src/HOL/Topological_Spaces.thy
author wenzelm
Tue, 03 Sep 2013 22:04:23 +0200
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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 "\<exists>z. x < z \<and> z < y")
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  case True
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  then obtain z where "x < z \<and> z < y" ..
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  then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}"
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    by auto
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  then show ?thesis by blast
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next
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  case False
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  with `x < y` have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}"
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    by auto
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  then show ?thesis by blast
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qed
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instance linorder_topology \<subseteq> t2_space
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proof
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  fix x y :: 'a
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  from less_separate[of x y] less_separate[of y x]
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  show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
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    by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+
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qed
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lemma (in linorder_topology) open_right:
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  assumes "open S" "x \<in> S" and gt_ex: "x < y" shows "\<exists>b>x. {x ..< b} \<subseteq> S"
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  using assms unfolding open_generated_order
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proof induction
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  case (Int A B)
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  then obtain a b where "a > x" "{x ..< a} \<subseteq> A"  "b > x" "{x ..< b} \<subseteq> B" by auto
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  then show ?case by (auto intro!: exI[of _ "min a b"])
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next
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  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] gt_ex)
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qed blast+
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lemma (in linorder_topology) open_left:
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  assumes "open S" "x \<in> S" and lt_ex: "y < x" shows "\<exists>b<x. {b <.. x} \<subseteq> S"
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  using assms unfolding open_generated_order
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proof induction
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  case (Int A B)
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  then obtain a b where "a < x" "{a <.. x} \<subseteq> A"  "b < x" "{b <.. x} \<subseteq> B" by auto
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  then show ?case by (auto intro!: exI[of _ "max a b"])
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next
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  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex)
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qed blast+
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subsection {* Filters *}
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text {*
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  This definition also allows non-proper filters.
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*}
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locale is_filter =
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  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
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  assumes True: "F (\<lambda>x. True)"
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  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
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  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
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typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
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proof
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  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
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qed
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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
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  using Rep_filter [of F] by simp
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lemma Abs_filter_inverse':
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  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
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  using assms by (simp add: Abs_filter_inverse)
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subsubsection {* Eventually *}
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
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  where "eventually P F \<longleftrightarrow> Rep_filter F P"
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lemma eventually_Abs_filter:
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  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
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  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
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lemma filter_eq_iff:
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  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
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  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
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  unfolding eventually_def
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  by (rule is_filter.True [OF is_filter_Rep_filter])
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
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proof -
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  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
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  thus "eventually P F" by simp
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qed
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lemma eventually_mono:
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  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
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  unfolding eventually_def
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  by (rule is_filter.mono [OF is_filter_Rep_filter])
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lemma eventually_conj:
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  assumes P: "eventually (\<lambda>x. P x) F"
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  assumes Q: "eventually (\<lambda>x. Q x) F"
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  shows "eventually (\<lambda>x. P x \<and> Q x) F"
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  using assms unfolding eventually_def
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   319
  by (rule is_filter.conj [OF is_filter_Rep_filter])
cad22a3cc09c move topological_space to its own theory
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parents:
diff changeset
   320
cad22a3cc09c move topological_space to its own theory
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parents:
diff changeset
   321
lemma eventually_Ball_finite:
cad22a3cc09c move topological_space to its own theory
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parents:
diff changeset
   322
  assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   323
  shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   324
using assms by (induct set: finite, simp, simp add: eventually_conj)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   325
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   326
lemma eventually_all_finite:
cad22a3cc09c move topological_space to its own theory
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parents:
diff changeset
   327
  fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   328
  assumes "\<And>y. eventually (\<lambda>x. P x y) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   329
  shows "eventually (\<lambda>x. \<forall>y. P x y) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   330
using eventually_Ball_finite [of UNIV P] assms by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   331
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   332
lemma eventually_mp:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   333
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   334
  assumes "eventually (\<lambda>x. P x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   335
  shows "eventually (\<lambda>x. Q x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   336
proof (rule eventually_mono)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   337
  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   338
  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   339
    using assms by (rule eventually_conj)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   340
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   341
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   342
lemma eventually_rev_mp:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   343
  assumes "eventually (\<lambda>x. P x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   344
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   345
  shows "eventually (\<lambda>x. Q x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   346
using assms(2) assms(1) by (rule eventually_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   347
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   348
lemma eventually_conj_iff:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   349
  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   350
  by (auto intro: eventually_conj elim: eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   351
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   352
lemma eventually_elim1:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   353
  assumes "eventually (\<lambda>i. P i) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   354
  assumes "\<And>i. P i \<Longrightarrow> Q i"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   355
  shows "eventually (\<lambda>i. Q i) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   356
  using assms by (auto elim!: eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   357
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   358
lemma eventually_elim2:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   359
  assumes "eventually (\<lambda>i. P i) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   360
  assumes "eventually (\<lambda>i. Q i) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   361
  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   362
  shows "eventually (\<lambda>i. R i) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   363
  using assms by (auto elim!: eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   364
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   365
lemma eventually_subst:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   366
  assumes "eventually (\<lambda>n. P n = Q n) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   367
  shows "eventually P F = eventually Q F" (is "?L = ?R")
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   368
proof -
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   369
  from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   370
      and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   371
    by (auto elim: eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   372
  then show ?thesis by (auto elim: eventually_elim2)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   373
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   374
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   375
ML {*
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   376
  fun eventually_elim_tac ctxt thms thm =
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
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   377
    let
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
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   378
      val thy = Proof_Context.theory_of ctxt
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   379
      val mp_thms = thms RL [@{thm eventually_rev_mp}]
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   380
      val raw_elim_thm =
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   381
        (@{thm allI} RS @{thm always_eventually})
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   382
        |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   383
        |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   384
      val cases_prop = prop_of (raw_elim_thm RS thm)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   385
      val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   386
    in
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   387
      CASES cases (rtac raw_elim_thm 1) thm
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   388
    end
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   389
*}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   390
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   391
method_setup eventually_elim = {*
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   392
  Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   393
*} "elimination of eventually quantifiers"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   394
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   395
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   396
subsubsection {* Finer-than relation *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   397
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   398
text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   399
filter @{term F'}. *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   400
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   401
instantiation filter :: (type) complete_lattice
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   402
begin
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   403
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   404
definition le_filter_def:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   405
  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   406
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   407
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   408
  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   409
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   410
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   411
  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   412
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   413
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   414
  "bot = Abs_filter (\<lambda>P. True)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   415
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   416
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   417
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   418
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   419
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   420
  "inf F F' = Abs_filter
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   421
      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   422
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   423
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   424
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   425
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   426
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   427
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   428
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   429
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   430
  unfolding top_filter_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   431
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   432
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   433
lemma eventually_bot [simp]: "eventually P bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   434
  unfolding bot_filter_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   435
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   436
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   437
lemma eventually_sup:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   438
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   439
  unfolding sup_filter_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   440
  by (rule eventually_Abs_filter, rule is_filter.intro)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   441
     (auto elim!: eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   442
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   443
lemma eventually_inf:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   444
  "eventually P (inf F F') \<longleftrightarrow>
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   445
   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   446
  unfolding inf_filter_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   447
  apply (rule eventually_Abs_filter, rule is_filter.intro)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   448
  apply (fast intro: eventually_True)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   449
  apply clarify
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   450
  apply (intro exI conjI)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   451
  apply (erule (1) eventually_conj)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   452
  apply (erule (1) eventually_conj)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   453
  apply simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   454
  apply auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   455
  done
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   456
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   457
lemma eventually_Sup:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   458
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   459
  unfolding Sup_filter_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   460
  apply (rule eventually_Abs_filter, rule is_filter.intro)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   461
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   462
  done
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   463
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   464
instance proof
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   465
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   466
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   467
    by (rule less_filter_def) }
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   468
  { show "F \<le> F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   469
    unfolding le_filter_def by simp }
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   470
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   471
    unfolding le_filter_def by simp }
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   472
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   473
    unfolding le_filter_def filter_eq_iff by fast }
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   474
  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   475
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   476
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   477
    unfolding le_filter_def eventually_inf
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   478
    by (auto elim!: eventually_mono intro: eventually_conj) }
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   479
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   480
    unfolding le_filter_def eventually_sup by simp_all }
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   481
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   482
    unfolding le_filter_def eventually_sup by simp }
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   483
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   484
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   485
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   486
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   487
  { assume "F \<in> S" thus "F \<le> Sup S"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   488
    unfolding le_filter_def eventually_Sup by simp }
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   489
  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   490
    unfolding le_filter_def eventually_Sup by simp }
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   491
  { show "Inf {} = (top::'a filter)"
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   492
    by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   493
      (metis (full_types) Topological_Spaces.top_filter_def always_eventually eventually_top) }
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   494
  { show "Sup {} = (bot::'a filter)"
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52265
diff changeset
   495
    by (auto simp: bot_filter_def Sup_filter_def) }
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   496
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   497
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   498
end
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   499
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   500
lemma filter_leD:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   501
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   502
  unfolding le_filter_def by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   503
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   504
lemma filter_leI:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   505
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   506
  unfolding le_filter_def by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   507
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   508
lemma eventually_False:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   509
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   510
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   511
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   512
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   513
  where "trivial_limit F \<equiv> F = bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   514
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   515
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   516
  by (rule eventually_False [symmetric])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   517
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   518
lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   519
  by (cases P) (simp_all add: eventually_False)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   520
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   521
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   522
subsubsection {* Map function for filters *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   523
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   524
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   525
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   526
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   527
lemma eventually_filtermap:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   528
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   529
  unfolding filtermap_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   530
  apply (rule eventually_Abs_filter)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   531
  apply (rule is_filter.intro)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   532
  apply (auto elim!: eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   533
  done
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   534
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   535
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   536
  by (simp add: filter_eq_iff eventually_filtermap)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   537
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   538
lemma filtermap_filtermap:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   539
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   540
  by (simp add: filter_eq_iff eventually_filtermap)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   541
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   542
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   543
  unfolding le_filter_def eventually_filtermap by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   544
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   545
lemma filtermap_bot [simp]: "filtermap f bot = bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   546
  by (simp add: filter_eq_iff eventually_filtermap)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   547
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   548
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   549
  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   550
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   551
subsubsection {* Order filters *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   552
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   553
definition at_top :: "('a::order) filter"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   554
  where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   555
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   556
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   557
  unfolding at_top_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   558
proof (rule eventually_Abs_filter, rule is_filter.intro)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   559
  fix P Q :: "'a \<Rightarrow> bool"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   560
  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   561
  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   562
  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   563
  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   564
qed auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   565
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   566
lemma eventually_ge_at_top:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   567
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   568
  unfolding eventually_at_top_linorder by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   569
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
   570
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::unbounded_dense_linorder. \<forall>n>N. P n)"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   571
  unfolding eventually_at_top_linorder
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   572
proof safe
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
   573
  fix N assume "\<forall>n\<ge>N. P n"
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
   574
  then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   575
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   576
  fix N assume "\<forall>n>N. P n"
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
   577
  moreover obtain y where "N < y" using gt_ex[of N] ..
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   578
  ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   579
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   580
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   581
lemma eventually_gt_at_top:
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
   582
  "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   583
  unfolding eventually_at_top_dense by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   584
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   585
definition at_bot :: "('a::order) filter"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   586
  where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   587
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   588
lemma eventually_at_bot_linorder:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   589
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   590
  unfolding at_bot_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   591
proof (rule eventually_Abs_filter, rule is_filter.intro)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   592
  fix P Q :: "'a \<Rightarrow> bool"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   593
  assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   594
  then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   595
  then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   596
  then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   597
qed auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   598
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   599
lemma eventually_le_at_bot:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   600
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   601
  unfolding eventually_at_bot_linorder by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   602
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   603
lemma eventually_at_bot_dense:
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
   604
  fixes P :: "'a::unbounded_dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   605
  unfolding eventually_at_bot_linorder
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   606
proof safe
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   607
  fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   608
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   609
  fix N assume "\<forall>n<N. P n" 
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
   610
  moreover obtain y where "y < N" using lt_ex[of N] ..
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   611
  ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   612
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   613
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   614
lemma eventually_gt_at_bot:
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
   615
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   616
  unfolding eventually_at_bot_dense by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   617
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   618
subsection {* Sequentially *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   619
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   620
abbreviation sequentially :: "nat filter"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   621
  where "sequentially == at_top"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   622
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   623
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   624
  unfolding at_top_def by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   625
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   626
lemma eventually_sequentially:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   627
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   628
  by (rule eventually_at_top_linorder)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   629
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   630
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   631
  unfolding filter_eq_iff eventually_sequentially by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   632
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   633
lemmas trivial_limit_sequentially = sequentially_bot
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   634
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   635
lemma eventually_False_sequentially [simp]:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   636
  "\<not> eventually (\<lambda>n. False) sequentially"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   637
  by (simp add: eventually_False)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   638
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   639
lemma le_sequentially:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   640
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   641
  unfolding le_filter_def eventually_sequentially
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   642
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   643
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   644
lemma eventually_sequentiallyI:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   645
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   646
  shows "eventually P sequentially"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   647
using assms by (auto simp: eventually_sequentially)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   648
51474
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   649
lemma eventually_sequentially_seg:
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   650
  "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   651
  unfolding eventually_sequentially
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   652
  apply safe
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   653
   apply (rule_tac x="N + k" in exI)
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   654
   apply rule
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   655
   apply (erule_tac x="n - k" in allE)
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   656
   apply auto []
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   657
  apply (rule_tac x=N in exI)
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   658
  apply auto []
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
   659
  done
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   660
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   661
subsubsection {* Standard filters *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   662
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   663
definition principal :: "'a set \<Rightarrow> 'a filter" where
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   664
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   665
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   666
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   667
  unfolding principal_def
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   668
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   669
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   670
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   671
  unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   672
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   673
lemma principal_UNIV[simp]: "principal UNIV = top"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   674
  by (auto simp: filter_eq_iff eventually_principal)
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   675
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   676
lemma principal_empty[simp]: "principal {} = bot"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   677
  by (auto simp: filter_eq_iff eventually_principal)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   678
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   679
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   680
  by (auto simp: le_filter_def eventually_principal)
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   681
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   682
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   683
  unfolding le_filter_def eventually_principal
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   684
  apply safe
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   685
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   686
  apply (auto elim: eventually_elim1)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   687
  done
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   688
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   689
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   690
  unfolding eq_iff by simp
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   691
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   692
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   693
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   694
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   695
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   696
  unfolding filter_eq_iff eventually_inf eventually_principal
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   697
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   698
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   699
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   700
  unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   701
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   702
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   703
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   704
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   705
subsubsection {* Topological filters *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   706
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   707
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   708
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   709
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   710
definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   711
  where "at a within s = inf (nhds a) (principal (s - {a}))"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   712
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   713
abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   714
  "at x \<equiv> at x within (CONST UNIV)"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   715
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
   716
abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   717
  "at_right x \<equiv> at x within {x <..}"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   718
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
   719
abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   720
  "at_left x \<equiv> at x within {..< x}"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   721
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
   722
lemma (in topological_space) eventually_nhds:
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   723
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   724
  unfolding nhds_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   725
proof (rule eventually_Abs_filter, rule is_filter.intro)
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
   726
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   727
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   728
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   729
  fix P Q
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   730
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   731
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   732
  then obtain S T where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   733
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   734
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   735
  hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   736
    by (simp add: open_Int)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   737
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   738
qed auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   739
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   740
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   741
  unfolding trivial_limit_def eventually_nhds by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   742
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   743
lemma eventually_at_filter:
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   744
  "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   745
  unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   746
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   747
lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   748
  unfolding at_within_def by (intro inf_mono) auto
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   749
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   750
lemma eventually_at_topological:
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   751
  "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))"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   752
  unfolding eventually_nhds eventually_at_filter by simp
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   753
51481
ef949192e5d6 move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents: 51480
diff changeset
   754
lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   755
  unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
51481
ef949192e5d6 move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents: 51480
diff changeset
   756
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   757
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   758
  unfolding trivial_limit_def eventually_at_topological
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   759
  by (safe, case_tac "S = {a}", simp, fast, fast)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   760
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   761
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   762
  by (simp add: at_eq_bot_iff not_open_singleton)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   763
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   764
lemma eventually_at_right:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   765
  fixes x :: "'a :: {no_top, linorder_topology}"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   766
  shows "eventually P (at_right x) \<longleftrightarrow> (\<exists>b. x < b \<and> (\<forall>z. x < z \<and> z < b \<longrightarrow> P z))"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   767
  unfolding eventually_at_topological
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   768
proof safe
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
   769
  obtain y where "x < y" using gt_ex[of x] ..
51480
3793c3a11378 move connected to HOL image; used to show intermediate value theorem
hoelzl
parents: 51479
diff changeset
   770
  moreover fix S assume "open S" "x \<in> S" note open_right[OF this, of y]
3793c3a11378 move connected to HOL image; used to show intermediate value theorem
hoelzl
parents: 51479
diff changeset
   771
  moreover note gt_ex[of x]
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   772
  moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {x<..} \<longrightarrow> P s"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   773
  ultimately show "\<exists>b>x. \<forall>z. x < z \<and> z < b \<longrightarrow> P z"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   774
    by (auto simp: subset_eq Ball_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   775
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   776
  fix b assume "x < b" "\<forall>z. x < z \<and> z < b \<longrightarrow> P z"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   777
  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)"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   778
    by (intro exI[of _ "{..< b}"]) auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   779
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   780
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   781
lemma eventually_at_left:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   782
  fixes x :: "'a :: {no_bot, linorder_topology}"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   783
  shows "eventually P (at_left x) \<longleftrightarrow> (\<exists>b. x > b \<and> (\<forall>z. b < z \<and> z < x \<longrightarrow> P z))"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   784
  unfolding eventually_at_topological
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   785
proof safe
53381
355a4cac5440 tuned proofs -- less guessing;
wenzelm
parents: 53374
diff changeset
   786
  obtain y where "y < x" using lt_ex[of x] ..
51480
3793c3a11378 move connected to HOL image; used to show intermediate value theorem
hoelzl
parents: 51479
diff changeset
   787
  moreover fix S assume "open S" "x \<in> S" note open_left[OF this, of y]
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   788
  moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {..<x} \<longrightarrow> P s"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   789
  ultimately show "\<exists>b<x. \<forall>z. b < z \<and> z < x \<longrightarrow> P z"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   790
    by (auto simp: subset_eq Ball_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   791
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   792
  fix b assume "b < x" "\<forall>z. b < z \<and> z < x \<longrightarrow> P z"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   793
  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)"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   794
    by (intro exI[of _ "{b <..}"]) auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   795
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   796
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   797
lemma trivial_limit_at_left_real [simp]:
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
   798
  "\<not> trivial_limit (at_left (x::'a::{no_bot, unbounded_dense_linorder, linorder_topology}))"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   799
  unfolding trivial_limit_def eventually_at_left by (auto dest: dense)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   800
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   801
lemma trivial_limit_at_right_real [simp]:
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
   802
  "\<not> trivial_limit (at_right (x::'a::{no_top, unbounded_dense_linorder, linorder_topology}))"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   803
  unfolding trivial_limit_def eventually_at_right by (auto dest: dense)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   804
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   805
lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   806
  by (auto simp: eventually_at_filter filter_eq_iff eventually_sup 
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   807
           elim: eventually_elim2 eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   808
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   809
lemma eventually_at_split:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   810
  "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   811
  by (subst at_eq_sup_left_right) (simp add: eventually_sup)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   812
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   813
subsection {* Limits *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   814
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   815
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   816
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   817
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   818
syntax
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   819
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   820
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   821
translations
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   822
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   823
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   824
lemma filterlim_iff:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   825
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   826
  unfolding filterlim_def le_filter_def eventually_filtermap ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   827
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   828
lemma filterlim_compose:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   829
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   830
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   831
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   832
lemma filterlim_mono:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   833
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   834
  unfolding filterlim_def by (metis filtermap_mono order_trans)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   835
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   836
lemma filterlim_ident: "LIM x F. x :> F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   837
  by (simp add: filterlim_def filtermap_ident)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   838
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   839
lemma filterlim_cong:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   840
  "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   841
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   842
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   843
lemma filterlim_principal:
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   844
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   845
  unfolding filterlim_def eventually_filtermap le_principal ..
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   846
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   847
lemma filterlim_inf:
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   848
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   849
  unfolding filterlim_def by simp
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   850
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   851
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   852
  unfolding filterlim_def filtermap_filtermap ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   853
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   854
lemma filterlim_sup:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   855
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   856
  unfolding filterlim_def filtermap_sup by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   857
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   858
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   859
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   860
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   861
subsubsection {* Tendsto *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   862
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   863
abbreviation (in topological_space)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   864
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   865
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   866
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   867
definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   868
  "Lim A f = (THE l. (f ---> l) A)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   869
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   870
lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   871
  by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   872
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   873
ML {*
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   874
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   875
structure Tendsto_Intros = Named_Thms
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   876
(
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   877
  val name = @{binding tendsto_intros}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   878
  val description = "introduction rules for tendsto"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   879
)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   880
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   881
*}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   882
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   883
setup {*
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   884
  Tendsto_Intros.setup #>
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   885
  Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   886
    map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])) o Tendsto_Intros.get o Context.proof_of);
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   887
*}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   888
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
   889
lemma (in topological_space) tendsto_def:
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
   890
   "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   891
  unfolding filterlim_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   892
proof safe
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   893
  fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   894
  then show "eventually (\<lambda>x. f x \<in> S) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   895
    unfolding eventually_nhds eventually_filtermap le_filter_def
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   896
    by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   897
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   898
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   899
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   900
  unfolding tendsto_def le_filter_def by fast
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   901
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   902
lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   903
  by (blast intro: tendsto_mono at_le)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   904
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   905
lemma filterlim_at:
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   906
  "(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)"
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   907
  by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   908
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
   909
lemma (in topological_space) topological_tendstoI:
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   910
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f ---> l) F"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   911
  unfolding tendsto_def by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   912
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
   913
lemma (in topological_space) topological_tendstoD:
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   914
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   915
  unfolding tendsto_def by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   916
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   917
lemma order_tendstoI:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   918
  fixes y :: "_ :: order_topology"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   919
  assumes "\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   920
  assumes "\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   921
  shows "(f ---> y) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   922
proof (rule topological_tendstoI)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   923
  fix S assume "open S" "y \<in> S"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   924
  then show "eventually (\<lambda>x. f x \<in> S) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   925
    unfolding open_generated_order
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   926
  proof induct
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   927
    case (UN K)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   928
    then obtain k where "y \<in> k" "k \<in> K" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   929
    with UN(2)[of k] show ?case
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   930
      by (auto elim: eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   931
  qed (insert assms, auto elim: eventually_elim2)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   932
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   933
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   934
lemma order_tendstoD:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   935
  fixes y :: "_ :: order_topology"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   936
  assumes y: "(f ---> y) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   937
  shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   938
    and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   939
  using topological_tendstoD[OF y, of "{..< a}"] topological_tendstoD[OF y, of "{a <..}"] by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   940
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   941
lemma order_tendsto_iff: 
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   942
  fixes f :: "_ \<Rightarrow> 'a :: order_topology"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   943
  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)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   944
  by (metis order_tendstoI order_tendstoD)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   945
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   946
lemma tendsto_bot [simp]: "(f ---> a) bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   947
  unfolding tendsto_def by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   948
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
   949
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a within s)"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   950
  unfolding tendsto_def eventually_at_topological by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   951
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   952
lemma (in topological_space) tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   953
  by (simp add: tendsto_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   954
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   955
lemma (in t2_space) tendsto_unique:
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   956
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   957
  shows "a = b"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   958
proof (rule ccontr)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   959
  assume "a \<noteq> b"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   960
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   961
    using hausdorff [OF `a \<noteq> b`] by fast
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   962
  have "eventually (\<lambda>x. f x \<in> U) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   963
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   964
  moreover
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   965
  have "eventually (\<lambda>x. f x \<in> V) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   966
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   967
  ultimately
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   968
  have "eventually (\<lambda>x. False) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   969
  proof eventually_elim
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   970
    case (elim x)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   971
    hence "f x \<in> U \<inter> V" by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   972
    with `U \<inter> V = {}` show ?case by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   973
  qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   974
  with `\<not> trivial_limit F` show "False"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   975
    by (simp add: trivial_limit_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   976
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   977
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   978
lemma (in t2_space) tendsto_const_iff:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
   979
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   980
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   981
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   982
lemma increasing_tendsto:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   983
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   984
  assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   985
      and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   986
  shows "(f ---> l) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   987
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   988
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   989
lemma decreasing_tendsto:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   990
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   991
  assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   992
      and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   993
  shows "(f ---> l) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   994
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   995
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   996
lemma tendsto_sandwich:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   997
  fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   998
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
   999
  assumes lim: "(f ---> c) net" "(h ---> c) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1000
  shows "(g ---> c) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1001
proof (rule order_tendstoI)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1002
  fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1003
    using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1004
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1005
  fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1006
    using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1007
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1008
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1009
lemma tendsto_le:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1010
  fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1011
  assumes F: "\<not> trivial_limit F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1012
  assumes x: "(f ---> x) F" and y: "(g ---> y) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1013
  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1014
  shows "y \<le> x"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1015
proof (rule ccontr)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1016
  assume "\<not> y \<le> x"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1017
  with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1018
    by (auto simp: not_le)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1019
  then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1020
    using x y by (auto intro: order_tendstoD)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1021
  with ev have "eventually (\<lambda>x. False) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1022
    by eventually_elim (insert xy, fastforce)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1023
  with F show False
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1024
    by (simp add: eventually_False)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1025
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1026
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1027
lemma tendsto_le_const:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1028
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1029
  assumes F: "\<not> trivial_limit F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1030
  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>x. a \<le> f x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1031
  shows "a \<le> x"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1032
  using F x tendsto_const a by (rule tendsto_le)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1033
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1034
subsubsection {* Rules about @{const Lim} *}
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1035
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1036
lemma (in t2_space) tendsto_Lim:
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1037
  "\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1038
  unfolding Lim_def using tendsto_unique[of net f] by auto
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1039
51641
cd05e9fcc63d remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents: 51518
diff changeset
  1040
lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1041
  by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1042
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1043
subsection {* Limits to @{const at_top} and @{const at_bot} *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1044
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1045
lemma filterlim_at_top:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1046
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1047
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1048
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1049
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1050
lemma filterlim_at_top_dense:
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
  1051
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1052
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1053
  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1054
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1055
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1056
lemma filterlim_at_top_ge:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1057
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1058
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1059
  unfolding filterlim_at_top
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1060
proof safe
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1061
  fix Z assume *: "\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1062
  with *[THEN spec, of "max Z c"] show "eventually (\<lambda>x. Z \<le> f x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1063
    by (auto elim!: eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1064
qed simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1065
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1066
lemma filterlim_at_top_at_top:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1067
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1068
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1069
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1070
  assumes Q: "eventually Q at_top"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1071
  assumes P: "eventually P at_top"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1072
  shows "filterlim f at_top at_top"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1073
proof -
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1074
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1075
    unfolding eventually_at_top_linorder by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1076
  show ?thesis
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1077
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1078
    fix z assume "x \<le> z"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1079
    with x have "P z" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1080
    have "eventually (\<lambda>x. g z \<le> x) at_top"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1081
      by (rule eventually_ge_at_top)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1082
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1083
      by eventually_elim (metis mono bij `P z`)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1084
  qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1085
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1086
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1087
lemma filterlim_at_top_gt:
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
  1088
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1089
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1090
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1091
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1092
lemma filterlim_at_bot: 
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1093
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1094
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1095
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1096
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1097
lemma filterlim_at_bot_le:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1098
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1099
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1100
  unfolding filterlim_at_bot
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1101
proof safe
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1102
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1103
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1104
    by (auto elim!: eventually_elim1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1105
qed simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1106
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1107
lemma filterlim_at_bot_lt:
53215
5e47c31c6f7c renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents: 52729
diff changeset
  1108
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1109
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1110
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1111
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1112
lemma filterlim_at_bot_at_right:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1113
  fixes f :: "'a::{no_top, linorder_topology} \<Rightarrow> 'b::linorder"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1114
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1115
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1116
  assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1117
  assumes P: "eventually P at_bot"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1118
  shows "filterlim f at_bot (at_right a)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1119
proof -
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1120
  from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1121
    unfolding eventually_at_bot_linorder by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1122
  show ?thesis
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1123
  proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1124
    fix z assume "z \<le> x"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1125
    with x have "P z" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1126
    have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1127
      using bound[OF bij(2)[OF `P z`]]
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1128
      unfolding eventually_at_right by (auto intro!: exI[of _ "g z"])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1129
    with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1130
      by eventually_elim (metis bij `P z` mono)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1131
  qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1132
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1133
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1134
lemma filterlim_at_top_at_left:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1135
  fixes f :: "'a::{no_bot, linorder_topology} \<Rightarrow> 'b::linorder"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1136
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1137
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1138
  assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1139
  assumes P: "eventually P at_top"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1140
  shows "filterlim f at_top (at_left a)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1141
proof -
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1142
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1143
    unfolding eventually_at_top_linorder by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1144
  show ?thesis
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1145
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1146
    fix z assume "x \<le> z"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1147
    with x have "P z" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1148
    have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1149
      using bound[OF bij(2)[OF `P z`]]
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1150
      unfolding eventually_at_left by (auto intro!: exI[of _ "g z"])
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1151
    with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1152
      by eventually_elim (metis bij `P z` mono)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1153
  qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1154
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1155
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1156
lemma filterlim_split_at:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1157
  "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1158
  by (subst at_eq_sup_left_right) (rule filterlim_sup)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1159
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1160
lemma filterlim_at_split:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1161
  "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1162
  by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1163
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1164
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1165
subsection {* Limits on sequences *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1166
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1167
abbreviation (in topological_space)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1168
  LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1169
    ("((_)/ ----> (_))" [60, 60] 60) where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1170
  "X ----> L \<equiv> (X ---> L) sequentially"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1171
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1172
abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1173
  "lim X \<equiv> Lim sequentially X"
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1174
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1175
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1176
  "convergent X = (\<exists>L. X ----> L)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1177
51478
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1178
lemma lim_def: "lim X = (THE L. X ----> L)"
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1179
  unfolding Lim_def ..
270b21f3ae0a move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents: 51474
diff changeset
  1180
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1181
subsubsection {* Monotone sequences and subsequences *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1182
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1183
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1184
  monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1185
    --{*Definition of monotonicity.
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1186
        The use of disjunction here complicates proofs considerably.
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1187
        One alternative is to add a Boolean argument to indicate the direction.
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1188
        Another is to develop the notions of increasing and decreasing first.*}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1189
  "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1190
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1191
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1192
  incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1193
    --{*Increasing sequence*}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1194
  "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1195
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1196
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1197
  decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1198
    --{*Decreasing sequence*}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1199
  "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1200
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1201
definition
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1202
  subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1203
    --{*Definition of subsequence*}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1204
  "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1205
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1206
lemma incseq_mono: "mono f \<longleftrightarrow> incseq f"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1207
  unfolding mono_def incseq_def by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1208
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1209
lemma incseq_SucI:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1210
  "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1211
  using lift_Suc_mono_le[of X]
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1212
  by (auto simp: incseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1213
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1214
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1215
  by (auto simp: incseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1216
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1217
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1218
  using incseqD[of A i "Suc i"] by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1219
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1220
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1221
  by (auto intro: incseq_SucI dest: incseq_SucD)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1222
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1223
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1224
  unfolding incseq_def by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1225
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1226
lemma decseq_SucI:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1227
  "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1228
  using order.lift_Suc_mono_le[OF dual_order, of X]
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1229
  by (auto simp: decseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1230
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1231
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1232
  by (auto simp: decseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1233
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1234
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1235
  using decseqD[of A i "Suc i"] by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1236
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1237
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1238
  by (auto intro: decseq_SucI dest: decseq_SucD)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1239
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1240
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1241
  unfolding decseq_def by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1242
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1243
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1244
  unfolding monoseq_def incseq_def decseq_def ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1245
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1246
lemma monoseq_Suc:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1247
  "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1248
  unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1249
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1250
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1251
by (simp add: monoseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1252
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1253
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1254
by (simp add: monoseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1255
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1256
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1257
by (simp add: monoseq_Suc)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1258
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1259
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1260
by (simp add: monoseq_Suc)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1261
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1262
lemma monoseq_minus:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1263
  fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1264
  assumes "monoseq a"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1265
  shows "monoseq (\<lambda> n. - a n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1266
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1267
  case True
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1268
  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1269
  thus ?thesis by (rule monoI2)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1270
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1271
  case False
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1272
  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1273
  thus ?thesis by (rule monoI1)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1274
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1275
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1276
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1277
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1278
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1279
apply (simp add: subseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1280
apply (auto dest!: less_imp_Suc_add)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1281
apply (induct_tac k)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1282
apply (auto intro: less_trans)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1283
done
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1284
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1285
text{* for any sequence, there is a monotonic subsequence *}
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1286
lemma seq_monosub:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1287
  fixes s :: "nat => 'a::linorder"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1288
  shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1289
proof cases
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1290
  let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1291
  assume *: "\<forall>n. \<exists>p. ?P p n"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1292
  def f \<equiv> "nat_rec (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1293
  have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1294
  have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1295
  have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1296
  have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1297
  then have "subseq f" unfolding subseq_Suc_iff by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1298
  moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1299
  proof (intro disjI2 allI)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1300
    fix n show "s (f (Suc n)) \<le> s (f n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1301
    proof (cases n)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1302
      case 0 with P_Suc[of 0] P_0 show ?thesis by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1303
    next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1304
      case (Suc m)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1305
      from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1306
      with P_Suc Suc show ?thesis by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1307
    qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1308
  qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1309
  ultimately show ?thesis by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1310
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1311
  let "?P p m" = "m < p \<and> s m < s p"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1312
  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1313
  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1314
  def f \<equiv> "nat_rec (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1315
  have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1316
  have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc ..
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1317
  have P_0: "?P (f 0) (Suc N)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1318
    unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1319
  { fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1320
      unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . }
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1321
  note P' = this
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1322
  { fix i have "N < f i \<and> ?P (f (Suc i)) (f i)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1323
      by (induct i) (insert P_0 P', auto) }
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1324
  then have "subseq f" "monoseq (\<lambda>x. s (f x))"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1325
    unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1326
  then show ?thesis by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1327
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1328
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1329
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1330
proof(induct n)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1331
  case 0 thus ?case by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1332
next
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1333
  case (Suc n)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1334
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1335
  have "n < f (Suc n)" by arith
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1336
  thus ?case by arith
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1337
qed
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1338
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1339
lemma eventually_subseq:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1340
  "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1341
  unfolding eventually_sequentially by (metis seq_suble le_trans)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1342
51473
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1343
lemma not_eventually_sequentiallyD:
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1344
  assumes P: "\<not> eventually P sequentially"
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1345
  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1346
proof -
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1347
  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1348
    unfolding eventually_sequentially by (simp add: not_less)
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1349
  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1350
    by (auto simp: choice_iff)
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1351
  then show ?thesis
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1352
    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1353
             simp: less_eq_Suc_le subseq_Suc_iff)
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1354
qed
1210309fddab move first_countable_topology to the HOL image
hoelzl
parents: 51471
diff changeset
  1355
51471
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1356
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1357
  unfolding filterlim_iff by (metis eventually_subseq)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1358
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1359
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1360
  unfolding subseq_def by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1361
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1362
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1363
  using assms by (auto simp: subseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1364
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1365
lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1366
  by (simp add: incseq_def monoseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1367
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1368
lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1369
  by (simp add: decseq_def monoseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1370
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1371
lemma decseq_eq_incseq:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1372
  fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" 
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1373
  by (simp add: decseq_def incseq_def)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1374
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1375
lemma INT_decseq_offset:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1376
  assumes "decseq F"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1377
  shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1378
proof safe
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1379
  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1380
  show "x \<in> F i"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1381
  proof cases
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1382
    from x have "x \<in> F n" by auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1383
    also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1384
      unfolding decseq_def by simp
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1385
    finally show ?thesis .
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1386
  qed (insert x, simp)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1387
qed auto
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1388
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1389
lemma LIMSEQ_const_iff:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1390
  fixes k l :: "'a::t2_space"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1391
  shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1392
  using trivial_limit_sequentially by (rule tendsto_const_iff)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1393
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1394
lemma LIMSEQ_SUP:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1395
  "incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1396
  by (intro increasing_tendsto)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1397
     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1398
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1399
lemma LIMSEQ_INF:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1400
  "decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1401
  by (intro decreasing_tendsto)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1402
     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1403
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1404
lemma LIMSEQ_ignore_initial_segment:
cad22a3cc09c move topological_space to its own theory
hoelzl
parents:
diff changeset
  1405
  "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
51474
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
  1406
  unfolding tendsto_def
1e9e68247ad1 generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents: 51473
diff changeset
  1407
  by (subst eventually_sequentially_seg[where k=k])