src/HOL/Topological_Spaces.thy
author hoelzl
Mon Mar 31 12:16:37 2014 +0200 (2014-03-31)
changeset 56329 9597a53b3429
parent 56289 d8d2a2b97168
child 56371 fb9ae0727548
permissions -rw-r--r--
add connected_local_const
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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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  using open_Union [of "B ` A"] by simp
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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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  using open_Inter [of "B ` A"] by simp
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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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  using closed_Union [of "B ` A"] by simp
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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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  using generate_topology.UN [of "K ` I"] by 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 "\<exists>z. x < z \<and> z < y")
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  case True
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  then obtain z where "x < z \<and> z < y" ..
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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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  case False
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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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proof -
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  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
hoelzl@51471
   307
qed
hoelzl@51471
   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"
hoelzl@51471
   311
  unfolding eventually_def
hoelzl@51471
   312
  by (rule is_filter.mono [OF is_filter_Rep_filter])
hoelzl@51471
   313
hoelzl@51471
   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"
hoelzl@51471
   317
  shows "eventually (\<lambda>x. P x \<and> Q x) F"
hoelzl@51471
   318
  using assms unfolding eventually_def
hoelzl@51471
   319
  by (rule is_filter.conj [OF is_filter_Rep_filter])
hoelzl@51471
   320
hoelzl@51471
   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"
hoelzl@51471
   323
  shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
hoelzl@51471
   324
using assms by (induct set: finite, simp, simp add: eventually_conj)
hoelzl@51471
   325
hoelzl@51471
   326
lemma eventually_all_finite:
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   327
  fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
hoelzl@51471
   328
  assumes "\<And>y. eventually (\<lambda>x. P x y) net"
hoelzl@51471
   329
  shows "eventually (\<lambda>x. \<forall>y. P x y) net"
hoelzl@51471
   330
using eventually_Ball_finite [of UNIV P] assms by simp
hoelzl@51471
   331
hoelzl@51471
   332
lemma eventually_mp:
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   333
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@51471
   334
  assumes "eventually (\<lambda>x. P x) F"
hoelzl@51471
   335
  shows "eventually (\<lambda>x. Q x) F"
hoelzl@51471
   336
proof (rule eventually_mono)
hoelzl@51471
   337
  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
hoelzl@51471
   338
  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
hoelzl@51471
   339
    using assms by (rule eventually_conj)
hoelzl@51471
   340
qed
hoelzl@51471
   341
hoelzl@51471
   342
lemma eventually_rev_mp:
hoelzl@51471
   343
  assumes "eventually (\<lambda>x. P x) F"
hoelzl@51471
   344
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@51471
   345
  shows "eventually (\<lambda>x. Q x) F"
hoelzl@51471
   346
using assms(2) assms(1) by (rule eventually_mp)
hoelzl@51471
   347
hoelzl@51471
   348
lemma eventually_conj_iff:
hoelzl@51471
   349
  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
hoelzl@51471
   350
  by (auto intro: eventually_conj elim: eventually_rev_mp)
hoelzl@51471
   351
hoelzl@51471
   352
lemma eventually_elim1:
hoelzl@51471
   353
  assumes "eventually (\<lambda>i. P i) F"
hoelzl@51471
   354
  assumes "\<And>i. P i \<Longrightarrow> Q i"
hoelzl@51471
   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:
hoelzl@51471
   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:
hoelzl@51471
   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 {*
wenzelm@56231
   376
  fun eventually_elim_tac ctxt thms = SUBGOAL_CASES (fn (_, _, st) =>
hoelzl@51471
   377
    let
hoelzl@51471
   378
      val thy = Proof_Context.theory_of ctxt
hoelzl@51471
   379
      val mp_thms = thms RL [@{thm eventually_rev_mp}]
hoelzl@51471
   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
wenzelm@56231
   384
      val cases_prop = prop_of (raw_elim_thm RS st)
hoelzl@51471
   385
      val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
hoelzl@51471
   386
    in
wenzelm@56231
   387
      CASES cases (rtac raw_elim_thm 1)
wenzelm@56231
   388
    end) 1
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 "inf F F' \<le> F" and "inf F F' \<le> F'"
hoelzl@51471
   475
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
hoelzl@51471
   476
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
hoelzl@51471
   477
    unfolding le_filter_def eventually_inf
hoelzl@51471
   478
    by (auto elim!: eventually_mono intro: eventually_conj) }
haftmann@52729
   479
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
haftmann@52729
   480
    unfolding le_filter_def eventually_sup by simp_all }
haftmann@52729
   481
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
haftmann@52729
   482
    unfolding le_filter_def eventually_sup by simp }
haftmann@52729
   483
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
haftmann@52729
   484
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
haftmann@52729
   485
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
haftmann@52729
   486
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
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 }
haftmann@52729
   491
  { show "Inf {} = (top::'a filter)"
haftmann@52729
   492
    by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
huffman@53859
   493
      (metis (full_types) top_filter_def always_eventually eventually_top) }
haftmann@52729
   494
  { show "Sup {} = (bot::'a filter)"
haftmann@52729
   495
    by (auto simp: bot_filter_def Sup_filter_def) }
hoelzl@51471
   496
qed
hoelzl@51471
   497
hoelzl@51471
   498
end
hoelzl@51471
   499
hoelzl@51471
   500
lemma filter_leD:
hoelzl@51471
   501
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
hoelzl@51471
   502
  unfolding le_filter_def by simp
hoelzl@51471
   503
hoelzl@51471
   504
lemma filter_leI:
hoelzl@51471
   505
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
hoelzl@51471
   506
  unfolding le_filter_def by simp
hoelzl@51471
   507
hoelzl@51471
   508
lemma eventually_False:
hoelzl@51471
   509
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
hoelzl@51471
   510
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
hoelzl@51471
   511
hoelzl@51471
   512
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
hoelzl@51471
   513
  where "trivial_limit F \<equiv> F = bot"
hoelzl@51471
   514
hoelzl@51471
   515
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
hoelzl@51471
   516
  by (rule eventually_False [symmetric])
hoelzl@51471
   517
hoelzl@51471
   518
lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"
hoelzl@51471
   519
  by (cases P) (simp_all add: eventually_False)
hoelzl@51471
   520
hoelzl@51471
   521
hoelzl@51471
   522
subsubsection {* Map function for filters *}
hoelzl@51471
   523
hoelzl@51471
   524
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
hoelzl@51471
   525
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
hoelzl@51471
   526
hoelzl@51471
   527
lemma eventually_filtermap:
hoelzl@51471
   528
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
hoelzl@51471
   529
  unfolding filtermap_def
hoelzl@51471
   530
  apply (rule eventually_Abs_filter)
hoelzl@51471
   531
  apply (rule is_filter.intro)
hoelzl@51471
   532
  apply (auto elim!: eventually_rev_mp)
hoelzl@51471
   533
  done
hoelzl@51471
   534
hoelzl@51471
   535
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
hoelzl@51471
   536
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   537
hoelzl@51471
   538
lemma filtermap_filtermap:
hoelzl@51471
   539
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
hoelzl@51471
   540
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   541
hoelzl@51471
   542
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
hoelzl@51471
   543
  unfolding le_filter_def eventually_filtermap by simp
hoelzl@51471
   544
hoelzl@51471
   545
lemma filtermap_bot [simp]: "filtermap f bot = bot"
hoelzl@51471
   546
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   547
hoelzl@51471
   548
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
hoelzl@51471
   549
  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
hoelzl@51471
   550
hoelzl@51471
   551
subsubsection {* Order filters *}
hoelzl@51471
   552
hoelzl@51471
   553
definition at_top :: "('a::order) filter"
hoelzl@51471
   554
  where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
hoelzl@51471
   555
hoelzl@51471
   556
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
hoelzl@51471
   557
  unfolding at_top_def
hoelzl@51471
   558
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   559
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@51471
   560
  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
hoelzl@51471
   561
  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
hoelzl@51471
   562
  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
hoelzl@51471
   563
  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
hoelzl@51471
   564
qed auto
hoelzl@51471
   565
hoelzl@51471
   566
lemma eventually_ge_at_top:
hoelzl@51471
   567
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
hoelzl@51471
   568
  unfolding eventually_at_top_linorder by auto
hoelzl@51471
   569
hoelzl@53215
   570
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::unbounded_dense_linorder. \<forall>n>N. P n)"
hoelzl@51471
   571
  unfolding eventually_at_top_linorder
hoelzl@51471
   572
proof safe
wenzelm@53381
   573
  fix N assume "\<forall>n\<ge>N. P n"
wenzelm@53381
   574
  then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])
hoelzl@51471
   575
next
hoelzl@51471
   576
  fix N assume "\<forall>n>N. P n"
wenzelm@53381
   577
  moreover obtain y where "N < y" using gt_ex[of N] ..
hoelzl@51471
   578
  ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])
hoelzl@51471
   579
qed
hoelzl@51471
   580
hoelzl@51471
   581
lemma eventually_gt_at_top:
hoelzl@53215
   582
  "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"
hoelzl@51471
   583
  unfolding eventually_at_top_dense by auto
hoelzl@51471
   584
hoelzl@51471
   585
definition at_bot :: "('a::order) filter"
hoelzl@51471
   586
  where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
hoelzl@51471
   587
hoelzl@51471
   588
lemma eventually_at_bot_linorder:
hoelzl@51471
   589
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@51471
   590
  unfolding at_bot_def
hoelzl@51471
   591
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   592
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@51471
   593
  assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
hoelzl@51471
   594
  then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
hoelzl@51471
   595
  then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
hoelzl@51471
   596
  then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
hoelzl@51471
   597
qed auto
hoelzl@51471
   598
hoelzl@51471
   599
lemma eventually_le_at_bot:
hoelzl@51471
   600
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
hoelzl@51471
   601
  unfolding eventually_at_bot_linorder by auto
hoelzl@51471
   602
hoelzl@51471
   603
lemma eventually_at_bot_dense:
hoelzl@53215
   604
  fixes P :: "'a::unbounded_dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
hoelzl@51471
   605
  unfolding eventually_at_bot_linorder
hoelzl@51471
   606
proof safe
hoelzl@51471
   607
  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
   608
next
hoelzl@51471
   609
  fix N assume "\<forall>n<N. P n" 
wenzelm@53381
   610
  moreover obtain y where "y < N" using lt_ex[of N] ..
hoelzl@51471
   611
  ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
hoelzl@51471
   612
qed
hoelzl@51471
   613
hoelzl@51471
   614
lemma eventually_gt_at_bot:
hoelzl@53215
   615
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
hoelzl@51471
   616
  unfolding eventually_at_bot_dense by auto
hoelzl@51471
   617
lp15@56289
   618
lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
lp15@56289
   619
  unfolding trivial_limit_def
lp15@56289
   620
  by (metis eventually_at_bot_linorder order_refl)
lp15@56289
   621
lp15@56289
   622
lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
lp15@56289
   623
  unfolding trivial_limit_def
lp15@56289
   624
  by (metis eventually_at_top_linorder order_refl)
lp15@56289
   625
hoelzl@51471
   626
subsection {* Sequentially *}
hoelzl@51471
   627
hoelzl@51471
   628
abbreviation sequentially :: "nat filter"
hoelzl@51471
   629
  where "sequentially == at_top"
hoelzl@51471
   630
hoelzl@51471
   631
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
hoelzl@51471
   632
  unfolding at_top_def by simp
hoelzl@51471
   633
hoelzl@51471
   634
lemma eventually_sequentially:
hoelzl@51471
   635
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@51471
   636
  by (rule eventually_at_top_linorder)
hoelzl@51471
   637
hoelzl@51471
   638
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
hoelzl@51471
   639
  unfolding filter_eq_iff eventually_sequentially by auto
hoelzl@51471
   640
hoelzl@51471
   641
lemmas trivial_limit_sequentially = sequentially_bot
hoelzl@51471
   642
hoelzl@51471
   643
lemma eventually_False_sequentially [simp]:
hoelzl@51471
   644
  "\<not> eventually (\<lambda>n. False) sequentially"
hoelzl@51471
   645
  by (simp add: eventually_False)
hoelzl@51471
   646
hoelzl@51471
   647
lemma le_sequentially:
hoelzl@51471
   648
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
hoelzl@51471
   649
  unfolding le_filter_def eventually_sequentially
hoelzl@51471
   650
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
hoelzl@51471
   651
hoelzl@51471
   652
lemma eventually_sequentiallyI:
hoelzl@51471
   653
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
hoelzl@51471
   654
  shows "eventually P sequentially"
hoelzl@51471
   655
using assms by (auto simp: eventually_sequentially)
hoelzl@51471
   656
hoelzl@51474
   657
lemma eventually_sequentially_seg:
hoelzl@51474
   658
  "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@51474
   659
  unfolding eventually_sequentially
hoelzl@51474
   660
  apply safe
hoelzl@51474
   661
   apply (rule_tac x="N + k" in exI)
hoelzl@51474
   662
   apply rule
hoelzl@51474
   663
   apply (erule_tac x="n - k" in allE)
hoelzl@51474
   664
   apply auto []
hoelzl@51474
   665
  apply (rule_tac x=N in exI)
hoelzl@51474
   666
  apply auto []
hoelzl@51474
   667
  done
hoelzl@51471
   668
hoelzl@51471
   669
subsubsection {* Standard filters *}
hoelzl@51471
   670
hoelzl@51641
   671
definition principal :: "'a set \<Rightarrow> 'a filter" where
hoelzl@51641
   672
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
hoelzl@51641
   673
hoelzl@51641
   674
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
hoelzl@51641
   675
  unfolding principal_def
hoelzl@51641
   676
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
hoelzl@51471
   677
hoelzl@51641
   678
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
hoelzl@51641
   679
  unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
hoelzl@51641
   680
hoelzl@51641
   681
lemma principal_UNIV[simp]: "principal UNIV = top"
hoelzl@51641
   682
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@51471
   683
hoelzl@51641
   684
lemma principal_empty[simp]: "principal {} = bot"
hoelzl@51641
   685
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@51641
   686
hoelzl@51641
   687
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
hoelzl@51641
   688
  by (auto simp: le_filter_def eventually_principal)
hoelzl@51471
   689
hoelzl@51641
   690
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
hoelzl@51641
   691
  unfolding le_filter_def eventually_principal
hoelzl@51641
   692
  apply safe
hoelzl@51641
   693
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
hoelzl@51641
   694
  apply (auto elim: eventually_elim1)
hoelzl@51641
   695
  done
hoelzl@51471
   696
hoelzl@51641
   697
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
hoelzl@51641
   698
  unfolding eq_iff by simp
hoelzl@51471
   699
hoelzl@51641
   700
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
hoelzl@51641
   701
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
hoelzl@51471
   702
hoelzl@51641
   703
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
hoelzl@51641
   704
  unfolding filter_eq_iff eventually_inf eventually_principal
hoelzl@51641
   705
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
hoelzl@51471
   706
hoelzl@51641
   707
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
hoelzl@51641
   708
  unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
hoelzl@51641
   709
hoelzl@51641
   710
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
hoelzl@51641
   711
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
hoelzl@51471
   712
hoelzl@51471
   713
subsubsection {* Topological filters *}
hoelzl@51471
   714
hoelzl@51471
   715
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
hoelzl@51471
   716
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
hoelzl@51471
   717
hoelzl@51641
   718
definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60)
hoelzl@51641
   719
  where "at a within s = inf (nhds a) (principal (s - {a}))"
hoelzl@51641
   720
hoelzl@51641
   721
abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where
hoelzl@51641
   722
  "at x \<equiv> at x within (CONST UNIV)"
hoelzl@51471
   723
hoelzl@51473
   724
abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   725
  "at_right x \<equiv> at x within {x <..}"
hoelzl@51471
   726
hoelzl@51473
   727
abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   728
  "at_left x \<equiv> at x within {..< x}"
hoelzl@51471
   729
hoelzl@51473
   730
lemma (in topological_space) eventually_nhds:
hoelzl@51471
   731
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
hoelzl@51471
   732
  unfolding nhds_def
hoelzl@51471
   733
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51473
   734
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
hoelzl@51471
   735
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
hoelzl@51471
   736
next
hoelzl@51471
   737
  fix P Q
hoelzl@51471
   738
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
hoelzl@51471
   739
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
hoelzl@51471
   740
  then obtain S T where
hoelzl@51471
   741
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
hoelzl@51471
   742
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
hoelzl@51471
   743
  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
   744
    by (simp add: open_Int)
hoelzl@51471
   745
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" ..
hoelzl@51471
   746
qed auto
hoelzl@51471
   747
hoelzl@51471
   748
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
hoelzl@51471
   749
  unfolding trivial_limit_def eventually_nhds by simp
hoelzl@51471
   750
hoelzl@51641
   751
lemma eventually_at_filter:
hoelzl@51641
   752
  "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
hoelzl@51641
   753
  unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)
hoelzl@51641
   754
hoelzl@51641
   755
lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
hoelzl@51641
   756
  unfolding at_within_def by (intro inf_mono) auto
hoelzl@51641
   757
hoelzl@51471
   758
lemma eventually_at_topological:
hoelzl@51641
   759
  "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
   760
  unfolding eventually_nhds eventually_at_filter by simp
hoelzl@51471
   761
hoelzl@51481
   762
lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
hoelzl@51641
   763
  unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
hoelzl@51481
   764
huffman@53859
   765
lemma at_within_empty [simp]: "at a within {} = bot"
huffman@53859
   766
  unfolding at_within_def by simp
huffman@53859
   767
huffman@53860
   768
lemma at_within_union: "at x within (S \<union> T) = sup (at x within S) (at x within T)"
huffman@53860
   769
  unfolding filter_eq_iff eventually_sup eventually_at_filter
huffman@53860
   770
  by (auto elim!: eventually_rev_mp)
huffman@53860
   771
hoelzl@51471
   772
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
hoelzl@51471
   773
  unfolding trivial_limit_def eventually_at_topological
hoelzl@51471
   774
  by (safe, case_tac "S = {a}", simp, fast, fast)
hoelzl@51471
   775
hoelzl@51471
   776
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
hoelzl@51471
   777
  by (simp add: at_eq_bot_iff not_open_singleton)
hoelzl@51471
   778
hoelzl@51471
   779
lemma eventually_at_right:
hoelzl@51471
   780
  fixes x :: "'a :: {no_top, linorder_topology}"
hoelzl@51471
   781
  shows "eventually P (at_right x) \<longleftrightarrow> (\<exists>b. x < b \<and> (\<forall>z. x < z \<and> z < b \<longrightarrow> P z))"
hoelzl@51641
   782
  unfolding eventually_at_topological
hoelzl@51471
   783
proof safe
wenzelm@53381
   784
  obtain y where "x < y" using gt_ex[of x] ..
hoelzl@51480
   785
  moreover fix S assume "open S" "x \<in> S" note open_right[OF this, of y]
hoelzl@51480
   786
  moreover note gt_ex[of x]
hoelzl@51641
   787
  moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {x<..} \<longrightarrow> P s"
hoelzl@51471
   788
  ultimately show "\<exists>b>x. \<forall>z. x < z \<and> z < b \<longrightarrow> P z"
hoelzl@51471
   789
    by (auto simp: subset_eq Ball_def)
hoelzl@51471
   790
next
hoelzl@51471
   791
  fix b assume "x < b" "\<forall>z. x < z \<and> z < b \<longrightarrow> P z"
hoelzl@51641
   792
  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
   793
    by (intro exI[of _ "{..< b}"]) auto
hoelzl@51471
   794
qed
hoelzl@51471
   795
hoelzl@51471
   796
lemma eventually_at_left:
hoelzl@51471
   797
  fixes x :: "'a :: {no_bot, linorder_topology}"
hoelzl@51471
   798
  shows "eventually P (at_left x) \<longleftrightarrow> (\<exists>b. x > b \<and> (\<forall>z. b < z \<and> z < x \<longrightarrow> P z))"
hoelzl@51641
   799
  unfolding eventually_at_topological
hoelzl@51471
   800
proof safe
wenzelm@53381
   801
  obtain y where "y < x" using lt_ex[of x] ..
hoelzl@51480
   802
  moreover fix S assume "open S" "x \<in> S" note open_left[OF this, of y]
hoelzl@51641
   803
  moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {..<x} \<longrightarrow> P s"
hoelzl@51471
   804
  ultimately show "\<exists>b<x. \<forall>z. b < z \<and> z < x \<longrightarrow> P z"
hoelzl@51471
   805
    by (auto simp: subset_eq Ball_def)
hoelzl@51471
   806
next
hoelzl@51471
   807
  fix b assume "b < x" "\<forall>z. b < z \<and> z < x \<longrightarrow> P z"
hoelzl@51641
   808
  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
   809
    by (intro exI[of _ "{b <..}"]) auto
hoelzl@51471
   810
qed
hoelzl@51471
   811
hoelzl@51471
   812
lemma trivial_limit_at_left_real [simp]:
hoelzl@53215
   813
  "\<not> trivial_limit (at_left (x::'a::{no_bot, unbounded_dense_linorder, linorder_topology}))"
hoelzl@51471
   814
  unfolding trivial_limit_def eventually_at_left by (auto dest: dense)
hoelzl@51471
   815
hoelzl@51471
   816
lemma trivial_limit_at_right_real [simp]:
hoelzl@53215
   817
  "\<not> trivial_limit (at_right (x::'a::{no_top, unbounded_dense_linorder, linorder_topology}))"
hoelzl@51471
   818
  unfolding trivial_limit_def eventually_at_right by (auto dest: dense)
hoelzl@51471
   819
hoelzl@51471
   820
lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"
hoelzl@51641
   821
  by (auto simp: eventually_at_filter filter_eq_iff eventually_sup 
hoelzl@51471
   822
           elim: eventually_elim2 eventually_elim1)
hoelzl@51471
   823
hoelzl@51471
   824
lemma eventually_at_split:
hoelzl@51471
   825
  "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
hoelzl@51471
   826
  by (subst at_eq_sup_left_right) (simp add: eventually_sup)
hoelzl@51471
   827
hoelzl@51471
   828
subsection {* Limits *}
hoelzl@51471
   829
hoelzl@51471
   830
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@51471
   831
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@51471
   832
hoelzl@51471
   833
syntax
hoelzl@51471
   834
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@51471
   835
hoelzl@51471
   836
translations
hoelzl@51471
   837
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@51471
   838
hoelzl@51471
   839
lemma filterlim_iff:
hoelzl@51471
   840
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
hoelzl@51471
   841
  unfolding filterlim_def le_filter_def eventually_filtermap ..
hoelzl@51471
   842
hoelzl@51471
   843
lemma filterlim_compose:
hoelzl@51471
   844
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@51471
   845
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@51471
   846
hoelzl@51471
   847
lemma filterlim_mono:
hoelzl@51471
   848
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@51471
   849
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@51471
   850
hoelzl@51471
   851
lemma filterlim_ident: "LIM x F. x :> F"
hoelzl@51471
   852
  by (simp add: filterlim_def filtermap_ident)
hoelzl@51471
   853
hoelzl@51471
   854
lemma filterlim_cong:
hoelzl@51471
   855
  "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
   856
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
hoelzl@51471
   857
hoelzl@51641
   858
lemma filterlim_principal:
hoelzl@51641
   859
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@51641
   860
  unfolding filterlim_def eventually_filtermap le_principal ..
hoelzl@51641
   861
hoelzl@51641
   862
lemma filterlim_inf:
hoelzl@51641
   863
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
hoelzl@51641
   864
  unfolding filterlim_def by simp
hoelzl@51471
   865
hoelzl@51471
   866
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
hoelzl@51471
   867
  unfolding filterlim_def filtermap_filtermap ..
hoelzl@51471
   868
hoelzl@51471
   869
lemma filterlim_sup:
hoelzl@51471
   870
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
hoelzl@51471
   871
  unfolding filterlim_def filtermap_sup by auto
hoelzl@51471
   872
hoelzl@51471
   873
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
hoelzl@51471
   874
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
hoelzl@51471
   875
hoelzl@51471
   876
subsubsection {* Tendsto *}
hoelzl@51471
   877
hoelzl@51471
   878
abbreviation (in topological_space)
hoelzl@51471
   879
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
hoelzl@51471
   880
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
hoelzl@51471
   881
hoelzl@51478
   882
definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
   883
  "Lim A f = (THE l. (f ---> l) A)"
hoelzl@51478
   884
hoelzl@51471
   885
lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F"
hoelzl@51471
   886
  by simp
hoelzl@51471
   887
hoelzl@51471
   888
ML {*
hoelzl@51471
   889
hoelzl@51471
   890
structure Tendsto_Intros = Named_Thms
hoelzl@51471
   891
(
hoelzl@51471
   892
  val name = @{binding tendsto_intros}
hoelzl@51471
   893
  val description = "introduction rules for tendsto"
hoelzl@51471
   894
)
hoelzl@51471
   895
hoelzl@51471
   896
*}
hoelzl@51471
   897
hoelzl@51471
   898
setup {*
hoelzl@51471
   899
  Tendsto_Intros.setup #>
hoelzl@51471
   900
  Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},
hoelzl@51641
   901
    map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])) o Tendsto_Intros.get o Context.proof_of);
hoelzl@51471
   902
*}
hoelzl@51471
   903
hoelzl@51473
   904
lemma (in topological_space) tendsto_def:
hoelzl@51473
   905
   "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@51471
   906
  unfolding filterlim_def
hoelzl@51471
   907
proof safe
hoelzl@51471
   908
  fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
hoelzl@51471
   909
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   910
    unfolding eventually_nhds eventually_filtermap le_filter_def
hoelzl@51471
   911
    by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
hoelzl@51471
   912
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
hoelzl@51471
   913
hoelzl@51471
   914
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
hoelzl@51471
   915
  unfolding tendsto_def le_filter_def by fast
hoelzl@51471
   916
hoelzl@51641
   917
lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)"
hoelzl@51641
   918
  by (blast intro: tendsto_mono at_le)
hoelzl@51641
   919
hoelzl@51641
   920
lemma filterlim_at:
hoelzl@51641
   921
  "(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
   922
  by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
hoelzl@51641
   923
hoelzl@51473
   924
lemma (in topological_space) topological_tendstoI:
hoelzl@51641
   925
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f ---> l) F"
hoelzl@51471
   926
  unfolding tendsto_def by auto
hoelzl@51471
   927
hoelzl@51473
   928
lemma (in topological_space) topological_tendstoD:
hoelzl@51471
   929
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   930
  unfolding tendsto_def by auto
hoelzl@51471
   931
hoelzl@51471
   932
lemma order_tendstoI:
hoelzl@51471
   933
  fixes y :: "_ :: order_topology"
hoelzl@51471
   934
  assumes "\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
hoelzl@51471
   935
  assumes "\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
hoelzl@51471
   936
  shows "(f ---> y) F"
hoelzl@51471
   937
proof (rule topological_tendstoI)
hoelzl@51471
   938
  fix S assume "open S" "y \<in> S"
hoelzl@51471
   939
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   940
    unfolding open_generated_order
hoelzl@51471
   941
  proof induct
hoelzl@51471
   942
    case (UN K)
hoelzl@51471
   943
    then obtain k where "y \<in> k" "k \<in> K" by auto
hoelzl@51471
   944
    with UN(2)[of k] show ?case
hoelzl@51471
   945
      by (auto elim: eventually_elim1)
hoelzl@51471
   946
  qed (insert assms, auto elim: eventually_elim2)
hoelzl@51471
   947
qed
hoelzl@51471
   948
hoelzl@51471
   949
lemma order_tendstoD:
hoelzl@51471
   950
  fixes y :: "_ :: order_topology"
hoelzl@51471
   951
  assumes y: "(f ---> y) F"
hoelzl@51471
   952
  shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
hoelzl@51471
   953
    and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
hoelzl@51471
   954
  using topological_tendstoD[OF y, of "{..< a}"] topological_tendstoD[OF y, of "{a <..}"] by auto
hoelzl@51471
   955
hoelzl@51471
   956
lemma order_tendsto_iff: 
hoelzl@51471
   957
  fixes f :: "_ \<Rightarrow> 'a :: order_topology"
hoelzl@51471
   958
  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
   959
  by (metis order_tendstoI order_tendstoD)
hoelzl@51471
   960
hoelzl@51471
   961
lemma tendsto_bot [simp]: "(f ---> a) bot"
hoelzl@51471
   962
  unfolding tendsto_def by simp
hoelzl@51471
   963
hoelzl@51641
   964
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a within s)"
hoelzl@51471
   965
  unfolding tendsto_def eventually_at_topological by auto
hoelzl@51471
   966
hoelzl@51478
   967
lemma (in topological_space) tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
hoelzl@51471
   968
  by (simp add: tendsto_def)
hoelzl@51471
   969
hoelzl@51478
   970
lemma (in t2_space) tendsto_unique:
hoelzl@51471
   971
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
hoelzl@51471
   972
  shows "a = b"
hoelzl@51471
   973
proof (rule ccontr)
hoelzl@51471
   974
  assume "a \<noteq> b"
hoelzl@51471
   975
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
hoelzl@51471
   976
    using hausdorff [OF `a \<noteq> b`] by fast
hoelzl@51471
   977
  have "eventually (\<lambda>x. f x \<in> U) F"
hoelzl@51471
   978
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
hoelzl@51471
   979
  moreover
hoelzl@51471
   980
  have "eventually (\<lambda>x. f x \<in> V) F"
hoelzl@51471
   981
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
hoelzl@51471
   982
  ultimately
hoelzl@51471
   983
  have "eventually (\<lambda>x. False) F"
hoelzl@51471
   984
  proof eventually_elim
hoelzl@51471
   985
    case (elim x)
hoelzl@51471
   986
    hence "f x \<in> U \<inter> V" by simp
hoelzl@51471
   987
    with `U \<inter> V = {}` show ?case by simp
hoelzl@51471
   988
  qed
hoelzl@51471
   989
  with `\<not> trivial_limit F` show "False"
hoelzl@51471
   990
    by (simp add: trivial_limit_def)
hoelzl@51471
   991
qed
hoelzl@51471
   992
hoelzl@51478
   993
lemma (in t2_space) tendsto_const_iff:
hoelzl@51478
   994
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"
hoelzl@51471
   995
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
hoelzl@51471
   996
hoelzl@51471
   997
lemma increasing_tendsto:
hoelzl@51471
   998
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
   999
  assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
hoelzl@51471
  1000
      and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
hoelzl@51471
  1001
  shows "(f ---> l) F"
hoelzl@51471
  1002
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
hoelzl@51471
  1003
hoelzl@51471
  1004
lemma decreasing_tendsto:
hoelzl@51471
  1005
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
  1006
  assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
hoelzl@51471
  1007
      and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
hoelzl@51471
  1008
  shows "(f ---> l) F"
hoelzl@51471
  1009
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
hoelzl@51471
  1010
hoelzl@51471
  1011
lemma tendsto_sandwich:
hoelzl@51471
  1012
  fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
hoelzl@51471
  1013
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
hoelzl@51471
  1014
  assumes lim: "(f ---> c) net" "(h ---> c) net"
hoelzl@51471
  1015
  shows "(g ---> c) net"
hoelzl@51471
  1016
proof (rule order_tendstoI)
hoelzl@51471
  1017
  fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
hoelzl@51471
  1018
    using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
  1019
next
hoelzl@51471
  1020
  fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
hoelzl@51471
  1021
    using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
  1022
qed
hoelzl@51471
  1023
hoelzl@51471
  1024
lemma tendsto_le:
hoelzl@51471
  1025
  fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
  1026
  assumes F: "\<not> trivial_limit F"
hoelzl@51471
  1027
  assumes x: "(f ---> x) F" and y: "(g ---> y) F"
hoelzl@51471
  1028
  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
hoelzl@51471
  1029
  shows "y \<le> x"
hoelzl@51471
  1030
proof (rule ccontr)
hoelzl@51471
  1031
  assume "\<not> y \<le> x"
hoelzl@51471
  1032
  with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
hoelzl@51471
  1033
    by (auto simp: not_le)
hoelzl@51471
  1034
  then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
hoelzl@51471
  1035
    using x y by (auto intro: order_tendstoD)
hoelzl@51471
  1036
  with ev have "eventually (\<lambda>x. False) F"
hoelzl@51471
  1037
    by eventually_elim (insert xy, fastforce)
hoelzl@51471
  1038
  with F show False
hoelzl@51471
  1039
    by (simp add: eventually_False)
hoelzl@51471
  1040
qed
hoelzl@51471
  1041
hoelzl@51471
  1042
lemma tendsto_le_const:
hoelzl@51471
  1043
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
  1044
  assumes F: "\<not> trivial_limit F"
lp15@56289
  1045
  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F"
hoelzl@51471
  1046
  shows "a \<le> x"
hoelzl@51471
  1047
  using F x tendsto_const a by (rule tendsto_le)
hoelzl@51471
  1048
lp15@56289
  1049
lemma tendsto_ge_const:
lp15@56289
  1050
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
lp15@56289
  1051
  assumes F: "\<not> trivial_limit F"
lp15@56289
  1052
  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F"
lp15@56289
  1053
  shows "a \<ge> x"
lp15@56289
  1054
  by (rule tendsto_le [OF F tendsto_const x a])
lp15@56289
  1055
hoelzl@51478
  1056
subsubsection {* Rules about @{const Lim} *}
hoelzl@51478
  1057
hoelzl@51478
  1058
lemma (in t2_space) tendsto_Lim:
hoelzl@51478
  1059
  "\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l"
hoelzl@51478
  1060
  unfolding Lim_def using tendsto_unique[of net f] by auto
hoelzl@51478
  1061
hoelzl@51641
  1062
lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
hoelzl@51478
  1063
  by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
hoelzl@51478
  1064
hoelzl@51471
  1065
subsection {* Limits to @{const at_top} and @{const at_bot} *}
hoelzl@51471
  1066
hoelzl@51471
  1067
lemma filterlim_at_top:
hoelzl@51471
  1068
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@51471
  1069
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1070
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
hoelzl@51471
  1071
hoelzl@51471
  1072
lemma filterlim_at_top_dense:
hoelzl@53215
  1073
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
hoelzl@51471
  1074
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@51471
  1075
  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
hoelzl@51471
  1076
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
hoelzl@51471
  1077
hoelzl@51471
  1078
lemma filterlim_at_top_ge:
hoelzl@51471
  1079
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@51471
  1080
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1081
  unfolding filterlim_at_top
hoelzl@51471
  1082
proof safe
hoelzl@51471
  1083
  fix Z assume *: "\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F"
hoelzl@51471
  1084
  with *[THEN spec, of "max Z c"] show "eventually (\<lambda>x. Z \<le> f x) F"
hoelzl@51471
  1085
    by (auto elim!: eventually_elim1)
hoelzl@51471
  1086
qed simp
hoelzl@51471
  1087
hoelzl@51471
  1088
lemma filterlim_at_top_at_top:
hoelzl@51471
  1089
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
hoelzl@51471
  1090
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1091
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1092
  assumes Q: "eventually Q at_top"
hoelzl@51471
  1093
  assumes P: "eventually P at_top"
hoelzl@51471
  1094
  shows "filterlim f at_top at_top"
hoelzl@51471
  1095
proof -
hoelzl@51471
  1096
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
  1097
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
  1098
  show ?thesis
hoelzl@51471
  1099
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
  1100
    fix z assume "x \<le> z"
hoelzl@51471
  1101
    with x have "P z" by auto
hoelzl@51471
  1102
    have "eventually (\<lambda>x. g z \<le> x) at_top"
hoelzl@51471
  1103
      by (rule eventually_ge_at_top)
hoelzl@51471
  1104
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
hoelzl@51471
  1105
      by eventually_elim (metis mono bij `P z`)
hoelzl@51471
  1106
  qed
hoelzl@51471
  1107
qed
hoelzl@51471
  1108
hoelzl@51471
  1109
lemma filterlim_at_top_gt:
hoelzl@53215
  1110
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@51471
  1111
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1112
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
hoelzl@51471
  1113
hoelzl@51471
  1114
lemma filterlim_at_bot: 
hoelzl@51471
  1115
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@51471
  1116
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
hoelzl@51471
  1117
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
hoelzl@51471
  1118
hoelzl@51471
  1119
lemma filterlim_at_bot_le:
hoelzl@51471
  1120
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@51471
  1121
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@51471
  1122
  unfolding filterlim_at_bot
hoelzl@51471
  1123
proof safe
hoelzl@51471
  1124
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@51471
  1125
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@51471
  1126
    by (auto elim!: eventually_elim1)
hoelzl@51471
  1127
qed simp
hoelzl@51471
  1128
hoelzl@51471
  1129
lemma filterlim_at_bot_lt:
hoelzl@53215
  1130
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@51471
  1131
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@51471
  1132
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
hoelzl@51471
  1133
hoelzl@51471
  1134
lemma filterlim_at_bot_at_right:
hoelzl@51471
  1135
  fixes f :: "'a::{no_top, linorder_topology} \<Rightarrow> 'b::linorder"
hoelzl@51471
  1136
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1137
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1138
  assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
hoelzl@51471
  1139
  assumes P: "eventually P at_bot"
hoelzl@51471
  1140
  shows "filterlim f at_bot (at_right a)"
hoelzl@51471
  1141
proof -
hoelzl@51471
  1142
  from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
hoelzl@51471
  1143
    unfolding eventually_at_bot_linorder by auto
hoelzl@51471
  1144
  show ?thesis
hoelzl@51471
  1145
  proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
hoelzl@51471
  1146
    fix z assume "z \<le> x"
hoelzl@51471
  1147
    with x have "P z" by auto
hoelzl@51471
  1148
    have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
hoelzl@51471
  1149
      using bound[OF bij(2)[OF `P z`]]
hoelzl@51471
  1150
      unfolding eventually_at_right by (auto intro!: exI[of _ "g z"])
hoelzl@51471
  1151
    with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
hoelzl@51471
  1152
      by eventually_elim (metis bij `P z` mono)
hoelzl@51471
  1153
  qed
hoelzl@51471
  1154
qed
hoelzl@51471
  1155
hoelzl@51471
  1156
lemma filterlim_at_top_at_left:
hoelzl@51471
  1157
  fixes f :: "'a::{no_bot, linorder_topology} \<Rightarrow> 'b::linorder"
hoelzl@51471
  1158
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1159
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1160
  assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
hoelzl@51471
  1161
  assumes P: "eventually P at_top"
hoelzl@51471
  1162
  shows "filterlim f at_top (at_left a)"
hoelzl@51471
  1163
proof -
hoelzl@51471
  1164
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
  1165
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
  1166
  show ?thesis
hoelzl@51471
  1167
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
  1168
    fix z assume "x \<le> z"
hoelzl@51471
  1169
    with x have "P z" by auto
hoelzl@51471
  1170
    have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
hoelzl@51471
  1171
      using bound[OF bij(2)[OF `P z`]]
hoelzl@51471
  1172
      unfolding eventually_at_left by (auto intro!: exI[of _ "g z"])
hoelzl@51471
  1173
    with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
hoelzl@51471
  1174
      by eventually_elim (metis bij `P z` mono)
hoelzl@51471
  1175
  qed
hoelzl@51471
  1176
qed
hoelzl@51471
  1177
hoelzl@51471
  1178
lemma filterlim_split_at:
hoelzl@51471
  1179
  "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"
hoelzl@51471
  1180
  by (subst at_eq_sup_left_right) (rule filterlim_sup)
hoelzl@51471
  1181
hoelzl@51471
  1182
lemma filterlim_at_split:
hoelzl@51471
  1183
  "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
hoelzl@51471
  1184
  by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
hoelzl@51471
  1185
hoelzl@51471
  1186
hoelzl@51471
  1187
subsection {* Limits on sequences *}
hoelzl@51471
  1188
hoelzl@51471
  1189
abbreviation (in topological_space)
hoelzl@51471
  1190
  LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
hoelzl@51471
  1191
    ("((_)/ ----> (_))" [60, 60] 60) where
hoelzl@51471
  1192
  "X ----> L \<equiv> (X ---> L) sequentially"
hoelzl@51471
  1193
hoelzl@51478
  1194
abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
  1195
  "lim X \<equiv> Lim sequentially X"
hoelzl@51471
  1196
hoelzl@51471
  1197
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
hoelzl@51471
  1198
  "convergent X = (\<exists>L. X ----> L)"
hoelzl@51471
  1199
hoelzl@51478
  1200
lemma lim_def: "lim X = (THE L. X ----> L)"
hoelzl@51478
  1201
  unfolding Lim_def ..
hoelzl@51478
  1202
hoelzl@51471
  1203
subsubsection {* Monotone sequences and subsequences *}
hoelzl@51471
  1204
hoelzl@51471
  1205
definition
hoelzl@51471
  1206
  monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@51471
  1207
    --{*Definition of monotonicity.
hoelzl@51471
  1208
        The use of disjunction here complicates proofs considerably.
hoelzl@51471
  1209
        One alternative is to add a Boolean argument to indicate the direction.
hoelzl@51471
  1210
        Another is to develop the notions of increasing and decreasing first.*}
hoelzl@56020
  1211
  "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
hoelzl@56020
  1212
hoelzl@56020
  1213
abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@56020
  1214
  "incseq X \<equiv> mono X"
hoelzl@56020
  1215
hoelzl@56020
  1216
lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)"
hoelzl@56020
  1217
  unfolding mono_def ..
hoelzl@56020
  1218
hoelzl@56020
  1219
abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@56020
  1220
  "decseq X \<equiv> antimono X"
hoelzl@56020
  1221
hoelzl@56020
  1222
lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
hoelzl@56020
  1223
  unfolding antimono_def ..
hoelzl@51471
  1224
hoelzl@51471
  1225
definition
hoelzl@51471
  1226
  subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
hoelzl@51471
  1227
    --{*Definition of subsequence*}
hoelzl@51471
  1228
  "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
hoelzl@51471
  1229
hoelzl@51471
  1230
lemma incseq_SucI:
hoelzl@51471
  1231
  "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
hoelzl@51471
  1232
  using lift_Suc_mono_le[of X]
hoelzl@51471
  1233
  by (auto simp: incseq_def)
hoelzl@51471
  1234
hoelzl@51471
  1235
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
hoelzl@51471
  1236
  by (auto simp: incseq_def)
hoelzl@51471
  1237
hoelzl@51471
  1238
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
hoelzl@51471
  1239
  using incseqD[of A i "Suc i"] by auto
hoelzl@51471
  1240
hoelzl@51471
  1241
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
hoelzl@51471
  1242
  by (auto intro: incseq_SucI dest: incseq_SucD)
hoelzl@51471
  1243
hoelzl@51471
  1244
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
hoelzl@51471
  1245
  unfolding incseq_def by auto
hoelzl@51471
  1246
hoelzl@51471
  1247
lemma decseq_SucI:
hoelzl@51471
  1248
  "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
hoelzl@51471
  1249
  using order.lift_Suc_mono_le[OF dual_order, of X]
hoelzl@51471
  1250
  by (auto simp: decseq_def)
hoelzl@51471
  1251
hoelzl@51471
  1252
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
hoelzl@51471
  1253
  by (auto simp: decseq_def)
hoelzl@51471
  1254
hoelzl@51471
  1255
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
hoelzl@51471
  1256
  using decseqD[of A i "Suc i"] by auto
hoelzl@51471
  1257
hoelzl@51471
  1258
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@51471
  1259
  by (auto intro: decseq_SucI dest: decseq_SucD)
hoelzl@51471
  1260
hoelzl@51471
  1261
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
hoelzl@51471
  1262
  unfolding decseq_def by auto
hoelzl@51471
  1263
hoelzl@51471
  1264
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
hoelzl@51471
  1265
  unfolding monoseq_def incseq_def decseq_def ..
hoelzl@51471
  1266
hoelzl@51471
  1267
lemma monoseq_Suc:
hoelzl@51471
  1268
  "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
hoelzl@51471
  1269
  unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
hoelzl@51471
  1270
hoelzl@51471
  1271
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
hoelzl@51471
  1272
by (simp add: monoseq_def)
hoelzl@51471
  1273
hoelzl@51471
  1274
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
hoelzl@51471
  1275
by (simp add: monoseq_def)
hoelzl@51471
  1276
hoelzl@51471
  1277
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
hoelzl@51471
  1278
by (simp add: monoseq_Suc)
hoelzl@51471
  1279
hoelzl@51471
  1280
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
hoelzl@51471
  1281
by (simp add: monoseq_Suc)
hoelzl@51471
  1282
hoelzl@51471
  1283
lemma monoseq_minus:
hoelzl@51471
  1284
  fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
hoelzl@51471
  1285
  assumes "monoseq a"
hoelzl@51471
  1286
  shows "monoseq (\<lambda> n. - a n)"
hoelzl@51471
  1287
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
hoelzl@51471
  1288
  case True
hoelzl@51471
  1289
  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
hoelzl@51471
  1290
  thus ?thesis by (rule monoI2)
hoelzl@51471
  1291
next
hoelzl@51471
  1292
  case False
hoelzl@51471
  1293
  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
hoelzl@51471
  1294
  thus ?thesis by (rule monoI1)
hoelzl@51471
  1295
qed
hoelzl@51471
  1296
hoelzl@51471
  1297
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
hoelzl@51471
  1298
hoelzl@51471
  1299
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
hoelzl@51471
  1300
apply (simp add: subseq_def)
hoelzl@51471
  1301
apply (auto dest!: less_imp_Suc_add)
hoelzl@51471
  1302
apply (induct_tac k)
hoelzl@51471
  1303
apply (auto intro: less_trans)
hoelzl@51471
  1304
done
hoelzl@51471
  1305
hoelzl@51471
  1306
text{* for any sequence, there is a monotonic subsequence *}
hoelzl@51471
  1307
lemma seq_monosub:
hoelzl@51471
  1308
  fixes s :: "nat => 'a::linorder"
hoelzl@51471
  1309
  shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
hoelzl@51471
  1310
proof cases
hoelzl@51471
  1311
  let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"
hoelzl@51471
  1312
  assume *: "\<forall>n. \<exists>p. ?P p n"
blanchet@55415
  1313
  def f \<equiv> "rec_nat (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"
hoelzl@51471
  1314
  have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
blanchet@55642
  1315
  have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat.rec(2) ..
hoelzl@51471
  1316
  have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
hoelzl@51471
  1317
  have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
hoelzl@51471
  1318
  then have "subseq f" unfolding subseq_Suc_iff by auto
hoelzl@51471
  1319
  moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc
hoelzl@51471
  1320
  proof (intro disjI2 allI)
hoelzl@51471
  1321
    fix n show "s (f (Suc n)) \<le> s (f n)"
hoelzl@51471
  1322
    proof (cases n)
hoelzl@51471
  1323
      case 0 with P_Suc[of 0] P_0 show ?thesis by auto
hoelzl@51471
  1324
    next
hoelzl@51471
  1325
      case (Suc m)
hoelzl@51471
  1326
      from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp
hoelzl@51471
  1327
      with P_Suc Suc show ?thesis by simp
hoelzl@51471
  1328
    qed
hoelzl@51471
  1329
  qed
hoelzl@51471
  1330
  ultimately show ?thesis by auto
hoelzl@51471
  1331
next
hoelzl@51471
  1332
  let "?P p m" = "m < p \<and> s m < s p"
hoelzl@51471
  1333
  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
hoelzl@51471
  1334
  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
blanchet@55415
  1335
  def f \<equiv> "rec_nat (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"
hoelzl@51471
  1336
  have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
blanchet@55642
  1337
  have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat.rec(2) ..
hoelzl@51471
  1338
  have P_0: "?P (f 0) (Suc N)"
hoelzl@51471
  1339
    unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto
hoelzl@51471
  1340
  { fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"
hoelzl@51471
  1341
      unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }
hoelzl@51471
  1342
  note P' = this
hoelzl@51471
  1343
  { fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"
hoelzl@51471
  1344
      by (induct i) (insert P_0 P', auto) }
hoelzl@51471
  1345
  then have "subseq f" "monoseq (\<lambda>x. s (f x))"
hoelzl@51471
  1346
    unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
hoelzl@51471
  1347
  then show ?thesis by auto
hoelzl@51471
  1348
qed
hoelzl@51471
  1349
hoelzl@51471
  1350
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
hoelzl@51471
  1351
proof(induct n)
hoelzl@51471
  1352
  case 0 thus ?case by simp
hoelzl@51471
  1353
next
hoelzl@51471
  1354
  case (Suc n)
hoelzl@51471
  1355
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
hoelzl@51471
  1356
  have "n < f (Suc n)" by arith
hoelzl@51471
  1357
  thus ?case by arith
hoelzl@51471
  1358
qed
hoelzl@51471
  1359
hoelzl@51471
  1360
lemma eventually_subseq:
hoelzl@51471
  1361
  "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
hoelzl@51471
  1362
  unfolding eventually_sequentially by (metis seq_suble le_trans)
hoelzl@51471
  1363
hoelzl@51473
  1364
lemma not_eventually_sequentiallyD:
hoelzl@51473
  1365
  assumes P: "\<not> eventually P sequentially"
hoelzl@51473
  1366
  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
hoelzl@51473
  1367
proof -
hoelzl@51473
  1368
  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
hoelzl@51473
  1369
    unfolding eventually_sequentially by (simp add: not_less)
hoelzl@51473
  1370
  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
hoelzl@51473
  1371
    by (auto simp: choice_iff)
hoelzl@51473
  1372
  then show ?thesis
hoelzl@51473
  1373
    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
hoelzl@51473
  1374
             simp: less_eq_Suc_le subseq_Suc_iff)
hoelzl@51473
  1375
qed
hoelzl@51473
  1376
hoelzl@51471
  1377
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
hoelzl@51471
  1378
  unfolding filterlim_iff by (metis eventually_subseq)
hoelzl@51471
  1379
hoelzl@51471
  1380
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
hoelzl@51471
  1381
  unfolding subseq_def by simp
hoelzl@51471
  1382
hoelzl@51471
  1383
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
hoelzl@51471
  1384
  using assms by (auto simp: subseq_def)
hoelzl@51471
  1385
hoelzl@51471
  1386
lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
hoelzl@51471
  1387
  by (simp add: incseq_def monoseq_def)
hoelzl@51471
  1388
hoelzl@51471
  1389
lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
hoelzl@51471
  1390
  by (simp add: decseq_def monoseq_def)
hoelzl@51471
  1391
hoelzl@51471
  1392
lemma decseq_eq_incseq:
hoelzl@51471
  1393
  fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" 
hoelzl@51471
  1394
  by (simp add: decseq_def incseq_def)
hoelzl@51471
  1395
hoelzl@51471
  1396
lemma INT_decseq_offset:
hoelzl@51471
  1397
  assumes "decseq F"
hoelzl@51471
  1398
  shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
  1399
proof safe
hoelzl@51471
  1400
  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
  1401
  show "x \<in> F i"
hoelzl@51471
  1402
  proof cases
hoelzl@51471
  1403
    from x have "x \<in> F n" by auto
hoelzl@51471
  1404
    also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
hoelzl@51471
  1405
      unfolding decseq_def by simp
hoelzl@51471
  1406
    finally show ?thesis .
hoelzl@51471
  1407
  qed (insert x, simp)
hoelzl@51471
  1408
qed auto
hoelzl@51471
  1409
hoelzl@51471
  1410
lemma LIMSEQ_const_iff:
hoelzl@51471
  1411
  fixes k l :: "'a::t2_space"
hoelzl@51471
  1412
  shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
hoelzl@51471
  1413
  using trivial_limit_sequentially by (rule tendsto_const_iff)
hoelzl@51471
  1414
hoelzl@51471
  1415
lemma LIMSEQ_SUP:
hoelzl@51471
  1416
  "incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1417
  by (intro increasing_tendsto)
hoelzl@51471
  1418
     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
hoelzl@51471
  1419
hoelzl@51471
  1420
lemma LIMSEQ_INF:
hoelzl@51471
  1421
  "decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1422
  by (intro decreasing_tendsto)
hoelzl@51471
  1423
     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
hoelzl@51471
  1424
hoelzl@51471
  1425
lemma LIMSEQ_ignore_initial_segment:
hoelzl@51471
  1426
  "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
hoelzl@51474
  1427
  unfolding tendsto_def
hoelzl@51474
  1428
  by (subst eventually_sequentially_seg[where k=k])
hoelzl@51471
  1429
hoelzl@51471
  1430
lemma LIMSEQ_offset:
hoelzl@51471
  1431
  "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
hoelzl@51474
  1432
  unfolding tendsto_def
hoelzl@51474
  1433
  by (subst (asm) eventually_sequentially_seg[where k=k])
hoelzl@51471
  1434
hoelzl@51471
  1435
lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
hoelzl@51471
  1436
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
hoelzl@51471
  1437
hoelzl@51471
  1438
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
hoelzl@51471
  1439
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
hoelzl@51471
  1440
hoelzl@51471
  1441
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
hoelzl@51471
  1442
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
hoelzl@51471
  1443
hoelzl@51471
  1444
lemma LIMSEQ_unique:
hoelzl@51471
  1445
  fixes a b :: "'a::t2_space"
hoelzl@51471
  1446
  shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
hoelzl@51471
  1447
  using trivial_limit_sequentially by (rule tendsto_unique)
hoelzl@51471
  1448
hoelzl@51471
  1449
lemma LIMSEQ_le_const:
hoelzl@51471
  1450
  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
hoelzl@51471
  1451
  using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
hoelzl@51471
  1452
hoelzl@51471
  1453
lemma LIMSEQ_le:
hoelzl@51471
  1454
  "\<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
  1455
  using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
hoelzl@51471
  1456
hoelzl@51471
  1457
lemma LIMSEQ_le_const2:
hoelzl@51471
  1458
  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
hoelzl@51471
  1459
  by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) (auto simp: tendsto_const)
hoelzl@51471
  1460
hoelzl@51471
  1461
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
hoelzl@51471
  1462
by (simp add: convergent_def)
hoelzl@51471
  1463
hoelzl@51471
  1464
lemma convergentI: "(X ----> L) ==> convergent X"
hoelzl@51471
  1465
by (auto simp add: convergent_def)
hoelzl@51471
  1466
hoelzl@51471
  1467
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
hoelzl@51471
  1468
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
hoelzl@51471
  1469
hoelzl@51471
  1470
lemma convergent_const: "convergent (\<lambda>n. c)"
hoelzl@51471
  1471
  by (rule convergentI, rule tendsto_const)
hoelzl@51471
  1472
hoelzl@51471
  1473
lemma monoseq_le:
hoelzl@51471
  1474
  "monoseq a \<Longrightarrow> a ----> (x::'a::linorder_topology) \<Longrightarrow>
hoelzl@51471
  1475
    ((\<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
  1476
  by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
hoelzl@51471
  1477
hoelzl@51471
  1478
lemma LIMSEQ_subseq_LIMSEQ:
hoelzl@51471
  1479
  "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
hoelzl@51471
  1480
  unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
hoelzl@51471
  1481
hoelzl@51471
  1482
lemma convergent_subseq_convergent:
hoelzl@51471
  1483
  "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
hoelzl@51471
  1484
  unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
hoelzl@51471
  1485
hoelzl@51471
  1486
lemma limI: "X ----> L ==> lim X = L"
hoelzl@51471
  1487
apply (simp add: lim_def)
hoelzl@51471
  1488
apply (blast intro: LIMSEQ_unique)
hoelzl@51471
  1489
done
hoelzl@51471
  1490
hoelzl@51471
  1491
lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"
hoelzl@51471
  1492
  using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
hoelzl@51471
  1493
hoelzl@51471
  1494
subsubsection{*Increasing and Decreasing Series*}
hoelzl@51471
  1495
hoelzl@51471
  1496
lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
hoelzl@51471
  1497
  by (metis incseq_def LIMSEQ_le_const)
hoelzl@51471
  1498
hoelzl@51471
  1499
lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
hoelzl@51471
  1500
  by (metis decseq_def LIMSEQ_le_const2)
hoelzl@51471
  1501
hoelzl@51473
  1502
subsection {* First countable topologies *}
hoelzl@51473
  1503
hoelzl@51473
  1504
class first_countable_topology = topological_space +
hoelzl@51473
  1505
  assumes first_countable_basis:
hoelzl@51473
  1506
    "\<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
  1507
hoelzl@51473
  1508
lemma (in first_countable_topology) countable_basis_at_decseq:
hoelzl@51473
  1509
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1510
    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
hoelzl@51473
  1511
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
hoelzl@51473
  1512
proof atomize_elim
hoelzl@51473
  1513
  from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1514
    nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1515
    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto
hoelzl@51473
  1516
  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i"
hoelzl@51473
  1517
  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
hoelzl@51473
  1518
      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
hoelzl@51473
  1519
  proof (safe intro!: exI[of _ F])
hoelzl@51473
  1520
    fix i
hoelzl@51480
  1521
    show "open (F i)" using nhds(1) by (auto simp: F_def)
hoelzl@51473
  1522
    show "x \<in> F i" using nhds(2) by (auto simp: F_def)
hoelzl@51473
  1523
  next
hoelzl@51473
  1524
    fix S assume "open S" "x \<in> S"
hoelzl@51473
  1525
    from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto
hoelzl@51473
  1526
    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
hoelzl@51473
  1527
      by (auto simp: F_def)
hoelzl@51473
  1528
    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
hoelzl@51473
  1529
      by (auto simp: eventually_sequentially)
hoelzl@51473
  1530
  qed
hoelzl@51473
  1531
qed
hoelzl@51473
  1532
hoelzl@51473
  1533
lemma (in first_countable_topology) countable_basis:
hoelzl@51473
  1534
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1535
    "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1536
    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
hoelzl@51473
  1537
proof atomize_elim
wenzelm@53381
  1538
  obtain A :: "nat \<Rightarrow> 'a set" where A:
wenzelm@53381
  1539
    "\<And>i. open (A i)"
wenzelm@53381
  1540
    "\<And>i. x \<in> A i"
wenzelm@53381
  1541
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
wenzelm@53381
  1542
    by (rule countable_basis_at_decseq) blast
wenzelm@53381
  1543
  {
wenzelm@53381
  1544
    fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
hoelzl@51473
  1545
    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
wenzelm@53381
  1546
      by (auto elim: eventually_elim1 simp: subset_eq)
wenzelm@53381
  1547
  }
hoelzl@51473
  1548
  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
  1549
    by (intro exI[of _ A]) (auto simp: tendsto_def)
hoelzl@51473
  1550
qed
hoelzl@51473
  1551
hoelzl@51473
  1552
lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
hoelzl@51473
  1553
  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51641
  1554
  shows "eventually P (inf (nhds a) (principal s))"
hoelzl@51473
  1555
proof (rule ccontr)
wenzelm@53381
  1556
  obtain A :: "nat \<Rightarrow> 'a set" where A:
wenzelm@53381
  1557
    "\<And>i. open (A i)"
wenzelm@53381
  1558
    "\<And>i. a \<in> A i"
wenzelm@53381
  1559
    "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F ----> a"
wenzelm@53381
  1560
    by (rule countable_basis) blast
wenzelm@53381
  1561
  assume "\<not> ?thesis"
hoelzl@51473
  1562
  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
  1563
    unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce
wenzelm@53381
  1564
  then obtain F where F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
wenzelm@53381
  1565
    by blast
hoelzl@51473
  1566
  with A have "F ----> a" by auto
hoelzl@51473
  1567
  hence "eventually (\<lambda>n. P (F n)) sequentially"
hoelzl@51473
  1568
    using assms F0 by simp
hoelzl@51473
  1569
  thus "False" by (simp add: F3)
hoelzl@51473
  1570
qed
hoelzl@51473
  1571
hoelzl@51473
  1572
lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
hoelzl@51641
  1573
  "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> 
hoelzl@51473
  1574
    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1575
proof (safe intro!: sequentially_imp_eventually_nhds_within)
hoelzl@51641
  1576
  assume "eventually P (inf (nhds a) (principal s))" 
hoelzl@51473
  1577
  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
hoelzl@51641
  1578
    by (auto simp: eventually_inf_principal eventually_nhds)
hoelzl@51473
  1579
  moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
hoelzl@51473
  1580
  ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51473
  1581
    by (auto dest!: topological_tendstoD elim: eventually_elim1)
hoelzl@51473
  1582
qed
hoelzl@51473
  1583
hoelzl@51473
  1584
lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
hoelzl@51473
  1585
  "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1586
  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
hoelzl@51473
  1587
hoelzl@51471
  1588
subsection {* Function limit at a point *}
hoelzl@51471
  1589
hoelzl@51471
  1590
abbreviation
hoelzl@51471
  1591
  LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
hoelzl@51471
  1592
        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
hoelzl@51471
  1593
  "f -- a --> L \<equiv> (f ---> L) (at a)"
hoelzl@51471
  1594
hoelzl@51481
  1595
lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l) (at a within S) \<longleftrightarrow> (f -- a --> l)"
hoelzl@51641
  1596
  unfolding tendsto_def by (simp add: at_within_open[where S=S])
hoelzl@51481
  1597
hoelzl@51471
  1598
lemma LIM_const_not_eq[tendsto_intros]:
hoelzl@51471
  1599
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1600
  fixes k L :: "'b::t2_space"
hoelzl@51471
  1601
  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
hoelzl@51471
  1602
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1603
hoelzl@51471
  1604
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
hoelzl@51471
  1605
hoelzl@51471
  1606
lemma LIM_const_eq:
hoelzl@51471
  1607
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1608
  fixes k L :: "'b::t2_space"
hoelzl@51471
  1609
  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
hoelzl@51471
  1610
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1611
hoelzl@51471
  1612
lemma LIM_unique:
hoelzl@51471
  1613
  fixes a :: "'a::perfect_space" and L M :: "'b::t2_space"
hoelzl@51471
  1614
  shows "f -- a --> L \<Longrightarrow> f -- a --> M \<Longrightarrow> L = M"
hoelzl@51471
  1615
  using at_neq_bot by (rule tendsto_unique)
hoelzl@51471
  1616
hoelzl@51471
  1617
text {* Limits are equal for functions equal except at limit point *}
hoelzl@51471
  1618
hoelzl@51471
  1619
lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- a --> l)"
hoelzl@51471
  1620
  unfolding tendsto_def eventually_at_topological by simp
hoelzl@51471
  1621
hoelzl@51471
  1622
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
  1623
  by (simp add: LIM_equal)
hoelzl@51471
  1624
hoelzl@51471
  1625
lemma LIM_cong_limit: "f -- x --> L \<Longrightarrow> K = L \<Longrightarrow> f -- x --> K"
hoelzl@51471
  1626
  by simp
hoelzl@51471
  1627
hoelzl@51471
  1628
lemma tendsto_at_iff_tendsto_nhds:
hoelzl@51471
  1629
  "g -- l --> g l \<longleftrightarrow> (g ---> g l) (nhds l)"
hoelzl@51641
  1630
  unfolding tendsto_def eventually_at_filter
hoelzl@51471
  1631
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@51471
  1632
hoelzl@51471
  1633
lemma tendsto_compose:
hoelzl@51471
  1634
  "g -- l --> g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@51471
  1635
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
hoelzl@51471
  1636
hoelzl@51471
  1637
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
hoelzl@51471
  1638
  unfolding o_def by (rule tendsto_compose)
hoelzl@51471
  1639
hoelzl@51471
  1640
lemma tendsto_compose_eventually:
hoelzl@51471
  1641
  "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
  1642
  by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
hoelzl@51471
  1643
hoelzl@51471
  1644
lemma LIM_compose_eventually:
hoelzl@51471
  1645
  assumes f: "f -- a --> b"
hoelzl@51471
  1646
  assumes g: "g -- b --> c"
hoelzl@51471
  1647
  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
hoelzl@51471
  1648
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51471
  1649
  using g f inj by (rule tendsto_compose_eventually)
hoelzl@51471
  1650
hoelzl@51473
  1651
subsubsection {* Relation of LIM and LIMSEQ *}
hoelzl@51473
  1652
hoelzl@51473
  1653
lemma (in first_countable_topology) sequentially_imp_eventually_within:
hoelzl@51473
  1654
  "(\<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
  1655
    eventually P (at a within s)"
hoelzl@51641
  1656
  unfolding at_within_def
hoelzl@51473
  1657
  by (intro sequentially_imp_eventually_nhds_within) auto
hoelzl@51473
  1658
hoelzl@51473
  1659
lemma (in first_countable_topology) sequentially_imp_eventually_at:
hoelzl@51473
  1660
  "(\<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
  1661
  using assms sequentially_imp_eventually_within [where s=UNIV] by simp
hoelzl@51473
  1662
hoelzl@51473
  1663
lemma LIMSEQ_SEQ_conv1:
hoelzl@51473
  1664
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
hoelzl@51473
  1665
  assumes f: "f -- a --> l"
hoelzl@51473
  1666
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
hoelzl@51473
  1667
  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
hoelzl@51473
  1668
hoelzl@51473
  1669
lemma LIMSEQ_SEQ_conv2:
hoelzl@51473
  1670
  fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
hoelzl@51473
  1671
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
hoelzl@51473
  1672
  shows "f -- a --> l"
hoelzl@51473
  1673
  using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
hoelzl@51473
  1674
hoelzl@51473
  1675
lemma LIMSEQ_SEQ_conv:
hoelzl@51473
  1676
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
hoelzl@51473
  1677
   (X -- a --> (L::'b::topological_space))"
hoelzl@51473
  1678
  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
hoelzl@51473
  1679
hoelzl@51471
  1680
subsection {* Continuity *}
hoelzl@51471
  1681
hoelzl@51478
  1682
subsubsection {* Continuity on a set *}
hoelzl@51478
  1683
hoelzl@51478
  1684
definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where
hoelzl@51478
  1685
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
hoelzl@51478
  1686
hoelzl@51481
  1687
lemma continuous_on_cong [cong]:
hoelzl@51481
  1688
  "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
  1689
  unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
hoelzl@51481
  1690
hoelzl@51478
  1691
lemma continuous_on_topological:
hoelzl@51478
  1692
  "continuous_on s f \<longleftrightarrow>
hoelzl@51478
  1693
    (\<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
  1694
  unfolding continuous_on_def tendsto_def eventually_at_topological by metis
hoelzl@51478
  1695
hoelzl@51478
  1696
lemma continuous_on_open_invariant:
hoelzl@51478
  1697
  "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  1698
proof safe
hoelzl@51478
  1699
  fix B :: "'b set" assume "continuous_on s f" "open B"
hoelzl@51478
  1700
  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
  1701
    by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
wenzelm@53381
  1702
  then obtain A where "\<forall>x\<in>f -` B \<inter> s. open (A x) \<and> x \<in> A x \<and> s \<inter> A x \<subseteq> f -` B"
wenzelm@53381
  1703
    unfolding bchoice_iff ..
hoelzl@51478
  1704
  then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"
hoelzl@51478
  1705
    by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto
hoelzl@51478
  1706
next
hoelzl@51478
  1707
  assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"
hoelzl@51478
  1708
  show "continuous_on s f"
hoelzl@51478
  1709
    unfolding continuous_on_topological
hoelzl@51478
  1710
  proof safe
hoelzl@51478
  1711
    fix x B assume "x \<in> s" "open B" "f x \<in> B"
hoelzl@51478
  1712
    with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto
hoelzl@51478
  1713
    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
  1714
      by (intro exI[of _ A]) auto
hoelzl@51478
  1715
  qed
hoelzl@51478
  1716
qed
hoelzl@51478
  1717
hoelzl@51481
  1718
lemma continuous_on_open_vimage:
hoelzl@51481
  1719
  "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
hoelzl@51481
  1720
  unfolding continuous_on_open_invariant
hoelzl@51481
  1721
  by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  1722
lp15@55734
  1723
corollary continuous_imp_open_vimage:
lp15@55734
  1724
  assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s"
lp15@55734
  1725
    shows "open (f -` B)"
lp15@55734
  1726
by (metis assms continuous_on_open_vimage le_iff_inf)
lp15@55734
  1727
lp15@55775
  1728
corollary open_vimage:
lp15@55775
  1729
  assumes "open s" and "continuous_on UNIV f"
lp15@55775
  1730
  shows "open (f -` s)"
lp15@55775
  1731
  using assms unfolding continuous_on_open_vimage [OF open_UNIV]
lp15@55775
  1732
  by simp
lp15@55775
  1733
hoelzl@51478
  1734
lemma continuous_on_closed_invariant:
hoelzl@51478
  1735
  "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  1736
proof -
hoelzl@51478
  1737
  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
  1738
    by (metis double_compl)
hoelzl@51478
  1739
  show ?thesis
hoelzl@51478
  1740
    unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric])
hoelzl@51478
  1741
qed
hoelzl@51478
  1742
hoelzl@51481
  1743
lemma continuous_on_closed_vimage:
hoelzl@51481
  1744
  "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
hoelzl@51481
  1745
  unfolding continuous_on_closed_invariant
hoelzl@51481
  1746
  by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  1747
hoelzl@51481
  1748
lemma continuous_on_open_Union:
hoelzl@51481
  1749
  "(\<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
  1750
  unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI)
hoelzl@51481
  1751
hoelzl@51481
  1752
lemma continuous_on_open_UN:
hoelzl@51481
  1753
  "(\<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
  1754
  unfolding Union_image_eq[symmetric] by (rule continuous_on_open_Union) auto
hoelzl@51481
  1755
hoelzl@51481
  1756
lemma continuous_on_closed_Un:
hoelzl@51481
  1757
  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
hoelzl@51481
  1758
  by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
hoelzl@51481
  1759
hoelzl@51481
  1760
lemma continuous_on_If:
hoelzl@51481
  1761
  assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g"
hoelzl@51481
  1762
    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
  1763
  shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h")
hoelzl@51481
  1764
proof-
hoelzl@51481
  1765
  from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
hoelzl@51481
  1766
    by auto
hoelzl@51481
  1767
  with cont have "continuous_on s ?h" "continuous_on t ?h"
hoelzl@51481
  1768
    by simp_all
hoelzl@51481
  1769
  with closed show ?thesis
hoelzl@51481
  1770
    by (rule continuous_on_closed_Un)
hoelzl@51481
  1771
qed
hoelzl@51481
  1772
hoelzl@51478
  1773
ML {*
hoelzl@51478
  1774
hoelzl@51478
  1775
structure Continuous_On_Intros = Named_Thms
hoelzl@51478
  1776
(
hoelzl@51478
  1777
  val name = @{binding continuous_on_intros}
hoelzl@51478
  1778
  val description = "Structural introduction rules for setwise continuity"
hoelzl@51478
  1779
)
hoelzl@51478
  1780
hoelzl@51478
  1781
*}
hoelzl@51478
  1782
hoelzl@51478
  1783
setup Continuous_On_Intros.setup
hoelzl@51478
  1784
hoelzl@51478
  1785
lemma continuous_on_id[continuous_on_intros]: "continuous_on s (\<lambda>x. x)"
hoelzl@51641
  1786
  unfolding continuous_on_def by (fast intro: tendsto_ident_at)
hoelzl@51478
  1787
hoelzl@51478
  1788
lemma continuous_on_const[continuous_on_intros]: "continuous_on s (\<lambda>x. c)"
hoelzl@51478
  1789
  unfolding continuous_on_def by (auto intro: tendsto_const)
hoelzl@51478
  1790
hoelzl@51478
  1791
lemma continuous_on_compose[continuous_on_intros]:
hoelzl@51478
  1792
  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
hoelzl@51478
  1793
  unfolding continuous_on_topological by simp metis
hoelzl@51478
  1794
hoelzl@51481
  1795
lemma continuous_on_compose2:
hoelzl@51481
  1796
  "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> t = f ` s \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
hoelzl@51481
  1797
  using continuous_on_compose[of s f g] by (simp add: comp_def)
hoelzl@51481
  1798
hoelzl@51478
  1799
subsubsection {* Continuity at a point *}
hoelzl@51478
  1800
hoelzl@51478
  1801
definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
hoelzl@51478
  1802
  "continuous F f \<longleftrightarrow> (f ---> f (Lim F (\<lambda>x. x))) F"
hoelzl@51478
  1803
hoelzl@51478
  1804
ML {*
hoelzl@51471
  1805
hoelzl@51478
  1806
structure Continuous_Intros = Named_Thms
hoelzl@51478
  1807
(
hoelzl@51478
  1808
  val name = @{binding continuous_intros}
hoelzl@51478
  1809
  val description = "Structural introduction rules for pointwise continuity"
hoelzl@51478
  1810
)
hoelzl@51478
  1811
hoelzl@51478
  1812
*}
hoelzl@51478
  1813
hoelzl@51478
  1814
setup Continuous_Intros.setup
hoelzl@51478
  1815
hoelzl@51478
  1816
lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
hoelzl@51478
  1817
  unfolding continuous_def by auto
hoelzl@51478
  1818
hoelzl@51478
  1819
lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
hoelzl@51478
  1820
  by simp
hoelzl@51478
  1821
hoelzl@51478
  1822
lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f x) (at x within s)"
hoelzl@51641
  1823
  by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
hoelzl@51478
  1824
hoelzl@51478
  1825
lemma continuous_within_topological:
hoelzl@51478
  1826
  "continuous (at x within s) f \<longleftrightarrow>
hoelzl@51478
  1827
    (\<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
  1828
  unfolding continuous_within tendsto_def eventually_at_topological by metis
hoelzl@51478
  1829
hoelzl@51478
  1830
lemma continuous_within_compose[continuous_intros]:
hoelzl@51478
  1831
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  1832
  continuous (at x within s) (g o f)"
hoelzl@51478
  1833
  by (simp add: continuous_within_topological) metis
hoelzl@51478
  1834
hoelzl@51478
  1835
lemma continuous_within_compose2:
hoelzl@51478
  1836
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  1837
  continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  1838
  using continuous_within_compose[of x s f g] by (simp add: comp_def)
hoelzl@51471
  1839
hoelzl@51478
  1840
lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f -- x --> f x"
hoelzl@51478
  1841
  using continuous_within[of x UNIV f] by simp
hoelzl@51478
  1842
hoelzl@51478
  1843
lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"
hoelzl@51641
  1844
  unfolding continuous_within by (rule tendsto_ident_at)
hoelzl@51478
  1845
hoelzl@51478
  1846
lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"
hoelzl@51478
  1847
  unfolding continuous_def by (rule tendsto_const)
hoelzl@51478
  1848
hoelzl@51478
  1849
lemma continuous_on_eq_continuous_within:
hoelzl@51478
  1850
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
hoelzl@51478
  1851
  unfolding continuous_on_def continuous_within ..
hoelzl@51478
  1852
hoelzl@51478
  1853
abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where
hoelzl@51478
  1854
  "isCont f a \<equiv> continuous (at a) f"
hoelzl@51478
  1855
hoelzl@51478
  1856
lemma isCont_def: "isCont f a \<longleftrightarrow> f -- a --> f a"
hoelzl@51478
  1857
  by (rule continuous_at)
hoelzl@51478
  1858
hoelzl@51478
  1859
lemma continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"
hoelzl@51641
  1860
  by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)
hoelzl@51478
  1861
hoelzl@51481
  1862
lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"
hoelzl@51641
  1863
  by (simp add: continuous_on_def continuous_at at_within_open[of _ s])
hoelzl@51481
  1864
hoelzl@51481
  1865
lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"
hoelzl@51481
  1866
  unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)
hoelzl@51481
  1867
hoelzl@51478
  1868
lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"
hoelzl@51478
  1869
  by (auto intro: continuous_at_within simp: continuous_on_eq_continuous_within)
hoelzl@51478
  1870
hoelzl@51478
  1871
lemma isContI_continuous: "continuous (at x within UNIV) f \<Longrightarrow> isCont f x"
hoelzl@51478
  1872
  by simp
hoelzl@51478
  1873
hoelzl@51478
  1874
lemma isCont_ident[continuous_intros, simp]: "isCont (\<lambda>x. x) a"
hoelzl@51478
  1875
  using continuous_ident by (rule isContI_continuous)
hoelzl@51478
  1876
hoelzl@51478
  1877
lemmas isCont_const = continuous_const
hoelzl@51478
  1878
hoelzl@51478
  1879
lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
hoelzl@51478
  1880
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51478
  1881
hoelzl@51478
  1882
lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
hoelzl@51478
  1883
  unfolding o_def by (rule isCont_o2)
hoelzl@51471
  1884
hoelzl@51471
  1885
lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@51471
  1886
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51471
  1887
hoelzl@51478
  1888
lemma continuous_within_compose3:
hoelzl@51478
  1889
  "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  1890
  using continuous_within_compose2[of x s f g] by (simp add: continuous_at_within)
hoelzl@51471
  1891
hoelzl@51479
  1892
subsubsection{* Open-cover compactness *}
hoelzl@51479
  1893
hoelzl@51479
  1894
context topological_space
hoelzl@51479
  1895
begin
hoelzl@51479
  1896
hoelzl@51479
  1897
definition compact :: "'a set \<Rightarrow> bool" where
hoelzl@51479
  1898
  compact_eq_heine_borel: -- "This name is used for backwards compatibility"
hoelzl@51479
  1899
    "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
  1900
hoelzl@51479
  1901
lemma compactI:
hoelzl@51479
  1902
  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
  1903
  shows "compact s"
hoelzl@51479
  1904
  unfolding compact_eq_heine_borel using assms by metis
hoelzl@51479
  1905
hoelzl@51479
  1906
lemma compact_empty[simp]: "compact {}"
hoelzl@51479
  1907
  by (auto intro!: compactI)
hoelzl@51479
  1908
hoelzl@51479
  1909
lemma compactE:
hoelzl@51479
  1910
  assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
hoelzl@51479
  1911
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
hoelzl@51479
  1912
  using assms unfolding compact_eq_heine_borel by metis
hoelzl@51479
  1913
hoelzl@51479
  1914
lemma compactE_image:
hoelzl@51479
  1915
  assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
hoelzl@51479
  1916
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
hoelzl@51479
  1917
  using assms unfolding ball_simps[symmetric] SUP_def
hoelzl@51479
  1918
  by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
hoelzl@51479
  1919
hoelzl@51481
  1920
lemma compact_inter_closed [intro]:
hoelzl@51481
  1921
  assumes "compact s" and "closed t"
hoelzl@51481
  1922
  shows "compact (s \<inter> t)"
hoelzl@51481
  1923
proof (rule compactI)
hoelzl@51481
  1924
  fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
hoelzl@51481
  1925
  from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto
hoelzl@51481
  1926
  moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
hoelzl@51481
  1927
  ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
hoelzl@51481
  1928
    using `compact s` unfolding compact_eq_heine_borel by auto
wenzelm@53381
  1929
  then obtain D where "D \<subseteq> C \<union> {- t} \<and> finite D \<and> s \<subseteq> \<Union>D" ..
hoelzl@51481
  1930
  then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
hoelzl@51481
  1931
    by (intro exI[of _ "D - {-t}"]) auto
hoelzl@51481
  1932
qed
hoelzl@51481
  1933
hoelzl@54797
  1934
lemma inj_setminus: "inj_on uminus (A::'a set set)"
hoelzl@54797
  1935
  by (auto simp: inj_on_def)
hoelzl@54797
  1936
hoelzl@54797
  1937
lemma compact_fip:
hoelzl@54797
  1938
  "compact U \<longleftrightarrow>
hoelzl@54797
  1939
    (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"
hoelzl@54797
  1940
  (is "_ \<longleftrightarrow> ?R")
hoelzl@54797
  1941
proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
hoelzl@54797
  1942
  fix A
hoelzl@54797
  1943
  assume "compact U"
hoelzl@54797
  1944
    and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
hoelzl@54797
  1945
    and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
hoelzl@54797
  1946
  from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"
hoelzl@54797
  1947
    by auto
hoelzl@54797
  1948
  with `compact U` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
hoelzl@54797
  1949
    unfolding compact_eq_heine_borel by (metis subset_image_iff)
hoelzl@54797
  1950
  with fi[THEN spec, of B] show False
hoelzl@54797
  1951
    by (auto dest: finite_imageD intro: inj_setminus)
hoelzl@54797
  1952
next
hoelzl@54797
  1953
  fix A
hoelzl@54797
  1954
  assume ?R
hoelzl@54797
  1955
  assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
hoelzl@54797
  1956
  then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
hoelzl@54797
  1957
    by auto
hoelzl@54797
  1958
  with `?R` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"
hoelzl@54797
  1959
    by (metis subset_image_iff)
hoelzl@54797
  1960
  then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
hoelzl@54797
  1961
    by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
hoelzl@54797
  1962
qed
hoelzl@54797
  1963
hoelzl@54797
  1964
lemma compact_imp_fip:
hoelzl@54797
  1965
  "compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>
hoelzl@54797
  1966
    s \<inter> (\<Inter> f) \<noteq> {}"
hoelzl@54797
  1967
  unfolding compact_fip by auto
hoelzl@54797
  1968
hoelzl@54797
  1969
lemma compact_imp_fip_image:
haftmann@56166
  1970
  assumes "compact s"
haftmann@56166
  1971
    and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)"
haftmann@56166
  1972
    and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})"
haftmann@56166
  1973
  shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}"
haftmann@56166
  1974
proof -
haftmann@56166
  1975
  note `compact s`
haftmann@56166
  1976
  moreover from P have "\<forall>i \<in> f ` I. closed i" by blast
haftmann@56166
  1977
  moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})"
haftmann@56166
  1978
  proof (rule, rule, erule conjE)
haftmann@56166
  1979
    fix A :: "'a set set"
haftmann@56166
  1980
    assume "finite A"
haftmann@56166
  1981
    moreover assume "A \<subseteq> f ` I"
haftmann@56166
  1982
    ultimately obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B"
haftmann@56166
  1983
      using finite_subset_image [of A f I] by blast
haftmann@56166
  1984
    with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}" by simp
haftmann@56166
  1985
  qed
haftmann@56166
  1986
  ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}" by (rule compact_imp_fip)
haftmann@56166
  1987
  then show ?thesis by simp
haftmann@56166
  1988
qed
hoelzl@54797
  1989
hoelzl@51471
  1990
end
hoelzl@51471
  1991
hoelzl@51481
  1992
lemma (in t2_space) compact_imp_closed:
hoelzl@51481
  1993
  assumes "compact s" shows "closed s"
hoelzl@51481
  1994
unfolding closed_def
hoelzl@51481
  1995
proof (rule openI)
hoelzl@51481
  1996
  fix y assume "y \<in> - s"
hoelzl@51481
  1997
  let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
hoelzl@51481
  1998
  note `compact s`
hoelzl@51481
  1999
  moreover have "\<forall>u\<in>?C. open u" by simp
hoelzl@51481
  2000
  moreover have "s \<subseteq> \<Union>?C"
hoelzl@51481
  2001
  proof
hoelzl@51481
  2002
    fix x assume "x \<in> s"
hoelzl@51481
  2003
    with `y \<in> - s` have "x \<noteq> y" by clarsimp
hoelzl@51481
  2004
    hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
hoelzl@51481
  2005
      by (rule hausdorff)
hoelzl@51481
  2006
    with `x \<in> s` show "x \<in> \<Union>?C"
hoelzl@51481
  2007
      unfolding eventually_nhds by auto
hoelzl@51481
  2008
  qed
hoelzl@51481
  2009
  ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
hoelzl@51481
  2010
    by (rule compactE)
hoelzl@51481
  2011
  from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
hoelzl@51481
  2012
  with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
hoelzl@51481
  2013
    by (simp add: eventually_Ball_finite)
hoelzl@51481
  2014
  with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
hoelzl@51481
  2015
    by (auto elim!: eventually_mono [rotated])
hoelzl@51481
  2016
  thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
hoelzl@51481
  2017
    by (simp add: eventually_nhds subset_eq)
hoelzl@51481
  2018
qed
hoelzl@51481
  2019
hoelzl@51481
  2020
lemma compact_continuous_image:
hoelzl@51481
  2021
  assumes f: "continuous_on s f" and s: "compact s"
hoelzl@51481
  2022
  shows "compact (f ` s)"
hoelzl@51481
  2023
proof (rule compactI)
hoelzl@51481
  2024
  fix C assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
hoelzl@51481
  2025
  with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"
hoelzl@51481
  2026
    unfolding continuous_on_open_invariant by blast
wenzelm@53381
  2027
  then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s"
wenzelm@53381
  2028
    unfolding bchoice_iff ..
hoelzl@51481
  2029
  with cover have "\<forall>c\<in>C. open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)"
hoelzl@51481
  2030
    by (fastforce simp add: subset_eq set_eq_iff)+
hoelzl@51481
  2031
  from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" .
hoelzl@51481
  2032
  with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D"
hoelzl@51481
  2033
    by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+
hoelzl@51481
  2034
qed
hoelzl@51481
  2035
hoelzl@51481
  2036
lemma continuous_on_inv:
hoelzl@51481
  2037
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
hoelzl@51481
  2038
  assumes "continuous_on s f"  "compact s"  "\<forall>x\<in>s. g (f x) = x"
hoelzl@51481
  2039
  shows "continuous_on (f ` s) g"
hoelzl@51481
  2040
unfolding continuous_on_topological
hoelzl@51481
  2041
proof (clarsimp simp add: assms(3))
hoelzl@51481
  2042
  fix x :: 'a and B :: "'a set"
hoelzl@51481
  2043
  assume "x \<in> s" and "open B" and "x \<in> B"
hoelzl@51481
  2044
  have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
hoelzl@51481
  2045
    using assms(3) by (auto, metis)
hoelzl@51481
  2046
  have "continuous_on (s - B) f"
hoelzl@51481
  2047
    using `continuous_on s f` Diff_subset
hoelzl@51481
  2048
    by (rule continuous_on_subset)
hoelzl@51481
  2049
  moreover have "compact (s - B)"
hoelzl@51481
  2050
    using `open B` and `compact s`
hoelzl@51481
  2051
    unfolding Diff_eq by (intro compact_inter_closed closed_Compl)
hoelzl@51481
  2052
  ultimately have "compact (f ` (s - B))"
hoelzl@51481
  2053
    by (rule compact_continuous_image)
hoelzl@51481
  2054
  hence "closed (f ` (s - B))"
hoelzl@51481
  2055
    by (rule compact_imp_closed)
hoelzl@51481
  2056
  hence "open (- f ` (s - B))"
hoelzl@51481
  2057
    by (rule open_Compl)
hoelzl@51481
  2058
  moreover have "f x \<in> - f ` (s - B)"
hoelzl@51481
  2059
    using `x \<in> s` and `x \<in> B` by (simp add: 1)
hoelzl@51481
  2060
  moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
hoelzl@51481
  2061
    by (simp add: 1)
hoelzl@51481
  2062
  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
  2063
    by fast
hoelzl@51481
  2064
qed
hoelzl@51481
  2065
hoelzl@51481
  2066
lemma continuous_on_inv_into:
hoelzl@51481
  2067
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
hoelzl@51481
  2068
  assumes s: "continuous_on s f" "compact s" and f: "inj_on f s"
hoelzl@51481
  2069
  shows "continuous_on (f ` s) (the_inv_into s f)"
hoelzl@51481
  2070
  by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])
hoelzl@51481
  2071
hoelzl@51479
  2072
lemma (in linorder_topology) compact_attains_sup:
hoelzl@51479
  2073
  assumes "compact S" "S \<noteq> {}"
hoelzl@51479
  2074
  shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"
hoelzl@51479
  2075
proof (rule classical)
hoelzl@51479
  2076
  assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"
hoelzl@51479
  2077
  then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"
hoelzl@51479
  2078
    by (metis not_le)
hoelzl@51479
  2079
  then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"
hoelzl@51479
  2080
    by auto
hoelzl@51479
  2081
  with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
hoelzl@51479
  2082
    by (erule compactE_image)
hoelzl@51479
  2083
  with `S \<noteq> {}` have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"
hoelzl@51479
  2084
    by (auto intro!: Max_in)
hoelzl@51479
  2085
  with C have "S \<subseteq> {..< Max (t`C)}"
hoelzl@51479
  2086
    by (auto intro: less_le_trans simp: subset_eq)
hoelzl@51479
  2087
  with t Max `C \<subseteq> S` show ?thesis
hoelzl@51479
  2088
    by fastforce
hoelzl@51479
  2089
qed
hoelzl@51479
  2090
hoelzl@51479
  2091
lemma (in linorder_topology) compact_attains_inf:
hoelzl@51479
  2092
  assumes "compact S" "S \<noteq> {}"
hoelzl@51479
  2093
  shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"
hoelzl@51479
  2094
proof (rule classical)
hoelzl@51479
  2095
  assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"
hoelzl@51479
  2096
  then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"
hoelzl@51479
  2097
    by (metis not_le)
hoelzl@51479
  2098
  then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"
hoelzl@51479
  2099
    by auto
hoelzl@51479
  2100
  with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
hoelzl@51479
  2101
    by (erule compactE_image)
hoelzl@51479
  2102
  with `S \<noteq> {}` have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"
hoelzl@51479
  2103
    by (auto intro!: Min_in)
hoelzl@51479
  2104
  with C have "S \<subseteq> {Min (t`C) <..}"
hoelzl@51479
  2105
    by (auto intro: le_less_trans simp: subset_eq)
hoelzl@51479
  2106
  with t Min `C \<subseteq> S` show ?thesis
hoelzl@51479
  2107
    by fastforce
hoelzl@51479
  2108
qed
hoelzl@51479
  2109
hoelzl@51479
  2110
lemma continuous_attains_sup:
hoelzl@51479
  2111
  fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
hoelzl@51479
  2112
  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
  2113
  using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto
hoelzl@51479
  2114
hoelzl@51479
  2115
lemma continuous_attains_inf:
hoelzl@51479
  2116
  fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
hoelzl@51479
  2117
  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
  2118
  using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
hoelzl@51479
  2119
hoelzl@51480
  2120
hoelzl@51480
  2121
subsection {* Connectedness *}
hoelzl@51480
  2122
hoelzl@51480
  2123
context topological_space
hoelzl@51480
  2124
begin
hoelzl@51480
  2125
hoelzl@51480
  2126
definition "connected S \<longleftrightarrow>
hoelzl@51480
  2127
  \<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
  2128
hoelzl@51480
  2129
lemma connectedI:
hoelzl@51480
  2130
  "(\<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
  2131
  \<Longrightarrow> connected U"
hoelzl@51480
  2132
  by (auto simp: connected_def)
hoelzl@51480
  2133
hoelzl@51480
  2134
lemma connected_empty[simp]: "connected {}"
hoelzl@51480
  2135
  by (auto intro!: connectedI)
hoelzl@51480
  2136
hoelzl@56329
  2137
lemma connectedD:
hoelzl@56329
  2138
  "connected A \<Longrightarrow> open U \<Longrightarrow> open V \<Longrightarrow> U \<inter> V \<inter> A = {} \<Longrightarrow> A \<subseteq> U \<union> V \<Longrightarrow> U \<inter> A = {} \<or> V \<inter> A = {}" 
hoelzl@56329
  2139
  by (auto simp: connected_def)
hoelzl@56329
  2140
hoelzl@51479
  2141
end
hoelzl@51479
  2142
hoelzl@56329
  2143
lemma connected_local_const:
hoelzl@56329
  2144
  assumes "connected A" "a \<in> A" "b \<in> A"
hoelzl@56329
  2145
  assumes *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)"
hoelzl@56329
  2146
  shows "f a = f b"
hoelzl@56329
  2147
proof -
hoelzl@56329
  2148
  obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)"
hoelzl@56329
  2149
    "\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x"
hoelzl@56329
  2150
    using * unfolding eventually_at_topological by metis
hoelzl@56329
  2151
hoelzl@56329
  2152
  let ?P = "\<Union>b\<in>{b\<in>A. f a = f b}. S b" and ?N = "\<Union>b\<in>{b\<in>A. f a \<noteq> f b}. S b"
hoelzl@56329
  2153
  have "?P \<inter> A = {} \<or> ?N \<inter> A = {}"
hoelzl@56329
  2154
    using `connected A` S `a\<in>A`
hoelzl@56329
  2155
    by (intro connectedD) (auto, metis)
hoelzl@56329
  2156
  then show "f a = f b"
hoelzl@56329
  2157
  proof
hoelzl@56329
  2158
    assume "?N \<inter> A = {}"
hoelzl@56329
  2159
    then have "\<forall>x\<in>A. f a = f x"
hoelzl@56329
  2160
      using S(1) by auto
hoelzl@56329
  2161
    with `b\<in>A` show ?thesis by auto
hoelzl@56329
  2162
  next
hoelzl@56329
  2163
    assume "?P \<inter> A = {}" then show ?thesis
hoelzl@56329
  2164
      using `a \<in> A` S(1)[of a] by auto
hoelzl@56329
  2165
  qed
hoelzl@56329
  2166
qed
hoelzl@56329
  2167
hoelzl@51480
  2168
lemma (in linorder_topology) connectedD_interval:
hoelzl@51480
  2169
  assumes "connected U" and xy: "x \<in> U" "y \<in> U" and "x \<le> z" "z \<le> y"
hoelzl@51480
  2170
  shows "z \<in> U"
hoelzl@51480
  2171
proof -
hoelzl@51480
  2172
  have eq: "{..<z} \<union> {z<..} = - {z}"
hoelzl@51480
  2173
    by auto
hoelzl@51480
  2174
  { assume "z \<notin> U" "x < z" "z < y"
hoelzl@51480
  2175
    with xy have "\<not> connected U"
hoelzl@51480
  2176
      unfolding connected_def simp_thms
hoelzl@51480
  2177
      apply (rule_tac exI[of _ "{..< z}"])
hoelzl@51480
  2178
      apply (rule_tac exI[of _ "{z <..}"])
hoelzl@51480
  2179
      apply (auto simp add: eq)
hoelzl@51480
  2180
      done }
hoelzl@51480
  2181
  with assms show "z \<in> U"
hoelzl@51480
  2182
    by (metis less_le)
hoelzl@51480
  2183
qed
hoelzl@51480
  2184
hoelzl@51480
  2185
lemma connected_continuous_image:
hoelzl@51480
  2186
  assumes *: "continuous_on s f"
hoelzl@51480
  2187
  assumes "connected s"
hoelzl@51480
  2188
  shows "connected (f ` s)"
hoelzl@51480
  2189
proof (rule connectedI)
hoelzl@51480
  2190
  fix A B assume A: "open A" "A \<inter> f ` s \<noteq> {}" and B: "open B" "B \<inter> f ` s \<noteq> {}" and
hoelzl@51480
  2191
    AB: "A \<inter> B \<inter> f ` s = {}" "f ` s \<subseteq> A \<union> B"
hoelzl@51480
  2192
  obtain A' where A': "open A'" "f -` A \<inter> s = A' \<inter> s"
hoelzl@51480
  2193
    using * `open A` unfolding continuous_on_open_invariant by metis
hoelzl@51480
  2194
  obtain B' where B': "open B'" "f -` B \<inter> s = B' \<inter> s"
hoelzl@51480
  2195
    using * `open B` unfolding continuous_on_open_invariant by metis
hoelzl@51480
  2196
hoelzl@51480
  2197
  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
  2198
  proof (rule exI[of _ A'], rule exI[of _ B'], intro conjI)
hoelzl@51480
  2199
    have "s \<subseteq> (f -` A \<inter> s) \<union> (f -` B \<inter> s)" using AB by auto
hoelzl@51480
  2200
    then show "s \<subseteq> A' \<union> B'" using A' B' by auto
hoelzl@51480
  2201
  next
hoelzl@51480
  2202
    have "(f -` A \<inter> s) \<inter> (f -` B \<inter> s) = {}" using AB by auto
hoelzl@51480
  2203
    then show "A' \<inter> B' \<inter> s = {}" using A' B' by auto
hoelzl@51480
  2204
  qed (insert A' B' A B, auto)
hoelzl@51480
  2205
  with `connected s` show False
hoelzl@51480
  2206
    unfolding connected_def by blast
hoelzl@51480
  2207
qed
hoelzl@51480
  2208
hoelzl@51518
  2209
hoelzl@51518
  2210
section {* Connectedness *}
hoelzl@51518
  2211
hoelzl@51775
  2212
class linear_continuum_topology = linorder_topology + linear_continuum
hoelzl@51518
  2213
begin
hoelzl@51518
  2214
hoelzl@51518
  2215
lemma Inf_notin_open:
hoelzl@51518
  2216
  assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a"
hoelzl@51518
  2217
  shows "Inf A \<notin> A"
hoelzl@51518
  2218
proof
hoelzl@51518
  2219
  assume "Inf A \<in> A"
hoelzl@51518
  2220
  then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A"
hoelzl@51518
  2221
    using open_left[of A "Inf A" x] assms by auto
hoelzl@51518
  2222
  with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
hoelzl@51518
  2223
    by (auto simp: subset_eq)
hoelzl@51518
  2224
  then show False
hoelzl@54258
  2225
    using cInf_lower[OF `c \<in> A`] bnd by (metis not_le less_imp_le bdd_belowI)
hoelzl@51518
  2226
qed
hoelzl@51518
  2227
hoelzl@51518
  2228
lemma Sup_notin_open:
hoelzl@51518
  2229
  assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x"
hoelzl@51518
  2230
  shows "Sup A \<notin> A"
hoelzl@51518
  2231
proof
hoelzl@51518
  2232
  assume "Sup A \<in> A"
hoelzl@51518
  2233
  then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
hoelzl@51518
  2234
    using open_right[of A "Sup A" x] assms by auto
hoelzl@51518
  2235
  with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
hoelzl@51518
  2236
    by (auto simp: subset_eq)
hoelzl@51518
  2237
  then show False
hoelzl@54258
  2238
    using cSup_upper[OF `c \<in> A`] bnd by (metis less_imp_le not_le bdd_aboveI)
hoelzl@51518
  2239
qed
hoelzl@51518
  2240
hoelzl@51480
  2241
end
hoelzl@51480
  2242
hoelzl@51775
  2243
instance linear_continuum_topology \<subseteq> perfect_space
hoelzl@51775
  2244
proof
hoelzl@51775
  2245
  fix x :: 'a
wenzelm@53381
  2246
  obtain y where "x < y \<or> y < x"
wenzelm@53381
  2247
    using ex_gt_or_lt [of x] ..
hoelzl@51775
  2248
  with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y]
hoelzl@51775
  2249
  show "\<not> open {x}"
hoelzl@51775
  2250
    by auto
hoelzl@51775
  2251
qed
hoelzl@51775
  2252
hoelzl@51518
  2253
lemma connectedI_interval:
hoelzl@51775
  2254
  fixes U :: "'a :: linear_continuum_topology set"
hoelzl@51518
  2255
  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
  2256
  shows "connected U"
hoelzl@51518
  2257
proof (rule connectedI)
hoelzl@51518
  2258
  { fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
hoelzl@51518
  2259
    fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
hoelzl@51518
  2260
hoelzl@51518
  2261
    let ?z = "Inf (B \<inter> {x <..})"
hoelzl@51518
  2262
hoelzl@51518
  2263
    have "x \<le> ?z" "?z \<le> y"
hoelzl@54258
  2264
      using `y \<in> B` `x < y` by (auto intro: cInf_lower cInf_greatest)
hoelzl@51518
  2265
    with `x \<in> U` `y \<in> U` have "?z \<in> U"
hoelzl@51518
  2266
      by (rule *)
hoelzl@51518
  2267
    moreover have "?z \<notin> B \<inter> {x <..}"
hoelzl@51518
  2268
      using `open B` by (intro Inf_notin_open) auto
hoelzl@51518
  2269
    ultimately have "?z \<in> A"
hoelzl@51518
  2270
      using `x \<le> ?z` `A \<inter> B \<inter> U = {}` `x \<in> A` `U \<subseteq> A \<union> B` by auto
hoelzl@51518
  2271
hoelzl@51518
  2272
    { assume "?z < y"
hoelzl@51518
  2273
      obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
hoelzl@51518
  2274
        using open_right[OF `open A` `?z \<in> A` `?z < y`] by auto
hoelzl@51518
  2275
      moreover obtain b where "b \<in> B" "x < b" "b < min a y"
hoelzl@54258
  2276
        using cInf_less_iff[of "B \<inter> {x <..}" "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B`
hoelzl@51518
  2277
        by (auto intro: less_imp_le)
wenzelm@53374
  2278
      moreover have "?z \<le> b"
wenzelm@53374
  2279
        using `b \<in> B` `x < b`
hoelzl@54258
  2280
        by (intro cInf_lower) auto
hoelzl@51518
  2281
      moreover have "b \<in> U"
hoelzl@51518
  2282
        using `x \<le> ?z` `?z \<le> b` `b < min a y`
hoelzl@51518
  2283
        by (intro *[OF `x \<in> U` `y \<in> U`]) (auto simp: less_imp_le)
hoelzl@51518
  2284
      ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U"
hoelzl@51518
  2285
        by (intro bexI[of _ b]) auto }
hoelzl@51518
  2286
    then have False
hoelzl@51518
  2287
      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
  2288
  note not_disjoint = this
hoelzl@51518
  2289
hoelzl@51518
  2290
  fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"
hoelzl@51518
  2291
  moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto
hoelzl@51518
  2292
  moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto
hoelzl@51518
  2293
  moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
hoelzl@51518
  2294
  ultimately show False by (cases x y rule: linorder_cases) auto
hoelzl@51518
  2295
qed
hoelzl@51518
  2296
hoelzl@51518
  2297
lemma connected_iff_interval:
hoelzl@51775
  2298
  fixes U :: "'a :: linear_continuum_topology set"
hoelzl@51518
  2299
  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
  2300
  by (auto intro: connectedI_interval dest: connectedD_interval)
hoelzl@51518
  2301
hoelzl@51775
  2302
lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
hoelzl@51518
  2303
  unfolding connected_iff_interval by auto
hoelzl@51518
  2304
hoelzl@51775
  2305
lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}"
hoelzl@51518
  2306
  unfolding connected_iff_interval by auto
hoelzl@51518
  2307
hoelzl@51775
  2308
lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}"
hoelzl@51518
  2309
  unfolding connected_iff_interval by auto
hoelzl@51518
  2310
hoelzl@51775
  2311
lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}"
hoelzl@51518
  2312
  unfolding connected_iff_interval by auto
hoelzl@51518
  2313
hoelzl@51775
  2314
lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}"
hoelzl@51518
  2315
  unfolding connected_iff_interval by auto
hoelzl@51518
  2316
hoelzl@51775
  2317
lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}"
hoelzl@51518
  2318
  unfolding connected_iff_interval by auto
hoelzl@51518
  2319
hoelzl@51775
  2320
lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}"
hoelzl@51518
  2321
  unfolding connected_iff_interval by auto
hoelzl@51518
  2322
hoelzl@51775
  2323
lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}"
hoelzl@51518
  2324
  unfolding connected_iff_interval by auto
hoelzl@51518
  2325
hoelzl@51775
  2326
lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}"
hoelzl@51518
  2327
  unfolding connected_iff_interval by auto
hoelzl@51518
  2328
hoelzl@51518
  2329
lemma connected_contains_Ioo: 
hoelzl@51518
  2330
  fixes A :: "'a :: linorder_topology set"
hoelzl@51518
  2331
  assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
hoelzl@51518
  2332
  using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le)
hoelzl@51518
  2333
hoelzl@51518
  2334
subsection {* Intermediate Value Theorem *}
hoelzl@51518
  2335
hoelzl@51518
  2336
lemma IVT':
hoelzl@51775
  2337
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2338
  assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"
hoelzl@51518
  2339
  assumes *: "continuous_on {a .. b} f"
hoelzl@51518
  2340
  shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2341
proof -
hoelzl@51518
  2342
  have "connected {a..b}"
hoelzl@51518
  2343
    unfolding connected_iff_interval by auto
hoelzl@51518
  2344
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y
hoelzl@51518
  2345
  show ?thesis
hoelzl@51518
  2346
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
hoelzl@51518
  2347
qed
hoelzl@51518
  2348