoelzl@51471
  1185
        One alternative is to add a Boolean argument to indicate the direction.
hoelzl@51471
  1186
        Another is to develop the notions of increasing and decreasing first.*}
hoelzl@51471
  1187
  "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
hoelzl@51471
  1188
hoelzl@51471
  1189
definition
hoelzl@51471
  1190
  incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@51471
  1191
    --{*Increasing sequence*}
hoelzl@51471
  1192
  "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
hoelzl@51471
  1193
hoelzl@51471
  1194
definition
hoelzl@51471
  1195
  decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@51471
  1196
    --{*Decreasing sequence*}
hoelzl@51471
  1197
  "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
hoelzl@51471
  1198
hoelzl@51471
  1199
definition
hoelzl@51471
  1200
  subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
hoelzl@51471
  1201
    --{*Definition of subsequence*}
hoelzl@51471
  1202
  "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
hoelzl@51471
  1203
hoelzl@51471
  1204
lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
hoelzl@51471
  1205
  unfolding mono_def incseq_def by auto
hoelzl@51471
  1206
hoelzl@51471
  1207
lemma incseq_SucI:
hoelzl@51471
  1208
  "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
hoelzl@51471
  1209
  using lift_Suc_mono_le[of X]
hoelzl@51471
  1210
  by (auto simp: incseq_def)
hoelzl@51471
  1211
hoelzl@51471
  1212
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
hoelzl@51471
  1213
  by (auto simp: incseq_def)
hoelzl@51471
  1214
hoelzl@51471
  1215
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
hoelzl@51471
  1216
  using incseqD[of A i "Suc i"] by auto
hoelzl@51471
  1217
hoelzl@51471
  1218
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
hoelzl@51471
  1219
  by (auto intro: incseq_SucI dest: incseq_SucD)
hoelzl@51471
  1220
hoelzl@51471
  1221
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
hoelzl@51471
  1222
  unfolding incseq_def by auto
hoelzl@51471
  1223
hoelzl@51471
  1224
lemma decseq_SucI:
hoelzl@51471
  1225
  "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
hoelzl@51471
  1226
  using order.lift_Suc_mono_le[OF dual_order, of X]
hoelzl@51471
  1227
  by (auto simp: decseq_def)
hoelzl@51471
  1228
hoelzl@51471
  1229
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
hoelzl@51471
  1230
  by (auto simp: decseq_def)
hoelzl@51471
  1231
hoelzl@51471
  1232
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
hoelzl@51471
  1233
  using decseqD[of A i "Suc i"] by auto
hoelzl@51471
  1234
hoelzl@51471
  1235
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@51471
  1236
  by (auto intro: decseq_SucI dest: decseq_SucD)
hoelzl@51471
  1237
hoelzl@51471
  1238
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
hoelzl@51471
  1239
  unfolding decseq_def by auto
hoelzl@51471
  1240
hoelzl@51471
  1241
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
hoelzl@51471
  1242
  unfolding monoseq_def incseq_def decseq_def ..
hoelzl@51471
  1243
hoelzl@51471
  1244
lemma monoseq_Suc:
hoelzl@51471
  1245
  "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
hoelzl@51471
  1246
  unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
hoelzl@51471
  1247
hoelzl@51471
  1248
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
hoelzl@51471
  1249
by (simp add: monoseq_def)
hoelzl@51471
  1250
hoelzl@51471
  1251
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
hoelzl@51471
  1252
by (simp add: monoseq_def)
hoelzl@51471
  1253
hoelzl@51471
  1254
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
hoelzl@51471
  1255
by (simp add: monoseq_Suc)
hoelzl@51471
  1256
hoelzl@51471
  1257
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
hoelzl@51471
  1258
by (simp add: monoseq_Suc)
hoelzl@51471
  1259
hoelzl@51471
  1260
lemma monoseq_minus:
hoelzl@51471
  1261
  fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
hoelzl@51471
  1262
  assumes "monoseq a"
hoelzl@51471
  1263
  shows "monoseq (\<lambda> n. - a n)"
hoelzl@51471
  1264
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
hoelzl@51471
  1265
  case True
hoelzl@51471
  1266
  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
hoelzl@51471
  1267
  thus ?thesis by (rule monoI2)
hoelzl@51471
  1268
next
hoelzl@51471
  1269
  case False
hoelzl@51471
  1270
  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
hoelzl@51471
  1271
  thus ?thesis by (rule monoI1)
hoelzl@51471
  1272
qed
hoelzl@51471
  1273
hoelzl@51471
  1274
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
hoelzl@51471
  1275
hoelzl@51471
  1276
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
hoelzl@51471
  1277
apply (simp add: subseq_def)
hoelzl@51471
  1278
apply (auto dest!: less_imp_Suc_add)
hoelzl@51471
  1279
apply (induct_tac k)
hoelzl@51471
  1280
apply (auto intro: less_trans)
hoelzl@51471
  1281
done
hoelzl@51471
  1282
hoelzl@51471
  1283
text{* for any sequence, there is a monotonic subsequence *}
hoelzl@51471
  1284
lemma seq_monosub:
hoelzl@51471
  1285
  fixes s :: "nat => 'a::linorder"
hoelzl@51471
  1286
  shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
hoelzl@51471
  1287
proof cases
hoelzl@51471
  1288
  let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"
hoelzl@51471
  1289
  assume *: "\<forall>n. \<exists>p. ?P p n"
hoelzl@51471
  1290
  def f \<equiv> "nat_rec (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"
hoelzl@51471
  1291
  have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
hoelzl@51471
  1292
  have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
hoelzl@51471
  1293
  have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
hoelzl@51471
  1294
  have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
hoelzl@51471
  1295
  then have "subseq f" unfolding subseq_Suc_iff by auto
hoelzl@51471
  1296
  moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc
hoelzl@51471
  1297
  proof (intro disjI2 allI)
hoelzl@51471
  1298
    fix n show "s (f (Suc n)) \<le> s (f n)"
hoelzl@51471
  1299
    proof (cases n)
hoelzl@51471
  1300
      case 0 with P_Suc[of 0] P_0 show ?thesis by auto
hoelzl@51471
  1301
    next
hoelzl@51471
  1302
      case (Suc m)
hoelzl@51471
  1303
      from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp
hoelzl@51471
  1304
      with P_Suc Suc show ?thesis by simp
hoelzl@51471
  1305
    qed
hoelzl@51471
  1306
  qed
hoelzl@51471
  1307
  ultimately show ?thesis by auto
hoelzl@51471
  1308
next
hoelzl@51471
  1309
  let "?P p m" = "m < p \<and> s m < s p"
hoelzl@51471
  1310
  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
hoelzl@51471
  1311
  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
hoelzl@51471
  1312
  def f \<equiv> "nat_rec (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"
hoelzl@51471
  1313
  have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
hoelzl@51471
  1314
  have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
hoelzl@51471
  1315
  have P_0: "?P (f 0) (Suc N)"
hoelzl@51471
  1316
    unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto
hoelzl@51471
  1317
  { fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"
hoelzl@51471
  1318
      unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }
hoelzl@51471
  1319
  note P' = this
hoelzl@51471
  1320
  { fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"
hoelzl@51471
  1321
      by (induct i) (insert P_0 P', auto) }
hoelzl@51471
  1322
  then have "subseq f" "monoseq (\<lambda>x. s (f x))"
hoelzl@51471
  1323
    unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
hoelzl@51471
  1324
  then show ?thesis by auto
hoelzl@51471
  1325
qed
hoelzl@51471
  1326
hoelzl@51471
  1327
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
hoelzl@51471
  1328
proof(induct n)
hoelzl@51471
  1329
  case 0 thus ?case by simp
hoelzl@51471
  1330
next
hoelzl@51471
  1331
  case (Suc n)
hoelzl@51471
  1332
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
hoelzl@51471
  1333
  have "n < f (Suc n)" by arith
hoelzl@51471
  1334
  thus ?case by arith
hoelzl@51471
  1335
qed
hoelzl@51471
  1336
hoelzl@51471
  1337
lemma eventually_subseq:
hoelzl@51471
  1338
  "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
hoelzl@51471
  1339
  unfolding eventually_sequentially by (metis seq_suble le_trans)
hoelzl@51471
  1340
hoelzl@51473
  1341
lemma not_eventually_sequentiallyD:
hoelzl@51473
  1342
  assumes P: "\<not> eventually P sequentially"
hoelzl@51473
  1343
  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
hoelzl@51473
  1344
proof -
hoelzl@51473
  1345
  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
hoelzl@51473
  1346
    unfolding eventually_sequentially by (simp add: not_less)
hoelzl@51473
  1347
  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
hoelzl@51473
  1348
    by (auto simp: choice_iff)
hoelzl@51473
  1349
  then show ?thesis
hoelzl@51473
  1350
    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
hoelzl@51473
  1351
             simp: less_eq_Suc_le subseq_Suc_iff)
hoelzl@51473
  1352
qed
hoelzl@51473
  1353
hoelzl@51471
  1354
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
hoelzl@51471
  1355
  unfolding filterlim_iff by (metis eventually_subseq)
hoelzl@51471
  1356
hoelzl@51471
  1357
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
hoelzl@51471
  1358
  unfolding subseq_def by simp
hoelzl@51471
  1359
hoelzl@51471
  1360
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
hoelzl@51471
  1361
  using assms by (auto simp: subseq_def)
hoelzl@51471
  1362
hoelzl@51471
  1363
lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
hoelzl@51471
  1364
  by (simp add: incseq_def monoseq_def)
hoelzl@51471
  1365
hoelzl@51471
  1366
lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
hoelzl@51471
  1367
  by (simp add: decseq_def monoseq_def)
hoelzl@51471
  1368
hoelzl@51471
  1369
lemma decseq_eq_incseq:
hoelzl@51471
  1370
  fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" 
hoelzl@51471
  1371
  by (simp add: decseq_def incseq_def)
hoelzl@51471
  1372
hoelzl@51471
  1373
lemma INT_decseq_offset:
hoelzl@51471
  1374
  assumes "decseq F"
hoelzl@51471
  1375
  shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
  1376
proof safe
hoelzl@51471
  1377
  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
  1378
  show "x \<in> F i"
hoelzl@51471
  1379
  proof cases
hoelzl@51471
  1380
    from x have "x \<in> F n" by auto
hoelzl@51471
  1381
    also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
hoelzl@51471
  1382
      unfolding decseq_def by simp
hoelzl@51471
  1383
    finally show ?thesis .
hoelzl@51471
  1384
  qed (insert x, simp)
hoelzl@51471
  1385
qed auto
hoelzl@51471
  1386
hoelzl@51471
  1387
lemma LIMSEQ_const_iff:
hoelzl@51471
  1388
  fixes k l :: "'a::t2_space"
hoelzl@51471
  1389
  shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
hoelzl@51471
  1390
  using trivial_limit_sequentially by (rule tendsto_const_iff)
hoelzl@51471
  1391
hoelzl@51471
  1392
lemma LIMSEQ_SUP:
hoelzl@51471
  1393
  "incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1394
  by (intro increasing_tendsto)
hoelzl@51471
  1395
     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
hoelzl@51471
  1396
hoelzl@51471
  1397
lemma LIMSEQ_INF:
hoelzl@51471
  1398
  "decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1399
  by (intro decreasing_tendsto)
hoelzl@51471
  1400
     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
hoelzl@51471
  1401
hoelzl@51471
  1402
lemma LIMSEQ_ignore_initial_segment:
hoelzl@51471
  1403
  "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
hoelzl@51474
  1404
  unfolding tendsto_def
hoelzl@51474
  1405
  by (subst eventually_sequentially_seg[where k=k])
hoelzl@51471
  1406
hoelzl@51471
  1407
lemma LIMSEQ_offset:
hoelzl@51471
  1408
  "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
hoelzl@51474
  1409
  unfolding tendsto_def
hoelzl@51474
  1410
  by (subst (asm) eventually_sequentially_seg[where k=k])
hoelzl@51471
  1411
hoelzl@51471
  1412
lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
hoelzl@51471
  1413
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
hoelzl@51471
  1414
hoelzl@51471
  1415
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
hoelzl@51471
  1416
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
hoelzl@51471
  1417
hoelzl@51471
  1418
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
hoelzl@51471
  1419
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
hoelzl@51471
  1420
hoelzl@51471
  1421
lemma LIMSEQ_unique:
hoelzl@51471
  1422
  fixes a b :: "'a::t2_space"
hoelzl@51471
  1423
  shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
hoelzl@51471
  1424
  using trivial_limit_sequentially by (rule tendsto_unique)
hoelzl@51471
  1425
hoelzl@51471
  1426
lemma LIMSEQ_le_const:
hoelzl@51471
  1427
  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
hoelzl@51471
  1428
  using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
hoelzl@51471
  1429
hoelzl@51471
  1430
lemma LIMSEQ_le:
hoelzl@51471
  1431
  "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"
hoelzl@51471
  1432
  using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
hoelzl@51471
  1433
hoelzl@51471
  1434
lemma LIMSEQ_le_const2:
hoelzl@51471
  1435
  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
hoelzl@51471
  1436
  by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) (auto simp: tendsto_const)
hoelzl@51471
  1437
hoelzl@51471
  1438
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
hoelzl@51471
  1439
by (simp add: convergent_def)
hoelzl@51471
  1440
hoelzl@51471
  1441
lemma convergentI: "(X ----> L) ==> convergent X"
hoelzl@51471
  1442
by (auto simp add: convergent_def)
hoelzl@51471
  1443
hoelzl@51471
  1444
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
hoelzl@51471
  1445
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
hoelzl@51471
  1446
hoelzl@51471
  1447
lemma convergent_const: "convergent (\<lambda>n. c)"
hoelzl@51471
  1448
  by (rule convergentI, rule tendsto_const)
hoelzl@51471
  1449
hoelzl@51471
  1450
lemma monoseq_le:
hoelzl@51471
  1451
  "monoseq a \<Longrightarrow> a ----> (x::'a::linorder_topology) \<Longrightarrow>
hoelzl@51471
  1452
    ((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
hoelzl@51471
  1453
  by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
hoelzl@51471
  1454
hoelzl@51471
  1455
lemma LIMSEQ_subseq_LIMSEQ:
hoelzl@51471
  1456
  "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
hoelzl@51471
  1457
  unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
hoelzl@51471
  1458
hoelzl@51471
  1459
lemma convergent_subseq_convergent:
hoelzl@51471
  1460
  "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
hoelzl@51471
  1461
  unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
hoelzl@51471
  1462
hoelzl@51471
  1463
lemma limI: "X ----> L ==> lim X = L"
hoelzl@51471
  1464
apply (simp add: lim_def)
hoelzl@51471
  1465
apply (blast intro: LIMSEQ_unique)
hoelzl@51471
  1466
done
hoelzl@51471
  1467
hoelzl@51471
  1468
lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"
hoelzl@51471
  1469
  using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
hoelzl@51471
  1470
hoelzl@51471
  1471
subsubsection{*Increasing and Decreasing Series*}
hoelzl@51471
  1472
hoelzl@51471
  1473
lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
hoelzl@51471
  1474
  by (metis incseq_def LIMSEQ_le_const)
hoelzl@51471
  1475
hoelzl@51471
  1476
lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
hoelzl@51471
  1477
  by (metis decseq_def LIMSEQ_le_const2)
hoelzl@51471
  1478
hoelzl@51473
  1479
subsection {* First countable topologies *}
hoelzl@51473
  1480
hoelzl@51473
  1481
class first_countable_topology = topological_space +
hoelzl@51473
  1482
  assumes first_countable_basis:
hoelzl@51473
  1483
    "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
hoelzl@51473
  1484
hoelzl@51473
  1485
lemma (in first_countable_topology) countable_basis_at_decseq:
hoelzl@51473
  1486
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1487
    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
hoelzl@51473
  1488
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
hoelzl@51473
  1489
proof atomize_elim
hoelzl@51473
  1490
  from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1491
    nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1492
    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto
hoelzl@51473
  1493
  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i"
hoelzl@51473
  1494
  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
hoelzl@51473
  1495
      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
hoelzl@51473
  1496
  proof (safe intro!: exI[of _ F])
hoelzl@51473
  1497
    fix i
hoelzl@51480
  1498
    show "open (F i)" using nhds(1) by (auto simp: F_def)
hoelzl@51473
  1499
    show "x \<in> F i" using nhds(2) by (auto simp: F_def)
hoelzl@51473
  1500
  next
hoelzl@51473
  1501
    fix S assume "open S" "x \<in> S"
hoelzl@51473
  1502
    from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto
hoelzl@51473
  1503
    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
hoelzl@51473
  1504
      by (auto simp: F_def)
hoelzl@51473
  1505
    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
hoelzl@51473
  1506
      by (auto simp: eventually_sequentially)
hoelzl@51473
  1507
  qed
hoelzl@51473
  1508
qed
hoelzl@51473
  1509
hoelzl@51473
  1510
lemma (in first_countable_topology) countable_basis:
hoelzl@51473
  1511
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1512
    "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1513
    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
hoelzl@51473
  1514
proof atomize_elim
hoelzl@51473
  1515
  from countable_basis_at_decseq[of x] guess A . note A = this
hoelzl@51473
  1516
  { fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
hoelzl@51473
  1517
    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
hoelzl@51473
  1518
      by (auto elim: eventually_elim1 simp: subset_eq) }
hoelzl@51473
  1519
  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)"
hoelzl@51473
  1520
    by (intro exI[of _ A]) (auto simp: tendsto_def)
hoelzl@51473
  1521
qed
hoelzl@51473
  1522
hoelzl@51473
  1523
lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
hoelzl@51473
  1524
  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51641
  1525
  shows "eventually P (inf (nhds a) (principal s))"
hoelzl@51473
  1526
proof (rule ccontr)
hoelzl@51473
  1527
  from countable_basis[of a] guess A . note A = this
hoelzl@51641
  1528
  assume "\<not> eventually P (inf (nhds a) (principal s))"
hoelzl@51473
  1529
  with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
hoelzl@51641
  1530
    unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce
hoelzl@51473
  1531
  then guess F ..
hoelzl@51473
  1532
  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@51473
  1533
    by fast+
hoelzl@51473
  1534
  with A have "F ----> a" by auto
hoelzl@51473
  1535
  hence "eventually (\<lambda>n. P (F n)) sequentially"
hoelzl@51473
  1536
    using assms F0 by simp
hoelzl@51473
  1537
  thus "False" by (simp add: F3)
hoelzl@51473
  1538
qed
hoelzl@51473
  1539
hoelzl@51473
  1540
lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
hoelzl@51641
  1541
  "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> 
hoelzl@51473
  1542
    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1543
proof (safe intro!: sequentially_imp_eventually_nhds_within)
hoelzl@51641
  1544
  assume "eventually P (inf (nhds a) (principal s))" 
hoelzl@51473
  1545
  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
hoelzl@51641
  1546
    by (auto simp: eventually_inf_principal eventually_nhds)
hoelzl@51473
  1547
  moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
hoelzl@51473
  1548
  ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51473
  1549
    by (auto dest!: topological_tendstoD elim: eventually_elim1)
hoelzl@51473
  1550
qed
hoelzl@51473
  1551
hoelzl@51473
  1552
lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
hoelzl@51473
  1553
  "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1554
  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
hoelzl@51473
  1555
hoelzl@51471
  1556
subsection {* Function limit at a point *}
hoelzl@51471
  1557
hoelzl@51471
  1558
abbreviation
hoelzl@51471
  1559
  LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
hoelzl@51471
  1560
        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
hoelzl@51471
  1561
  "f -- a --> L \<equiv> (f ---> L) (at a)"
hoelzl@51471
  1562
hoelzl@51481
  1563
lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l) (at a within S) \<longleftrightarrow> (f -- a --> l)"
hoelzl@51641
  1564
  unfolding tendsto_def by (simp add: at_within_open[where S=S])
hoelzl@51481
  1565
hoelzl@51471
  1566
lemma LIM_const_not_eq[tendsto_intros]:
hoelzl@51471
  1567
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1568
  fixes k L :: "'b::t2_space"
hoelzl@51471
  1569
  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
hoelzl@51471
  1570
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1571
hoelzl@51471
  1572
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
hoelzl@51471
  1573
hoelzl@51471
  1574
lemma LIM_const_eq:
hoelzl@51471
  1575
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1576
  fixes k L :: "'b::t2_space"
hoelzl@51471
  1577
  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
hoelzl@51471
  1578
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1579
hoelzl@51471
  1580
lemma LIM_unique:
hoelzl@51471
  1581
  fixes a :: "'a::perfect_space" and L M :: "'b::t2_space"
hoelzl@51471
  1582
  shows "f -- a --> L \<Longrightarrow> f -- a --> M \<Longrightarrow> L = M"
hoelzl@51471
  1583
  using at_neq_bot by (rule tendsto_unique)
hoelzl@51471
  1584
hoelzl@51471
  1585
text {* Limits are equal for functions equal except at limit point *}
hoelzl@51471
  1586
hoelzl@51471
  1587
lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- a --> l)"
hoelzl@51471
  1588
  unfolding tendsto_def eventually_at_topological by simp
hoelzl@51471
  1589
hoelzl@51471
  1590
lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- b --> m)"
hoelzl@51471
  1591
  by (simp add: LIM_equal)
hoelzl@51471
  1592
hoelzl@51471
  1593
lemma LIM_cong_limit: "f -- x --> L \<Longrightarrow> K = L \<Longrightarrow> f -- x --> K"
hoelzl@51471
  1594
  by simp
hoelzl@51471
  1595
hoelzl@51471
  1596
lemma tendsto_at_iff_tendsto_nhds:
hoelzl@51471
  1597
  "g -- l --> g l \<longleftrightarrow> (g ---> g l) (nhds l)"
hoelzl@51641
  1598
  unfolding tendsto_def eventually_at_filter
hoelzl@51471
  1599
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@51471
  1600
hoelzl@51471
  1601
lemma tendsto_compose:
hoelzl@51471
  1602
  "g -- l --> g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@51471
  1603
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
hoelzl@51471
  1604
hoelzl@51471
  1605
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
hoelzl@51471
  1606
  unfolding o_def by (rule tendsto_compose)
hoelzl@51471
  1607
hoelzl@51471
  1608
lemma tendsto_compose_eventually:
hoelzl@51471
  1609
  "g -- l --> m \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F"
hoelzl@51471
  1610
  by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
hoelzl@51471
  1611
hoelzl@51471
  1612
lemma LIM_compose_eventually:
hoelzl@51471
  1613
  assumes f: "f -- a --> b"
hoelzl@51471
  1614
  assumes g: "g -- b --> c"
hoelzl@51471
  1615
  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
hoelzl@51471
  1616
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51471
  1617
  using g f inj by (rule tendsto_compose_eventually)
hoelzl@51471
  1618
hoelzl@51473
  1619
subsubsection {* Relation of LIM and LIMSEQ *}
hoelzl@51473
  1620
hoelzl@51473
  1621
lemma (in first_countable_topology) sequentially_imp_eventually_within:
hoelzl@51473
  1622
  "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
hoelzl@51473
  1623
    eventually P (at a within s)"
hoelzl@51641
  1624
  unfolding at_within_def
hoelzl@51473
  1625
  by (intro sequentially_imp_eventually_nhds_within) auto
hoelzl@51473
  1626
hoelzl@51473
  1627
lemma (in first_countable_topology) sequentially_imp_eventually_at:
hoelzl@51473
  1628
  "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"
hoelzl@51473
  1629
  using assms sequentially_imp_eventually_within [where s=UNIV] by simp
hoelzl@51473
  1630
hoelzl@51473
  1631
lemma LIMSEQ_SEQ_conv1:
hoelzl@51473
  1632
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
hoelzl@51473
  1633
  assumes f: "f -- a --> l"
hoelzl@51473
  1634
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
hoelzl@51473
  1635
  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
hoelzl@51473
  1636
hoelzl@51473
  1637
lemma LIMSEQ_SEQ_conv2:
hoelzl@51473
  1638
  fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
hoelzl@51473
  1639
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
hoelzl@51473
  1640
  shows "f -- a --> l"
hoelzl@51473
  1641
  using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
hoelzl@51473
  1642
hoelzl@51473
  1643
lemma LIMSEQ_SEQ_conv:
hoelzl@51473
  1644
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
hoelzl@51473
  1645
   (X -- a --> (L::'b::topological_space))"
hoelzl@51473
  1646
  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
hoelzl@51473
  1647
hoelzl@51471
  1648
subsection {* Continuity *}
hoelzl@51471
  1649
hoelzl@51478
  1650
subsubsection {* Continuity on a set *}
hoelzl@51478
  1651
hoelzl@51478
  1652
definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where
hoelzl@51478
  1653
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
hoelzl@51478
  1654
hoelzl@51481
  1655
lemma continuous_on_cong [cong]:
hoelzl@51481
  1656
  "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
hoelzl@51641
  1657
  unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
hoelzl@51481
  1658
hoelzl@51478
  1659
lemma continuous_on_topological:
hoelzl@51478
  1660
  "continuous_on s f \<longleftrightarrow>
hoelzl@51478
  1661
    (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
hoelzl@51641
  1662
  unfolding continuous_on_def tendsto_def eventually_at_topological by metis
hoelzl@51478
  1663
hoelzl@51478
  1664
lemma continuous_on_open_invariant:
hoelzl@51478
  1665
  "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  1666
proof safe
hoelzl@51478
  1667
  fix B :: "'b set" assume "continuous_on s f" "open B"
hoelzl@51478
  1668
  then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)"
hoelzl@51478
  1669
    by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
hoelzl@51478
  1670
  then guess A unfolding bchoice_iff ..
hoelzl@51478
  1671
  then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"
hoelzl@51478
  1672
    by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto
hoelzl@51478
  1673
next
hoelzl@51478
  1674
  assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"
hoelzl@51478
  1675
  show "continuous_on s f"
hoelzl@51478
  1676
    unfolding continuous_on_topological
hoelzl@51478
  1677
  proof safe
hoelzl@51478
  1678
    fix x B assume "x \<in> s" "open B" "f x \<in> B"
hoelzl@51478
  1679
    with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto
hoelzl@51478
  1680
    with `x \<in> s` `f x \<in> B` show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
hoelzl@51478
  1681
      by (intro exI[of _ A]) auto
hoelzl@51478
  1682
  qed
hoelzl@51478
  1683
qed
hoelzl@51478
  1684
hoelzl@51481
  1685
lemma continuous_on_open_vimage:
hoelzl@51481
  1686
  "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
hoelzl@51481
  1687
  unfolding continuous_on_open_invariant
hoelzl@51481
  1688
  by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  1689
hoelzl@51478
  1690
lemma continuous_on_closed_invariant:
hoelzl@51478
  1691
  "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  1692
proof -
hoelzl@51478
  1693
  have *: "\<And>P Q::'b set\<Rightarrow>bool. (\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)"
hoelzl@51478
  1694
    by (metis double_compl)
hoelzl@51478
  1695
  show ?thesis
hoelzl@51478
  1696
    unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric])
hoelzl@51478
  1697
qed
hoelzl@51478
  1698
hoelzl@51481
  1699
lemma continuous_on_closed_vimage:
hoelzl@51481
  1700
  "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
hoelzl@51481
  1701
  unfolding continuous_on_closed_invariant
hoelzl@51481
  1702
  by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  1703
hoelzl@51481
  1704
lemma continuous_on_open_Union:
hoelzl@51481
  1705
  "(\<And>s. s \<in> S \<Longrightarrow> open s) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on s f) \<Longrightarrow> continuous_on (\<Union>S) f"
hoelzl@51641
  1706
  unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI)
hoelzl@51481
  1707
hoelzl@51481
  1708
lemma continuous_on_open_UN:
hoelzl@51481
  1709
  "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow> continuous_on (\<Union>s\<in>S. A s) f"
hoelzl@51481
  1710
  unfolding Union_image_eq[symmetric] by (rule continuous_on_open_Union) auto
hoelzl@51481
  1711
hoelzl@51481
  1712
lemma continuous_on_closed_Un:
hoelzl@51481
  1713
  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
hoelzl@51481
  1714
  by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
hoelzl@51481
  1715
hoelzl@51481
  1716
lemma continuous_on_If:
hoelzl@51481
  1717
  assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g"
hoelzl@51481
  1718
    and P: "\<And>x. x \<in> s \<Longrightarrow> \<not> P x \<Longrightarrow> f x = g x" "\<And>x. x \<in> t \<Longrightarrow> P x \<Longrightarrow> f x = g x"
hoelzl@51481
  1719
  shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h")
hoelzl@51481
  1720
proof-
hoelzl@51481
  1721
  from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
hoelzl@51481
  1722
    by auto
hoelzl@51481
  1723
  with cont have "continuous_on s ?h" "continuous_on t ?h"
hoelzl@51481
  1724
    by simp_all
hoelzl@51481
  1725
  with closed show ?thesis
hoelzl@51481
  1726
    by (rule continuous_on_closed_Un)
hoelzl@51481
  1727
qed
hoelzl@51481
  1728
hoelzl@51478
  1729
ML {*
hoelzl@51478
  1730
hoelzl@51478
  1731
structure Continuous_On_Intros = Named_Thms
hoelzl@51478
  1732
(
hoelzl@51478
  1733
  val name = @{binding continuous_on_intros}
hoelzl@51478
  1734
  val description = "Structural introduction rules for setwise continuity"
hoelzl@51478
  1735
)
hoelzl@51478
  1736
hoelzl@51478
  1737
*}
hoelzl@51478
  1738
hoelzl@51478
  1739
setup Continuous_On_Intros.setup
hoelzl@51478
  1740
hoelzl@51478
  1741
lemma continuous_on_id[continuous_on_intros]: "continuous_on s (\<lambda>x. x)"
hoelzl@51641
  1742
  unfolding continuous_on_def by (fast intro: tendsto_ident_at)
hoelzl@51478
  1743
hoelzl@51478
  1744
lemma continuous_on_const[continuous_on_intros]: "continuous_on s (\<lambda>x. c)"
hoelzl@51478
  1745
  unfolding continuous_on_def by (auto intro: tendsto_const)
hoelzl@51478
  1746
hoelzl@51478
  1747
lemma continuous_on_compose[continuous_on_intros]:
hoelzl@51478
  1748
  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
hoelzl@51478
  1749
  unfolding continuous_on_topological by simp metis
hoelzl@51478
  1750
hoelzl@51481
  1751
lemma continuous_on_compose2:
hoelzl@51481
  1752
  "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> t = f ` s \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
hoelzl@51481
  1753
  using continuous_on_compose[of s f g] by (simp add: comp_def)
hoelzl@51481
  1754
hoelzl@51478
  1755
subsubsection {* Continuity at a point *}
hoelzl@51478
  1756
hoelzl@51478
  1757
definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
hoelzl@51478
  1758
  "continuous F f \<longleftrightarrow> (f ---> f (Lim F (\<lambda>x. x))) F"
hoelzl@51478
  1759
hoelzl@51478
  1760
ML {*
hoelzl@51471
  1761
hoelzl@51478
  1762
structure Continuous_Intros = Named_Thms
hoelzl@51478
  1763
(
hoelzl@51478
  1764
  val name = @{binding continuous_intros}
hoelzl@51478
  1765
  val description = "Structural introduction rules for pointwise continuity"
hoelzl@51478
  1766
)
hoelzl@51478
  1767
hoelzl@51478
  1768
*}
hoelzl@51478
  1769
hoelzl@51478
  1770
setup Continuous_Intros.setup
hoelzl@51478
  1771
hoelzl@51478
  1772
lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
hoelzl@51478
  1773
  unfolding continuous_def by auto
hoelzl@51478
  1774
hoelzl@51478
  1775
lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
hoelzl@51478
  1776
  by simp
hoelzl@51478
  1777
hoelzl@51478
  1778
lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f x) (at x within s)"
hoelzl@51641
  1779
  by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
hoelzl@51478
  1780
hoelzl@51478
  1781
lemma continuous_within_topological:
hoelzl@51478
  1782
  "continuous (at x within s) f \<longleftrightarrow>
hoelzl@51478
  1783
    (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
hoelzl@51641
  1784
  unfolding continuous_within tendsto_def eventually_at_topological by metis
hoelzl@51478
  1785
hoelzl@51478
  1786
lemma continuous_within_compose[continuous_intros]:
hoelzl@51478
  1787
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  1788
  continuous (at x within s) (g o f)"
hoelzl@51478
  1789
  by (simp add: continuous_within_topological) metis
hoelzl@51478
  1790
hoelzl@51478
  1791
lemma continuous_within_compose2:
hoelzl@51478
  1792
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  1793
  continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  1794
  using continuous_within_compose[of x s f g] by (simp add: comp_def)
hoelzl@51471
  1795
hoelzl@51478
  1796
lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f -- x --> f x"
hoelzl@51478
  1797
  using continuous_within[of x UNIV f] by simp
hoelzl@51478
  1798
hoelzl@51478
  1799
lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"
hoelzl@51641
  1800
  unfolding continuous_within by (rule tendsto_ident_at)
hoelzl@51478
  1801
hoelzl@51478
  1802
lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"
hoelzl@51478
  1803
  unfolding continuous_def by (rule tendsto_const)
hoelzl@51478
  1804
hoelzl@51478
  1805
lemma continuous_on_eq_continuous_within:
hoelzl@51478
  1806
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
hoelzl@51478
  1807
  unfolding continuous_on_def continuous_within ..
hoelzl@51478
  1808
hoelzl@51478
  1809
abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where
hoelzl@51478
  1810
  "isCont f a \<equiv> continuous (at a) f"
hoelzl@51478
  1811
hoelzl@51478
  1812
lemma isCont_def: "isCont f a \<longleftrightarrow> f -- a --> f a"
hoelzl@51478
  1813
  by (rule continuous_at)
hoelzl@51478
  1814
hoelzl@51478
  1815
lemma continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"
hoelzl@51641
  1816
  by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)
hoelzl@51478
  1817
hoelzl@51481
  1818
lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"
hoelzl@51641
  1819
  by (simp add: continuous_on_def continuous_at at_within_open[of _ s])
hoelzl@51481
  1820
hoelzl@51481
  1821
lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"
hoelzl@51481
  1822
  unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)
hoelzl@51481
  1823
hoelzl@51478
  1824
lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"
hoelzl@51478
  1825
  by (auto intro: continuous_at_within simp: continuous_on_eq_continuous_within)
hoelzl@51478
  1826
hoelzl@51478
  1827
lemma isContI_continuous: "continuous (at x within UNIV) f \<Longrightarrow> isCont f x"
hoelzl@51478
  1828
  by simp
hoelzl@51478
  1829
hoelzl@51478
  1830
lemma isCont_ident[continuous_intros, simp]: "isCont (\<lambda>x. x) a"
hoelzl@51478
  1831
  using continuous_ident by (rule isContI_continuous)
hoelzl@51478
  1832
hoelzl@51478
  1833
lemmas isCont_const = continuous_const
hoelzl@51478
  1834
hoelzl@51478
  1835
lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
hoelzl@51478
  1836
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51478
  1837
hoelzl@51478
  1838
lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
hoelzl@51478
  1839
  unfolding o_def by (rule isCont_o2)
hoelzl@51471
  1840
hoelzl@51471
  1841
lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@51471
  1842
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51471
  1843
hoelzl@51478
  1844
lemma continuous_within_compose3:
hoelzl@51478
  1845
  "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  1846
  using continuous_within_compose2[of x s f g] by (simp add: continuous_at_within)
hoelzl@51471
  1847
hoelzl@51479
  1848
subsubsection{* Open-cover compactness *}
hoelzl@51479
  1849
hoelzl@51479
  1850
context topological_space
hoelzl@51479
  1851
begin
hoelzl@51479
  1852
hoelzl@51479
  1853
definition compact :: "'a set \<Rightarrow> bool" where
hoelzl@51479
  1854
  compact_eq_heine_borel: -- "This name is used for backwards compatibility"
hoelzl@51479
  1855
    "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
hoelzl@51479
  1856
hoelzl@51479
  1857
lemma compactI:
hoelzl@51479
  1858
  assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union> C \<Longrightarrow> \<exists>C'. C' \<subseteq> C \<and> finite C' \<and> s \<subseteq> \<Union> C'"
hoelzl@51479
  1859
  shows "compact s"
hoelzl@51479
  1860
  unfolding compact_eq_heine_borel using assms by metis
hoelzl@51479
  1861
hoelzl@51479
  1862
lemma compact_empty[simp]: "compact {}"
hoelzl@51479
  1863
  by (auto intro!: compactI)
hoelzl@51479
  1864
hoelzl@51479
  1865
lemma compactE:
hoelzl@51479
  1866
  assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
hoelzl@51479
  1867
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
hoelzl@51479
  1868
  using assms unfolding compact_eq_heine_borel by metis
hoelzl@51479
  1869
hoelzl@51479
  1870
lemma compactE_image:
hoelzl@51479
  1871
  assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
hoelzl@51479
  1872
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
hoelzl@51479
  1873
  using assms unfolding ball_simps[symmetric] SUP_def
hoelzl@51479
  1874
  by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
hoelzl@51479
  1875
hoelzl@51481
  1876
lemma compact_inter_closed [intro]:
hoelzl@51481
  1877
  assumes "compact s" and "closed t"
hoelzl@51481
  1878
  shows "compact (s \<inter> t)"
hoelzl@51481
  1879
proof (rule compactI)
hoelzl@51481
  1880
  fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
hoelzl@51481
  1881
  from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto
hoelzl@51481
  1882
  moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
hoelzl@51481
  1883
  ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
hoelzl@51481
  1884
    using `compact s` unfolding compact_eq_heine_borel by auto
hoelzl@51481
  1885
  then guess D ..
hoelzl@51481
  1886
  then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
hoelzl@51481
  1887
    by (intro exI[of _ "D - {-t}"]) auto
hoelzl@51481
  1888
qed
hoelzl@51481
  1889
hoelzl@51471
  1890
end
hoelzl@51471
  1891
hoelzl@51481
  1892
lemma (in t2_space) compact_imp_closed:
hoelzl@51481
  1893
  assumes "compact s" shows "closed s"
hoelzl@51481
  1894
unfolding closed_def
hoelzl@51481
  1895
proof (rule openI)
hoelzl@51481
  1896
  fix y assume "y \<in> - s"
hoelzl@51481
  1897
  let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
hoelzl@51481
  1898
  note `compact s`
hoelzl@51481
  1899
  moreover have "\<forall>u\<in>?C. open u" by simp
hoelzl@51481
  1900
  moreover have "s \<subseteq> \<Union>?C"
hoelzl@51481
  1901
  proof
hoelzl@51481
  1902
    fix x assume "x \<in> s"
hoelzl@51481
  1903
    with `y \<in> - s` have "x \<noteq> y" by clarsimp
hoelzl@51481
  1904
    hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
hoelzl@51481
  1905
      by (rule hausdorff)
hoelzl@51481
  1906
    with `x \<in> s` show "x \<in> \<Union>?C"
hoelzl@51481
  1907
      unfolding eventually_nhds by auto
hoelzl@51481
  1908
  qed
hoelzl@51481
  1909
  ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
hoelzl@51481
  1910
    by (rule compactE)
hoelzl@51481
  1911
  from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
hoelzl@51481
  1912
  with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
hoelzl@51481
  1913
    by (simp add: eventually_Ball_finite)
hoelzl@51481
  1914
  with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
hoelzl@51481
  1915
    by (auto elim!: eventually_mono [rotated])
hoelzl@51481
  1916
  thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
hoelzl@51481
  1917
    by (simp add: eventually_nhds subset_eq)
hoelzl@51481
  1918
qed
hoelzl@51481
  1919
hoelzl@51481
  1920
lemma compact_continuous_image:
hoelzl@51481
  1921
  assumes f: "continuous_on s f" and s: "compact s"
hoelzl@51481
  1922
  shows "compact (f ` s)"
hoelzl@51481
  1923
proof (rule compactI)
hoelzl@51481
  1924
  fix C assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
hoelzl@51481
  1925
  with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"
hoelzl@51481
  1926
    unfolding continuous_on_open_invariant by blast
hoelzl@51481
  1927
  then guess A unfolding bchoice_iff .. note A = this
hoelzl@51481
  1928
  with cover have "\<forall>c\<in>C. open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)"
hoelzl@51481
  1929
    by (fastforce simp add: subset_eq set_eq_iff)+
hoelzl@51481
  1930
  from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" .
hoelzl@51481
  1931
  with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D"
hoelzl@51481
  1932
    by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+
hoelzl@51481
  1933
qed
hoelzl@51481
  1934
hoelzl@51481
  1935
lemma continuous_on_inv:
hoelzl@51481
  1936
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
hoelzl@51481
  1937
  assumes "continuous_on s f"  "compact s"  "\<forall>x\<in>s. g (f x) = x"
hoelzl@51481
  1938
  shows "continuous_on (f ` s) g"
hoelzl@51481
  1939
unfolding continuous_on_topological
hoelzl@51481
  1940
proof (clarsimp simp add: assms(3))
hoelzl@51481
  1941
  fix x :: 'a and B :: "'a set"
hoelzl@51481
  1942
  assume "x \<in> s" and "open B" and "x \<in> B"
hoelzl@51481
  1943
  have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
hoelzl@51481
  1944
    using assms(3) by (auto, metis)
hoelzl@51481
  1945
  have "continuous_on (s - B) f"
hoelzl@51481
  1946
    using `continuous_on s f` Diff_subset
hoelzl@51481
  1947
    by (rule continuous_on_subset)
hoelzl@51481
  1948
  moreover have "compact (s - B)"
hoelzl@51481
  1949
    using `open B` and `compact s`
hoelzl@51481
  1950
    unfolding Diff_eq by (intro compact_inter_closed closed_Compl)
hoelzl@51481
  1951
  ultimately have "compact (f ` (s - B))"
hoelzl@51481
  1952
    by (rule compact_continuous_image)
hoelzl@51481
  1953
  hence "closed (f ` (s - B))"
hoelzl@51481
  1954
    by (rule compact_imp_closed)
hoelzl@51481
  1955
  hence "open (- f ` (s - B))"
hoelzl@51481
  1956
    by (rule open_Compl)
hoelzl@51481
  1957
  moreover have "f x \<in> - f ` (s - B)"
hoelzl@51481
  1958
    using `x \<in> s` and `x \<in> B` by (simp add: 1)
hoelzl@51481
  1959
  moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
hoelzl@51481
  1960
    by (simp add: 1)
hoelzl@51481
  1961
  ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"
hoelzl@51481
  1962
    by fast
hoelzl@51481
  1963
qed
hoelzl@51481
  1964
hoelzl@51481
  1965
lemma continuous_on_inv_into:
hoelzl@51481
  1966
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
hoelzl@51481
  1967
  assumes s: "continuous_on s f" "compact s" and f: "inj_on f s"
hoelzl@51481
  1968
  shows "continuous_on (f ` s) (the_inv_into s f)"
hoelzl@51481
  1969
  by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])
hoelzl@51481
  1970
hoelzl@51479
  1971
lemma (in linorder_topology) compact_attains_sup:
hoelzl@51479
  1972
  assumes "compact S" "S \<noteq> {}"
hoelzl@51479
  1973
  shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"
hoelzl@51479
  1974
proof (rule classical)
hoelzl@51479
  1975
  assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"
hoelzl@51479
  1976
  then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"
hoelzl@51479
  1977
    by (metis not_le)
hoelzl@51479
  1978
  then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"
hoelzl@51479
  1979
    by auto
hoelzl@51479
  1980
  with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
hoelzl@51479
  1981
    by (erule compactE_image)
hoelzl@51479
  1982
  with `S \<noteq> {}` have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"
hoelzl@51479
  1983
    by (auto intro!: Max_in)
hoelzl@51479
  1984
  with C have "S \<subseteq> {..< Max (t`C)}"
hoelzl@51479
  1985
    by (auto intro: less_le_trans simp: subset_eq)
hoelzl@51479
  1986
  with t Max `C \<subseteq> S` show ?thesis
hoelzl@51479
  1987
    by fastforce
hoelzl@51479
  1988
qed
hoelzl@51479
  1989
hoelzl@51479
  1990
lemma (in linorder_topology) compact_attains_inf:
hoelzl@51479
  1991
  assumes "compact S" "S \<noteq> {}"
hoelzl@51479
  1992
  shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"
hoelzl@51479
  1993
proof (rule classical)
hoelzl@51479
  1994
  assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"
hoelzl@51479
  1995
  then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"
hoelzl@51479
  1996
    by (metis not_le)
hoelzl@51479
  1997
  then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"
hoelzl@51479
  1998
    by auto
hoelzl@51479
  1999
  with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
hoelzl@51479
  2000
    by (erule compactE_image)
hoelzl@51479
  2001
  with `S \<noteq> {}` have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"
hoelzl@51479
  2002
    by (auto intro!: Min_in)
hoelzl@51479
  2003
  with C have "S \<subseteq> {Min (t`C) <..}"
hoelzl@51479
  2004
    by (auto intro: le_less_trans simp: subset_eq)
hoelzl@51479
  2005
  with t Min `C \<subseteq> S` show ?thesis
hoelzl@51479
  2006
    by fastforce
hoelzl@51479
  2007
qed
hoelzl@51479
  2008
hoelzl@51479
  2009
lemma continuous_attains_sup:
hoelzl@51479
  2010
  fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
hoelzl@51479
  2011
  shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s.  f y \<le> f x)"
hoelzl@51479
  2012
  using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto
hoelzl@51479
  2013
hoelzl@51479
  2014
lemma continuous_attains_inf:
hoelzl@51479
  2015
  fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
hoelzl@51479
  2016
  shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y)"
hoelzl@51479
  2017
  using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
hoelzl@51479
  2018
hoelzl@51480
  2019
hoelzl@51480
  2020
subsection {* Connectedness *}
hoelzl@51480
  2021
hoelzl@51480
  2022
context topological_space
hoelzl@51480
  2023
begin
hoelzl@51480
  2024
hoelzl@51480
  2025
definition "connected S \<longleftrightarrow>
hoelzl@51480
  2026
  \<not> (\<exists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"
hoelzl@51480
  2027
hoelzl@51480
  2028
lemma connectedI:
hoelzl@51480
  2029
  "(\<And>A B. open A \<Longrightarrow> open B \<Longrightarrow> A \<inter> U \<noteq> {} \<Longrightarrow> B \<inter> U \<noteq> {} \<Longrightarrow> A \<inter> B \<inter> U = {} \<Longrightarrow> U \<subseteq> A \<union> B \<Longrightarrow> False)
hoelzl@51480
  2030
  \<Longrightarrow> connected U"
hoelzl@51480
  2031
  by (auto simp: connected_def)
hoelzl@51480
  2032
hoelzl@51480
  2033
lemma connected_empty[simp]: "connected {}"
hoelzl@51480
  2034
  by (auto intro!: connectedI)
hoelzl@51480
  2035
hoelzl@51479
  2036
end
hoelzl@51479
  2037
hoelzl@51480
  2038
lemma (in linorder_topology) connectedD_interval:
hoelzl@51480
  2039
  assumes "connected U" and xy: "x \<in> U" "y \<in> U" and "x \<le> z" "z \<le> y"
hoelzl@51480
  2040
  shows "z \<in> U"
hoelzl@51480
  2041
proof -
hoelzl@51480
  2042
  have eq: "{..<z} \<union> {z<..} = - {z}"
hoelzl@51480
  2043
    by auto
hoelzl@51480
  2044
  { assume "z \<notin> U" "x < z" "z < y"
hoelzl@51480
  2045
    with xy have "\<not> connected U"
hoelzl@51480
  2046
      unfolding connected_def simp_thms
hoelzl@51480
  2047
      apply (rule_tac exI[of _ "{..< z}"])
hoelzl@51480
  2048
      apply (rule_tac exI[of _ "{z <..}"])
hoelzl@51480
  2049
      apply (auto simp add: eq)
hoelzl@51480
  2050
      done }
hoelzl@51480
  2051
  with assms show "z \<in> U"
hoelzl@51480
  2052
    by (metis less_le)
hoelzl@51480
  2053
qed
hoelzl@51480
  2054
hoelzl@51480
  2055
lemma connected_continuous_image:
hoelzl@51480
  2056
  assumes *: "continuous_on s f"
hoelzl@51480
  2057
  assumes "connected s"
hoelzl@51480
  2058
  shows "connected (f ` s)"
hoelzl@51480
  2059
proof (rule connectedI)
hoelzl@51480
  2060
  fix A B assume A: "open A" "A \<inter> f ` s \<noteq> {}" and B: "open B" "B \<inter> f ` s \<noteq> {}" and
hoelzl@51480
  2061
    AB: "A \<inter> B \<inter> f ` s = {}" "f ` s \<subseteq> A \<union> B"
hoelzl@51480
  2062
  obtain A' where A': "open A'" "f -` A \<inter> s = A' \<inter> s"
hoelzl@51480
  2063
    using * `open A` unfolding continuous_on_open_invariant by metis
hoelzl@51480
  2064
  obtain B' where B': "open B'" "f -` B \<inter> s = B' \<inter> s"
hoelzl@51480
  2065
    using * `open B` unfolding continuous_on_open_invariant by metis
hoelzl@51480
  2066
hoelzl@51480
  2067
  have "\<exists>A B. open A \<and> open B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {}"
hoelzl@51480
  2068
  proof (rule exI[of _ A'], rule exI[of _ B'], intro conjI)
hoelzl@51480
  2069
    have "s \<subseteq> (f -` A \<inter> s) \<union> (f -` B \<inter> s)" using AB by auto
hoelzl@51480
  2070
    then show "s \<subseteq> A' \<union> B'" using A' B' by auto
hoelzl@51480
  2071
  next
hoelzl@51480
  2072
    have "(f -` A \<inter> s) \<inter> (f -` B \<inter> s) = {}" using AB by auto
hoelzl@51480
  2073
    then show "A' \<inter> B' \<inter> s = {}" using A' B' by auto
hoelzl@51480
  2074
  qed (insert A' B' A B, auto)
hoelzl@51480
  2075
  with `connected s` show False
hoelzl@51480
  2076
    unfolding connected_def by blast
hoelzl@51480
  2077
qed
hoelzl@51480
  2078
hoelzl@51518
  2079
hoelzl@51518
  2080
section {* Connectedness *}
hoelzl@51518
  2081
hoelzl@51775
  2082
class linear_continuum_topology = linorder_topology + linear_continuum
hoelzl@51518
  2083
begin
hoelzl@51518
  2084
hoelzl@51518
  2085
lemma Inf_notin_open:
hoelzl@51518
  2086
  assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a"
hoelzl@51518
  2087
  shows "Inf A \<notin> A"
hoelzl@51518
  2088
proof
hoelzl@51518
  2089
  assume "Inf A \<in> A"
hoelzl@51518
  2090
  then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A"
hoelzl@51518
  2091
    using open_left[of A "Inf A" x] assms by auto
hoelzl@51518
  2092
  with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
hoelzl@51518
  2093
    by (auto simp: subset_eq)
hoelzl@51518
  2094
  then show False
hoelzl@51518
  2095
    using cInf_lower[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le)
hoelzl@51518
  2096
qed
hoelzl@51518
  2097
hoelzl@51518
  2098
lemma Sup_notin_open:
hoelzl@51518
  2099
  assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x"
hoelzl@51518
  2100
  shows "Sup A \<notin> A"
hoelzl@51518
  2101
proof
hoelzl@51518
  2102
  assume "Sup A \<in> A"
hoelzl@51518
  2103
  then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
hoelzl@51518
  2104
    using open_right[of A "Sup A" x] assms by auto
hoelzl@51518
  2105
  with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
hoelzl@51518
  2106
    by (auto simp: subset_eq)
hoelzl@51518
  2107
  then show False
hoelzl@51518
  2108
    using cSup_upper[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le)
hoelzl@51518
  2109
qed
hoelzl@51518
  2110
hoelzl@51480
  2111
end
hoelzl@51480
  2112
hoelzl@51775
  2113
instance linear_continuum_topology \<subseteq> perfect_space
hoelzl@51775
  2114
proof
hoelzl@51775
  2115
  fix x :: 'a
hoelzl@51775
  2116
  from ex_gt_or_lt [of x] guess y ..
hoelzl@51775
  2117
  with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y]
hoelzl@51775
  2118
  show "\<not> open {x}"
hoelzl@51775
  2119
    by auto
hoelzl@51775
  2120
qed
hoelzl@51775
  2121
hoelzl@51518
  2122
lemma connectedI_interval:
hoelzl@51775
  2123
  fixes U :: "'a :: linear_continuum_topology set"
hoelzl@51518
  2124
  assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U"
hoelzl@51518
  2125
  shows "connected U"
hoelzl@51518
  2126
proof (rule connectedI)
hoelzl@51518
  2127
  { fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
hoelzl@51518
  2128
    fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
hoelzl@51518
  2129
hoelzl@51518
  2130
    let ?z = "Inf (B \<inter> {x <..})"
hoelzl@51518
  2131
hoelzl@51518
  2132
    have "x \<le> ?z" "?z \<le> y"
hoelzl@51518
  2133
      using `y \<in> B` `x < y` by (auto intro: cInf_lower[where z=x] cInf_greatest)
hoelzl@51518
  2134
    with `x \<in> U` `y \<in> U` have "?z \<in> U"
hoelzl@51518
  2135
      by (rule *)
hoelzl@51518
  2136
    moreover have "?z \<notin> B \<inter> {x <..}"
hoelzl@51518
  2137
      using `open B` by (intro Inf_notin_open) auto
hoelzl@51518
  2138
    ultimately have "?z \<in> A"
hoelzl@51518
  2139
      using `x \<le> ?z` `A \<inter> B \<inter> U = {}` `x \<in> A` `U \<subseteq> A \<union> B` by auto
hoelzl@51518
  2140
hoelzl@51518
  2141
    { assume "?z < y"
hoelzl@51518
  2142
      obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
hoelzl@51518
  2143
        using open_right[OF `open A` `?z \<in> A` `?z < y`] by auto
hoelzl@51518
  2144
      moreover obtain b where "b \<in> B" "x < b" "b < min a y"
hoelzl@51518
  2145
        using cInf_less_iff[of "B \<inter> {x <..}" x "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B`
hoelzl@51518
  2146
        by (auto intro: less_imp_le)
hoelzl@51518
  2147
      moreover then have "?z \<le> b"
hoelzl@51518
  2148
        by (intro cInf_lower[where z=x]) auto
hoelzl@51518
  2149
      moreover have "b \<in> U"
hoelzl@51518
  2150
        using `x \<le> ?z` `?z \<le> b` `b < min a y`
hoelzl@51518
  2151
        by (intro *[OF `x \<in> U` `y \<in> U`]) (auto simp: less_imp_le)
hoelzl@51518
  2152
      ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U"
hoelzl@51518
  2153
        by (intro bexI[of _ b]) auto }
hoelzl@51518
  2154
    then have False
hoelzl@51518
  2155
      using `?z \<le> y` `?z \<in> A` `y \<in> B` `y \<in> U` `A \<inter> B \<inter> U = {}` unfolding le_less by blast }
hoelzl@51518
  2156
  note not_disjoint = this
hoelzl@51518
  2157
hoelzl@51518
  2158
  fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"
hoelzl@51518
  2159
  moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto
hoelzl@51518
  2160
  moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto
hoelzl@51518
  2161
  moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
hoelzl@51518
  2162
  ultimately show False by (cases x y rule: linorder_cases) auto
hoelzl@51518
  2163
qed
hoelzl@51518
  2164
hoelzl@51518
  2165
lemma connected_iff_interval:
hoelzl@51775
  2166
  fixes U :: "'a :: linear_continuum_topology set"
hoelzl@51518
  2167
  shows "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)"
hoelzl@51518
  2168
  by (auto intro: connectedI_interval dest: connectedD_interval)
hoelzl@51518
  2169
hoelzl@51775
  2170
lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
hoelzl@51518
  2171
  unfolding connected_iff_interval by auto
hoelzl@51518
  2172
hoelzl@51775
  2173
lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}"
hoelzl@51518
  2174
  unfolding connected_iff_interval by auto
hoelzl@51518
  2175
hoelzl@51775
  2176
lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}"
hoelzl@51518
  2177
  unfolding connected_iff_interval by auto
hoelzl@51518
  2178
hoelzl@51775
  2179
lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}"
hoelzl@51518
  2180
  unfolding connected_iff_interval by auto
hoelzl@51518
  2181
hoelzl@51775
  2182
lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}"
hoelzl@51518
  2183
  unfolding connected_iff_interval by auto
hoelzl@51518
  2184
hoelzl@51775
  2185
lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}"
hoelzl@51518
  2186
  unfolding connected_iff_interval by auto
hoelzl@51518
  2187
hoelzl@51775
  2188
lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}"
hoelzl@51518
  2189
  unfolding connected_iff_interval by auto
hoelzl@51518
  2190
hoelzl@51775
  2191
lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}"
hoelzl@51518
  2192
  unfolding connected_iff_interval by auto
hoelzl@51518
  2193
hoelzl@51775
  2194
lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}"
hoelzl@51518
  2195
  unfolding connected_iff_interval by auto
hoelzl@51518
  2196
hoelzl@51518
  2197
lemma connected_contains_Ioo: 
hoelzl@51518
  2198
  fixes A :: "'a :: linorder_topology set"
hoelzl@51518
  2199
  assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
hoelzl@51518
  2200
  using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le)
hoelzl@51518
  2201
hoelzl@51518
  2202
subsection {* Intermediate Value Theorem *}
hoelzl@51518
  2203
hoelzl@51518
  2204
lemma IVT':
hoelzl@51775
  2205
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2206
  assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"
hoelzl@51518
  2207
  assumes *: "continuous_on {a .. b} f"
hoelzl@51518
  2208
  shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2209
proof -
hoelzl@51518
  2210
  have "connected {a..b}"
hoelzl@51518
  2211
    unfolding connected_iff_interval by auto
hoelzl@51518
  2212
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y
hoelzl@51518
  2213
  show ?thesis
hoelzl@51518
  2214
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
hoelzl@51518
  2215
qed
hoelzl@51518
  2216
hoelzl@51518
  2217
lemma IVT2':
hoelzl@51775
  2218
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2219
  assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b"
hoelzl@51518
  2220
  assumes *: "continuous_on {a .. b} f"
hoelzl@51518
  2221
  shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2222
proof -
hoelzl@51518
  2223
  have "connected {a..b}"
hoelzl@51518
  2224
    unfolding connected_iff_interval by auto
hoelzl@51518
  2225
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y
hoelzl@51518
  2226
  show ?thesis
hoelzl@51518
  2227
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
hoelzl@51518
  2228
qed
hoelzl@51518
  2229
hoelzl@51518
  2230
lemma IVT:
hoelzl@51775
  2231
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2232
  shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2233
  by (rule IVT') (auto intro: continuous_at_imp_continuous_on)
hoelzl@51518
  2234
hoelzl@51518
  2235
lemma IVT2:
hoelzl@51775
  2236
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2237
  shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2238
  by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)
hoelzl@51518
  2239
hoelzl@51518
  2240
lemma continuous_inj_imp_mono:
hoelzl@51775
  2241
  fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2242
  assumes x: "a < x" "x < b"
hoelzl@51518
  2243
  assumes cont: "continuous_on {a..b} f"
hoelzl@51518
  2244
  assumes inj: "inj_on f {a..b}"
hoelzl@51518
  2245
  shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)"
hoelzl@51518
  2246
proof -
hoelzl@51518
  2247
  note I = inj_on_iff[OF inj]
hoelzl@51518
  2248
  { assume "f x < f a" "f x < f b"
hoelzl@51518
  2249
    then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f x < f s"
hoelzl@51518
  2250
      using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x
hoelzl@51518
  2251
      by (auto simp: continuous_on_subset[OF cont] less_imp_le)
hoelzl@51518
  2252
    with x I have False by auto }
hoelzl@51518
  2253
  moreover
hoelzl@51518
  2254
  { assume "f a < f x" "f b < f x"
hoelzl@51518
  2255
    then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f s < f x"
hoelzl@51518
  2256
      using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x
hoelzl@51518
  2257
      by (auto simp: continuous_on_subset[OF cont] less_imp_le)
hoelzl@51518
  2258
    with x I have False by auto }
hoelzl@51518
  2259
  ultimately show ?thesis
hoelzl@51518
  2260
    using I[of a x] I[of x b] x less_trans[OF x] by (auto simp add: le_less less_imp_neq neq_iff)
hoelzl@51518
  2261
qed
hoelzl@51518
  2262
hoelzl@51518
  2263
end
hoelzl@51518
  2264