src/HOL/Topological_Spaces.thy
author wenzelm
Tue Sep 03 01:12:40 2013 +0200 (2013-09-03)
changeset 53374 a14d2a854c02
parent 53215 5e47c31c6f7c
child 53381 355a4cac5440
permissions -rw-r--r--
tuned proofs -- clarified flow of facts wrt. calculation;
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(*  Title:      HOL/Topological_Spaces.thy
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    Author:     Brian Huffman
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    Author:     Johannes Hölzl
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*)
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header {* Topological Spaces *}
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theory Topological_Spaces
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imports Main Conditionally_Complete_Lattices
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begin
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subsection {* Topological space *}
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class "open" =
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  fixes "open" :: "'a set \<Rightarrow> bool"
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class topological_space = "open" +
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  assumes open_UNIV [simp, intro]: "open UNIV"
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  assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)"
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  assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)"
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begin
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definition
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  closed :: "'a set \<Rightarrow> bool" where
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  "closed S \<longleftrightarrow> open (- S)"
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lemma open_empty [intro, simp]: "open {}"
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  using open_Union [of "{}"] by simp
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lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)"
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  using open_Union [of "{S, T}"] by simp
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lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
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  unfolding SUP_def by (rule open_Union) auto
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lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)"
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  by (induct set: finite) auto
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lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
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  unfolding INF_def by (rule open_Inter) auto
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lemma openI:
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  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
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  shows "open S"
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proof -
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  have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
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  moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
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  ultimately show "open S" by simp
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qed
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lemma closed_empty [intro, simp]:  "closed {}"
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  unfolding closed_def by simp
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lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)"
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  unfolding closed_def by auto
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lemma closed_UNIV [intro, simp]: "closed UNIV"
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  unfolding closed_def by simp
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lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)"
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  unfolding closed_def by auto
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lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)"
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  unfolding closed_def by auto
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lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)"
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  unfolding closed_def uminus_Inf by auto
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lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)"
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  by (induct set: finite) auto
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lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
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  unfolding SUP_def by (rule closed_Union) auto
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lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
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  unfolding closed_def by simp
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lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
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  unfolding closed_def by simp
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lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)"
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  unfolding closed_open Diff_eq by (rule open_Int)
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lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)"
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  unfolding open_closed Diff_eq by (rule closed_Int)
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lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)"
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  unfolding closed_open .
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lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)"
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  unfolding open_closed .
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end
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subsection{* Hausdorff and other separation properties *}
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class t0_space = topological_space +
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  assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)"
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class t1_space = topological_space +
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  assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U"
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instance t1_space \<subseteq> t0_space
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proof qed (fast dest: t1_space)
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lemma separation_t1:
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  fixes x y :: "'a::t1_space"
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  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)"
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  using t1_space[of x y] by blast
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lemma closed_singleton:
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  fixes a :: "'a::t1_space"
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  shows "closed {a}"
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proof -
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  let ?T = "\<Union>{S. open S \<and> a \<notin> S}"
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  have "open ?T" by (simp add: open_Union)
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  also have "?T = - {a}"
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    by (simp add: set_eq_iff separation_t1, auto)
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  finally show "closed {a}" unfolding closed_def .
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qed
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lemma closed_insert [simp]:
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  fixes a :: "'a::t1_space"
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  assumes "closed S" shows "closed (insert a S)"
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proof -
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  from closed_singleton assms
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  have "closed ({a} \<union> S)" by (rule closed_Un)
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  thus "closed (insert a S)" by simp
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qed
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lemma finite_imp_closed:
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  fixes S :: "'a::t1_space set"
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  shows "finite S \<Longrightarrow> closed S"
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by (induct set: finite, simp_all)
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text {* T2 spaces are also known as Hausdorff spaces. *}
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class t2_space = topological_space +
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  assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
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instance t2_space \<subseteq> t1_space
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proof qed (fast dest: hausdorff)
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lemma separation_t2:
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  fixes x y :: "'a::t2_space"
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  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
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  using hausdorff[of x y] by blast
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lemma separation_t0:
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  fixes x y :: "'a::t0_space"
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  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
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  using t0_space[of x y] by blast
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text {* A perfect space is a topological space with no isolated points. *}
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class perfect_space = topological_space +
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  assumes not_open_singleton: "\<not> open {x}"
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subsection {* Generators for toplogies *}
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inductive generate_topology for S where
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  UNIV: "generate_topology S UNIV"
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| Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)"
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| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)"
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| Basis: "s \<in> S \<Longrightarrow> generate_topology S s"
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hide_fact (open) UNIV Int UN Basis 
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lemma generate_topology_Union: 
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  "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"
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  unfolding SUP_def by (intro generate_topology.UN) auto
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lemma topological_space_generate_topology:
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  "class.topological_space (generate_topology S)"
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  by default (auto intro: generate_topology.intros)
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subsection {* Order topologies *}
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class order_topology = order + "open" +
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  assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
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begin
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subclass topological_space
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  unfolding open_generated_order
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  by (rule topological_space_generate_topology)
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lemma open_greaterThan [simp]: "open {a <..}"
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  unfolding open_generated_order by (auto intro: generate_topology.Basis)
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lemma open_lessThan [simp]: "open {..< a}"
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  unfolding open_generated_order by (auto intro: generate_topology.Basis)
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lemma open_greaterThanLessThan [simp]: "open {a <..< b}"
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   unfolding greaterThanLessThan_eq by (simp add: open_Int)
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end
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class linorder_topology = linorder + order_topology
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lemma closed_atMost [simp]: "closed {.. a::'a::linorder_topology}"
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  by (simp add: closed_open)
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lemma closed_atLeast [simp]: "closed {a::'a::linorder_topology ..}"
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  by (simp add: closed_open)
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lemma closed_atLeastAtMost [simp]: "closed {a::'a::linorder_topology .. b}"
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proof -
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  have "{a .. b} = {a ..} \<inter> {.. b}"
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    by auto
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  then show ?thesis
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    by (simp add: closed_Int)
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qed
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lemma (in linorder) less_separate:
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  assumes "x < y"
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  shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}"
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proof cases
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  assume "\<exists>z. x < z \<and> z < y"
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  then guess z ..
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  then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}"
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    by auto
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  then show ?thesis by blast
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next
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  assume "\<not> (\<exists>z. x < z \<and> z < y)"
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  with `x < y` have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}"
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    by auto
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  then show ?thesis by blast
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qed
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instance linorder_topology \<subseteq> t2_space
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proof
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  fix x y :: 'a
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  from less_separate[of x y] less_separate[of y x]
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  show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
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    by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+
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qed
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lemma (in linorder_topology) open_right:
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  assumes "open S" "x \<in> S" and gt_ex: "x < y" shows "\<exists>b>x. {x ..< b} \<subseteq> S"
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  using assms unfolding open_generated_order
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proof induction
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  case (Int A B)
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  then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B" by auto
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  then show ?case by (auto intro!: exI[of _ "min a b"])
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next
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  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] gt_ex)
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qed blast+
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lemma (in linorder_topology) open_left:
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  assumes "open S" "x \<in> S" and lt_ex: "y < x" shows "\<exists>b<x. {b <.. x} \<subseteq> S"
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  using assms unfolding open_generated_order
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proof induction
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  case (Int A B)
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  then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B" by auto
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  then show ?case by (auto intro!: exI[of _ "max a b"])
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next
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  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex)
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qed blast+
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subsection {* Filters *}
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text {*
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  This definition also allows non-proper filters.
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*}
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locale is_filter =
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  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
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  assumes True: "F (\<lambda>x. True)"
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  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
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  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
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typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
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proof
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  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
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qed
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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
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  using Rep_filter [of F] by simp
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lemma Abs_filter_inverse':
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  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
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  using assms by (simp add: Abs_filter_inverse)
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subsubsection {* Eventually *}
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
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  where "eventually P F \<longleftrightarrow> Rep_filter F P"
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lemma eventually_Abs_filter:
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  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
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  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
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lemma filter_eq_iff:
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  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
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  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
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  unfolding eventually_def
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  by (rule is_filter.True [OF is_filter_Rep_filter])
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
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proof -
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   305
  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
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   306
  thus "eventually P F" by simp
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"
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   311
  unfolding eventually_def
hoelzl@51471
   312
  by (rule is_filter.mono [OF is_filter_Rep_filter])
hoelzl@51471
   313
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   314
lemma eventually_conj:
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   315
  assumes P: "eventually (\<lambda>x. P x) F"
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   316
  assumes Q: "eventually (\<lambda>x. Q x) F"
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 {*
hoelzl@51471
   376
  fun eventually_elim_tac ctxt thms thm =
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
hoelzl@51471
   384
      val cases_prop = prop_of (raw_elim_thm RS thm)
hoelzl@51471
   385
      val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
hoelzl@51471
   386
    in
hoelzl@51471
   387
      CASES cases (rtac raw_elim_thm 1) thm
hoelzl@51471
   388
    end
hoelzl@51471
   389
*}
hoelzl@51471
   390
hoelzl@51471
   391
method_setup eventually_elim = {*
hoelzl@51471
   392
  Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
hoelzl@51471
   393
*} "elimination of eventually quantifiers"
hoelzl@51471
   394
hoelzl@51471
   395
hoelzl@51471
   396
subsubsection {* Finer-than relation *}
hoelzl@51471
   397
hoelzl@51471
   398
text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
hoelzl@51471
   399
filter @{term F'}. *}
hoelzl@51471
   400
hoelzl@51471
   401
instantiation filter :: (type) complete_lattice
hoelzl@51471
   402
begin
hoelzl@51471
   403
hoelzl@51471
   404
definition le_filter_def:
hoelzl@51471
   405
  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
hoelzl@51471
   406
hoelzl@51471
   407
definition
hoelzl@51471
   408
  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@51471
   409
hoelzl@51471
   410
definition
hoelzl@51471
   411
  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
hoelzl@51471
   412
hoelzl@51471
   413
definition
hoelzl@51471
   414
  "bot = Abs_filter (\<lambda>P. True)"
hoelzl@51471
   415
hoelzl@51471
   416
definition
hoelzl@51471
   417
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
hoelzl@51471
   418
hoelzl@51471
   419
definition
hoelzl@51471
   420
  "inf F F' = Abs_filter
hoelzl@51471
   421
      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@51471
   422
hoelzl@51471
   423
definition
hoelzl@51471
   424
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
hoelzl@51471
   425
hoelzl@51471
   426
definition
hoelzl@51471
   427
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
hoelzl@51471
   428
hoelzl@51471
   429
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
hoelzl@51471
   430
  unfolding top_filter_def
hoelzl@51471
   431
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@51471
   432
hoelzl@51471
   433
lemma eventually_bot [simp]: "eventually P bot"
hoelzl@51471
   434
  unfolding bot_filter_def
hoelzl@51471
   435
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@51471
   436
hoelzl@51471
   437
lemma eventually_sup:
hoelzl@51471
   438
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
hoelzl@51471
   439
  unfolding sup_filter_def
hoelzl@51471
   440
  by (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   441
     (auto elim!: eventually_rev_mp)
hoelzl@51471
   442
hoelzl@51471
   443
lemma eventually_inf:
hoelzl@51471
   444
  "eventually P (inf F F') \<longleftrightarrow>
hoelzl@51471
   445
   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@51471
   446
  unfolding inf_filter_def
hoelzl@51471
   447
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   448
  apply (fast intro: eventually_True)
hoelzl@51471
   449
  apply clarify
hoelzl@51471
   450
  apply (intro exI conjI)
hoelzl@51471
   451
  apply (erule (1) eventually_conj)
hoelzl@51471
   452
  apply (erule (1) eventually_conj)
hoelzl@51471
   453
  apply simp
hoelzl@51471
   454
  apply auto
hoelzl@51471
   455
  done
hoelzl@51471
   456
hoelzl@51471
   457
lemma eventually_Sup:
hoelzl@51471
   458
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
hoelzl@51471
   459
  unfolding Sup_filter_def
hoelzl@51471
   460
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   461
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
hoelzl@51471
   462
  done
hoelzl@51471
   463
hoelzl@51471
   464
instance proof
hoelzl@51471
   465
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
hoelzl@51471
   466
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@51471
   467
    by (rule less_filter_def) }
hoelzl@51471
   468
  { show "F \<le> F"
hoelzl@51471
   469
    unfolding le_filter_def by simp }
hoelzl@51471
   470
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
hoelzl@51471
   471
    unfolding le_filter_def by simp }
hoelzl@51471
   472
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
hoelzl@51471
   473
    unfolding le_filter_def filter_eq_iff by fast }
hoelzl@51471
   474
  { show "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)
haftmann@52729
   493
      (metis (full_types) Topological_Spaces.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
hoelzl@51471
   573
  fix N assume "\<forall>n\<ge>N. P n" then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])
hoelzl@51471
   574
next
hoelzl@51471
   575
  fix N assume "\<forall>n>N. P n"
hoelzl@51471
   576
  moreover from gt_ex[of N] guess y ..
hoelzl@51471
   577
  ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])
hoelzl@51471
   578
qed
hoelzl@51471
   579
hoelzl@51471
   580
lemma eventually_gt_at_top:
hoelzl@53215
   581
  "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"
hoelzl@51471
   582
  unfolding eventually_at_top_dense by auto
hoelzl@51471
   583
hoelzl@51471
   584
definition at_bot :: "('a::order) filter"
hoelzl@51471
   585
  where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
hoelzl@51471
   586
hoelzl@51471
   587
lemma eventually_at_bot_linorder:
hoelzl@51471
   588
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@51471
   589
  unfolding at_bot_def
hoelzl@51471
   590
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   591
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@51471
   592
  assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
hoelzl@51471
   593
  then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
hoelzl@51471
   594
  then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
hoelzl@51471
   595
  then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
hoelzl@51471
   596
qed auto
hoelzl@51471
   597
hoelzl@51471
   598
lemma eventually_le_at_bot:
hoelzl@51471
   599
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
hoelzl@51471
   600
  unfolding eventually_at_bot_linorder by auto
hoelzl@51471
   601
hoelzl@51471
   602
lemma eventually_at_bot_dense:
hoelzl@53215
   603
  fixes P :: "'a::unbounded_dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
hoelzl@51471
   604
  unfolding eventually_at_bot_linorder
hoelzl@51471
   605
proof safe
hoelzl@51471
   606
  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
   607
next
hoelzl@51471
   608
  fix N assume "\<forall>n<N. P n" 
hoelzl@51471
   609
  moreover from lt_ex[of N] guess y ..
hoelzl@51471
   610
  ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
hoelzl@51471
   611
qed
hoelzl@51471
   612
hoelzl@51471
   613
lemma eventually_gt_at_bot:
hoelzl@53215
   614
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
hoelzl@51471
   615
  unfolding eventually_at_bot_dense by auto
hoelzl@51471
   616
hoelzl@51471
   617
subsection {* Sequentially *}
hoelzl@51471
   618
hoelzl@51471
   619
abbreviation sequentially :: "nat filter"
hoelzl@51471
   620
  where "sequentially == at_top"
hoelzl@51471
   621
hoelzl@51471
   622
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
hoelzl@51471
   623
  unfolding at_top_def by simp
hoelzl@51471
   624
hoelzl@51471
   625
lemma eventually_sequentially:
hoelzl@51471
   626
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@51471
   627
  by (rule eventually_at_top_linorder)
hoelzl@51471
   628
hoelzl@51471
   629
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
hoelzl@51471
   630
  unfolding filter_eq_iff eventually_sequentially by auto
hoelzl@51471
   631
hoelzl@51471
   632
lemmas trivial_limit_sequentially = sequentially_bot
hoelzl@51471
   633
hoelzl@51471
   634
lemma eventually_False_sequentially [simp]:
hoelzl@51471
   635
  "\<not> eventually (\<lambda>n. False) sequentially"
hoelzl@51471
   636
  by (simp add: eventually_False)
hoelzl@51471
   637
hoelzl@51471
   638
lemma le_sequentially:
hoelzl@51471
   639
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
hoelzl@51471
   640
  unfolding le_filter_def eventually_sequentially
hoelzl@51471
   641
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
hoelzl@51471
   642
hoelzl@51471
   643
lemma eventually_sequentiallyI:
hoelzl@51471
   644
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
hoelzl@51471
   645
  shows "eventually P sequentially"
hoelzl@51471
   646
using assms by (auto simp: eventually_sequentially)
hoelzl@51471
   647
hoelzl@51474
   648
lemma eventually_sequentially_seg:
hoelzl@51474
   649
  "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@51474
   650
  unfolding eventually_sequentially
hoelzl@51474
   651
  apply safe
hoelzl@51474
   652
   apply (rule_tac x="N + k" in exI)
hoelzl@51474
   653
   apply rule
hoelzl@51474
   654
   apply (erule_tac x="n - k" in allE)
hoelzl@51474
   655
   apply auto []
hoelzl@51474
   656
  apply (rule_tac x=N in exI)
hoelzl@51474
   657
  apply auto []
hoelzl@51474
   658
  done
hoelzl@51471
   659
hoelzl@51471
   660
subsubsection {* Standard filters *}
hoelzl@51471
   661
hoelzl@51641
   662
definition principal :: "'a set \<Rightarrow> 'a filter" where
hoelzl@51641
   663
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
hoelzl@51641
   664
hoelzl@51641
   665
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
hoelzl@51641
   666
  unfolding principal_def
hoelzl@51641
   667
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
hoelzl@51471
   668
hoelzl@51641
   669
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
hoelzl@51641
   670
  unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
hoelzl@51641
   671
hoelzl@51641
   672
lemma principal_UNIV[simp]: "principal UNIV = top"
hoelzl@51641
   673
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@51471
   674
hoelzl@51641
   675
lemma principal_empty[simp]: "principal {} = bot"
hoelzl@51641
   676
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@51641
   677
hoelzl@51641
   678
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
hoelzl@51641
   679
  by (auto simp: le_filter_def eventually_principal)
hoelzl@51471
   680
hoelzl@51641
   681
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
hoelzl@51641
   682
  unfolding le_filter_def eventually_principal
hoelzl@51641
   683
  apply safe
hoelzl@51641
   684
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
hoelzl@51641
   685
  apply (auto elim: eventually_elim1)
hoelzl@51641
   686
  done
hoelzl@51471
   687
hoelzl@51641
   688
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
hoelzl@51641
   689
  unfolding eq_iff by simp
hoelzl@51471
   690
hoelzl@51641
   691
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
hoelzl@51641
   692
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
hoelzl@51471
   693
hoelzl@51641
   694
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
hoelzl@51641
   695
  unfolding filter_eq_iff eventually_inf eventually_principal
hoelzl@51641
   696
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
hoelzl@51471
   697
hoelzl@51641
   698
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
hoelzl@51641
   699
  unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
hoelzl@51641
   700
hoelzl@51641
   701
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
hoelzl@51641
   702
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
hoelzl@51471
   703
hoelzl@51471
   704
subsubsection {* Topological filters *}
hoelzl@51471
   705
hoelzl@51471
   706
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
hoelzl@51471
   707
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
hoelzl@51471
   708
hoelzl@51641
   709
definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60)
hoelzl@51641
   710
  where "at a within s = inf (nhds a) (principal (s - {a}))"
hoelzl@51641
   711
hoelzl@51641
   712
abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where
hoelzl@51641
   713
  "at x \<equiv> at x within (CONST UNIV)"
hoelzl@51471
   714
hoelzl@51473
   715
abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   716
  "at_right x \<equiv> at x within {x <..}"
hoelzl@51471
   717
hoelzl@51473
   718
abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   719
  "at_left x \<equiv> at x within {..< x}"
hoelzl@51471
   720
hoelzl@51473
   721
lemma (in topological_space) eventually_nhds:
hoelzl@51471
   722
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
hoelzl@51471
   723
  unfolding nhds_def
hoelzl@51471
   724
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51473
   725
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
hoelzl@51471
   726
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
hoelzl@51471
   727
next
hoelzl@51471
   728
  fix P Q
hoelzl@51471
   729
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
hoelzl@51471
   730
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
hoelzl@51471
   731
  then obtain S T where
hoelzl@51471
   732
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
hoelzl@51471
   733
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
hoelzl@51471
   734
  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
   735
    by (simp add: open_Int)
hoelzl@51471
   736
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" ..
hoelzl@51471
   737
qed auto
hoelzl@51471
   738
hoelzl@51471
   739
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
hoelzl@51471
   740
  unfolding trivial_limit_def eventually_nhds by simp
hoelzl@51471
   741
hoelzl@51641
   742
lemma eventually_at_filter:
hoelzl@51641
   743
  "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
hoelzl@51641
   744
  unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)
hoelzl@51641
   745
hoelzl@51641
   746
lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
hoelzl@51641
   747
  unfolding at_within_def by (intro inf_mono) auto
hoelzl@51641
   748
hoelzl@51471
   749
lemma eventually_at_topological:
hoelzl@51641
   750
  "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
   751
  unfolding eventually_nhds eventually_at_filter by simp
hoelzl@51471
   752
hoelzl@51481
   753
lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
hoelzl@51641
   754
  unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
hoelzl@51481
   755
hoelzl@51471
   756
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
hoelzl@51471
   757
  unfolding trivial_limit_def eventually_at_topological
hoelzl@51471
   758
  by (safe, case_tac "S = {a}", simp, fast, fast)
hoelzl@51471
   759
hoelzl@51471
   760
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
hoelzl@51471
   761
  by (simp add: at_eq_bot_iff not_open_singleton)
hoelzl@51471
   762
hoelzl@51471
   763
lemma eventually_at_right:
hoelzl@51471
   764
  fixes x :: "'a :: {no_top, linorder_topology}"
hoelzl@51471
   765
  shows "eventually P (at_right x) \<longleftrightarrow> (\<exists>b. x < b \<and> (\<forall>z. x < z \<and> z < b \<longrightarrow> P z))"
hoelzl@51641
   766
  unfolding eventually_at_topological
hoelzl@51471
   767
proof safe
hoelzl@51480
   768
  from gt_ex[of x] guess y ..
hoelzl@51480
   769
  moreover fix S assume "open S" "x \<in> S" note open_right[OF this, of y]
hoelzl@51480
   770
  moreover note gt_ex[of x]
hoelzl@51641
   771
  moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {x<..} \<longrightarrow> P s"
hoelzl@51471
   772
  ultimately show "\<exists>b>x. \<forall>z. x < z \<and> z < b \<longrightarrow> P z"
hoelzl@51471
   773
    by (auto simp: subset_eq Ball_def)
hoelzl@51471
   774
next
hoelzl@51471
   775
  fix b assume "x < b" "\<forall>z. x < z \<and> z < b \<longrightarrow> P z"
hoelzl@51641
   776
  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
   777
    by (intro exI[of _ "{..< b}"]) auto
hoelzl@51471
   778
qed
hoelzl@51471
   779
hoelzl@51471
   780
lemma eventually_at_left:
hoelzl@51471
   781
  fixes x :: "'a :: {no_bot, linorder_topology}"
hoelzl@51471
   782
  shows "eventually P (at_left x) \<longleftrightarrow> (\<exists>b. x > b \<and> (\<forall>z. b < z \<and> z < x \<longrightarrow> P z))"
hoelzl@51641
   783
  unfolding eventually_at_topological
hoelzl@51471
   784
proof safe
hoelzl@51480
   785
  from lt_ex[of x] guess y ..
hoelzl@51480
   786
  moreover fix S assume "open S" "x \<in> S" note open_left[OF this, of y]
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. b < z \<and> z < x \<longrightarrow> P z"
hoelzl@51471
   789
    by (auto simp: subset_eq Ball_def)
hoelzl@51471
   790
next
hoelzl@51471
   791
  fix b assume "b < x" "\<forall>z. b < z \<and> z < x \<longrightarrow> P z"
hoelzl@51641
   792
  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
   793
    by (intro exI[of _ "{b <..}"]) auto
hoelzl@51471
   794
qed
hoelzl@51471
   795
hoelzl@51471
   796
lemma trivial_limit_at_left_real [simp]:
hoelzl@53215
   797
  "\<not> trivial_limit (at_left (x::'a::{no_bot, unbounded_dense_linorder, linorder_topology}))"
hoelzl@51471
   798
  unfolding trivial_limit_def eventually_at_left by (auto dest: dense)
hoelzl@51471
   799
hoelzl@51471
   800
lemma trivial_limit_at_right_real [simp]:
hoelzl@53215
   801
  "\<not> trivial_limit (at_right (x::'a::{no_top, unbounded_dense_linorder, linorder_topology}))"
hoelzl@51471
   802
  unfolding trivial_limit_def eventually_at_right by (auto dest: dense)
hoelzl@51471
   803
hoelzl@51471
   804
lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"
hoelzl@51641
   805
  by (auto simp: eventually_at_filter filter_eq_iff eventually_sup 
hoelzl@51471
   806
           elim: eventually_elim2 eventually_elim1)
hoelzl@51471
   807
hoelzl@51471
   808
lemma eventually_at_split:
hoelzl@51471
   809
  "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
hoelzl@51471
   810
  by (subst at_eq_sup_left_right) (simp add: eventually_sup)
hoelzl@51471
   811
hoelzl@51471
   812
subsection {* Limits *}
hoelzl@51471
   813
hoelzl@51471
   814
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@51471
   815
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@51471
   816
hoelzl@51471
   817
syntax
hoelzl@51471
   818
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@51471
   819
hoelzl@51471
   820
translations
hoelzl@51471
   821
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@51471
   822
hoelzl@51471
   823
lemma filterlim_iff:
hoelzl@51471
   824
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
hoelzl@51471
   825
  unfolding filterlim_def le_filter_def eventually_filtermap ..
hoelzl@51471
   826
hoelzl@51471
   827
lemma filterlim_compose:
hoelzl@51471
   828
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@51471
   829
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@51471
   830
hoelzl@51471
   831
lemma filterlim_mono:
hoelzl@51471
   832
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@51471
   833
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@51471
   834
hoelzl@51471
   835
lemma filterlim_ident: "LIM x F. x :> F"
hoelzl@51471
   836
  by (simp add: filterlim_def filtermap_ident)
hoelzl@51471
   837
hoelzl@51471
   838
lemma filterlim_cong:
hoelzl@51471
   839
  "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
   840
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
hoelzl@51471
   841
hoelzl@51641
   842
lemma filterlim_principal:
hoelzl@51641
   843
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@51641
   844
  unfolding filterlim_def eventually_filtermap le_principal ..
hoelzl@51641
   845
hoelzl@51641
   846
lemma filterlim_inf:
hoelzl@51641
   847
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
hoelzl@51641
   848
  unfolding filterlim_def by simp
hoelzl@51471
   849
hoelzl@51471
   850
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
hoelzl@51471
   851
  unfolding filterlim_def filtermap_filtermap ..
hoelzl@51471
   852
hoelzl@51471
   853
lemma filterlim_sup:
hoelzl@51471
   854
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
hoelzl@51471
   855
  unfolding filterlim_def filtermap_sup by auto
hoelzl@51471
   856
hoelzl@51471
   857
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
hoelzl@51471
   858
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
hoelzl@51471
   859
hoelzl@51471
   860
subsubsection {* Tendsto *}
hoelzl@51471
   861
hoelzl@51471
   862
abbreviation (in topological_space)
hoelzl@51471
   863
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
hoelzl@51471
   864
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
hoelzl@51471
   865
hoelzl@51478
   866
definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
   867
  "Lim A f = (THE l. (f ---> l) A)"
hoelzl@51478
   868
hoelzl@51471
   869
lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F"
hoelzl@51471
   870
  by simp
hoelzl@51471
   871
hoelzl@51471
   872
ML {*
hoelzl@51471
   873
hoelzl@51471
   874
structure Tendsto_Intros = Named_Thms
hoelzl@51471
   875
(
hoelzl@51471
   876
  val name = @{binding tendsto_intros}
hoelzl@51471
   877
  val description = "introduction rules for tendsto"
hoelzl@51471
   878
)
hoelzl@51471
   879
hoelzl@51471
   880
*}
hoelzl@51471
   881
hoelzl@51471
   882
setup {*
hoelzl@51471
   883
  Tendsto_Intros.setup #>
hoelzl@51471
   884
  Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},
hoelzl@51641
   885
    map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])) o Tendsto_Intros.get o Context.proof_of);
hoelzl@51471
   886
*}
hoelzl@51471
   887
hoelzl@51473
   888
lemma (in topological_space) tendsto_def:
hoelzl@51473
   889
   "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@51471
   890
  unfolding filterlim_def
hoelzl@51471
   891
proof safe
hoelzl@51471
   892
  fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
hoelzl@51471
   893
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   894
    unfolding eventually_nhds eventually_filtermap le_filter_def
hoelzl@51471
   895
    by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
hoelzl@51471
   896
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
hoelzl@51471
   897
hoelzl@51471
   898
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
hoelzl@51471
   899
  unfolding tendsto_def le_filter_def by fast
hoelzl@51471
   900
hoelzl@51641
   901
lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)"
hoelzl@51641
   902
  by (blast intro: tendsto_mono at_le)
hoelzl@51641
   903
hoelzl@51641
   904
lemma filterlim_at:
hoelzl@51641
   905
  "(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
   906
  by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
hoelzl@51641
   907
hoelzl@51473
   908
lemma (in topological_space) topological_tendstoI:
hoelzl@51641
   909
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f ---> l) F"
hoelzl@51471
   910
  unfolding tendsto_def by auto
hoelzl@51471
   911
hoelzl@51473
   912
lemma (in topological_space) topological_tendstoD:
hoelzl@51471
   913
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   914
  unfolding tendsto_def by auto
hoelzl@51471
   915
hoelzl@51471
   916
lemma order_tendstoI:
hoelzl@51471
   917
  fixes y :: "_ :: order_topology"
hoelzl@51471
   918
  assumes "\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
hoelzl@51471
   919
  assumes "\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
hoelzl@51471
   920
  shows "(f ---> y) F"
hoelzl@51471
   921
proof (rule topological_tendstoI)
hoelzl@51471
   922
  fix S assume "open S" "y \<in> S"
hoelzl@51471
   923
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   924
    unfolding open_generated_order
hoelzl@51471
   925
  proof induct
hoelzl@51471
   926
    case (UN K)
hoelzl@51471
   927
    then obtain k where "y \<in> k" "k \<in> K" by auto
hoelzl@51471
   928
    with UN(2)[of k] show ?case
hoelzl@51471
   929
      by (auto elim: eventually_elim1)
hoelzl@51471
   930
  qed (insert assms, auto elim: eventually_elim2)
hoelzl@51471
   931
qed
hoelzl@51471
   932
hoelzl@51471
   933
lemma order_tendstoD:
hoelzl@51471
   934
  fixes y :: "_ :: order_topology"
hoelzl@51471
   935
  assumes y: "(f ---> y) F"
hoelzl@51471
   936
  shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
hoelzl@51471
   937
    and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
hoelzl@51471
   938
  using topological_tendstoD[OF y, of "{..< a}"] topological_tendstoD[OF y, of "{a <..}"] by auto
hoelzl@51471
   939
hoelzl@51471
   940
lemma order_tendsto_iff: 
hoelzl@51471
   941
  fixes f :: "_ \<Rightarrow> 'a :: order_topology"
hoelzl@51471
   942
  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
   943
  by (metis order_tendstoI order_tendstoD)
hoelzl@51471
   944
hoelzl@51471
   945
lemma tendsto_bot [simp]: "(f ---> a) bot"
hoelzl@51471
   946
  unfolding tendsto_def by simp
hoelzl@51471
   947
hoelzl@51641
   948
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a within s)"
hoelzl@51471
   949
  unfolding tendsto_def eventually_at_topological by auto
hoelzl@51471
   950
hoelzl@51478
   951
lemma (in topological_space) tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
hoelzl@51471
   952
  by (simp add: tendsto_def)
hoelzl@51471
   953
hoelzl@51478
   954
lemma (in t2_space) tendsto_unique:
hoelzl@51471
   955
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
hoelzl@51471
   956
  shows "a = b"
hoelzl@51471
   957
proof (rule ccontr)
hoelzl@51471
   958
  assume "a \<noteq> b"
hoelzl@51471
   959
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
hoelzl@51471
   960
    using hausdorff [OF `a \<noteq> b`] by fast
hoelzl@51471
   961
  have "eventually (\<lambda>x. f x \<in> U) F"
hoelzl@51471
   962
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
hoelzl@51471
   963
  moreover
hoelzl@51471
   964
  have "eventually (\<lambda>x. f x \<in> V) F"
hoelzl@51471
   965
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
hoelzl@51471
   966
  ultimately
hoelzl@51471
   967
  have "eventually (\<lambda>x. False) F"
hoelzl@51471
   968
  proof eventually_elim
hoelzl@51471
   969
    case (elim x)
hoelzl@51471
   970
    hence "f x \<in> U \<inter> V" by simp
hoelzl@51471
   971
    with `U \<inter> V = {}` show ?case by simp
hoelzl@51471
   972
  qed
hoelzl@51471
   973
  with `\<not> trivial_limit F` show "False"
hoelzl@51471
   974
    by (simp add: trivial_limit_def)
hoelzl@51471
   975
qed
hoelzl@51471
   976
hoelzl@51478
   977
lemma (in t2_space) tendsto_const_iff:
hoelzl@51478
   978
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"
hoelzl@51471
   979
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
hoelzl@51471
   980
hoelzl@51471
   981
lemma increasing_tendsto:
hoelzl@51471
   982
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
   983
  assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
hoelzl@51471
   984
      and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
hoelzl@51471
   985
  shows "(f ---> l) F"
hoelzl@51471
   986
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
hoelzl@51471
   987
hoelzl@51471
   988
lemma decreasing_tendsto:
hoelzl@51471
   989
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
   990
  assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
hoelzl@51471
   991
      and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
hoelzl@51471
   992
  shows "(f ---> l) F"
hoelzl@51471
   993
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
hoelzl@51471
   994
hoelzl@51471
   995
lemma tendsto_sandwich:
hoelzl@51471
   996
  fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
hoelzl@51471
   997
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
hoelzl@51471
   998
  assumes lim: "(f ---> c) net" "(h ---> c) net"
hoelzl@51471
   999
  shows "(g ---> c) net"
hoelzl@51471
  1000
proof (rule order_tendstoI)
hoelzl@51471
  1001
  fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
hoelzl@51471
  1002
    using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
  1003
next
hoelzl@51471
  1004
  fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
hoelzl@51471
  1005
    using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
  1006
qed
hoelzl@51471
  1007
hoelzl@51471
  1008
lemma tendsto_le:
hoelzl@51471
  1009
  fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
  1010
  assumes F: "\<not> trivial_limit F"
hoelzl@51471
  1011
  assumes x: "(f ---> x) F" and y: "(g ---> y) F"
hoelzl@51471
  1012
  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
hoelzl@51471
  1013
  shows "y \<le> x"
hoelzl@51471
  1014
proof (rule ccontr)
hoelzl@51471
  1015
  assume "\<not> y \<le> x"
hoelzl@51471
  1016
  with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
hoelzl@51471
  1017
    by (auto simp: not_le)
hoelzl@51471
  1018
  then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
hoelzl@51471
  1019
    using x y by (auto intro: order_tendstoD)
hoelzl@51471
  1020
  with ev have "eventually (\<lambda>x. False) F"
hoelzl@51471
  1021
    by eventually_elim (insert xy, fastforce)
hoelzl@51471
  1022
  with F show False
hoelzl@51471
  1023
    by (simp add: eventually_False)
hoelzl@51471
  1024
qed
hoelzl@51471
  1025
hoelzl@51471
  1026
lemma tendsto_le_const:
hoelzl@51471
  1027
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
  1028
  assumes F: "\<not> trivial_limit F"
hoelzl@51471
  1029
  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>x. a \<le> f x) F"
hoelzl@51471
  1030
  shows "a \<le> x"
hoelzl@51471
  1031
  using F x tendsto_const a by (rule tendsto_le)
hoelzl@51471
  1032
hoelzl@51478
  1033
subsubsection {* Rules about @{const Lim} *}
hoelzl@51478
  1034
hoelzl@51478
  1035
lemma (in t2_space) tendsto_Lim:
hoelzl@51478
  1036
  "\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l"
hoelzl@51478
  1037
  unfolding Lim_def using tendsto_unique[of net f] by auto
hoelzl@51478
  1038
hoelzl@51641
  1039
lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
hoelzl@51478
  1040
  by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
hoelzl@51478
  1041
hoelzl@51471
  1042
subsection {* Limits to @{const at_top} and @{const at_bot} *}
hoelzl@51471
  1043
hoelzl@51471
  1044
lemma filterlim_at_top:
hoelzl@51471
  1045
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@51471
  1046
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1047
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
hoelzl@51471
  1048
hoelzl@51471
  1049
lemma filterlim_at_top_dense:
hoelzl@53215
  1050
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
hoelzl@51471
  1051
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@51471
  1052
  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
hoelzl@51471
  1053
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
hoelzl@51471
  1054
hoelzl@51471
  1055
lemma filterlim_at_top_ge:
hoelzl@51471
  1056
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@51471
  1057
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1058
  unfolding filterlim_at_top
hoelzl@51471
  1059
proof safe
hoelzl@51471
  1060
  fix Z assume *: "\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F"
hoelzl@51471
  1061
  with *[THEN spec, of "max Z c"] show "eventually (\<lambda>x. Z \<le> f x) F"
hoelzl@51471
  1062
    by (auto elim!: eventually_elim1)
hoelzl@51471
  1063
qed simp
hoelzl@51471
  1064
hoelzl@51471
  1065
lemma filterlim_at_top_at_top:
hoelzl@51471
  1066
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
hoelzl@51471
  1067
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1068
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1069
  assumes Q: "eventually Q at_top"
hoelzl@51471
  1070
  assumes P: "eventually P at_top"
hoelzl@51471
  1071
  shows "filterlim f at_top at_top"
hoelzl@51471
  1072
proof -
hoelzl@51471
  1073
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
  1074
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
  1075
  show ?thesis
hoelzl@51471
  1076
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
  1077
    fix z assume "x \<le> z"
hoelzl@51471
  1078
    with x have "P z" by auto
hoelzl@51471
  1079
    have "eventually (\<lambda>x. g z \<le> x) at_top"
hoelzl@51471
  1080
      by (rule eventually_ge_at_top)
hoelzl@51471
  1081
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
hoelzl@51471
  1082
      by eventually_elim (metis mono bij `P z`)
hoelzl@51471
  1083
  qed
hoelzl@51471
  1084
qed
hoelzl@51471
  1085
hoelzl@51471
  1086
lemma filterlim_at_top_gt:
hoelzl@53215
  1087
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@51471
  1088
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1089
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
hoelzl@51471
  1090
hoelzl@51471
  1091
lemma filterlim_at_bot: 
hoelzl@51471
  1092
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@51471
  1093
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
hoelzl@51471
  1094
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
hoelzl@51471
  1095
hoelzl@51471
  1096
lemma filterlim_at_bot_le:
hoelzl@51471
  1097
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@51471
  1098
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@51471
  1099
  unfolding filterlim_at_bot
hoelzl@51471
  1100
proof safe
hoelzl@51471
  1101
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@51471
  1102
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@51471
  1103
    by (auto elim!: eventually_elim1)
hoelzl@51471
  1104
qed simp
hoelzl@51471
  1105
hoelzl@51471
  1106
lemma filterlim_at_bot_lt:
hoelzl@53215
  1107
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@51471
  1108
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@51471
  1109
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
hoelzl@51471
  1110
hoelzl@51471
  1111
lemma filterlim_at_bot_at_right:
hoelzl@51471
  1112
  fixes f :: "'a::{no_top, linorder_topology} \<Rightarrow> 'b::linorder"
hoelzl@51471
  1113
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1114
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1115
  assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
hoelzl@51471
  1116
  assumes P: "eventually P at_bot"
hoelzl@51471
  1117
  shows "filterlim f at_bot (at_right a)"
hoelzl@51471
  1118
proof -
hoelzl@51471
  1119
  from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
hoelzl@51471
  1120
    unfolding eventually_at_bot_linorder by auto
hoelzl@51471
  1121
  show ?thesis
hoelzl@51471
  1122
  proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
hoelzl@51471
  1123
    fix z assume "z \<le> x"
hoelzl@51471
  1124
    with x have "P z" by auto
hoelzl@51471
  1125
    have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
hoelzl@51471
  1126
      using bound[OF bij(2)[OF `P z`]]
hoelzl@51471
  1127
      unfolding eventually_at_right by (auto intro!: exI[of _ "g z"])
hoelzl@51471
  1128
    with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
hoelzl@51471
  1129
      by eventually_elim (metis bij `P z` mono)
hoelzl@51471
  1130
  qed
hoelzl@51471
  1131
qed
hoelzl@51471
  1132
hoelzl@51471
  1133
lemma filterlim_at_top_at_left:
hoelzl@51471
  1134
  fixes f :: "'a::{no_bot, linorder_topology} \<Rightarrow> 'b::linorder"
hoelzl@51471
  1135
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1136
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1137
  assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
hoelzl@51471
  1138
  assumes P: "eventually P at_top"
hoelzl@51471
  1139
  shows "filterlim f at_top (at_left a)"
hoelzl@51471
  1140
proof -
hoelzl@51471
  1141
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
  1142
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
  1143
  show ?thesis
hoelzl@51471
  1144
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
  1145
    fix z assume "x \<le> z"
hoelzl@51471
  1146
    with x have "P z" by auto
hoelzl@51471
  1147
    have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
hoelzl@51471
  1148
      using bound[OF bij(2)[OF `P z`]]
hoelzl@51471
  1149
      unfolding eventually_at_left by (auto intro!: exI[of _ "g z"])
hoelzl@51471
  1150
    with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
hoelzl@51471
  1151
      by eventually_elim (metis bij `P z` mono)
hoelzl@51471
  1152
  qed
hoelzl@51471
  1153
qed
hoelzl@51471
  1154
hoelzl@51471
  1155
lemma filterlim_split_at:
hoelzl@51471
  1156
  "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"
hoelzl@51471
  1157
  by (subst at_eq_sup_left_right) (rule filterlim_sup)
hoelzl@51471
  1158
hoelzl@51471
  1159
lemma filterlim_at_split:
hoelzl@51471
  1160
  "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
hoelzl@51471
  1161
  by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
hoelzl@51471
  1162
hoelzl@51471
  1163
hoelzl@51471
  1164
subsection {* Limits on sequences *}
hoelzl@51471
  1165
hoelzl@51471
  1166
abbreviation (in topological_space)
hoelzl@51471
  1167
  LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
hoelzl@51471
  1168
    ("((_)/ ----> (_))" [60, 60] 60) where
hoelzl@51471
  1169
  "X ----> L \<equiv> (X ---> L) sequentially"
hoelzl@51471
  1170
hoelzl@51478
  1171
abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
  1172
  "lim X \<equiv> Lim sequentially X"
hoelzl@51471
  1173
hoelzl@51471
  1174
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
hoelzl@51471
  1175
  "convergent X = (\<exists>L. X ----> L)"
hoelzl@51471
  1176
hoelzl@51478
  1177
lemma lim_def: "lim X = (THE L. X ----> L)"
hoelzl@51478
  1178
  unfolding Lim_def ..
hoelzl@51478
  1179
hoelzl@51471
  1180
subsubsection {* Monotone sequences and subsequences *}
hoelzl@51471
  1181
hoelzl@51471
  1182
definition
hoelzl@51471
  1183
  monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@51471
  1184
    --{*Definition of monotonicity.
hoelzl@51471
  1185
        The use of disjunction here complicates proofs considerably.
hoelzl@51471
  1186
        One alternative is to add a Boolean argument to indicate the direction.
hoelzl@51471
  1187
        Another is to develop the notions of increasing and decreasing first.*}
hoelzl@51471
  1188
  "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
hoelzl@51471
  1189
hoelzl@51471
  1190
definition
hoelzl@51471
  1191
  incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@51471
  1192
    --{*Increasing sequence*}
hoelzl@51471
  1193
  "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
hoelzl@51471
  1194
hoelzl@51471
  1195
definition
hoelzl@51471
  1196
  decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@51471
  1197
    --{*Decreasing sequence*}
hoelzl@51471
  1198
  "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
hoelzl@51471
  1199
hoelzl@51471
  1200
definition
hoelzl@51471
  1201
  subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
hoelzl@51471
  1202
    --{*Definition of subsequence*}
hoelzl@51471
  1203
  "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
hoelzl@51471
  1204
hoelzl@51471
  1205
lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
hoelzl@51471
  1206
  unfolding mono_def incseq_def by auto
hoelzl@51471
  1207
hoelzl@51471
  1208
lemma incseq_SucI:
hoelzl@51471
  1209
  "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
hoelzl@51471
  1210
  using lift_Suc_mono_le[of X]
hoelzl@51471
  1211
  by (auto simp: incseq_def)
hoelzl@51471
  1212
hoelzl@51471
  1213
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
hoelzl@51471
  1214
  by (auto simp: incseq_def)
hoelzl@51471
  1215
hoelzl@51471
  1216
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
hoelzl@51471
  1217
  using incseqD[of A i "Suc i"] by auto
hoelzl@51471
  1218
hoelzl@51471
  1219
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
hoelzl@51471
  1220
  by (auto intro: incseq_SucI dest: incseq_SucD)
hoelzl@51471
  1221
hoelzl@51471
  1222
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
hoelzl@51471
  1223
  unfolding incseq_def by auto
hoelzl@51471
  1224
hoelzl@51471
  1225
lemma decseq_SucI:
hoelzl@51471
  1226
  "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
hoelzl@51471
  1227
  using order.lift_Suc_mono_le[OF dual_order, of X]
hoelzl@51471
  1228
  by (auto simp: decseq_def)
hoelzl@51471
  1229
hoelzl@51471
  1230
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
hoelzl@51471
  1231
  by (auto simp: decseq_def)
hoelzl@51471
  1232
hoelzl@51471
  1233
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
hoelzl@51471
  1234
  using decseqD[of A i "Suc i"] by auto
hoelzl@51471
  1235
hoelzl@51471
  1236
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@51471
  1237
  by (auto intro: decseq_SucI dest: decseq_SucD)
hoelzl@51471
  1238
hoelzl@51471
  1239
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
hoelzl@51471
  1240
  unfolding decseq_def by auto
hoelzl@51471
  1241
hoelzl@51471
  1242
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
hoelzl@51471
  1243
  unfolding monoseq_def incseq_def decseq_def ..
hoelzl@51471
  1244
hoelzl@51471
  1245
lemma monoseq_Suc:
hoelzl@51471
  1246
  "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
hoelzl@51471
  1247
  unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
hoelzl@51471
  1248
hoelzl@51471
  1249
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
hoelzl@51471
  1250
by (simp add: monoseq_def)
hoelzl@51471
  1251
hoelzl@51471
  1252
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
hoelzl@51471
  1253
by (simp add: monoseq_def)
hoelzl@51471
  1254
hoelzl@51471
  1255
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
hoelzl@51471
  1256
by (simp add: monoseq_Suc)
hoelzl@51471
  1257
hoelzl@51471
  1258
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
hoelzl@51471
  1259
by (simp add: monoseq_Suc)
hoelzl@51471
  1260
hoelzl@51471
  1261
lemma monoseq_minus:
hoelzl@51471
  1262
  fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
hoelzl@51471
  1263
  assumes "monoseq a"
hoelzl@51471
  1264
  shows "monoseq (\<lambda> n. - a n)"
hoelzl@51471
  1265
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
hoelzl@51471
  1266
  case True
hoelzl@51471
  1267
  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
hoelzl@51471
  1268
  thus ?thesis by (rule monoI2)
hoelzl@51471
  1269
next
hoelzl@51471
  1270
  case False
hoelzl@51471
  1271
  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
hoelzl@51471
  1272
  thus ?thesis by (rule monoI1)
hoelzl@51471
  1273
qed
hoelzl@51471
  1274
hoelzl@51471
  1275
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
hoelzl@51471
  1276
hoelzl@51471
  1277
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
hoelzl@51471
  1278
apply (simp add: subseq_def)
hoelzl@51471
  1279
apply (auto dest!: less_imp_Suc_add)
hoelzl@51471
  1280
apply (induct_tac k)
hoelzl@51471
  1281
apply (auto intro: less_trans)
hoelzl@51471
  1282
done
hoelzl@51471
  1283
hoelzl@51471
  1284
text{* for any sequence, there is a monotonic subsequence *}
hoelzl@51471
  1285
lemma seq_monosub:
hoelzl@51471
  1286
  fixes s :: "nat => 'a::linorder"
hoelzl@51471
  1287
  shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
hoelzl@51471
  1288
proof cases
hoelzl@51471
  1289
  let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"
hoelzl@51471
  1290
  assume *: "\<forall>n. \<exists>p. ?P p n"
hoelzl@51471
  1291
  def f \<equiv> "nat_rec (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"
hoelzl@51471
  1292
  have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
hoelzl@51471
  1293
  have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
hoelzl@51471
  1294
  have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
hoelzl@51471
  1295
  have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
hoelzl@51471
  1296
  then have "subseq f" unfolding subseq_Suc_iff by auto
hoelzl@51471
  1297
  moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc
hoelzl@51471
  1298
  proof (intro disjI2 allI)
hoelzl@51471
  1299
    fix n show "s (f (Suc n)) \<le> s (f n)"
hoelzl@51471
  1300
    proof (cases n)
hoelzl@51471
  1301
      case 0 with P_Suc[of 0] P_0 show ?thesis by auto
hoelzl@51471
  1302
    next
hoelzl@51471
  1303
      case (Suc m)
hoelzl@51471
  1304
      from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp
hoelzl@51471
  1305
      with P_Suc Suc show ?thesis by simp
hoelzl@51471
  1306
    qed
hoelzl@51471
  1307
  qed
hoelzl@51471
  1308
  ultimately show ?thesis by auto
hoelzl@51471
  1309
next
hoelzl@51471
  1310
  let "?P p m" = "m < p \<and> s m < s p"
hoelzl@51471
  1311
  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
hoelzl@51471
  1312
  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
hoelzl@51471
  1313
  def f \<equiv> "nat_rec (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"
hoelzl@51471
  1314
  have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
hoelzl@51471
  1315
  have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
hoelzl@51471
  1316
  have P_0: "?P (f 0) (Suc N)"
hoelzl@51471
  1317
    unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto
hoelzl@51471
  1318
  { fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"
hoelzl@51471
  1319
      unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }
hoelzl@51471
  1320
  note P' = this
hoelzl@51471
  1321
  { fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"
hoelzl@51471
  1322
      by (induct i) (insert P_0 P', auto) }
hoelzl@51471
  1323
  then have "subseq f" "monoseq (\<lambda>x. s (f x))"
hoelzl@51471
  1324
    unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
hoelzl@51471
  1325
  then show ?thesis by auto
hoelzl@51471
  1326
qed
hoelzl@51471
  1327
hoelzl@51471
  1328
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
hoelzl@51471
  1329
proof(induct n)
hoelzl@51471
  1330
  case 0 thus ?case by simp
hoelzl@51471
  1331
next
hoelzl@51471
  1332
  case (Suc n)
hoelzl@51471
  1333
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
hoelzl@51471
  1334
  have "n < f (Suc n)" by arith
hoelzl@51471
  1335
  thus ?case by arith
hoelzl@51471
  1336
qed
hoelzl@51471
  1337
hoelzl@51471
  1338
lemma eventually_subseq:
hoelzl@51471
  1339
  "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
hoelzl@51471
  1340
  unfolding eventually_sequentially by (metis seq_suble le_trans)
hoelzl@51471
  1341
hoelzl@51473
  1342
lemma not_eventually_sequentiallyD:
hoelzl@51473
  1343
  assumes P: "\<not> eventually P sequentially"
hoelzl@51473
  1344
  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
hoelzl@51473
  1345
proof -
hoelzl@51473
  1346
  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
hoelzl@51473
  1347
    unfolding eventually_sequentially by (simp add: not_less)
hoelzl@51473
  1348
  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
hoelzl@51473
  1349
    by (auto simp: choice_iff)
hoelzl@51473
  1350
  then show ?thesis
hoelzl@51473
  1351
    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
hoelzl@51473
  1352
             simp: less_eq_Suc_le subseq_Suc_iff)
hoelzl@51473
  1353
qed
hoelzl@51473
  1354
hoelzl@51471
  1355
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
hoelzl@51471
  1356
  unfolding filterlim_iff by (metis eventually_subseq)
hoelzl@51471
  1357
hoelzl@51471
  1358
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
hoelzl@51471
  1359
  unfolding subseq_def by simp
hoelzl@51471
  1360
hoelzl@51471
  1361
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
hoelzl@51471
  1362
  using assms by (auto simp: subseq_def)
hoelzl@51471
  1363
hoelzl@51471
  1364
lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
hoelzl@51471
  1365
  by (simp add: incseq_def monoseq_def)
hoelzl@51471
  1366
hoelzl@51471
  1367
lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
hoelzl@51471
  1368
  by (simp add: decseq_def monoseq_def)
hoelzl@51471
  1369
hoelzl@51471
  1370
lemma decseq_eq_incseq:
hoelzl@51471
  1371
  fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" 
hoelzl@51471
  1372
  by (simp add: decseq_def incseq_def)
hoelzl@51471
  1373
hoelzl@51471
  1374
lemma INT_decseq_offset:
hoelzl@51471
  1375
  assumes "decseq F"
hoelzl@51471
  1376
  shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
  1377
proof safe
hoelzl@51471
  1378
  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
  1379
  show "x \<in> F i"
hoelzl@51471
  1380
  proof cases
hoelzl@51471
  1381
    from x have "x \<in> F n" by auto
hoelzl@51471
  1382
    also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
hoelzl@51471
  1383
      unfolding decseq_def by simp
hoelzl@51471
  1384
    finally show ?thesis .
hoelzl@51471
  1385
  qed (insert x, simp)
hoelzl@51471
  1386
qed auto
hoelzl@51471
  1387
hoelzl@51471
  1388
lemma LIMSEQ_const_iff:
hoelzl@51471
  1389
  fixes k l :: "'a::t2_space"
hoelzl@51471
  1390
  shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
hoelzl@51471
  1391
  using trivial_limit_sequentially by (rule tendsto_const_iff)
hoelzl@51471
  1392
hoelzl@51471
  1393
lemma LIMSEQ_SUP:
hoelzl@51471
  1394
  "incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1395
  by (intro increasing_tendsto)
hoelzl@51471
  1396
     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
hoelzl@51471
  1397
hoelzl@51471
  1398
lemma LIMSEQ_INF:
hoelzl@51471
  1399
  "decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1400
  by (intro decreasing_tendsto)
hoelzl@51471
  1401
     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
hoelzl@51471
  1402
hoelzl@51471
  1403
lemma LIMSEQ_ignore_initial_segment:
hoelzl@51471
  1404
  "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
hoelzl@51474
  1405
  unfolding tendsto_def
hoelzl@51474
  1406
  by (subst eventually_sequentially_seg[where k=k])
hoelzl@51471
  1407
hoelzl@51471
  1408
lemma LIMSEQ_offset:
hoelzl@51471
  1409
  "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
hoelzl@51474
  1410
  unfolding tendsto_def
hoelzl@51474
  1411
  by (subst (asm) eventually_sequentially_seg[where k=k])
hoelzl@51471
  1412
hoelzl@51471
  1413
lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
hoelzl@51471
  1414
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
hoelzl@51471
  1415
hoelzl@51471
  1416
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
hoelzl@51471
  1417
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
hoelzl@51471
  1418
hoelzl@51471
  1419
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
hoelzl@51471
  1420
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
hoelzl@51471
  1421
hoelzl@51471
  1422
lemma LIMSEQ_unique:
hoelzl@51471
  1423
  fixes a b :: "'a::t2_space"
hoelzl@51471
  1424
  shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
hoelzl@51471
  1425
  using trivial_limit_sequentially by (rule tendsto_unique)
hoelzl@51471
  1426
hoelzl@51471
  1427
lemma LIMSEQ_le_const:
hoelzl@51471
  1428
  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
hoelzl@51471
  1429
  using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
hoelzl@51471
  1430
hoelzl@51471
  1431
lemma LIMSEQ_le:
hoelzl@51471
  1432
  "\<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
  1433
  using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
hoelzl@51471
  1434
hoelzl@51471
  1435
lemma LIMSEQ_le_const2:
hoelzl@51471
  1436
  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
hoelzl@51471
  1437
  by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) (auto simp: tendsto_const)
hoelzl@51471
  1438
hoelzl@51471
  1439
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
hoelzl@51471
  1440
by (simp add: convergent_def)
hoelzl@51471
  1441
hoelzl@51471
  1442
lemma convergentI: "(X ----> L) ==> convergent X"
hoelzl@51471
  1443
by (auto simp add: convergent_def)
hoelzl@51471
  1444
hoelzl@51471
  1445
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
hoelzl@51471
  1446
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
hoelzl@51471
  1447
hoelzl@51471
  1448
lemma convergent_const: "convergent (\<lambda>n. c)"
hoelzl@51471
  1449
  by (rule convergentI, rule tendsto_const)
hoelzl@51471
  1450
hoelzl@51471
  1451
lemma monoseq_le:
hoelzl@51471
  1452
  "monoseq a \<Longrightarrow> a ----> (x::'a::linorder_topology) \<Longrightarrow>
hoelzl@51471
  1453
    ((\<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
  1454
  by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
hoelzl@51471
  1455
hoelzl@51471
  1456
lemma LIMSEQ_subseq_LIMSEQ:
hoelzl@51471
  1457
  "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
hoelzl@51471
  1458
  unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
hoelzl@51471
  1459
hoelzl@51471
  1460
lemma convergent_subseq_convergent:
hoelzl@51471
  1461
  "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
hoelzl@51471
  1462
  unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
hoelzl@51471
  1463
hoelzl@51471
  1464
lemma limI: "X ----> L ==> lim X = L"
hoelzl@51471
  1465
apply (simp add: lim_def)
hoelzl@51471
  1466
apply (blast intro: LIMSEQ_unique)
hoelzl@51471
  1467
done
hoelzl@51471
  1468
hoelzl@51471
  1469
lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"
hoelzl@51471
  1470
  using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
hoelzl@51471
  1471
hoelzl@51471
  1472
subsubsection{*Increasing and Decreasing Series*}
hoelzl@51471
  1473
hoelzl@51471
  1474
lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
hoelzl@51471
  1475
  by (metis incseq_def LIMSEQ_le_const)
hoelzl@51471
  1476
hoelzl@51471
  1477
lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
hoelzl@51471
  1478
  by (metis decseq_def LIMSEQ_le_const2)
hoelzl@51471
  1479
hoelzl@51473
  1480
subsection {* First countable topologies *}
hoelzl@51473
  1481
hoelzl@51473
  1482
class first_countable_topology = topological_space +
hoelzl@51473
  1483
  assumes first_countable_basis:
hoelzl@51473
  1484
    "\<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
  1485
hoelzl@51473
  1486
lemma (in first_countable_topology) countable_basis_at_decseq:
hoelzl@51473
  1487
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1488
    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
hoelzl@51473
  1489
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
hoelzl@51473
  1490
proof atomize_elim
hoelzl@51473
  1491
  from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1492
    nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1493
    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto
hoelzl@51473
  1494
  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i"
hoelzl@51473
  1495
  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
hoelzl@51473
  1496
      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
hoelzl@51473
  1497
  proof (safe intro!: exI[of _ F])
hoelzl@51473
  1498
    fix i
hoelzl@51480
  1499
    show "open (F i)" using nhds(1) by (auto simp: F_def)
hoelzl@51473
  1500
    show "x \<in> F i" using nhds(2) by (auto simp: F_def)
hoelzl@51473
  1501
  next
hoelzl@51473
  1502
    fix S assume "open S" "x \<in> S"
hoelzl@51473
  1503
    from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto
hoelzl@51473
  1504
    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
hoelzl@51473
  1505
      by (auto simp: F_def)
hoelzl@51473
  1506
    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
hoelzl@51473
  1507
      by (auto simp: eventually_sequentially)
hoelzl@51473
  1508
  qed
hoelzl@51473
  1509
qed
hoelzl@51473
  1510
hoelzl@51473
  1511
lemma (in first_countable_topology) countable_basis:
hoelzl@51473
  1512
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1513
    "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1514
    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
hoelzl@51473
  1515
proof atomize_elim
hoelzl@51473
  1516
  from countable_basis_at_decseq[of x] guess A . note A = this
hoelzl@51473
  1517
  { fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
hoelzl@51473
  1518
    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
hoelzl@51473
  1519
      by (auto elim: eventually_elim1 simp: subset_eq) }
hoelzl@51473
  1520
  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
  1521
    by (intro exI[of _ A]) (auto simp: tendsto_def)
hoelzl@51473
  1522
qed
hoelzl@51473
  1523
hoelzl@51473
  1524
lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
hoelzl@51473
  1525
  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51641
  1526
  shows "eventually P (inf (nhds a) (principal s))"
hoelzl@51473
  1527
proof (rule ccontr)
hoelzl@51473
  1528
  from countable_basis[of a] guess A . note A = this
hoelzl@51641
  1529
  assume "\<not> eventually P (inf (nhds a) (principal s))"
hoelzl@51473
  1530
  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
  1531
    unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce
hoelzl@51473
  1532
  then guess F ..
hoelzl@51473
  1533
  hence F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
hoelzl@51473
  1534
    by fast+
hoelzl@51473
  1535
  with A have "F ----> a" by auto
hoelzl@51473
  1536
  hence "eventually (\<lambda>n. P (F n)) sequentially"
hoelzl@51473
  1537
    using assms F0 by simp
hoelzl@51473
  1538
  thus "False" by (simp add: F3)
hoelzl@51473
  1539
qed
hoelzl@51473
  1540
hoelzl@51473
  1541
lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
hoelzl@51641
  1542
  "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> 
hoelzl@51473
  1543
    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1544
proof (safe intro!: sequentially_imp_eventually_nhds_within)
hoelzl@51641
  1545
  assume "eventually P (inf (nhds a) (principal s))" 
hoelzl@51473
  1546
  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
hoelzl@51641
  1547
    by (auto simp: eventually_inf_principal eventually_nhds)
hoelzl@51473
  1548
  moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
hoelzl@51473
  1549
  ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51473
  1550
    by (auto dest!: topological_tendstoD elim: eventually_elim1)
hoelzl@51473
  1551
qed
hoelzl@51473
  1552
hoelzl@51473
  1553
lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
hoelzl@51473
  1554
  "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1555
  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
hoelzl@51473
  1556
hoelzl@51471
  1557
subsection {* Function limit at a point *}
hoelzl@51471
  1558
hoelzl@51471
  1559
abbreviation
hoelzl@51471
  1560
  LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
hoelzl@51471
  1561
        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
hoelzl@51471
  1562
  "f -- a --> L \<equiv> (f ---> L) (at a)"
hoelzl@51471
  1563
hoelzl@51481
  1564
lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l) (at a within S) \<longleftrightarrow> (f -- a --> l)"
hoelzl@51641
  1565
  unfolding tendsto_def by (simp add: at_within_open[where S=S])
hoelzl@51481
  1566
hoelzl@51471
  1567
lemma LIM_const_not_eq[tendsto_intros]:
hoelzl@51471
  1568
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1569
  fixes k L :: "'b::t2_space"
hoelzl@51471
  1570
  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
hoelzl@51471
  1571
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1572
hoelzl@51471
  1573
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
hoelzl@51471
  1574
hoelzl@51471
  1575
lemma LIM_const_eq:
hoelzl@51471
  1576
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1577
  fixes k L :: "'b::t2_space"
hoelzl@51471
  1578
  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
hoelzl@51471
  1579
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1580
hoelzl@51471
  1581
lemma LIM_unique:
hoelzl@51471
  1582
  fixes a :: "'a::perfect_space" and L M :: "'b::t2_space"
hoelzl@51471
  1583
  shows "f -- a --> L \<Longrightarrow> f -- a --> M \<Longrightarrow> L = M"
hoelzl@51471
  1584
  using at_neq_bot by (rule tendsto_unique)
hoelzl@51471
  1585
hoelzl@51471
  1586
text {* Limits are equal for functions equal except at limit point *}
hoelzl@51471
  1587
hoelzl@51471
  1588
lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- a --> l)"
hoelzl@51471
  1589
  unfolding tendsto_def eventually_at_topological by simp
hoelzl@51471
  1590
hoelzl@51471
  1591
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
  1592
  by (simp add: LIM_equal)
hoelzl@51471
  1593
hoelzl@51471
  1594
lemma LIM_cong_limit: "f -- x --> L \<Longrightarrow> K = L \<Longrightarrow> f -- x --> K"
hoelzl@51471
  1595
  by simp
hoelzl@51471
  1596
hoelzl@51471
  1597
lemma tendsto_at_iff_tendsto_nhds:
hoelzl@51471
  1598
  "g -- l --> g l \<longleftrightarrow> (g ---> g l) (nhds l)"
hoelzl@51641
  1599
  unfolding tendsto_def eventually_at_filter
hoelzl@51471
  1600
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@51471
  1601
hoelzl@51471
  1602
lemma tendsto_compose:
hoelzl@51471
  1603
  "g -- l --> g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@51471
  1604
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
hoelzl@51471
  1605
hoelzl@51471
  1606
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
hoelzl@51471
  1607
  unfolding o_def by (rule tendsto_compose)
hoelzl@51471
  1608
hoelzl@51471
  1609
lemma tendsto_compose_eventually:
hoelzl@51471
  1610
  "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
  1611
  by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
hoelzl@51471
  1612
hoelzl@51471
  1613
lemma LIM_compose_eventually:
hoelzl@51471
  1614
  assumes f: "f -- a --> b"
hoelzl@51471
  1615
  assumes g: "g -- b --> c"
hoelzl@51471
  1616
  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
hoelzl@51471
  1617
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51471
  1618
  using g f inj by (rule tendsto_compose_eventually)
hoelzl@51471
  1619
hoelzl@51473
  1620
subsubsection {* Relation of LIM and LIMSEQ *}
hoelzl@51473
  1621
hoelzl@51473
  1622
lemma (in first_countable_topology) sequentially_imp_eventually_within:
hoelzl@51473
  1623
  "(\<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
  1624
    eventually P (at a within s)"
hoelzl@51641
  1625
  unfolding at_within_def
hoelzl@51473
  1626
  by (intro sequentially_imp_eventually_nhds_within) auto
hoelzl@51473
  1627
hoelzl@51473
  1628
lemma (in first_countable_topology) sequentially_imp_eventually_at:
hoelzl@51473
  1629
  "(\<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
  1630
  using assms sequentially_imp_eventually_within [where s=UNIV] by simp
hoelzl@51473
  1631
hoelzl@51473
  1632
lemma LIMSEQ_SEQ_conv1:
hoelzl@51473
  1633
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
hoelzl@51473
  1634
  assumes f: "f -- a --> l"
hoelzl@51473
  1635
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
hoelzl@51473
  1636
  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
hoelzl@51473
  1637
hoelzl@51473
  1638
lemma LIMSEQ_SEQ_conv2:
hoelzl@51473
  1639
  fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
hoelzl@51473
  1640
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
hoelzl@51473
  1641
  shows "f -- a --> l"
hoelzl@51473
  1642
  using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
hoelzl@51473
  1643
hoelzl@51473
  1644
lemma LIMSEQ_SEQ_conv:
hoelzl@51473
  1645
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
hoelzl@51473
  1646
   (X -- a --> (L::'b::topological_space))"
hoelzl@51473
  1647
  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
hoelzl@51473
  1648
hoelzl@51471
  1649
subsection {* Continuity *}
hoelzl@51471
  1650
hoelzl@51478
  1651
subsubsection {* Continuity on a set *}
hoelzl@51478
  1652
hoelzl@51478
  1653
definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where
hoelzl@51478
  1654
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
hoelzl@51478
  1655
hoelzl@51481
  1656
lemma continuous_on_cong [cong]:
hoelzl@51481
  1657
  "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
  1658
  unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
hoelzl@51481
  1659
hoelzl@51478
  1660
lemma continuous_on_topological:
hoelzl@51478
  1661
  "continuous_on s f \<longleftrightarrow>
hoelzl@51478
  1662
    (\<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
  1663
  unfolding continuous_on_def tendsto_def eventually_at_topological by metis
hoelzl@51478
  1664
hoelzl@51478
  1665
lemma continuous_on_open_invariant:
hoelzl@51478
  1666
  "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  1667
proof safe
hoelzl@51478
  1668
  fix B :: "'b set" assume "continuous_on s f" "open B"
hoelzl@51478
  1669
  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
  1670
    by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
hoelzl@51478
  1671
  then guess A unfolding bchoice_iff ..
hoelzl@51478
  1672
  then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"
hoelzl@51478
  1673
    by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto
hoelzl@51478
  1674
next
hoelzl@51478
  1675
  assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"
hoelzl@51478
  1676
  show "continuous_on s f"
hoelzl@51478
  1677
    unfolding continuous_on_topological
hoelzl@51478
  1678
  proof safe
hoelzl@51478
  1679
    fix x B assume "x \<in> s" "open B" "f x \<in> B"
hoelzl@51478
  1680
    with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto
hoelzl@51478
  1681
    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
  1682
      by (intro exI[of _ A]) auto
hoelzl@51478
  1683
  qed
hoelzl@51478
  1684
qed
hoelzl@51478
  1685
hoelzl@51481
  1686
lemma continuous_on_open_vimage:
hoelzl@51481
  1687
  "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
hoelzl@51481
  1688
  unfolding continuous_on_open_invariant
hoelzl@51481
  1689
  by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  1690
hoelzl@51478
  1691
lemma continuous_on_closed_invariant:
hoelzl@51478
  1692
  "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  1693
proof -
hoelzl@51478
  1694
  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
  1695
    by (metis double_compl)
hoelzl@51478
  1696
  show ?thesis
hoelzl@51478
  1697
    unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric])
hoelzl@51478
  1698
qed
hoelzl@51478
  1699
hoelzl@51481
  1700
lemma continuous_on_closed_vimage:
hoelzl@51481
  1701
  "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
hoelzl@51481
  1702
  unfolding continuous_on_closed_invariant
hoelzl@51481
  1703
  by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  1704
hoelzl@51481
  1705
lemma continuous_on_open_Union:
hoelzl@51481
  1706
  "(\<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
  1707
  unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI)
hoelzl@51481
  1708
hoelzl@51481
  1709
lemma continuous_on_open_UN:
hoelzl@51481
  1710
  "(\<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
  1711
  unfolding Union_image_eq[symmetric] by (rule continuous_on_open_Union) auto
hoelzl@51481
  1712
hoelzl@51481
  1713
lemma continuous_on_closed_Un:
hoelzl@51481
  1714
  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
hoelzl@51481
  1715
  by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
hoelzl@51481
  1716
hoelzl@51481
  1717
lemma continuous_on_If:
hoelzl@51481
  1718
  assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g"
hoelzl@51481
  1719
    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
  1720
  shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h")
hoelzl@51481
  1721
proof-
hoelzl@51481
  1722
  from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
hoelzl@51481
  1723
    by auto
hoelzl@51481
  1724
  with cont have "continuous_on s ?h" "continuous_on t ?h"
hoelzl@51481
  1725
    by simp_all
hoelzl@51481
  1726
  with closed show ?thesis
hoelzl@51481
  1727
    by (rule continuous_on_closed_Un)
hoelzl@51481
  1728
qed
hoelzl@51481
  1729
hoelzl@51478
  1730
ML {*
hoelzl@51478
  1731
hoelzl@51478
  1732
structure Continuous_On_Intros = Named_Thms
hoelzl@51478
  1733
(
hoelzl@51478
  1734
  val name = @{binding continuous_on_intros}
hoelzl@51478
  1735
  val description = "Structural introduction rules for setwise continuity"
hoelzl@51478
  1736
)
hoelzl@51478
  1737
hoelzl@51478
  1738
*}
hoelzl@51478
  1739
hoelzl@51478
  1740
setup Continuous_On_Intros.setup
hoelzl@51478
  1741
hoelzl@51478
  1742
lemma continuous_on_id[continuous_on_intros]: "continuous_on s (\<lambda>x. x)"
hoelzl@51641
  1743
  unfolding continuous_on_def by (fast intro: tendsto_ident_at)
hoelzl@51478
  1744
hoelzl@51478
  1745
lemma continuous_on_const[continuous_on_intros]: "continuous_on s (\<lambda>x. c)"
hoelzl@51478
  1746
  unfolding continuous_on_def by (auto intro: tendsto_const)
hoelzl@51478
  1747
hoelzl@51478
  1748
lemma continuous_on_compose[continuous_on_intros]:
hoelzl@51478
  1749
  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
hoelzl@51478
  1750
  unfolding continuous_on_topological by simp metis
hoelzl@51478
  1751
hoelzl@51481
  1752
lemma continuous_on_compose2:
hoelzl@51481
  1753
  "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> t = f ` s \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
hoelzl@51481
  1754
  using continuous_on_compose[of s f g] by (simp add: comp_def)
hoelzl@51481
  1755
hoelzl@51478
  1756
subsubsection {* Continuity at a point *}
hoelzl@51478
  1757
hoelzl@51478
  1758
definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
hoelzl@51478
  1759
  "continuous F f \<longleftrightarrow> (f ---> f (Lim F (\<lambda>x. x))) F"
hoelzl@51478
  1760
hoelzl@51478
  1761
ML {*
hoelzl@51471
  1762
hoelzl@51478
  1763
structure Continuous_Intros = Named_Thms
hoelzl@51478
  1764
(
hoelzl@51478
  1765
  val name = @{binding continuous_intros}
hoelzl@51478
  1766
  val description = "Structural introduction rules for pointwise continuity"
hoelzl@51478
  1767
)
hoelzl@51478
  1768
hoelzl@51478
  1769
*}
hoelzl@51478
  1770
hoelzl@51478
  1771
setup Continuous_Intros.setup
hoelzl@51478
  1772
hoelzl@51478
  1773
lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
hoelzl@51478
  1774
  unfolding continuous_def by auto
hoelzl@51478
  1775
hoelzl@51478
  1776
lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
hoelzl@51478
  1777
  by simp
hoelzl@51478
  1778
hoelzl@51478
  1779
lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f x) (at x within s)"
hoelzl@51641
  1780
  by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
hoelzl@51478
  1781
hoelzl@51478
  1782
lemma continuous_within_topological:
hoelzl@51478
  1783
  "continuous (at x within s) f \<longleftrightarrow>
hoelzl@51478
  1784
    (\<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
  1785
  unfolding continuous_within tendsto_def eventually_at_topological by metis
hoelzl@51478
  1786
hoelzl@51478
  1787
lemma continuous_within_compose[continuous_intros]:
hoelzl@51478
  1788
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  1789
  continuous (at x within s) (g o f)"
hoelzl@51478
  1790
  by (simp add: continuous_within_topological) metis
hoelzl@51478
  1791
hoelzl@51478
  1792
lemma continuous_within_compose2:
hoelzl@51478
  1793
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  1794
  continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  1795
  using continuous_within_compose[of x s f g] by (simp add: comp_def)
hoelzl@51471
  1796
hoelzl@51478
  1797
lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f -- x --> f x"
hoelzl@51478
  1798
  using continuous_within[of x UNIV f] by simp
hoelzl@51478
  1799
hoelzl@51478
  1800
lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"
hoelzl@51641
  1801
  unfolding continuous_within by (rule tendsto_ident_at)
hoelzl@51478
  1802
hoelzl@51478
  1803
lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"
hoelzl@51478
  1804
  unfolding continuous_def by (rule tendsto_const)
hoelzl@51478
  1805
hoelzl@51478
  1806
lemma continuous_on_eq_continuous_within:
hoelzl@51478
  1807
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
hoelzl@51478
  1808
  unfolding continuous_on_def continuous_within ..
hoelzl@51478
  1809
hoelzl@51478
  1810
abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where
hoelzl@51478
  1811
  "isCont f a \<equiv> continuous (at a) f"
hoelzl@51478
  1812
hoelzl@51478
  1813
lemma isCont_def: "isCont f a \<longleftrightarrow> f -- a --> f a"
hoelzl@51478
  1814
  by (rule continuous_at)
hoelzl@51478
  1815
hoelzl@51478
  1816
lemma continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"
hoelzl@51641
  1817
  by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)
hoelzl@51478
  1818
hoelzl@51481
  1819
lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"
hoelzl@51641
  1820
  by (simp add: continuous_on_def continuous_at at_within_open[of _ s])
hoelzl@51481
  1821
hoelzl@51481
  1822
lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"
hoelzl@51481
  1823
  unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)
hoelzl@51481
  1824
hoelzl@51478
  1825
lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"
hoelzl@51478
  1826
  by (auto intro: continuous_at_within simp: continuous_on_eq_continuous_within)
hoelzl@51478
  1827
hoelzl@51478
  1828
lemma isContI_continuous: "continuous (at x within UNIV) f \<Longrightarrow> isCont f x"
hoelzl@51478
  1829
  by simp
hoelzl@51478
  1830
hoelzl@51478
  1831
lemma isCont_ident[continuous_intros, simp]: "isCont (\<lambda>x. x) a"
hoelzl@51478
  1832
  using continuous_ident by (rule isContI_continuous)
hoelzl@51478
  1833
hoelzl@51478
  1834
lemmas isCont_const = continuous_const
hoelzl@51478
  1835
hoelzl@51478
  1836
lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
hoelzl@51478
  1837
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51478
  1838
hoelzl@51478
  1839
lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
hoelzl@51478
  1840
  unfolding o_def by (rule isCont_o2)
hoelzl@51471
  1841
hoelzl@51471
  1842
lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@51471
  1843
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51471
  1844
hoelzl@51478
  1845
lemma continuous_within_compose3:
hoelzl@51478
  1846
  "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  1847
  using continuous_within_compose2[of x s f g] by (simp add: continuous_at_within)
hoelzl@51471
  1848
hoelzl@51479
  1849
subsubsection{* Open-cover compactness *}
hoelzl@51479
  1850
hoelzl@51479
  1851
context topological_space
hoelzl@51479
  1852
begin
hoelzl@51479
  1853
hoelzl@51479
  1854
definition compact :: "'a set \<Rightarrow> bool" where
hoelzl@51479
  1855
  compact_eq_heine_borel: -- "This name is used for backwards compatibility"
hoelzl@51479
  1856
    "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
  1857
hoelzl@51479
  1858
lemma compactI:
hoelzl@51479
  1859
  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
  1860
  shows "compact s"
hoelzl@51479
  1861
  unfolding compact_eq_heine_borel using assms by metis
hoelzl@51479
  1862
hoelzl@51479
  1863
lemma compact_empty[simp]: "compact {}"
hoelzl@51479
  1864
  by (auto intro!: compactI)
hoelzl@51479
  1865
hoelzl@51479
  1866
lemma compactE:
hoelzl@51479
  1867
  assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
hoelzl@51479
  1868
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
hoelzl@51479
  1869
  using assms unfolding compact_eq_heine_borel by metis
hoelzl@51479
  1870
hoelzl@51479
  1871
lemma compactE_image:
hoelzl@51479
  1872
  assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
hoelzl@51479
  1873
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
hoelzl@51479
  1874
  using assms unfolding ball_simps[symmetric] SUP_def
hoelzl@51479
  1875
  by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
hoelzl@51479
  1876
hoelzl@51481
  1877
lemma compact_inter_closed [intro]:
hoelzl@51481
  1878
  assumes "compact s" and "closed t"
hoelzl@51481
  1879
  shows "compact (s \<inter> t)"
hoelzl@51481
  1880
proof (rule compactI)
hoelzl@51481
  1881
  fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
hoelzl@51481
  1882
  from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto
hoelzl@51481
  1883
  moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
hoelzl@51481
  1884
  ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
hoelzl@51481
  1885
    using `compact s` unfolding compact_eq_heine_borel by auto
hoelzl@51481
  1886
  then guess D ..
hoelzl@51481
  1887
  then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
hoelzl@51481
  1888
    by (intro exI[of _ "D - {-t}"]) auto
hoelzl@51481
  1889
qed
hoelzl@51481
  1890
hoelzl@51471
  1891
end
hoelzl@51471
  1892
hoelzl@51481
  1893
lemma (in t2_space) compact_imp_closed:
hoelzl@51481
  1894
  assumes "compact s" shows "closed s"
hoelzl@51481
  1895
unfolding closed_def
hoelzl@51481
  1896
proof (rule openI)
hoelzl@51481
  1897
  fix y assume "y \<in> - s"
hoelzl@51481
  1898
  let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
hoelzl@51481
  1899
  note `compact s`
hoelzl@51481
  1900
  moreover have "\<forall>u\<in>?C. open u" by simp
hoelzl@51481
  1901
  moreover have "s \<subseteq> \<Union>?C"
hoelzl@51481
  1902
  proof
hoelzl@51481
  1903
    fix x assume "x \<in> s"
hoelzl@51481
  1904
    with `y \<in> - s` have "x \<noteq> y" by clarsimp
hoelzl@51481
  1905
    hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
hoelzl@51481
  1906
      by (rule hausdorff)
hoelzl@51481
  1907
    with `x \<in> s` show "x \<in> \<Union>?C"
hoelzl@51481
  1908
      unfolding eventually_nhds by auto
hoelzl@51481
  1909
  qed
hoelzl@51481
  1910
  ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
hoelzl@51481
  1911
    by (rule compactE)
hoelzl@51481
  1912
  from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
hoelzl@51481
  1913
  with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
hoelzl@51481
  1914
    by (simp add: eventually_Ball_finite)
hoelzl@51481
  1915
  with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
hoelzl@51481
  1916
    by (auto elim!: eventually_mono [rotated])
hoelzl@51481
  1917
  thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
hoelzl@51481
  1918
    by (simp add: eventually_nhds subset_eq)
hoelzl@51481
  1919
qed
hoelzl@51481
  1920
hoelzl@51481
  1921
lemma compact_continuous_image:
hoelzl@51481
  1922
  assumes f: "continuous_on s f" and s: "compact s"
hoelzl@51481
  1923
  shows "compact (f ` s)"
hoelzl@51481
  1924
proof (rule compactI)
hoelzl@51481
  1925
  fix C assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
hoelzl@51481
  1926
  with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"
hoelzl@51481
  1927
    unfolding continuous_on_open_invariant by blast
hoelzl@51481
  1928
  then guess A unfolding bchoice_iff .. note A = this
hoelzl@51481
  1929
  with cover have "\<forall>c\<in>C. open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)"
hoelzl@51481
  1930
    by (fastforce simp add: subset_eq set_eq_iff)+
hoelzl@51481
  1931
  from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" .
hoelzl@51481
  1932
  with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D"
hoelzl@51481
  1933
    by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+
hoelzl@51481
  1934
qed
hoelzl@51481
  1935
hoelzl@51481
  1936
lemma continuous_on_inv:
hoelzl@51481
  1937
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
hoelzl@51481
  1938
  assumes "continuous_on s f"  "compact s"  "\<forall>x\<in>s. g (f x) = x"
hoelzl@51481
  1939
  shows "continuous_on (f ` s) g"
hoelzl@51481
  1940
unfolding continuous_on_topological
hoelzl@51481
  1941
proof (clarsimp simp add: assms(3))
hoelzl@51481
  1942
  fix x :: 'a and B :: "'a set"
hoelzl@51481
  1943
  assume "x \<in> s" and "open B" and "x \<in> B"
hoelzl@51481
  1944
  have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
hoelzl@51481
  1945
    using assms(3) by (auto, metis)
hoelzl@51481
  1946
  have "continuous_on (s - B) f"
hoelzl@51481
  1947
    using `continuous_on s f` Diff_subset
hoelzl@51481
  1948
    by (rule continuous_on_subset)
hoelzl@51481
  1949
  moreover have "compact (s - B)"
hoelzl@51481
  1950
    using `open B` and `compact s`
hoelzl@51481
  1951
    unfolding Diff_eq by (intro compact_inter_closed closed_Compl)
hoelzl@51481
  1952
  ultimately have "compact (f ` (s - B))"
hoelzl@51481
  1953
    by (rule compact_continuous_image)
hoelzl@51481
  1954
  hence "closed (f ` (s - B))"
hoelzl@51481
  1955
    by (rule compact_imp_closed)
hoelzl@51481
  1956
  hence "open (- f ` (s - B))"
hoelzl@51481
  1957
    by (rule open_Compl)
hoelzl@51481
  1958
  moreover have "f x \<in> - f ` (s - B)"
hoelzl@51481
  1959
    using `x \<in> s` and `x \<in> B` by (simp add: 1)
hoelzl@51481
  1960
  moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
hoelzl@51481
  1961
    by (simp add: 1)
hoelzl@51481
  1962
  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
  1963
    by fast
hoelzl@51481
  1964
qed
hoelzl@51481
  1965
hoelzl@51481
  1966
lemma continuous_on_inv_into:
hoelzl@51481
  1967
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
hoelzl@51481
  1968
  assumes s: "continuous_on s f" "compact s" and f: "inj_on f s"
hoelzl@51481
  1969
  shows "continuous_on (f ` s) (the_inv_into s f)"
hoelzl@51481
  1970
  by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])
hoelzl@51481
  1971
hoelzl@51479
  1972
lemma (in linorder_topology) compact_attains_sup:
hoelzl@51479
  1973
  assumes "compact S" "S \<noteq> {}"
hoelzl@51479
  1974
  shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"
hoelzl@51479
  1975
proof (rule classical)
hoelzl@51479
  1976
  assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"
hoelzl@51479
  1977
  then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"
hoelzl@51479
  1978
    by (metis not_le)
hoelzl@51479
  1979
  then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"
hoelzl@51479
  1980
    by auto
hoelzl@51479
  1981
  with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
hoelzl@51479
  1982
    by (erule compactE_image)
hoelzl@51479
  1983
  with `S \<noteq> {}` have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"
hoelzl@51479
  1984
    by (auto intro!: Max_in)
hoelzl@51479
  1985
  with C have "S \<subseteq> {..< Max (t`C)}"
hoelzl@51479
  1986
    by (auto intro: less_le_trans simp: subset_eq)
hoelzl@51479
  1987
  with t Max `C \<subseteq> S` show ?thesis
hoelzl@51479
  1988
    by fastforce
hoelzl@51479
  1989
qed
hoelzl@51479
  1990
hoelzl@51479
  1991
lemma (in linorder_topology) compact_attains_inf:
hoelzl@51479
  1992
  assumes "compact S" "S \<noteq> {}"
hoelzl@51479
  1993
  shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"
hoelzl@51479
  1994
proof (rule classical)
hoelzl@51479
  1995
  assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"
hoelzl@51479
  1996
  then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"
hoelzl@51479
  1997
    by (metis not_le)
hoelzl@51479
  1998
  then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"
hoelzl@51479
  1999
    by auto
hoelzl@51479
  2000
  with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
hoelzl@51479
  2001
    by (erule compactE_image)
hoelzl@51479
  2002
  with `S \<noteq> {}` have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"
hoelzl@51479
  2003
    by (auto intro!: Min_in)
hoelzl@51479
  2004
  with C have "S \<subseteq> {Min (t`C) <..}"
hoelzl@51479
  2005
    by (auto intro: le_less_trans simp: subset_eq)
hoelzl@51479
  2006
  with t Min `C \<subseteq> S` show ?thesis
hoelzl@51479
  2007
    by fastforce
hoelzl@51479
  2008
qed
hoelzl@51479
  2009
hoelzl@51479
  2010
lemma continuous_attains_sup:
hoelzl@51479
  2011
  fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
hoelzl@51479
  2012
  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
  2013
  using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto
hoelzl@51479
  2014
hoelzl@51479
  2015
lemma continuous_attains_inf:
hoelzl@51479
  2016
  fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
hoelzl@51479
  2017
  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
  2018
  using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
hoelzl@51479
  2019
hoelzl@51480
  2020
hoelzl@51480
  2021
subsection {* Connectedness *}
hoelzl@51480
  2022
hoelzl@51480
  2023
context topological_space
hoelzl@51480
  2024
begin
hoelzl@51480
  2025
hoelzl@51480
  2026
definition "connected S \<longleftrightarrow>
hoelzl@51480
  2027
  \<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
  2028
hoelzl@51480
  2029
lemma connectedI:
hoelzl@51480
  2030
  "(\<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
  2031
  \<Longrightarrow> connected U"
hoelzl@51480
  2032
  by (auto simp: connected_def)
hoelzl@51480
  2033
hoelzl@51480
  2034
lemma connected_empty[simp]: "connected {}"
hoelzl@51480
  2035
  by (auto intro!: connectedI)
hoelzl@51480
  2036
hoelzl@51479
  2037
end
hoelzl@51479
  2038
hoelzl@51480
  2039
lemma (in linorder_topology) connectedD_interval:
hoelzl@51480
  2040
  assumes "connected U" and xy: "x \<in> U" "y \<in> U" and "x \<le> z" "z \<le> y"
hoelzl@51480
  2041
  shows "z \<in> U"
hoelzl@51480
  2042
proof -
hoelzl@51480
  2043
  have eq: "{..<z} \<union> {z<..} = - {z}"
hoelzl@51480
  2044
    by auto
hoelzl@51480
  2045
  { assume "z \<notin> U" "x < z" "z < y"
hoelzl@51480
  2046
    with xy have "\<not> connected U"
hoelzl@51480
  2047
      unfolding connected_def simp_thms
hoelzl@51480
  2048
      apply (rule_tac exI[of _ "{..< z}"])
hoelzl@51480
  2049
      apply (rule_tac exI[of _ "{z <..}"])
hoelzl@51480
  2050
      apply (auto simp add: eq)
hoelzl@51480
  2051
      done }
hoelzl@51480
  2052
  with assms show "z \<in> U"
hoelzl@51480
  2053
    by (metis less_le)
hoelzl@51480
  2054
qed
hoelzl@51480
  2055
hoelzl@51480
  2056
lemma connected_continuous_image:
hoelzl@51480
  2057
  assumes *: "continuous_on s f"
hoelzl@51480
  2058
  assumes "connected s"
hoelzl@51480
  2059
  shows "connected (f ` s)"
hoelzl@51480
  2060
proof (rule connectedI)
hoelzl@51480
  2061
  fix A B assume A: "open A" "A \<inter> f ` s \<noteq> {}" and B: "open B" "B \<inter> f ` s \<noteq> {}" and
hoelzl@51480
  2062
    AB: "A \<inter> B \<inter> f ` s = {}" "f ` s \<subseteq> A \<union> B"
hoelzl@51480
  2063
  obtain A' where A': "open A'" "f -` A \<inter> s = A' \<inter> s"
hoelzl@51480
  2064
    using * `open A` unfolding continuous_on_open_invariant by metis
hoelzl@51480
  2065
  obtain B' where B': "open B'" "f -` B \<inter> s = B' \<inter> s"
hoelzl@51480
  2066
    using * `open B` unfolding continuous_on_open_invariant by metis
hoelzl@51480
  2067
hoelzl@51480
  2068
  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
  2069
  proof (rule exI[of _ A'], rule exI[of _ B'], intro conjI)
hoelzl@51480
  2070
    have "s \<subseteq> (f -` A \<inter> s) \<union> (f -` B \<inter> s)" using AB by auto
hoelzl@51480
  2071
    then show "s \<subseteq> A' \<union> B'" using A' B' by auto
hoelzl@51480
  2072
  next
hoelzl@51480
  2073
    have "(f -` A \<inter> s) \<inter> (f -` B \<inter> s) = {}" using AB by auto
hoelzl@51480
  2074
    then show "A' \<inter> B' \<inter> s = {}" using A' B' by auto
hoelzl@51480
  2075
  qed (insert A' B' A B, auto)
hoelzl@51480
  2076
  with `connected s` show False
hoelzl@51480
  2077
    unfolding connected_def by blast
hoelzl@51480
  2078
qed
hoelzl@51480
  2079
hoelzl@51518
  2080
hoelzl@51518
  2081
section {* Connectedness *}
hoelzl@51518
  2082
hoelzl@51775
  2083
class linear_continuum_topology = linorder_topology + linear_continuum
hoelzl@51518
  2084
begin
hoelzl@51518
  2085
hoelzl@51518
  2086
lemma Inf_notin_open:
hoelzl@51518
  2087
  assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a"
hoelzl@51518
  2088
  shows "Inf A \<notin> A"
hoelzl@51518
  2089
proof
hoelzl@51518
  2090
  assume "Inf A \<in> A"
hoelzl@51518
  2091
  then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A"
hoelzl@51518
  2092
    using open_left[of A "Inf A" x] assms by auto
hoelzl@51518
  2093
  with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
hoelzl@51518
  2094
    by (auto simp: subset_eq)
hoelzl@51518
  2095
  then show False
hoelzl@51518
  2096
    using cInf_lower[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le)
hoelzl@51518
  2097
qed
hoelzl@51518
  2098
hoelzl@51518
  2099
lemma Sup_notin_open:
hoelzl@51518
  2100
  assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x"
hoelzl@51518
  2101
  shows "Sup A \<notin> A"
hoelzl@51518
  2102
proof
hoelzl@51518
  2103
  assume "Sup A \<in> A"
hoelzl@51518
  2104
  then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
hoelzl@51518
  2105
    using open_right[of A "Sup A" x] assms by auto
hoelzl@51518
  2106
  with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
hoelzl@51518
  2107
    by (auto simp: subset_eq)
hoelzl@51518
  2108
  then show False
hoelzl@51518
  2109
    using cSup_upper[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le)
hoelzl@51518
  2110
qed
hoelzl@51518
  2111
hoelzl@51480
  2112
end
hoelzl@51480
  2113
hoelzl@51775
  2114
instance linear_continuum_topology \<subseteq> perfect_space
hoelzl@51775
  2115
proof
hoelzl@51775
  2116
  fix x :: 'a
hoelzl@51775
  2117
  from ex_gt_or_lt [of x] guess y ..
hoelzl@51775
  2118
  with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y]
hoelzl@51775
  2119
  show "\<not> open {x}"
hoelzl@51775
  2120
    by auto
hoelzl@51775
  2121
qed
hoelzl@51775
  2122
hoelzl@51518
  2123
lemma connectedI_interval:
hoelzl@51775
  2124
  fixes U :: "'a :: linear_continuum_topology set"
hoelzl@51518
  2125
  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
  2126
  shows "connected U"
hoelzl@51518
  2127
proof (rule connectedI)
hoelzl@51518
  2128
  { fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
hoelzl@51518
  2129
    fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
hoelzl@51518
  2130
hoelzl@51518
  2131
    let ?z = "Inf (B \<inter> {x <..})"
hoelzl@51518
  2132
hoelzl@51518
  2133
    have "x \<le> ?z" "?z \<le> y"
hoelzl@51518
  2134
      using `y \<in> B` `x < y` by (auto intro: cInf_lower[where z=x] cInf_greatest)
hoelzl@51518
  2135
    with `x \<in> U` `y \<in> U` have "?z \<in> U"
hoelzl@51518
  2136
      by (rule *)
hoelzl@51518
  2137
    moreover have "?z \<notin> B \<inter> {x <..}"
hoelzl@51518
  2138
      using `open B` by (intro Inf_notin_open) auto
hoelzl@51518
  2139
    ultimately have "?z \<in> A"
hoelzl@51518
  2140
      using `x \<le> ?z` `A \<inter> B \<inter> U = {}` `x \<in> A` `U \<subseteq> A \<union> B` by auto
hoelzl@51518
  2141
hoelzl@51518
  2142
    { assume "?z < y"
hoelzl@51518
  2143
      obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
hoelzl@51518
  2144
        using open_right[OF `open A` `?z \<in> A` `?z < y`] by auto
hoelzl@51518
  2145
      moreover obtain b where "b \<in> B" "x < b" "b < min a y"
hoelzl@51518
  2146
        using cInf_less_iff[of "B \<inter> {x <..}" x "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B`
hoelzl@51518
  2147
        by (auto intro: less_imp_le)
wenzelm@53374
  2148
      moreover have "?z \<le> b"
wenzelm@53374
  2149
        using `b \<in> B` `x < b`
hoelzl@51518
  2150
        by (intro cInf_lower[where z=x]) auto
hoelzl@51518
  2151
      moreover have "b \<in> U"
hoelzl@51518
  2152
        using `x \<le> ?z` `?z \<le> b` `b < min a y`
hoelzl@51518
  2153
        by (intro *[OF `x \<in> U` `y \<in> U`]) (auto simp: less_imp_le)
hoelzl@51518
  2154
      ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U"
hoelzl@51518
  2155
        by (intro bexI[of _ b]) auto }
hoelzl@51518
  2156
    then have False
hoelzl@51518
  2157
      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
  2158
  note not_disjoint = this
hoelzl@51518
  2159
hoelzl@51518
  2160
  fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"
hoelzl@51518
  2161
  moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto
hoelzl@51518
  2162
  moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto
hoelzl@51518
  2163
  moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
hoelzl@51518
  2164
  ultimately show False by (cases x y rule: linorder_cases) auto
hoelzl@51518
  2165
qed
hoelzl@51518
  2166
hoelzl@51518
  2167
lemma connected_iff_interval:
hoelzl@51775
  2168
  fixes U :: "'a :: linear_continuum_topology set"
hoelzl@51518
  2169
  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
  2170
  by (auto intro: connectedI_interval dest: connectedD_interval)
hoelzl@51518
  2171
hoelzl@51775
  2172
lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
hoelzl@51518
  2173
  unfolding connected_iff_interval by auto
hoelzl@51518
  2174
hoelzl@51775
  2175
lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}"
hoelzl@51518
  2176
  unfolding connected_iff_interval by auto
hoelzl@51518
  2177
hoelzl@51775
  2178
lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}"
hoelzl@51518
  2179
  unfolding connected_iff_interval by auto
hoelzl@51518
  2180
hoelzl@51775
  2181
lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}"
hoelzl@51518
  2182
  unfolding connected_iff_interval by auto
hoelzl@51518
  2183
hoelzl@51775
  2184
lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}"
hoelzl@51518
  2185
  unfolding connected_iff_interval by auto
hoelzl@51518
  2186
hoelzl@51775
  2187
lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}"
hoelzl@51518
  2188
  unfolding connected_iff_interval by auto
hoelzl@51518
  2189
hoelzl@51775
  2190
lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}"
hoelzl@51518
  2191
  unfolding connected_iff_interval by auto
hoelzl@51518
  2192
hoelzl@51775
  2193
lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}"
hoelzl@51518
  2194
  unfolding connected_iff_interval by auto
hoelzl@51518
  2195
hoelzl@51775
  2196
lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}"
hoelzl@51518
  2197
  unfolding connected_iff_interval by auto
hoelzl@51518
  2198
hoelzl@51518
  2199
lemma connected_contains_Ioo: 
hoelzl@51518
  2200
  fixes A :: "'a :: linorder_topology set"
hoelzl@51518
  2201
  assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
hoelzl@51518
  2202
  using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le)
hoelzl@51518
  2203
hoelzl@51518
  2204
subsection {* Intermediate Value Theorem *}
hoelzl@51518
  2205
hoelzl@51518
  2206
lemma IVT':
hoelzl@51775
  2207
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2208
  assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"
hoelzl@51518
  2209
  assumes *: "continuous_on {a .. b} f"
hoelzl@51518
  2210
  shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2211
proof -
hoelzl@51518
  2212
  have "connected {a..b}"
hoelzl@51518
  2213
    unfolding connected_iff_interval by auto
hoelzl@51518
  2214
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y
hoelzl@51518
  2215
  show ?thesis
hoelzl@51518
  2216
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
hoelzl@51518
  2217
qed
hoelzl@51518
  2218
hoelzl@51518
  2219
lemma IVT2':
hoelzl@51775
  2220
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2221
  assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b"
hoelzl@51518
  2222
  assumes *: "continuous_on {a .. b} f"
hoelzl@51518
  2223
  shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2224
proof -
hoelzl@51518
  2225
  have "connected {a..b}"
hoelzl@51518
  2226
    unfolding connected_iff_interval by auto
hoelzl@51518
  2227
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y
hoelzl@51518
  2228
  show ?thesis
hoelzl@51518
  2229
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
hoelzl@51518
  2230
qed
hoelzl@51518
  2231
hoelzl@51518
  2232
lemma IVT:
hoelzl@51775
  2233
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2234
  shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2235
  by (rule IVT') (auto intro: continuous_at_imp_continuous_on)
hoelzl@51518
  2236
hoelzl@51518
  2237
lemma IVT2:
hoelzl@51775
  2238
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2239
  shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2240
  by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)
hoelzl@51518
  2241
hoelzl@51518
  2242
lemma continuous_inj_imp_mono:
hoelzl@51775
  2243
  fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2244
  assumes x: "a < x" "x < b"
hoelzl@51518
  2245
  assumes cont: "continuous_on {a..b} f"
hoelzl@51518
  2246
  assumes inj: "inj_on f {a..b}"
hoelzl@51518
  2247
  shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)"
hoelzl@51518
  2248
proof -
hoelzl@51518
  2249
  note I = inj_on_iff[OF inj]
hoelzl@51518
  2250
  { assume "f x < f a" "f x < f b"
hoelzl@51518
  2251
    then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f x < f s"
hoelzl@51518
  2252
      using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x
hoelzl@51518
  2253
      by (auto simp: continuous_on_subset[OF cont] less_imp_le)
hoelzl@51518
  2254
    with x I have False by auto }
hoelzl@51518
  2255
  moreover
hoelzl@51518
  2256
  { assume "f a < f x" "f b < f x"
hoelzl@51518
  2257
    then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f s < f x"
hoelzl@51518
  2258
      using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x
hoelzl@51518
  2259
      by (auto simp: continuous_on_subset[OF cont] less_imp_le)
hoelzl@51518
  2260
    with x I have False by auto }
hoelzl@51518
  2261
  ultimately show ?thesis
hoelzl@51518
  2262
    using I[of a x] I[of x b] x less_trans[OF x] by (auto simp add: le_less less_imp_neq neq_iff)
hoelzl@51518
  2263
qed
hoelzl@51518
  2264
hoelzl@51518
  2265
end
hoelzl@51518
  2266