src/HOL/Topological_Spaces.thy
author hoelzl
Mon Dec 08 14:32:11 2014 +0100 (2014-12-08)
changeset 59106 af691e67f71f
parent 58889 5b7a9633cfa8
child 59452 2538b2c51769
permissions -rw-r--r--
instance bool and enat as topologies
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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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section {* 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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named_theorems continuous_intros "structural introduction rules for continuity"
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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 [continuous_intros, intro, simp]: "open {}"
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  using open_Union [of "{}"] by simp
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lemma open_Un [continuous_intros, 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 [continuous_intros, intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)"
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  using open_Union [of "B ` A"] by simp
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lemma open_Inter [continuous_intros, 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 [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)"
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  using open_Inter [of "B ` A"] by simp
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lemma openI:
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  assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S"
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  shows "open S"
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proof -
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  have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto
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  moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms)
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  ultimately show "open S" by simp
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qed
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lemma closed_empty [continuous_intros, intro, simp]:  "closed {}"
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  unfolding closed_def by simp
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lemma closed_Un [continuous_intros, 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 [continuous_intros, intro, simp]: "closed UNIV"
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  unfolding closed_def by simp
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lemma closed_Int [continuous_intros, 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 [continuous_intros, 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 [continuous_intros, 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 [continuous_intros, 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 [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)"
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  using closed_Union [of "B ` A"] by simp
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lemma open_closed: "open S \<longleftrightarrow> closed (- S)"
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  unfolding closed_def by simp
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lemma closed_open: "closed S \<longleftrightarrow> open (- S)"
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  unfolding closed_def by simp
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lemma open_Diff [continuous_intros, 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 [continuous_intros, 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 [continuous_intros, intro]: "closed S \<Longrightarrow> open (- S)"
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  unfolding closed_open .
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lemma closed_Compl [continuous_intros, intro]: "open S \<Longrightarrow> closed (- S)"
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  unfolding open_closed .
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lemma open_Collect_neg: "closed {x. P x} \<Longrightarrow> open {x. \<not> P x}"
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  unfolding Collect_neg_eq by (rule open_Compl)
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lemma open_Collect_conj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<and> Q x}"
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  using open_Int[OF assms] by (simp add: Int_def)
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lemma open_Collect_disj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<or> Q x}"
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  using open_Un[OF assms] by (simp add: Un_def)
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lemma open_Collect_ex: "(\<And>i. open {x. P i x}) \<Longrightarrow> open {x. \<exists>i. P i x}"
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  using open_UN[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_ex_eq by simp 
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lemma open_Collect_imp: "closed {x. P x} \<Longrightarrow> open {x. Q x} \<Longrightarrow> open {x. P x \<longrightarrow> Q x}"
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  unfolding imp_conv_disj by (intro open_Collect_disj open_Collect_neg)
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lemma open_Collect_const: "open {x. P}"
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  by (cases P) auto
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lemma closed_Collect_neg: "open {x. P x} \<Longrightarrow> closed {x. \<not> P x}"
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  unfolding Collect_neg_eq by (rule closed_Compl)
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lemma closed_Collect_conj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<and> Q x}"
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  using closed_Int[OF assms] by (simp add: Int_def)
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lemma closed_Collect_disj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<or> Q x}"
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  using closed_Un[OF assms] by (simp add: Un_def)
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lemma closed_Collect_all: "(\<And>i. closed {x. P i x}) \<Longrightarrow> closed {x. \<forall>i. P i x}"
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  using closed_INT[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_all_eq by simp 
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lemma closed_Collect_imp: "open {x. P x} \<Longrightarrow> closed {x. Q x} \<Longrightarrow> closed {x. P x \<longrightarrow> Q x}"
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  unfolding imp_conv_disj by (intro closed_Collect_disj closed_Collect_neg)
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lemma closed_Collect_const: "closed {x. P}"
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  by (cases P) auto
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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 [continuous_intros, simp]:
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  fixes a :: "'a::t1_space"
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  assumes "closed S" shows "closed (insert a S)"
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proof -
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  from closed_singleton assms
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  have "closed ({a} \<union> S)" by (rule closed_Un)
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  thus "closed (insert a S)" by simp
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qed
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lemma finite_imp_closed:
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  fixes S :: "'a::t1_space set"
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  shows "finite S \<Longrightarrow> closed S"
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by (induct set: finite, simp_all)
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text {* T2 spaces are also known as Hausdorff spaces. *}
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class t2_space = topological_space +
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  assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}"
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instance t2_space \<subseteq> t1_space
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proof qed (fast dest: hausdorff)
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lemma separation_t2:
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  fixes x y :: "'a::t2_space"
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  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})"
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  using hausdorff[of x y] by blast
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lemma separation_t0:
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  fixes x y :: "'a::t0_space"
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  shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))"
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  using t0_space[of x y] by blast
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text {* A perfect space is a topological space with no isolated points. *}
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class perfect_space = topological_space +
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  assumes not_open_singleton: "\<not> open {x}"
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subsection {* Generators for toplogies *}
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inductive generate_topology for S where
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  UNIV: "generate_topology S UNIV"
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| Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)"
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| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)"
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| Basis: "s \<in> S \<Longrightarrow> generate_topology S s"
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hide_fact (open) UNIV Int UN Basis 
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lemma generate_topology_Union: 
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  "(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)"
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  using generate_topology.UN [of "K ` I"] by auto
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lemma topological_space_generate_topology:
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  "class.topological_space (generate_topology S)"
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  by default (auto intro: generate_topology.intros)
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subsection {* Order topologies *}
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class order_topology = order + "open" +
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  assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
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begin
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subclass topological_space
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  unfolding open_generated_order
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  by (rule topological_space_generate_topology)
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lemma open_greaterThan [continuous_intros, simp]: "open {a <..}"
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  unfolding open_generated_order by (auto intro: generate_topology.Basis)
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lemma open_lessThan [continuous_intros, simp]: "open {..< a}"
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  unfolding open_generated_order by (auto intro: generate_topology.Basis)
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lemma open_greaterThanLessThan [continuous_intros, 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 [continuous_intros, simp]: "closed {.. a::'a::linorder_topology}"
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  by (simp add: closed_open)
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lemma closed_atLeast [continuous_intros, simp]: "closed {a::'a::linorder_topology ..}"
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  by (simp add: closed_open)
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lemma closed_atLeastAtMost [continuous_intros, 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
hoelzl@51471
   295
  then show ?case by (auto intro!: exI[of _ "max a b"])
hoelzl@51471
   296
next
hoelzl@51480
   297
  case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex)
hoelzl@51480
   298
qed blast+
hoelzl@51471
   299
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   300
subsubsection {* Boolean is an order topology *}
hoelzl@59106
   301
hoelzl@59106
   302
text {* It also is a discrete topology, but don't have a type class for it (yet). *}
hoelzl@59106
   303
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   304
instantiation bool :: order_topology
hoelzl@59106
   305
begin
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   306
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   307
definition open_bool :: "bool set \<Rightarrow> bool" where
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   308
  "open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
hoelzl@59106
   309
hoelzl@59106
   310
instance
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   311
  proof qed (rule open_bool_def)
hoelzl@59106
   312
hoelzl@59106
   313
end
hoelzl@59106
   314
hoelzl@59106
   315
lemma open_bool[simp, intro!]: "open (A::bool set)"
hoelzl@59106
   316
proof -
hoelzl@59106
   317
  have *: "{False <..} = {True}" "{..< True} = {False}"
hoelzl@59106
   318
    by auto
hoelzl@59106
   319
  have "A = UNIV \<or> A = {} \<or> A = {False <..} \<or> A = {..< True}"
hoelzl@59106
   320
    using subset_UNIV[of A] unfolding UNIV_bool * by auto
hoelzl@59106
   321
  then show "open A"
hoelzl@59106
   322
    by auto
hoelzl@59106
   323
qed
hoelzl@59106
   324
hoelzl@51471
   325
subsection {* Filters *}
hoelzl@51471
   326
hoelzl@51471
   327
text {*
hoelzl@51471
   328
  This definition also allows non-proper filters.
hoelzl@51471
   329
*}
hoelzl@51471
   330
hoelzl@51471
   331
locale is_filter =
hoelzl@51471
   332
  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
hoelzl@51471
   333
  assumes True: "F (\<lambda>x. True)"
hoelzl@51471
   334
  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
hoelzl@51471
   335
  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
hoelzl@51471
   336
hoelzl@51471
   337
typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
hoelzl@51471
   338
proof
hoelzl@51471
   339
  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
hoelzl@51471
   340
qed
hoelzl@51471
   341
hoelzl@51471
   342
lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
hoelzl@51471
   343
  using Rep_filter [of F] by simp
hoelzl@51471
   344
hoelzl@51471
   345
lemma Abs_filter_inverse':
hoelzl@51471
   346
  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
hoelzl@51471
   347
  using assms by (simp add: Abs_filter_inverse)
hoelzl@51471
   348
hoelzl@51471
   349
hoelzl@51471
   350
subsubsection {* Eventually *}
hoelzl@51471
   351
hoelzl@51471
   352
definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
hoelzl@51471
   353
  where "eventually P F \<longleftrightarrow> Rep_filter F P"
hoelzl@51471
   354
hoelzl@51471
   355
lemma eventually_Abs_filter:
hoelzl@51471
   356
  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
hoelzl@51471
   357
  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
hoelzl@51471
   358
hoelzl@51471
   359
lemma filter_eq_iff:
hoelzl@51471
   360
  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
hoelzl@51471
   361
  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
hoelzl@51471
   362
hoelzl@51471
   363
lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
hoelzl@51471
   364
  unfolding eventually_def
hoelzl@51471
   365
  by (rule is_filter.True [OF is_filter_Rep_filter])
hoelzl@51471
   366
hoelzl@51471
   367
lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
hoelzl@51471
   368
proof -
hoelzl@51471
   369
  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
hoelzl@51471
   370
  thus "eventually P F" by simp
hoelzl@51471
   371
qed
hoelzl@51471
   372
hoelzl@51471
   373
lemma eventually_mono:
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   374
  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
hoelzl@51471
   375
  unfolding eventually_def
hoelzl@51471
   376
  by (rule is_filter.mono [OF is_filter_Rep_filter])
hoelzl@51471
   377
hoelzl@51471
   378
lemma eventually_conj:
hoelzl@51471
   379
  assumes P: "eventually (\<lambda>x. P x) F"
hoelzl@51471
   380
  assumes Q: "eventually (\<lambda>x. Q x) F"
hoelzl@51471
   381
  shows "eventually (\<lambda>x. P x \<and> Q x) F"
hoelzl@51471
   382
  using assms unfolding eventually_def
hoelzl@51471
   383
  by (rule is_filter.conj [OF is_filter_Rep_filter])
hoelzl@51471
   384
hoelzl@51471
   385
lemma eventually_Ball_finite:
hoelzl@51471
   386
  assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net"
hoelzl@51471
   387
  shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net"
hoelzl@51471
   388
using assms by (induct set: finite, simp, simp add: eventually_conj)
hoelzl@51471
   389
hoelzl@51471
   390
lemma eventually_all_finite:
hoelzl@51471
   391
  fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool"
hoelzl@51471
   392
  assumes "\<And>y. eventually (\<lambda>x. P x y) net"
hoelzl@51471
   393
  shows "eventually (\<lambda>x. \<forall>y. P x y) net"
hoelzl@51471
   394
using eventually_Ball_finite [of UNIV P] assms by simp
hoelzl@51471
   395
hoelzl@51471
   396
lemma eventually_mp:
hoelzl@51471
   397
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@51471
   398
  assumes "eventually (\<lambda>x. P x) F"
hoelzl@51471
   399
  shows "eventually (\<lambda>x. Q x) F"
hoelzl@51471
   400
proof (rule eventually_mono)
hoelzl@51471
   401
  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
hoelzl@51471
   402
  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
hoelzl@51471
   403
    using assms by (rule eventually_conj)
hoelzl@51471
   404
qed
hoelzl@51471
   405
hoelzl@51471
   406
lemma eventually_rev_mp:
hoelzl@51471
   407
  assumes "eventually (\<lambda>x. P x) F"
hoelzl@51471
   408
  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@51471
   409
  shows "eventually (\<lambda>x. Q x) F"
hoelzl@51471
   410
using assms(2) assms(1) by (rule eventually_mp)
hoelzl@51471
   411
hoelzl@51471
   412
lemma eventually_conj_iff:
hoelzl@51471
   413
  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
hoelzl@51471
   414
  by (auto intro: eventually_conj elim: eventually_rev_mp)
hoelzl@51471
   415
hoelzl@51471
   416
lemma eventually_elim1:
hoelzl@51471
   417
  assumes "eventually (\<lambda>i. P i) F"
hoelzl@51471
   418
  assumes "\<And>i. P i \<Longrightarrow> Q i"
hoelzl@51471
   419
  shows "eventually (\<lambda>i. Q i) F"
hoelzl@51471
   420
  using assms by (auto elim!: eventually_rev_mp)
hoelzl@51471
   421
hoelzl@51471
   422
lemma eventually_elim2:
hoelzl@51471
   423
  assumes "eventually (\<lambda>i. P i) F"
hoelzl@51471
   424
  assumes "eventually (\<lambda>i. Q i) F"
hoelzl@51471
   425
  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
hoelzl@51471
   426
  shows "eventually (\<lambda>i. R i) F"
hoelzl@51471
   427
  using assms by (auto elim!: eventually_rev_mp)
hoelzl@51471
   428
hoelzl@57447
   429
lemma not_eventually_impI: "eventually P F \<Longrightarrow> \<not> eventually Q F \<Longrightarrow> \<not> eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@57447
   430
  by (auto intro: eventually_mp)
hoelzl@57447
   431
hoelzl@57447
   432
lemma not_eventuallyD: "\<not> eventually P F \<Longrightarrow> \<exists>x. \<not> P x"
hoelzl@57447
   433
  by (metis always_eventually)
hoelzl@57447
   434
hoelzl@51471
   435
lemma eventually_subst:
hoelzl@51471
   436
  assumes "eventually (\<lambda>n. P n = Q n) F"
hoelzl@51471
   437
  shows "eventually P F = eventually Q F" (is "?L = ?R")
hoelzl@51471
   438
proof -
hoelzl@51471
   439
  from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
hoelzl@51471
   440
      and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
hoelzl@51471
   441
    by (auto elim: eventually_elim1)
hoelzl@51471
   442
  then show ?thesis by (auto elim: eventually_elim2)
hoelzl@51471
   443
qed
hoelzl@51471
   444
hoelzl@51471
   445
ML {*
wenzelm@56231
   446
  fun eventually_elim_tac ctxt thms = SUBGOAL_CASES (fn (_, _, st) =>
hoelzl@51471
   447
    let
hoelzl@51471
   448
      val thy = Proof_Context.theory_of ctxt
hoelzl@51471
   449
      val mp_thms = thms RL [@{thm eventually_rev_mp}]
hoelzl@51471
   450
      val raw_elim_thm =
hoelzl@51471
   451
        (@{thm allI} RS @{thm always_eventually})
hoelzl@51471
   452
        |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
hoelzl@51471
   453
        |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
wenzelm@56231
   454
      val cases_prop = prop_of (raw_elim_thm RS st)
hoelzl@51471
   455
      val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
hoelzl@51471
   456
    in
wenzelm@56231
   457
      CASES cases (rtac raw_elim_thm 1)
wenzelm@56231
   458
    end) 1
hoelzl@51471
   459
*}
hoelzl@51471
   460
hoelzl@51471
   461
method_setup eventually_elim = {*
hoelzl@51471
   462
  Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
hoelzl@51471
   463
*} "elimination of eventually quantifiers"
hoelzl@51471
   464
hoelzl@51471
   465
hoelzl@51471
   466
subsubsection {* Finer-than relation *}
hoelzl@51471
   467
hoelzl@51471
   468
text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
hoelzl@51471
   469
filter @{term F'}. *}
hoelzl@51471
   470
hoelzl@51471
   471
instantiation filter :: (type) complete_lattice
hoelzl@51471
   472
begin
hoelzl@51471
   473
hoelzl@51471
   474
definition le_filter_def:
hoelzl@51471
   475
  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
hoelzl@51471
   476
hoelzl@51471
   477
definition
hoelzl@51471
   478
  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@51471
   479
hoelzl@51471
   480
definition
hoelzl@51471
   481
  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
hoelzl@51471
   482
hoelzl@51471
   483
definition
hoelzl@51471
   484
  "bot = Abs_filter (\<lambda>P. True)"
hoelzl@51471
   485
hoelzl@51471
   486
definition
hoelzl@51471
   487
  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
hoelzl@51471
   488
hoelzl@51471
   489
definition
hoelzl@51471
   490
  "inf F F' = Abs_filter
hoelzl@51471
   491
      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@51471
   492
hoelzl@51471
   493
definition
hoelzl@51471
   494
  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
hoelzl@51471
   495
hoelzl@51471
   496
definition
hoelzl@51471
   497
  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
hoelzl@51471
   498
hoelzl@51471
   499
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
hoelzl@51471
   500
  unfolding top_filter_def
hoelzl@51471
   501
  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@51471
   502
hoelzl@51471
   503
lemma eventually_bot [simp]: "eventually P bot"
hoelzl@51471
   504
  unfolding bot_filter_def
hoelzl@51471
   505
  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
hoelzl@51471
   506
hoelzl@51471
   507
lemma eventually_sup:
hoelzl@51471
   508
  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
hoelzl@51471
   509
  unfolding sup_filter_def
hoelzl@51471
   510
  by (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   511
     (auto elim!: eventually_rev_mp)
hoelzl@51471
   512
hoelzl@51471
   513
lemma eventually_inf:
hoelzl@51471
   514
  "eventually P (inf F F') \<longleftrightarrow>
hoelzl@51471
   515
   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
hoelzl@51471
   516
  unfolding inf_filter_def
hoelzl@51471
   517
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   518
  apply (fast intro: eventually_True)
hoelzl@51471
   519
  apply clarify
hoelzl@51471
   520
  apply (intro exI conjI)
hoelzl@51471
   521
  apply (erule (1) eventually_conj)
hoelzl@51471
   522
  apply (erule (1) eventually_conj)
hoelzl@51471
   523
  apply simp
hoelzl@51471
   524
  apply auto
hoelzl@51471
   525
  done
hoelzl@51471
   526
hoelzl@51471
   527
lemma eventually_Sup:
hoelzl@51471
   528
  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
hoelzl@51471
   529
  unfolding Sup_filter_def
hoelzl@51471
   530
  apply (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@51471
   531
  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
hoelzl@51471
   532
  done
hoelzl@51471
   533
hoelzl@51471
   534
instance proof
hoelzl@51471
   535
  fix F F' F'' :: "'a filter" and S :: "'a filter set"
hoelzl@51471
   536
  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
hoelzl@51471
   537
    by (rule less_filter_def) }
hoelzl@51471
   538
  { show "F \<le> F"
hoelzl@51471
   539
    unfolding le_filter_def by simp }
hoelzl@51471
   540
  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
hoelzl@51471
   541
    unfolding le_filter_def by simp }
hoelzl@51471
   542
  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
hoelzl@51471
   543
    unfolding le_filter_def filter_eq_iff by fast }
hoelzl@51471
   544
  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
hoelzl@51471
   545
    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
hoelzl@51471
   546
  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
hoelzl@51471
   547
    unfolding le_filter_def eventually_inf
hoelzl@51471
   548
    by (auto elim!: eventually_mono intro: eventually_conj) }
haftmann@52729
   549
  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
haftmann@52729
   550
    unfolding le_filter_def eventually_sup by simp_all }
haftmann@52729
   551
  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
haftmann@52729
   552
    unfolding le_filter_def eventually_sup by simp }
haftmann@52729
   553
  { assume "F'' \<in> S" thus "Inf S \<le> F''"
haftmann@52729
   554
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
haftmann@52729
   555
  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
haftmann@52729
   556
    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
hoelzl@51471
   557
  { assume "F \<in> S" thus "F \<le> Sup S"
hoelzl@51471
   558
    unfolding le_filter_def eventually_Sup by simp }
hoelzl@51471
   559
  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
hoelzl@51471
   560
    unfolding le_filter_def eventually_Sup by simp }
haftmann@52729
   561
  { show "Inf {} = (top::'a filter)"
haftmann@52729
   562
    by (auto simp: top_filter_def Inf_filter_def Sup_filter_def)
huffman@53859
   563
      (metis (full_types) top_filter_def always_eventually eventually_top) }
haftmann@52729
   564
  { show "Sup {} = (bot::'a filter)"
haftmann@52729
   565
    by (auto simp: bot_filter_def Sup_filter_def) }
hoelzl@51471
   566
qed
hoelzl@51471
   567
hoelzl@51471
   568
end
hoelzl@51471
   569
hoelzl@51471
   570
lemma filter_leD:
hoelzl@51471
   571
  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
hoelzl@51471
   572
  unfolding le_filter_def by simp
hoelzl@51471
   573
hoelzl@51471
   574
lemma filter_leI:
hoelzl@51471
   575
  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
hoelzl@51471
   576
  unfolding le_filter_def by simp
hoelzl@51471
   577
hoelzl@51471
   578
lemma eventually_False:
hoelzl@51471
   579
  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
hoelzl@51471
   580
  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
hoelzl@51471
   581
hoelzl@51471
   582
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
hoelzl@51471
   583
  where "trivial_limit F \<equiv> F = bot"
hoelzl@51471
   584
hoelzl@51471
   585
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
hoelzl@51471
   586
  by (rule eventually_False [symmetric])
hoelzl@51471
   587
hoelzl@51471
   588
lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P"
hoelzl@51471
   589
  by (cases P) (simp_all add: eventually_False)
hoelzl@51471
   590
hoelzl@57276
   591
lemma eventually_Inf: "eventually P (Inf B) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X))"
hoelzl@57276
   592
proof -
hoelzl@57276
   593
  let ?F = "\<lambda>P. \<exists>X\<subseteq>B. finite X \<and> eventually P (Inf X)"
hoelzl@57276
   594
  
hoelzl@57276
   595
  { fix P have "eventually P (Abs_filter ?F) \<longleftrightarrow> ?F P"
hoelzl@57276
   596
    proof (rule eventually_Abs_filter is_filter.intro)+
hoelzl@57276
   597
      show "?F (\<lambda>x. True)"
hoelzl@57276
   598
        by (rule exI[of _ "{}"]) (simp add: le_fun_def)
hoelzl@57276
   599
    next
hoelzl@57276
   600
      fix P Q
hoelzl@57276
   601
      assume "?F P" then guess X ..
hoelzl@57276
   602
      moreover
hoelzl@57276
   603
      assume "?F Q" then guess Y ..
hoelzl@57276
   604
      ultimately show "?F (\<lambda>x. P x \<and> Q x)"
hoelzl@57276
   605
        by (intro exI[of _ "X \<union> Y"])
hoelzl@57276
   606
           (auto simp: Inf_union_distrib eventually_inf)
hoelzl@57276
   607
    next
hoelzl@57276
   608
      fix P Q
hoelzl@57276
   609
      assume "?F P" then guess X ..
hoelzl@57276
   610
      moreover assume "\<forall>x. P x \<longrightarrow> Q x"
hoelzl@57276
   611
      ultimately show "?F Q"
hoelzl@57276
   612
        by (intro exI[of _ X]) (auto elim: eventually_elim1)
hoelzl@57276
   613
    qed }
hoelzl@57276
   614
  note eventually_F = this
hoelzl@57276
   615
hoelzl@57276
   616
  have "Inf B = Abs_filter ?F"
hoelzl@57276
   617
  proof (intro antisym Inf_greatest)
hoelzl@57276
   618
    show "Inf B \<le> Abs_filter ?F"
hoelzl@57276
   619
      by (auto simp: le_filter_def eventually_F dest: Inf_superset_mono)
hoelzl@57276
   620
  next
hoelzl@57276
   621
    fix F assume "F \<in> B" then show "Abs_filter ?F \<le> F"
hoelzl@57276
   622
      by (auto simp add: le_filter_def eventually_F intro!: exI[of _ "{F}"])
hoelzl@57276
   623
  qed
hoelzl@57276
   624
  then show ?thesis
hoelzl@57276
   625
    by (simp add: eventually_F)
hoelzl@57276
   626
qed
hoelzl@57276
   627
hoelzl@57276
   628
lemma eventually_INF: "eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>X\<subseteq>B. finite X \<and> eventually P (INF b:X. F b))"
hoelzl@57276
   629
  unfolding INF_def[of B] eventually_Inf[of P "F`B"]
hoelzl@57276
   630
  by (metis Inf_image_eq finite_imageI image_mono finite_subset_image)
hoelzl@57276
   631
hoelzl@57276
   632
lemma Inf_filter_not_bot:
hoelzl@57276
   633
  fixes B :: "'a filter set"
hoelzl@57276
   634
  shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> Inf X \<noteq> bot) \<Longrightarrow> Inf B \<noteq> bot"
hoelzl@57276
   635
  unfolding trivial_limit_def eventually_Inf[of _ B]
hoelzl@57276
   636
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
hoelzl@57276
   637
hoelzl@57276
   638
lemma INF_filter_not_bot:
hoelzl@57276
   639
  fixes F :: "'i \<Rightarrow> 'a filter"
hoelzl@57276
   640
  shows "(\<And>X. X \<subseteq> B \<Longrightarrow> finite X \<Longrightarrow> (INF b:X. F b) \<noteq> bot) \<Longrightarrow> (INF b:B. F b) \<noteq> bot"
hoelzl@57276
   641
  unfolding trivial_limit_def eventually_INF[of _ B]
hoelzl@57276
   642
    bot_bool_def [symmetric] bot_fun_def [symmetric] bot_unique by simp
hoelzl@57276
   643
hoelzl@57276
   644
lemma eventually_Inf_base:
hoelzl@57276
   645
  assumes "B \<noteq> {}" and base: "\<And>F G. F \<in> B \<Longrightarrow> G \<in> B \<Longrightarrow> \<exists>x\<in>B. x \<le> inf F G"
hoelzl@57276
   646
  shows "eventually P (Inf B) \<longleftrightarrow> (\<exists>b\<in>B. eventually P b)"
hoelzl@57276
   647
proof (subst eventually_Inf, safe)
hoelzl@57276
   648
  fix X assume "finite X" "X \<subseteq> B"
hoelzl@57276
   649
  then have "\<exists>b\<in>B. \<forall>x\<in>X. b \<le> x"
hoelzl@57276
   650
  proof induct
hoelzl@57276
   651
    case empty then show ?case
hoelzl@57276
   652
      using `B \<noteq> {}` by auto
hoelzl@57276
   653
  next
hoelzl@57276
   654
    case (insert x X)
hoelzl@57276
   655
    then obtain b where "b \<in> B" "\<And>x. x \<in> X \<Longrightarrow> b \<le> x"
hoelzl@57276
   656
      by auto
hoelzl@57276
   657
    with `insert x X \<subseteq> B` base[of b x] show ?case
hoelzl@57276
   658
      by (auto intro: order_trans)
hoelzl@57276
   659
  qed
hoelzl@57276
   660
  then obtain b where "b \<in> B" "b \<le> Inf X"
hoelzl@57276
   661
    by (auto simp: le_Inf_iff)
hoelzl@57276
   662
  then show "eventually P (Inf X) \<Longrightarrow> Bex B (eventually P)"
hoelzl@57276
   663
    by (intro bexI[of _ b]) (auto simp: le_filter_def)
hoelzl@57276
   664
qed (auto intro!: exI[of _ "{x}" for x])
hoelzl@57276
   665
hoelzl@57276
   666
lemma eventually_INF_base:
hoelzl@57276
   667
  "B \<noteq> {} \<Longrightarrow> (\<And>a b. a \<in> B \<Longrightarrow> b \<in> B \<Longrightarrow> \<exists>x\<in>B. F x \<le> inf (F a) (F b)) \<Longrightarrow>
hoelzl@57276
   668
    eventually P (INF b:B. F b) \<longleftrightarrow> (\<exists>b\<in>B. eventually P (F b))"
hoelzl@57276
   669
  unfolding INF_def by (subst eventually_Inf_base) auto
hoelzl@57276
   670
hoelzl@51471
   671
hoelzl@51471
   672
subsubsection {* Map function for filters *}
hoelzl@51471
   673
hoelzl@51471
   674
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
hoelzl@51471
   675
  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
hoelzl@51471
   676
hoelzl@51471
   677
lemma eventually_filtermap:
hoelzl@51471
   678
  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
hoelzl@51471
   679
  unfolding filtermap_def
hoelzl@51471
   680
  apply (rule eventually_Abs_filter)
hoelzl@51471
   681
  apply (rule is_filter.intro)
hoelzl@51471
   682
  apply (auto elim!: eventually_rev_mp)
hoelzl@51471
   683
  done
hoelzl@51471
   684
hoelzl@51471
   685
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
hoelzl@51471
   686
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   687
hoelzl@51471
   688
lemma filtermap_filtermap:
hoelzl@51471
   689
  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
hoelzl@51471
   690
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   691
hoelzl@51471
   692
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
hoelzl@51471
   693
  unfolding le_filter_def eventually_filtermap by simp
hoelzl@51471
   694
hoelzl@51471
   695
lemma filtermap_bot [simp]: "filtermap f bot = bot"
hoelzl@51471
   696
  by (simp add: filter_eq_iff eventually_filtermap)
hoelzl@51471
   697
hoelzl@51471
   698
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)"
hoelzl@51471
   699
  by (auto simp: filter_eq_iff eventually_filtermap eventually_sup)
hoelzl@51471
   700
hoelzl@57276
   701
lemma filtermap_inf: "filtermap f (inf F1 F2) \<le> inf (filtermap f F1) (filtermap f F2)"
hoelzl@57276
   702
  by (auto simp: le_filter_def eventually_filtermap eventually_inf)
hoelzl@57276
   703
hoelzl@57276
   704
lemma filtermap_INF: "filtermap f (INF b:B. F b) \<le> (INF b:B. filtermap f (F b))"
hoelzl@57276
   705
proof -
hoelzl@57276
   706
  { fix X :: "'c set" assume "finite X"
hoelzl@57276
   707
    then have "filtermap f (INFIMUM X F) \<le> (INF b:X. filtermap f (F b))"
hoelzl@57276
   708
    proof induct
hoelzl@57276
   709
      case (insert x X)
hoelzl@57276
   710
      have "filtermap f (INF a:insert x X. F a) \<le> inf (filtermap f (F x)) (filtermap f (INF a:X. F a))"
hoelzl@57276
   711
        by (rule order_trans[OF _ filtermap_inf]) simp
hoelzl@57276
   712
      also have "\<dots> \<le> inf (filtermap f (F x)) (INF a:X. filtermap f (F a))"
hoelzl@57276
   713
        by (intro inf_mono insert order_refl)
hoelzl@57276
   714
      finally show ?case
hoelzl@57276
   715
        by simp
hoelzl@57276
   716
    qed simp }
hoelzl@57276
   717
  then show ?thesis
hoelzl@57276
   718
    unfolding le_filter_def eventually_filtermap
hoelzl@57276
   719
    by (subst (1 2) eventually_INF) auto
hoelzl@51471
   720
qed
hoelzl@51471
   721
subsubsection {* Standard filters *}
hoelzl@51471
   722
hoelzl@51641
   723
definition principal :: "'a set \<Rightarrow> 'a filter" where
hoelzl@51641
   724
  "principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)"
hoelzl@51641
   725
hoelzl@51641
   726
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)"
hoelzl@51641
   727
  unfolding principal_def
hoelzl@51641
   728
  by (rule eventually_Abs_filter, rule is_filter.intro) auto
hoelzl@51471
   729
hoelzl@51641
   730
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F"
hoelzl@51641
   731
  unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1)
hoelzl@51641
   732
hoelzl@51641
   733
lemma principal_UNIV[simp]: "principal UNIV = top"
hoelzl@51641
   734
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@51471
   735
hoelzl@51641
   736
lemma principal_empty[simp]: "principal {} = bot"
hoelzl@51641
   737
  by (auto simp: filter_eq_iff eventually_principal)
hoelzl@51641
   738
hoelzl@57276
   739
lemma principal_eq_bot_iff: "principal X = bot \<longleftrightarrow> X = {}"
hoelzl@57276
   740
  by (auto simp add: filter_eq_iff eventually_principal)
hoelzl@57276
   741
hoelzl@51641
   742
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B"
hoelzl@51641
   743
  by (auto simp: le_filter_def eventually_principal)
hoelzl@51471
   744
hoelzl@51641
   745
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F"
hoelzl@51641
   746
  unfolding le_filter_def eventually_principal
hoelzl@51641
   747
  apply safe
hoelzl@51641
   748
  apply (erule_tac x="\<lambda>x. x \<in> A" in allE)
hoelzl@51641
   749
  apply (auto elim: eventually_elim1)
hoelzl@51641
   750
  done
hoelzl@51471
   751
hoelzl@51641
   752
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B"
hoelzl@51641
   753
  unfolding eq_iff by simp
hoelzl@51471
   754
hoelzl@51641
   755
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)"
hoelzl@51641
   756
  unfolding filter_eq_iff eventually_sup eventually_principal by auto
hoelzl@51471
   757
hoelzl@51641
   758
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)"
hoelzl@51641
   759
  unfolding filter_eq_iff eventually_inf eventually_principal
hoelzl@51641
   760
  by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"])
hoelzl@51471
   761
hoelzl@51641
   762
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)"
hoelzl@51641
   763
  unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal)
hoelzl@51641
   764
hoelzl@57276
   765
lemma INF_principal_finite: "finite X \<Longrightarrow> (INF x:X. principal (f x)) = principal (\<Inter>x\<in>X. f x)"
hoelzl@57276
   766
  by (induct X rule: finite_induct) auto
hoelzl@57276
   767
hoelzl@51641
   768
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)"
hoelzl@51641
   769
  unfolding filter_eq_iff eventually_filtermap eventually_principal by simp
hoelzl@51471
   770
hoelzl@57276
   771
subsubsection {* Order filters *}
hoelzl@57276
   772
hoelzl@57276
   773
definition at_top :: "('a::order) filter"
hoelzl@57276
   774
  where "at_top = (INF k. principal {k ..})"
hoelzl@57276
   775
hoelzl@57448
   776
lemma at_top_sub: "at_top = (INF k:{c::'a::linorder..}. principal {k ..})"
hoelzl@57448
   777
  by (auto intro!: INF_eq max.cobounded1 max.cobounded2 simp: at_top_def)
hoelzl@57448
   778
hoelzl@57276
   779
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)"
hoelzl@57276
   780
  unfolding at_top_def
hoelzl@57276
   781
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
hoelzl@57276
   782
hoelzl@57276
   783
lemma eventually_ge_at_top:
hoelzl@57276
   784
  "eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top"
hoelzl@57276
   785
  unfolding eventually_at_top_linorder by auto
hoelzl@57276
   786
hoelzl@57276
   787
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::{no_top, linorder}. \<forall>n>N. P n)"
hoelzl@57276
   788
proof -
hoelzl@57276
   789
  have "eventually P (INF k. principal {k <..}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n>N. P n)"
hoelzl@57276
   790
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: max.cobounded1 max.cobounded2)
hoelzl@57276
   791
  also have "(INF k. principal {k::'a <..}) = at_top"
hoelzl@57276
   792
    unfolding at_top_def 
hoelzl@57276
   793
    by (intro INF_eq) (auto intro: less_imp_le simp: Ici_subset_Ioi_iff gt_ex)
hoelzl@57276
   794
  finally show ?thesis .
hoelzl@57276
   795
qed
hoelzl@57276
   796
hoelzl@57276
   797
lemma eventually_gt_at_top:
hoelzl@57276
   798
  "eventually (\<lambda>x. (c::_::unbounded_dense_linorder) < x) at_top"
hoelzl@57276
   799
  unfolding eventually_at_top_dense by auto
hoelzl@57276
   800
hoelzl@57276
   801
definition at_bot :: "('a::order) filter"
hoelzl@57276
   802
  where "at_bot = (INF k. principal {.. k})"
hoelzl@57276
   803
hoelzl@57448
   804
lemma at_bot_sub: "at_bot = (INF k:{.. c::'a::linorder}. principal {.. k})"
hoelzl@57448
   805
  by (auto intro!: INF_eq min.cobounded1 min.cobounded2 simp: at_bot_def)
hoelzl@57448
   806
hoelzl@57276
   807
lemma eventually_at_bot_linorder:
hoelzl@57276
   808
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@57276
   809
  unfolding at_bot_def
hoelzl@57276
   810
  by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
hoelzl@57276
   811
hoelzl@57276
   812
lemma eventually_le_at_bot:
hoelzl@57276
   813
  "eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot"
hoelzl@57276
   814
  unfolding eventually_at_bot_linorder by auto
hoelzl@57276
   815
hoelzl@57276
   816
lemma eventually_at_bot_dense: "eventually P at_bot \<longleftrightarrow> (\<exists>N::'a::{no_bot, linorder}. \<forall>n<N. P n)"
hoelzl@57276
   817
proof -
hoelzl@57276
   818
  have "eventually P (INF k. principal {..< k}) \<longleftrightarrow> (\<exists>N::'a. \<forall>n<N. P n)"
hoelzl@57276
   819
    by (subst eventually_INF_base) (auto simp: eventually_principal intro: min.cobounded1 min.cobounded2)
hoelzl@57276
   820
  also have "(INF k. principal {..< k::'a}) = at_bot"
hoelzl@57276
   821
    unfolding at_bot_def 
hoelzl@57276
   822
    by (intro INF_eq) (auto intro: less_imp_le simp: Iic_subset_Iio_iff lt_ex)
hoelzl@57276
   823
  finally show ?thesis .
hoelzl@57276
   824
qed
hoelzl@57276
   825
hoelzl@57276
   826
lemma eventually_gt_at_bot:
hoelzl@57276
   827
  "eventually (\<lambda>x. x < (c::_::unbounded_dense_linorder)) at_bot"
hoelzl@57276
   828
  unfolding eventually_at_bot_dense by auto
hoelzl@57276
   829
hoelzl@57276
   830
lemma trivial_limit_at_bot_linorder: "\<not> trivial_limit (at_bot ::('a::linorder) filter)"
hoelzl@57276
   831
  unfolding trivial_limit_def
hoelzl@57276
   832
  by (metis eventually_at_bot_linorder order_refl)
hoelzl@57276
   833
hoelzl@57276
   834
lemma trivial_limit_at_top_linorder: "\<not> trivial_limit (at_top ::('a::linorder) filter)"
hoelzl@57276
   835
  unfolding trivial_limit_def
hoelzl@57276
   836
  by (metis eventually_at_top_linorder order_refl)
hoelzl@57276
   837
hoelzl@57276
   838
subsection {* Sequentially *}
hoelzl@57276
   839
hoelzl@57276
   840
abbreviation sequentially :: "nat filter"
hoelzl@57276
   841
  where "sequentially \<equiv> at_top"
hoelzl@57276
   842
hoelzl@57276
   843
lemma eventually_sequentially:
hoelzl@57276
   844
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@57276
   845
  by (rule eventually_at_top_linorder)
hoelzl@57276
   846
hoelzl@57276
   847
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
hoelzl@57276
   848
  unfolding filter_eq_iff eventually_sequentially by auto
hoelzl@57276
   849
hoelzl@57276
   850
lemmas trivial_limit_sequentially = sequentially_bot
hoelzl@57276
   851
hoelzl@57276
   852
lemma eventually_False_sequentially [simp]:
hoelzl@57276
   853
  "\<not> eventually (\<lambda>n. False) sequentially"
hoelzl@57276
   854
  by (simp add: eventually_False)
hoelzl@57276
   855
hoelzl@57276
   856
lemma le_sequentially:
hoelzl@57276
   857
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
hoelzl@57276
   858
  by (simp add: at_top_def le_INF_iff le_principal)
hoelzl@57276
   859
hoelzl@57276
   860
lemma eventually_sequentiallyI:
hoelzl@57276
   861
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
hoelzl@57276
   862
  shows "eventually P sequentially"
hoelzl@57276
   863
using assms by (auto simp: eventually_sequentially)
hoelzl@57276
   864
hoelzl@57276
   865
lemma eventually_sequentially_seg:
hoelzl@57276
   866
  "eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@57276
   867
  unfolding eventually_sequentially
hoelzl@57276
   868
  apply safe
hoelzl@57276
   869
   apply (rule_tac x="N + k" in exI)
hoelzl@57276
   870
   apply rule
hoelzl@57276
   871
   apply (erule_tac x="n - k" in allE)
hoelzl@57276
   872
   apply auto []
hoelzl@57276
   873
  apply (rule_tac x=N in exI)
hoelzl@57276
   874
  apply auto []
hoelzl@57276
   875
  done
hoelzl@57276
   876
hoelzl@51471
   877
subsubsection {* Topological filters *}
hoelzl@51471
   878
hoelzl@51471
   879
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
hoelzl@57276
   880
  where "nhds a = (INF S:{S. open S \<and> a \<in> S}. principal S)"
hoelzl@51471
   881
hoelzl@51641
   882
definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60)
hoelzl@51641
   883
  where "at a within s = inf (nhds a) (principal (s - {a}))"
hoelzl@51641
   884
hoelzl@51641
   885
abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where
hoelzl@51641
   886
  "at x \<equiv> at x within (CONST UNIV)"
hoelzl@51471
   887
hoelzl@51473
   888
abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   889
  "at_right x \<equiv> at x within {x <..}"
hoelzl@51471
   890
hoelzl@51473
   891
abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   892
  "at_left x \<equiv> at x within {..< x}"
hoelzl@51471
   893
hoelzl@57448
   894
lemma (in topological_space) nhds_generated_topology:
hoelzl@57448
   895
  "open = generate_topology T \<Longrightarrow> nhds x = (INF S:{S\<in>T. x \<in> S}. principal S)"
hoelzl@57448
   896
  unfolding nhds_def
hoelzl@57448
   897
proof (safe intro!: antisym INF_greatest)
hoelzl@57448
   898
  fix S assume "generate_topology T S" "x \<in> S"
hoelzl@57448
   899
  then show "(INF S:{S \<in> T. x \<in> S}. principal S) \<le> principal S"
hoelzl@57448
   900
    by induction 
hoelzl@57448
   901
       (auto intro: INF_lower order_trans simp add: inf_principal[symmetric] simp del: inf_principal)
hoelzl@57448
   902
qed (auto intro!: INF_lower intro: generate_topology.intros)
hoelzl@57448
   903
hoelzl@51473
   904
lemma (in topological_space) eventually_nhds:
hoelzl@51471
   905
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
hoelzl@57276
   906
  unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal)
hoelzl@51471
   907
hoelzl@51471
   908
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
hoelzl@51471
   909
  unfolding trivial_limit_def eventually_nhds by simp
hoelzl@51471
   910
hoelzl@57448
   911
lemma at_within_eq: "at x within s = (INF S:{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))"
hoelzl@57448
   912
  unfolding nhds_def at_within_def by (subst INF_inf_const2[symmetric]) (auto simp add: Diff_Int_distrib)
hoelzl@57448
   913
hoelzl@51641
   914
lemma eventually_at_filter:
hoelzl@51641
   915
  "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
hoelzl@51641
   916
  unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)
hoelzl@51641
   917
hoelzl@51641
   918
lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
hoelzl@51641
   919
  unfolding at_within_def by (intro inf_mono) auto
hoelzl@51641
   920
hoelzl@51471
   921
lemma eventually_at_topological:
hoelzl@51641
   922
  "eventually P (at a within s) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))"
hoelzl@51641
   923
  unfolding eventually_nhds eventually_at_filter by simp
hoelzl@51471
   924
hoelzl@51481
   925
lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
hoelzl@51641
   926
  unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
hoelzl@51481
   927
huffman@53859
   928
lemma at_within_empty [simp]: "at a within {} = bot"
huffman@53859
   929
  unfolding at_within_def by simp
huffman@53859
   930
huffman@53860
   931
lemma at_within_union: "at x within (S \<union> T) = sup (at x within S) (at x within T)"
huffman@53860
   932
  unfolding filter_eq_iff eventually_sup eventually_at_filter
huffman@53860
   933
  by (auto elim!: eventually_rev_mp)
huffman@53860
   934
hoelzl@51471
   935
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
hoelzl@51471
   936
  unfolding trivial_limit_def eventually_at_topological
hoelzl@51471
   937
  by (safe, case_tac "S = {a}", simp, fast, fast)
hoelzl@51471
   938
hoelzl@51471
   939
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
hoelzl@51471
   940
  by (simp add: at_eq_bot_iff not_open_singleton)
hoelzl@51471
   941
hoelzl@57448
   942
lemma (in order_topology) nhds_order: "nhds x =
hoelzl@57448
   943
  inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})"
hoelzl@57448
   944
proof -
hoelzl@57448
   945
  have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} = 
hoelzl@57448
   946
      (\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}"
hoelzl@57448
   947
    by auto
hoelzl@57448
   948
  show ?thesis
hoelzl@57448
   949
    unfolding nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def ..
hoelzl@51471
   950
qed
hoelzl@51471
   951
hoelzl@57448
   952
lemma (in linorder_topology) at_within_order: "UNIV \<noteq> {x} \<Longrightarrow> 
hoelzl@57448
   953
  at x within s = inf (INF a:{x <..}. principal ({..< a} \<inter> s - {x}))
hoelzl@57448
   954
                      (INF a:{..< x}. principal ({a <..} \<inter> s - {x}))"
hoelzl@57448
   955
proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split[case_product case_split])
hoelzl@57448
   956
  assume "UNIV \<noteq> {x}" "{x<..} = {}" "{..< x} = {}"
hoelzl@57448
   957
  moreover have "UNIV = {..< x} \<union> {x} \<union> {x <..}"
hoelzl@57448
   958
    by auto
hoelzl@57448
   959
  ultimately show ?thesis
hoelzl@57448
   960
    by auto
hoelzl@57448
   961
qed (auto simp: at_within_def nhds_order Int_Diff inf_principal[symmetric] INF_inf_const2
hoelzl@57448
   962
                inf_sup_aci[where 'a="'a filter"]
hoelzl@57448
   963
          simp del: inf_principal)
hoelzl@57448
   964
hoelzl@57448
   965
lemma (in linorder_topology) at_left_eq:
hoelzl@57448
   966
  "y < x \<Longrightarrow> at_left x = (INF a:{..< x}. principal {a <..< x})"
hoelzl@57448
   967
  by (subst at_within_order)
hoelzl@57448
   968
     (auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant
hoelzl@57448
   969
           intro!: INF_lower2 inf_absorb2)
hoelzl@57448
   970
hoelzl@57448
   971
lemma (in linorder_topology) eventually_at_left:
hoelzl@57448
   972
  "y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
hoelzl@57448
   973
  unfolding at_left_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
hoelzl@57448
   974
hoelzl@57448
   975
lemma (in linorder_topology) at_right_eq:
hoelzl@57448
   976
  "x < y \<Longrightarrow> at_right x = (INF a:{x <..}. principal {x <..< a})"
hoelzl@57448
   977
  by (subst at_within_order)
hoelzl@57448
   978
     (auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute
hoelzl@57448
   979
           intro!: INF_lower2 inf_absorb1)
hoelzl@57448
   980
hoelzl@57448
   981
lemma (in linorder_topology) eventually_at_right:
hoelzl@57448
   982
  "x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
hoelzl@57448
   983
  unfolding at_right_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
hoelzl@51471
   984
hoelzl@57275
   985
lemma trivial_limit_at_right_top: "at_right (top::_::{order_top, linorder_topology}) = bot"
hoelzl@57275
   986
  unfolding filter_eq_iff eventually_at_topological by auto
hoelzl@57275
   987
hoelzl@57275
   988
lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot, linorder_topology}) = bot"
hoelzl@57275
   989
  unfolding filter_eq_iff eventually_at_topological by auto
hoelzl@57275
   990
hoelzl@51471
   991
lemma trivial_limit_at_left_real [simp]:
hoelzl@57275
   992
  "\<not> trivial_limit (at_left (x::'a::{no_bot, dense_order, linorder_topology}))"
hoelzl@57275
   993
  using lt_ex[of x]
hoelzl@57275
   994
  by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense)
hoelzl@51471
   995
hoelzl@51471
   996
lemma trivial_limit_at_right_real [simp]:
hoelzl@57275
   997
  "\<not> trivial_limit (at_right (x::'a::{no_top, dense_order, linorder_topology}))"
hoelzl@57275
   998
  using gt_ex[of x]
hoelzl@57275
   999
  by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense)
hoelzl@51471
  1000
hoelzl@51471
  1001
lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"
hoelzl@51641
  1002
  by (auto simp: eventually_at_filter filter_eq_iff eventually_sup 
hoelzl@51471
  1003
           elim: eventually_elim2 eventually_elim1)
hoelzl@51471
  1004
hoelzl@51471
  1005
lemma eventually_at_split:
hoelzl@51471
  1006
  "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
hoelzl@51471
  1007
  by (subst at_eq_sup_left_right) (simp add: eventually_sup)
hoelzl@51471
  1008
hoelzl@51471
  1009
subsection {* Limits *}
hoelzl@51471
  1010
hoelzl@51471
  1011
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@51471
  1012
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@51471
  1013
hoelzl@51471
  1014
syntax
hoelzl@51471
  1015
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@51471
  1016
hoelzl@51471
  1017
translations
hoelzl@51471
  1018
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@51471
  1019
hoelzl@51471
  1020
lemma filterlim_iff:
hoelzl@51471
  1021
  "(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)"
hoelzl@51471
  1022
  unfolding filterlim_def le_filter_def eventually_filtermap ..
hoelzl@51471
  1023
hoelzl@51471
  1024
lemma filterlim_compose:
hoelzl@51471
  1025
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@51471
  1026
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@51471
  1027
hoelzl@51471
  1028
lemma filterlim_mono:
hoelzl@51471
  1029
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@51471
  1030
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@51471
  1031
hoelzl@51471
  1032
lemma filterlim_ident: "LIM x F. x :> F"
hoelzl@51471
  1033
  by (simp add: filterlim_def filtermap_ident)
hoelzl@51471
  1034
hoelzl@51471
  1035
lemma filterlim_cong:
hoelzl@51471
  1036
  "F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'"
hoelzl@51471
  1037
  by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2)
hoelzl@51471
  1038
hoelzl@57275
  1039
lemma filterlim_mono_eventually:
hoelzl@57275
  1040
  assumes "filterlim f F G" and ord: "F \<le> F'" "G' \<le> G"
hoelzl@57275
  1041
  assumes eq: "eventually (\<lambda>x. f x = f' x) G'"
hoelzl@57275
  1042
  shows "filterlim f' F' G'"
hoelzl@57275
  1043
  apply (rule filterlim_cong[OF refl refl eq, THEN iffD1])
hoelzl@57275
  1044
  apply (rule filterlim_mono[OF _ ord])
hoelzl@57275
  1045
  apply fact
hoelzl@57275
  1046
  done
hoelzl@57275
  1047
hoelzl@57447
  1048
lemma filtermap_mono_strong: "inj f \<Longrightarrow> filtermap f F \<le> filtermap f G \<longleftrightarrow> F \<le> G"
hoelzl@57447
  1049
  apply (auto intro!: filtermap_mono) []
hoelzl@57447
  1050
  apply (auto simp: le_filter_def eventually_filtermap)
hoelzl@57447
  1051
  apply (erule_tac x="\<lambda>x. P (inv f x)" in allE)
hoelzl@57447
  1052
  apply auto
hoelzl@57447
  1053
  done
hoelzl@57447
  1054
hoelzl@57447
  1055
lemma filtermap_eq_strong: "inj f \<Longrightarrow> filtermap f F = filtermap f G \<longleftrightarrow> F = G"
hoelzl@57447
  1056
  by (simp add: filtermap_mono_strong eq_iff)
hoelzl@57447
  1057
hoelzl@51641
  1058
lemma filterlim_principal:
hoelzl@51641
  1059
  "(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@51641
  1060
  unfolding filterlim_def eventually_filtermap le_principal ..
hoelzl@51641
  1061
hoelzl@51641
  1062
lemma filterlim_inf:
hoelzl@51641
  1063
  "(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))"
hoelzl@51641
  1064
  unfolding filterlim_def by simp
hoelzl@51471
  1065
hoelzl@57276
  1066
lemma filterlim_INF:
hoelzl@57276
  1067
  "(LIM x F. f x :> (INF b:B. G b)) \<longleftrightarrow> (\<forall>b\<in>B. LIM x F. f x :> G b)"
hoelzl@57276
  1068
  unfolding filterlim_def le_INF_iff ..
hoelzl@57276
  1069
hoelzl@57448
  1070
lemma filterlim_INF_INF:
hoelzl@57448
  1071
  "(\<And>m. m \<in> J \<Longrightarrow> \<exists>i\<in>I. filtermap f (F i) \<le> G m) \<Longrightarrow> LIM x (INF i:I. F i). f x :> (INF j:J. G j)"
hoelzl@57448
  1072
  unfolding filterlim_def by (rule order_trans[OF filtermap_INF INF_mono])
hoelzl@57448
  1073
hoelzl@57448
  1074
lemma filterlim_base:
hoelzl@57448
  1075
  "(\<And>m x. m \<in> J \<Longrightarrow> i m \<in> I) \<Longrightarrow> (\<And>m x. m \<in> J \<Longrightarrow> x \<in> F (i m) \<Longrightarrow> f x \<in> G m) \<Longrightarrow> 
hoelzl@57448
  1076
    LIM x (INF i:I. principal (F i)). f x :> (INF j:J. principal (G j))"
hoelzl@57448
  1077
  by (force intro!: filterlim_INF_INF simp: image_subset_iff)
hoelzl@57448
  1078
hoelzl@57448
  1079
lemma filterlim_base_iff: 
hoelzl@57448
  1080
  assumes "I \<noteq> {}" and chain: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> F i \<subseteq> F j \<or> F j \<subseteq> F i"
hoelzl@57448
  1081
  shows "(LIM x (INF i:I. principal (F i)). f x :> INF j:J. principal (G j)) \<longleftrightarrow>
hoelzl@57448
  1082
    (\<forall>j\<in>J. \<exists>i\<in>I. \<forall>x\<in>F i. f x \<in> G j)"
hoelzl@57448
  1083
  unfolding filterlim_INF filterlim_principal
hoelzl@57448
  1084
proof (subst eventually_INF_base)
hoelzl@57448
  1085
  fix i j assume "i \<in> I" "j \<in> I"
hoelzl@57448
  1086
  with chain[OF this] show "\<exists>x\<in>I. principal (F x) \<le> inf (principal (F i)) (principal (F j))"
hoelzl@57448
  1087
    by auto
hoelzl@57448
  1088
qed (auto simp: eventually_principal `I \<noteq> {}`)
hoelzl@57448
  1089
hoelzl@51471
  1090
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2"
hoelzl@51471
  1091
  unfolding filterlim_def filtermap_filtermap ..
hoelzl@51471
  1092
hoelzl@51471
  1093
lemma filterlim_sup:
hoelzl@51471
  1094
  "filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)"
hoelzl@51471
  1095
  unfolding filterlim_def filtermap_sup by auto
hoelzl@51471
  1096
hoelzl@57447
  1097
lemma eventually_sequentially_Suc: "eventually (\<lambda>i. P (Suc i)) sequentially \<longleftrightarrow> eventually P sequentially"
hoelzl@57447
  1098
  unfolding eventually_sequentially by (metis Suc_le_D Suc_le_mono le_Suc_eq)
hoelzl@57447
  1099
hoelzl@57447
  1100
lemma filterlim_sequentially_Suc:
hoelzl@57447
  1101
  "(LIM x sequentially. f (Suc x) :> F) \<longleftrightarrow> (LIM x sequentially. f x :> F)"
hoelzl@57447
  1102
  unfolding filterlim_iff by (subst eventually_sequentially_Suc) simp
hoelzl@57447
  1103
hoelzl@51471
  1104
lemma filterlim_Suc: "filterlim Suc sequentially sequentially"
hoelzl@51471
  1105
  by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq)
hoelzl@51471
  1106
hoelzl@51471
  1107
subsubsection {* Tendsto *}
hoelzl@51471
  1108
hoelzl@51471
  1109
abbreviation (in topological_space)
hoelzl@51471
  1110
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
hoelzl@51471
  1111
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
hoelzl@51471
  1112
hoelzl@51478
  1113
definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
  1114
  "Lim A f = (THE l. (f ---> l) A)"
hoelzl@51478
  1115
hoelzl@51471
  1116
lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F"
hoelzl@51471
  1117
  by simp
hoelzl@51471
  1118
wenzelm@57953
  1119
named_theorems tendsto_intros "introduction rules for tendsto"
hoelzl@51471
  1120
setup {*
hoelzl@51471
  1121
  Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},
wenzelm@57953
  1122
    fn context =>
wenzelm@57953
  1123
      Named_Theorems.get (Context.proof_of context) @{named_theorems tendsto_intros}
wenzelm@57953
  1124
      |> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])))
hoelzl@51471
  1125
*}
hoelzl@51471
  1126
hoelzl@51473
  1127
lemma (in topological_space) tendsto_def:
hoelzl@51473
  1128
   "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@57276
  1129
   unfolding nhds_def filterlim_INF filterlim_principal by auto
hoelzl@51471
  1130
hoelzl@51471
  1131
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
hoelzl@51471
  1132
  unfolding tendsto_def le_filter_def by fast
hoelzl@51471
  1133
hoelzl@51641
  1134
lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)"
hoelzl@51641
  1135
  by (blast intro: tendsto_mono at_le)
hoelzl@51641
  1136
hoelzl@51641
  1137
lemma filterlim_at:
hoelzl@51641
  1138
  "(LIM x F. f x :> at b within s) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f ---> b) F)"
hoelzl@51641
  1139
  by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
hoelzl@51641
  1140
hoelzl@51473
  1141
lemma (in topological_space) topological_tendstoI:
hoelzl@51641
  1142
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f ---> l) F"
hoelzl@51471
  1143
  unfolding tendsto_def by auto
hoelzl@51471
  1144
hoelzl@51473
  1145
lemma (in topological_space) topological_tendstoD:
hoelzl@51471
  1146
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
  1147
  unfolding tendsto_def by auto
hoelzl@51471
  1148
hoelzl@57448
  1149
lemma (in order_topology) order_tendsto_iff:
hoelzl@57448
  1150
  "(f ---> x) F \<longleftrightarrow> (\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)"
hoelzl@57448
  1151
  unfolding nhds_order filterlim_inf filterlim_INF filterlim_principal by auto
hoelzl@57448
  1152
hoelzl@57448
  1153
lemma (in order_topology) order_tendstoI:
hoelzl@57448
  1154
  "(\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F) \<Longrightarrow> (\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F) \<Longrightarrow>
hoelzl@57448
  1155
    (f ---> y) F"
hoelzl@57448
  1156
  unfolding order_tendsto_iff by auto
hoelzl@57448
  1157
hoelzl@57448
  1158
lemma (in order_topology) order_tendstoD:
hoelzl@57448
  1159
  assumes "(f ---> y) F"
hoelzl@51471
  1160
  shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
hoelzl@51471
  1161
    and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
hoelzl@57448
  1162
  using assms unfolding order_tendsto_iff by auto
hoelzl@51471
  1163
hoelzl@51471
  1164
lemma tendsto_bot [simp]: "(f ---> a) bot"
hoelzl@51471
  1165
  unfolding tendsto_def by simp
hoelzl@51471
  1166
hoelzl@57448
  1167
lemma (in linorder_topology) tendsto_max:
hoelzl@56949
  1168
  assumes X: "(X ---> x) net"
hoelzl@56949
  1169
  assumes Y: "(Y ---> y) net"
hoelzl@56949
  1170
  shows "((\<lambda>x. max (X x) (Y x)) ---> max x y) net"
hoelzl@56949
  1171
proof (rule order_tendstoI)
hoelzl@56949
  1172
  fix a assume "a < max x y"
hoelzl@56949
  1173
  then show "eventually (\<lambda>x. a < max (X x) (Y x)) net"
hoelzl@56949
  1174
    using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
hoelzl@56949
  1175
    by (auto simp: less_max_iff_disj elim: eventually_elim1)
hoelzl@56949
  1176
next
hoelzl@56949
  1177
  fix a assume "max x y < a"
hoelzl@56949
  1178
  then show "eventually (\<lambda>x. max (X x) (Y x) < a) net"
hoelzl@56949
  1179
    using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
hoelzl@56949
  1180
    by (auto simp: eventually_conj_iff)
hoelzl@56949
  1181
qed
hoelzl@56949
  1182
hoelzl@57448
  1183
lemma (in linorder_topology) tendsto_min:
hoelzl@56949
  1184
  assumes X: "(X ---> x) net"
hoelzl@56949
  1185
  assumes Y: "(Y ---> y) net"
hoelzl@56949
  1186
  shows "((\<lambda>x. min (X x) (Y x)) ---> min x y) net"
hoelzl@56949
  1187
proof (rule order_tendstoI)
hoelzl@56949
  1188
  fix a assume "a < min x y"
hoelzl@56949
  1189
  then show "eventually (\<lambda>x. a < min (X x) (Y x)) net"
hoelzl@56949
  1190
    using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
hoelzl@56949
  1191
    by (auto simp: eventually_conj_iff)
hoelzl@56949
  1192
next
hoelzl@56949
  1193
  fix a assume "min x y < a"
hoelzl@56949
  1194
  then show "eventually (\<lambda>x. min (X x) (Y x) < a) net"
hoelzl@56949
  1195
    using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
hoelzl@56949
  1196
    by (auto simp: min_less_iff_disj elim: eventually_elim1)
hoelzl@56949
  1197
qed
hoelzl@56949
  1198
hoelzl@58729
  1199
lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) ---> a) (at a within s)"
hoelzl@51471
  1200
  unfolding tendsto_def eventually_at_topological by auto
hoelzl@51471
  1201
hoelzl@58729
  1202
lemma (in topological_space) tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) ---> k) F"
hoelzl@51471
  1203
  by (simp add: tendsto_def)
hoelzl@51471
  1204
hoelzl@51478
  1205
lemma (in t2_space) tendsto_unique:
hoelzl@57448
  1206
  assumes "F \<noteq> bot" and "(f ---> a) F" and "(f ---> b) F"
hoelzl@51471
  1207
  shows "a = b"
hoelzl@51471
  1208
proof (rule ccontr)
hoelzl@51471
  1209
  assume "a \<noteq> b"
hoelzl@51471
  1210
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
hoelzl@51471
  1211
    using hausdorff [OF `a \<noteq> b`] by fast
hoelzl@51471
  1212
  have "eventually (\<lambda>x. f x \<in> U) F"
hoelzl@51471
  1213
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
hoelzl@51471
  1214
  moreover
hoelzl@51471
  1215
  have "eventually (\<lambda>x. f x \<in> V) F"
hoelzl@51471
  1216
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
hoelzl@51471
  1217
  ultimately
hoelzl@51471
  1218
  have "eventually (\<lambda>x. False) F"
hoelzl@51471
  1219
  proof eventually_elim
hoelzl@51471
  1220
    case (elim x)
hoelzl@51471
  1221
    hence "f x \<in> U \<inter> V" by simp
hoelzl@51471
  1222
    with `U \<inter> V = {}` show ?case by simp
hoelzl@51471
  1223
  qed
hoelzl@51471
  1224
  with `\<not> trivial_limit F` show "False"
hoelzl@51471
  1225
    by (simp add: trivial_limit_def)
hoelzl@51471
  1226
qed
hoelzl@51471
  1227
hoelzl@51478
  1228
lemma (in t2_space) tendsto_const_iff:
hoelzl@51478
  1229
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b"
hoelzl@58729
  1230
  by (auto intro!: tendsto_unique [OF assms tendsto_const])
hoelzl@51471
  1231
hoelzl@51471
  1232
lemma increasing_tendsto:
hoelzl@51471
  1233
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
  1234
  assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
hoelzl@51471
  1235
      and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
hoelzl@51471
  1236
  shows "(f ---> l) F"
hoelzl@51471
  1237
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
hoelzl@51471
  1238
hoelzl@51471
  1239
lemma decreasing_tendsto:
hoelzl@51471
  1240
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
  1241
  assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
hoelzl@51471
  1242
      and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
hoelzl@51471
  1243
  shows "(f ---> l) F"
hoelzl@51471
  1244
  using assms by (intro order_tendstoI) (auto elim!: eventually_elim1)
hoelzl@51471
  1245
hoelzl@51471
  1246
lemma tendsto_sandwich:
hoelzl@51471
  1247
  fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
hoelzl@51471
  1248
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
hoelzl@51471
  1249
  assumes lim: "(f ---> c) net" "(h ---> c) net"
hoelzl@51471
  1250
  shows "(g ---> c) net"
hoelzl@51471
  1251
proof (rule order_tendstoI)
hoelzl@51471
  1252
  fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
hoelzl@51471
  1253
    using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
  1254
next
hoelzl@51471
  1255
  fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
hoelzl@51471
  1256
    using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
  1257
qed
hoelzl@51471
  1258
hoelzl@51471
  1259
lemma tendsto_le:
hoelzl@51471
  1260
  fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
  1261
  assumes F: "\<not> trivial_limit F"
hoelzl@51471
  1262
  assumes x: "(f ---> x) F" and y: "(g ---> y) F"
hoelzl@51471
  1263
  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
hoelzl@51471
  1264
  shows "y \<le> x"
hoelzl@51471
  1265
proof (rule ccontr)
hoelzl@51471
  1266
  assume "\<not> y \<le> x"
hoelzl@51471
  1267
  with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
hoelzl@51471
  1268
    by (auto simp: not_le)
hoelzl@51471
  1269
  then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
hoelzl@51471
  1270
    using x y by (auto intro: order_tendstoD)
hoelzl@51471
  1271
  with ev have "eventually (\<lambda>x. False) F"
hoelzl@51471
  1272
    by eventually_elim (insert xy, fastforce)
hoelzl@51471
  1273
  with F show False
hoelzl@51471
  1274
    by (simp add: eventually_False)
hoelzl@51471
  1275
qed
hoelzl@51471
  1276
hoelzl@51471
  1277
lemma tendsto_le_const:
hoelzl@51471
  1278
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
  1279
  assumes F: "\<not> trivial_limit F"
lp15@56289
  1280
  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F"
hoelzl@51471
  1281
  shows "a \<le> x"
hoelzl@51471
  1282
  using F x tendsto_const a by (rule tendsto_le)
hoelzl@51471
  1283
lp15@56289
  1284
lemma tendsto_ge_const:
lp15@56289
  1285
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
lp15@56289
  1286
  assumes F: "\<not> trivial_limit F"
lp15@56289
  1287
  assumes x: "(f ---> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F"
lp15@56289
  1288
  shows "a \<ge> x"
lp15@56289
  1289
  by (rule tendsto_le [OF F tendsto_const x a])
lp15@56289
  1290
hoelzl@51478
  1291
subsubsection {* Rules about @{const Lim} *}
hoelzl@51478
  1292
hoelzl@57276
  1293
lemma tendsto_Lim:
hoelzl@51478
  1294
  "\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l"
hoelzl@51478
  1295
  unfolding Lim_def using tendsto_unique[of net f] by auto
hoelzl@51478
  1296
hoelzl@51641
  1297
lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
hoelzl@51478
  1298
  by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
hoelzl@51478
  1299
hoelzl@51471
  1300
subsection {* Limits to @{const at_top} and @{const at_bot} *}
hoelzl@51471
  1301
hoelzl@51471
  1302
lemma filterlim_at_top:
hoelzl@51471
  1303
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@51471
  1304
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1305
  by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1)
hoelzl@51471
  1306
hoelzl@57448
  1307
lemma filterlim_at_top_mono:
hoelzl@57448
  1308
  "LIM x F. f x :> at_top \<Longrightarrow> eventually (\<lambda>x. f x \<le> (g x::'a::linorder)) F \<Longrightarrow>
hoelzl@57448
  1309
    LIM x F. g x :> at_top"
hoelzl@57448
  1310
  by (auto simp: filterlim_at_top elim: eventually_elim2 intro: order_trans)
hoelzl@57448
  1311
hoelzl@51471
  1312
lemma filterlim_at_top_dense:
hoelzl@53215
  1313
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)"
hoelzl@51471
  1314
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@51471
  1315
  by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le
hoelzl@51471
  1316
            filterlim_at_top[of f F] filterlim_iff[of f at_top F])
hoelzl@51471
  1317
hoelzl@51471
  1318
lemma filterlim_at_top_ge:
hoelzl@51471
  1319
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@51471
  1320
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@57448
  1321
  unfolding at_top_sub[of c] filterlim_INF by (auto simp add: filterlim_principal)
hoelzl@51471
  1322
hoelzl@51471
  1323
lemma filterlim_at_top_at_top:
hoelzl@51471
  1324
  fixes f :: "'a::linorder \<Rightarrow> 'b::linorder"
hoelzl@51471
  1325
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1326
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1327
  assumes Q: "eventually Q at_top"
hoelzl@51471
  1328
  assumes P: "eventually P at_top"
hoelzl@51471
  1329
  shows "filterlim f at_top at_top"
hoelzl@51471
  1330
proof -
hoelzl@51471
  1331
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
  1332
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
  1333
  show ?thesis
hoelzl@51471
  1334
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
  1335
    fix z assume "x \<le> z"
hoelzl@51471
  1336
    with x have "P z" by auto
hoelzl@51471
  1337
    have "eventually (\<lambda>x. g z \<le> x) at_top"
hoelzl@51471
  1338
      by (rule eventually_ge_at_top)
hoelzl@51471
  1339
    with Q show "eventually (\<lambda>x. z \<le> f x) at_top"
hoelzl@51471
  1340
      by eventually_elim (metis mono bij `P z`)
hoelzl@51471
  1341
  qed
hoelzl@51471
  1342
qed
hoelzl@51471
  1343
hoelzl@51471
  1344
lemma filterlim_at_top_gt:
hoelzl@53215
  1345
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@51471
  1346
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)"
hoelzl@51471
  1347
  by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge)
hoelzl@51471
  1348
hoelzl@51471
  1349
lemma filterlim_at_bot: 
hoelzl@51471
  1350
  fixes f :: "'a \<Rightarrow> ('b::linorder)"
hoelzl@51471
  1351
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)"
hoelzl@51471
  1352
  by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1)
hoelzl@51471
  1353
hoelzl@57447
  1354
lemma filterlim_at_bot_dense:
hoelzl@57447
  1355
  fixes f :: "'a \<Rightarrow> ('b::{dense_linorder, no_bot})"
hoelzl@57447
  1356
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
hoelzl@57447
  1357
proof (auto simp add: filterlim_at_bot[of f F])
hoelzl@57447
  1358
  fix Z :: 'b
hoelzl@57447
  1359
  from lt_ex [of Z] obtain Z' where 1: "Z' < Z" ..
hoelzl@57447
  1360
  assume "\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F"
hoelzl@57447
  1361
  hence "eventually (\<lambda>x. f x \<le> Z') F" by auto
hoelzl@57447
  1362
  thus "eventually (\<lambda>x. f x < Z) F"
hoelzl@57447
  1363
    apply (rule eventually_mono[rotated])
hoelzl@57447
  1364
    using 1 by auto
hoelzl@57447
  1365
  next 
hoelzl@57447
  1366
    fix Z :: 'b 
hoelzl@57447
  1367
    show "\<forall>Z. eventually (\<lambda>x. f x < Z) F \<Longrightarrow> eventually (\<lambda>x. f x \<le> Z) F"
hoelzl@57447
  1368
      by (drule spec [of _ Z], erule eventually_mono[rotated], auto simp add: less_imp_le)
hoelzl@57447
  1369
qed
hoelzl@57447
  1370
hoelzl@51471
  1371
lemma filterlim_at_bot_le:
hoelzl@51471
  1372
  fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b"
hoelzl@51471
  1373
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@51471
  1374
  unfolding filterlim_at_bot
hoelzl@51471
  1375
proof safe
hoelzl@51471
  1376
  fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@51471
  1377
  with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F"
hoelzl@51471
  1378
    by (auto elim!: eventually_elim1)
hoelzl@51471
  1379
qed simp
hoelzl@51471
  1380
hoelzl@51471
  1381
lemma filterlim_at_bot_lt:
hoelzl@53215
  1382
  fixes f :: "'a \<Rightarrow> ('b::unbounded_dense_linorder)" and c :: "'b"
hoelzl@51471
  1383
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)"
hoelzl@51471
  1384
  by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans)
hoelzl@51471
  1385
hoelzl@51471
  1386
lemma filterlim_at_bot_at_right:
hoelzl@57275
  1387
  fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
hoelzl@51471
  1388
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1389
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1390
  assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
hoelzl@51471
  1391
  assumes P: "eventually P at_bot"
hoelzl@51471
  1392
  shows "filterlim f at_bot (at_right a)"
hoelzl@51471
  1393
proof -
hoelzl@51471
  1394
  from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
hoelzl@51471
  1395
    unfolding eventually_at_bot_linorder by auto
hoelzl@51471
  1396
  show ?thesis
hoelzl@51471
  1397
  proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
hoelzl@51471
  1398
    fix z assume "z \<le> x"
hoelzl@51471
  1399
    with x have "P z" by auto
hoelzl@51471
  1400
    have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
hoelzl@51471
  1401
      using bound[OF bij(2)[OF `P z`]]
hoelzl@57275
  1402
      unfolding eventually_at_right[OF bound[OF bij(2)[OF `P z`]]] by (auto intro!: exI[of _ "g z"])
hoelzl@51471
  1403
    with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
hoelzl@51471
  1404
      by eventually_elim (metis bij `P z` mono)
hoelzl@51471
  1405
  qed
hoelzl@51471
  1406
qed
hoelzl@51471
  1407
hoelzl@51471
  1408
lemma filterlim_at_top_at_left:
hoelzl@57275
  1409
  fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
hoelzl@51471
  1410
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
  1411
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
  1412
  assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
hoelzl@51471
  1413
  assumes P: "eventually P at_top"
hoelzl@51471
  1414
  shows "filterlim f at_top (at_left a)"
hoelzl@51471
  1415
proof -
hoelzl@51471
  1416
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
  1417
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
  1418
  show ?thesis
hoelzl@51471
  1419
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
  1420
    fix z assume "x \<le> z"
hoelzl@51471
  1421
    with x have "P z" by auto
hoelzl@51471
  1422
    have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
hoelzl@51471
  1423
      using bound[OF bij(2)[OF `P z`]]
hoelzl@57275
  1424
      unfolding eventually_at_left[OF bound[OF bij(2)[OF `P z`]]] by (auto intro!: exI[of _ "g z"])
hoelzl@51471
  1425
    with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
hoelzl@51471
  1426
      by eventually_elim (metis bij `P z` mono)
hoelzl@51471
  1427
  qed
hoelzl@51471
  1428
qed
hoelzl@51471
  1429
hoelzl@51471
  1430
lemma filterlim_split_at:
hoelzl@51471
  1431
  "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"
hoelzl@51471
  1432
  by (subst at_eq_sup_left_right) (rule filterlim_sup)
hoelzl@51471
  1433
hoelzl@51471
  1434
lemma filterlim_at_split:
hoelzl@51471
  1435
  "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
hoelzl@51471
  1436
  by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
hoelzl@51471
  1437
hoelzl@57025
  1438
lemma eventually_nhds_top:
hoelzl@57025
  1439
  fixes P :: "'a :: {order_top, linorder_topology} \<Rightarrow> bool"
hoelzl@57025
  1440
  assumes "(b::'a) < top"
hoelzl@57025
  1441
  shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))"
hoelzl@57025
  1442
  unfolding eventually_nhds
hoelzl@57025
  1443
proof safe
hoelzl@57025
  1444
  fix S :: "'a set" assume "open S" "top \<in> S"
hoelzl@57025
  1445
  note open_left[OF this `b < top`]
hoelzl@57025
  1446
  moreover assume "\<forall>s\<in>S. P s"
hoelzl@57025
  1447
  ultimately show "\<exists>b<top. \<forall>z>b. P z"
hoelzl@57025
  1448
    by (auto simp: subset_eq Ball_def)
hoelzl@57025
  1449
next
hoelzl@57025
  1450
  fix b assume "b < top" "\<forall>z>b. P z"
hoelzl@57025
  1451
  then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)"
hoelzl@57025
  1452
    by (intro exI[of _ "{b <..}"]) auto
hoelzl@57025
  1453
qed
hoelzl@51471
  1454
hoelzl@57447
  1455
lemma tendsto_at_within_iff_tendsto_nhds:
hoelzl@57447
  1456
  "(g ---> g l) (at l within S) \<longleftrightarrow> (g ---> g l) (inf (nhds l) (principal S))"
hoelzl@57447
  1457
  unfolding tendsto_def eventually_at_filter eventually_inf_principal
hoelzl@57447
  1458
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@57447
  1459
hoelzl@51471
  1460
subsection {* Limits on sequences *}
hoelzl@51471
  1461
hoelzl@51471
  1462
abbreviation (in topological_space)
hoelzl@51471
  1463
  LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
hoelzl@51471
  1464
    ("((_)/ ----> (_))" [60, 60] 60) where
hoelzl@51471
  1465
  "X ----> L \<equiv> (X ---> L) sequentially"
hoelzl@51471
  1466
hoelzl@51478
  1467
abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
  1468
  "lim X \<equiv> Lim sequentially X"
hoelzl@51471
  1469
hoelzl@51471
  1470
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
hoelzl@51471
  1471
  "convergent X = (\<exists>L. X ----> L)"
hoelzl@51471
  1472
hoelzl@51478
  1473
lemma lim_def: "lim X = (THE L. X ----> L)"
hoelzl@51478
  1474
  unfolding Lim_def ..
hoelzl@51478
  1475
hoelzl@51471
  1476
subsubsection {* Monotone sequences and subsequences *}
hoelzl@51471
  1477
hoelzl@51471
  1478
definition
hoelzl@51471
  1479
  monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@51471
  1480
    --{*Definition of monotonicity.
hoelzl@51471
  1481
        The use of disjunction here complicates proofs considerably.
hoelzl@51471
  1482
        One alternative is to add a Boolean argument to indicate the direction.
hoelzl@51471
  1483
        Another is to develop the notions of increasing and decreasing first.*}
hoelzl@56020
  1484
  "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
hoelzl@56020
  1485
hoelzl@56020
  1486
abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@56020
  1487
  "incseq X \<equiv> mono X"
hoelzl@56020
  1488
hoelzl@56020
  1489
lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)"
hoelzl@56020
  1490
  unfolding mono_def ..
hoelzl@56020
  1491
hoelzl@56020
  1492
abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@56020
  1493
  "decseq X \<equiv> antimono X"
hoelzl@56020
  1494
hoelzl@56020
  1495
lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
hoelzl@56020
  1496
  unfolding antimono_def ..
hoelzl@51471
  1497
hoelzl@51471
  1498
definition
hoelzl@51471
  1499
  subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
hoelzl@51471
  1500
    --{*Definition of subsequence*}
hoelzl@51471
  1501
  "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
hoelzl@51471
  1502
hoelzl@51471
  1503
lemma incseq_SucI:
hoelzl@51471
  1504
  "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
hoelzl@51471
  1505
  using lift_Suc_mono_le[of X]
hoelzl@51471
  1506
  by (auto simp: incseq_def)
hoelzl@51471
  1507
hoelzl@51471
  1508
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
hoelzl@51471
  1509
  by (auto simp: incseq_def)
hoelzl@51471
  1510
hoelzl@51471
  1511
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
hoelzl@51471
  1512
  using incseqD[of A i "Suc i"] by auto
hoelzl@51471
  1513
hoelzl@51471
  1514
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
hoelzl@51471
  1515
  by (auto intro: incseq_SucI dest: incseq_SucD)
hoelzl@51471
  1516
hoelzl@51471
  1517
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
hoelzl@51471
  1518
  unfolding incseq_def by auto
hoelzl@51471
  1519
hoelzl@51471
  1520
lemma decseq_SucI:
hoelzl@51471
  1521
  "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
hoelzl@51471
  1522
  using order.lift_Suc_mono_le[OF dual_order, of X]
hoelzl@51471
  1523
  by (auto simp: decseq_def)
hoelzl@51471
  1524
hoelzl@51471
  1525
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
hoelzl@51471
  1526
  by (auto simp: decseq_def)
hoelzl@51471
  1527
hoelzl@51471
  1528
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
hoelzl@51471
  1529
  using decseqD[of A i "Suc i"] by auto
hoelzl@51471
  1530
hoelzl@51471
  1531
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@51471
  1532
  by (auto intro: decseq_SucI dest: decseq_SucD)
hoelzl@51471
  1533
hoelzl@51471
  1534
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
hoelzl@51471
  1535
  unfolding decseq_def by auto
hoelzl@51471
  1536
hoelzl@51471
  1537
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
hoelzl@51471
  1538
  unfolding monoseq_def incseq_def decseq_def ..
hoelzl@51471
  1539
hoelzl@51471
  1540
lemma monoseq_Suc:
hoelzl@51471
  1541
  "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
hoelzl@51471
  1542
  unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
hoelzl@51471
  1543
hoelzl@51471
  1544
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
hoelzl@51471
  1545
by (simp add: monoseq_def)
hoelzl@51471
  1546
hoelzl@51471
  1547
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
hoelzl@51471
  1548
by (simp add: monoseq_def)
hoelzl@51471
  1549
hoelzl@51471
  1550
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
hoelzl@51471
  1551
by (simp add: monoseq_Suc)
hoelzl@51471
  1552
hoelzl@51471
  1553
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
hoelzl@51471
  1554
by (simp add: monoseq_Suc)
hoelzl@51471
  1555
hoelzl@51471
  1556
lemma monoseq_minus:
hoelzl@51471
  1557
  fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
hoelzl@51471
  1558
  assumes "monoseq a"
hoelzl@51471
  1559
  shows "monoseq (\<lambda> n. - a n)"
hoelzl@51471
  1560
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
hoelzl@51471
  1561
  case True
hoelzl@51471
  1562
  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
hoelzl@51471
  1563
  thus ?thesis by (rule monoI2)
hoelzl@51471
  1564
next
hoelzl@51471
  1565
  case False
hoelzl@51471
  1566
  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
hoelzl@51471
  1567
  thus ?thesis by (rule monoI1)
hoelzl@51471
  1568
qed
hoelzl@51471
  1569
hoelzl@51471
  1570
text{*Subsequence (alternative definition, (e.g. Hoskins)*}
hoelzl@51471
  1571
hoelzl@51471
  1572
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
hoelzl@51471
  1573
apply (simp add: subseq_def)
hoelzl@51471
  1574
apply (auto dest!: less_imp_Suc_add)
hoelzl@51471
  1575
apply (induct_tac k)
hoelzl@51471
  1576
apply (auto intro: less_trans)
hoelzl@51471
  1577
done
hoelzl@51471
  1578
hoelzl@51471
  1579
text{* for any sequence, there is a monotonic subsequence *}
hoelzl@51471
  1580
lemma seq_monosub:
hoelzl@51471
  1581
  fixes s :: "nat => 'a::linorder"
hoelzl@57448
  1582
  shows "\<exists>f. subseq f \<and> monoseq (\<lambda>n. (s (f n)))"
hoelzl@51471
  1583
proof cases
hoelzl@57448
  1584
  assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p"
hoelzl@57448
  1585
  then have "\<exists>f. \<forall>n. (\<forall>m\<ge>f n. s m \<le> s (f n)) \<and> f n < f (Suc n)"
hoelzl@57448
  1586
    by (intro dependent_nat_choice) (auto simp: conj_commute)
hoelzl@57448
  1587
  then obtain f where "subseq f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)"
hoelzl@57448
  1588
    by (auto simp: subseq_Suc_iff)
hoelzl@57448
  1589
  moreover 
hoelzl@57448
  1590
  then have "incseq f"
hoelzl@57448
  1591
    unfolding subseq_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le)
hoelzl@57448
  1592
  then have "monoseq (\<lambda>n. s (f n))"
hoelzl@57448
  1593
    by (auto simp add: incseq_def intro!: mono monoI2)
hoelzl@57448
  1594
  ultimately show ?thesis
hoelzl@57448
  1595
    by auto
hoelzl@51471
  1596
next
hoelzl@51471
  1597
  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
hoelzl@51471
  1598
  then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less)
hoelzl@57448
  1599
  have "\<exists>f. \<forall>n. N < f n \<and> f n < f (Suc n) \<and> s (f n) \<le> s (f (Suc n))"
hoelzl@57448
  1600
  proof (intro dependent_nat_choice)
hoelzl@57448
  1601
    fix x assume "N < x" with N[of x] show "\<exists>y>N. x < y \<and> s x \<le> s y"
hoelzl@57448
  1602
      by (auto intro: less_trans)
hoelzl@57448
  1603
  qed auto
hoelzl@57448
  1604
  then show ?thesis
hoelzl@57448
  1605
    by (auto simp: monoseq_iff incseq_Suc_iff subseq_Suc_iff)
hoelzl@51471
  1606
qed
hoelzl@51471
  1607
hoelzl@51471
  1608
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
hoelzl@51471
  1609
proof(induct n)
hoelzl@51471
  1610
  case 0 thus ?case by simp
hoelzl@51471
  1611
next
hoelzl@51471
  1612
  case (Suc n)
hoelzl@51471
  1613
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
hoelzl@51471
  1614
  have "n < f (Suc n)" by arith
hoelzl@51471
  1615
  thus ?case by arith
hoelzl@51471
  1616
qed
hoelzl@51471
  1617
hoelzl@51471
  1618
lemma eventually_subseq:
hoelzl@51471
  1619
  "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
hoelzl@51471
  1620
  unfolding eventually_sequentially by (metis seq_suble le_trans)
hoelzl@51471
  1621
hoelzl@51473
  1622
lemma not_eventually_sequentiallyD:
hoelzl@51473
  1623
  assumes P: "\<not> eventually P sequentially"
hoelzl@51473
  1624
  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
hoelzl@51473
  1625
proof -
hoelzl@51473
  1626
  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
hoelzl@51473
  1627
    unfolding eventually_sequentially by (simp add: not_less)
hoelzl@51473
  1628
  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
hoelzl@51473
  1629
    by (auto simp: choice_iff)
hoelzl@51473
  1630
  then show ?thesis
hoelzl@51473
  1631
    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
hoelzl@51473
  1632
             simp: less_eq_Suc_le subseq_Suc_iff)
hoelzl@51473
  1633
qed
hoelzl@51473
  1634
hoelzl@51471
  1635
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
hoelzl@51471
  1636
  unfolding filterlim_iff by (metis eventually_subseq)
hoelzl@51471
  1637
hoelzl@51471
  1638
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
hoelzl@51471
  1639
  unfolding subseq_def by simp
hoelzl@51471
  1640
hoelzl@51471
  1641
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
hoelzl@51471
  1642
  using assms by (auto simp: subseq_def)
hoelzl@51471
  1643
hoelzl@51471
  1644
lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
hoelzl@51471
  1645
  by (simp add: incseq_def monoseq_def)
hoelzl@51471
  1646
hoelzl@51471
  1647
lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
hoelzl@51471
  1648
  by (simp add: decseq_def monoseq_def)
hoelzl@51471
  1649
hoelzl@51471
  1650
lemma decseq_eq_incseq:
hoelzl@51471
  1651
  fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" 
hoelzl@51471
  1652
  by (simp add: decseq_def incseq_def)
hoelzl@51471
  1653
hoelzl@51471
  1654
lemma INT_decseq_offset:
hoelzl@51471
  1655
  assumes "decseq F"
hoelzl@51471
  1656
  shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
  1657
proof safe
hoelzl@51471
  1658
  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
  1659
  show "x \<in> F i"
hoelzl@51471
  1660
  proof cases
hoelzl@51471
  1661
    from x have "x \<in> F n" by auto
hoelzl@51471
  1662
    also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i"
hoelzl@51471
  1663
      unfolding decseq_def by simp
hoelzl@51471
  1664
    finally show ?thesis .
hoelzl@51471
  1665
  qed (insert x, simp)
hoelzl@51471
  1666
qed auto
hoelzl@51471
  1667
hoelzl@51471
  1668
lemma LIMSEQ_const_iff:
hoelzl@51471
  1669
  fixes k l :: "'a::t2_space"
hoelzl@51471
  1670
  shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
hoelzl@51471
  1671
  using trivial_limit_sequentially by (rule tendsto_const_iff)
hoelzl@51471
  1672
hoelzl@51471
  1673
lemma LIMSEQ_SUP:
hoelzl@51471
  1674
  "incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1675
  by (intro increasing_tendsto)
hoelzl@51471
  1676
     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
hoelzl@51471
  1677
hoelzl@51471
  1678
lemma LIMSEQ_INF:
hoelzl@51471
  1679
  "decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1680
  by (intro decreasing_tendsto)
hoelzl@51471
  1681
     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
hoelzl@51471
  1682
hoelzl@51471
  1683
lemma LIMSEQ_ignore_initial_segment:
hoelzl@51471
  1684
  "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
hoelzl@51474
  1685
  unfolding tendsto_def
hoelzl@51474
  1686
  by (subst eventually_sequentially_seg[where k=k])
hoelzl@51471
  1687
hoelzl@51471
  1688
lemma LIMSEQ_offset:
hoelzl@51471
  1689
  "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
hoelzl@51474
  1690
  unfolding tendsto_def
hoelzl@51474
  1691
  by (subst (asm) eventually_sequentially_seg[where k=k])
hoelzl@51471
  1692
hoelzl@51471
  1693
lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
hoelzl@51471
  1694
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
hoelzl@51471
  1695
hoelzl@51471
  1696
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
hoelzl@51471
  1697
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
hoelzl@51471
  1698
hoelzl@51471
  1699
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
hoelzl@51471
  1700
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
hoelzl@51471
  1701
hoelzl@51471
  1702
lemma LIMSEQ_unique:
hoelzl@51471
  1703
  fixes a b :: "'a::t2_space"
hoelzl@51471
  1704
  shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
hoelzl@51471
  1705
  using trivial_limit_sequentially by (rule tendsto_unique)
hoelzl@51471
  1706
hoelzl@51471
  1707
lemma LIMSEQ_le_const:
hoelzl@51471
  1708
  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
hoelzl@51471
  1709
  using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
hoelzl@51471
  1710
hoelzl@51471
  1711
lemma LIMSEQ_le:
hoelzl@51471
  1712
  "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"
hoelzl@51471
  1713
  using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
hoelzl@51471
  1714
hoelzl@51471
  1715
lemma LIMSEQ_le_const2:
hoelzl@51471
  1716
  "\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
hoelzl@58729
  1717
  by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) auto
hoelzl@51471
  1718
hoelzl@51471
  1719
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
hoelzl@51471
  1720
by (simp add: convergent_def)
hoelzl@51471
  1721
hoelzl@51471
  1722
lemma convergentI: "(X ----> L) ==> convergent X"
hoelzl@51471
  1723
by (auto simp add: convergent_def)
hoelzl@51471
  1724
hoelzl@51471
  1725
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
hoelzl@51471
  1726
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
hoelzl@51471
  1727
hoelzl@51471
  1728
lemma convergent_const: "convergent (\<lambda>n. c)"
hoelzl@51471
  1729
  by (rule convergentI, rule tendsto_const)
hoelzl@51471
  1730
hoelzl@51471
  1731
lemma monoseq_le:
hoelzl@51471
  1732
  "monoseq a \<Longrightarrow> a ----> (x::'a::linorder_topology) \<Longrightarrow>
hoelzl@51471
  1733
    ((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
hoelzl@51471
  1734
  by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
hoelzl@51471
  1735
hoelzl@51471
  1736
lemma LIMSEQ_subseq_LIMSEQ:
hoelzl@51471
  1737
  "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
hoelzl@51471
  1738
  unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
hoelzl@51471
  1739
hoelzl@51471
  1740
lemma convergent_subseq_convergent:
hoelzl@51471
  1741
  "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
hoelzl@51471
  1742
  unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
hoelzl@51471
  1743
hoelzl@51471
  1744
lemma limI: "X ----> L ==> lim X = L"
hoelzl@57276
  1745
  by (rule tendsto_Lim) (rule trivial_limit_sequentially)
hoelzl@51471
  1746
hoelzl@51471
  1747
lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"
hoelzl@51471
  1748
  using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
hoelzl@51471
  1749
hoelzl@51471
  1750
subsubsection{*Increasing and Decreasing Series*}
hoelzl@51471
  1751
hoelzl@51471
  1752
lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
hoelzl@51471
  1753
  by (metis incseq_def LIMSEQ_le_const)
hoelzl@51471
  1754
hoelzl@51471
  1755
lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
hoelzl@51471
  1756
  by (metis decseq_def LIMSEQ_le_const2)
hoelzl@51471
  1757
hoelzl@51473
  1758
subsection {* First countable topologies *}
hoelzl@51473
  1759
hoelzl@51473
  1760
class first_countable_topology = topological_space +
hoelzl@51473
  1761
  assumes first_countable_basis:
hoelzl@51473
  1762
    "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
hoelzl@51473
  1763
hoelzl@51473
  1764
lemma (in first_countable_topology) countable_basis_at_decseq:
hoelzl@51473
  1765
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1766
    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
hoelzl@51473
  1767
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
hoelzl@51473
  1768
proof atomize_elim
hoelzl@51473
  1769
  from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1770
    nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1771
    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto
hoelzl@51473
  1772
  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i"
hoelzl@51473
  1773
  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
hoelzl@51473
  1774
      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
hoelzl@51473
  1775
  proof (safe intro!: exI[of _ F])
hoelzl@51473
  1776
    fix i
hoelzl@51480
  1777
    show "open (F i)" using nhds(1) by (auto simp: F_def)
hoelzl@51473
  1778
    show "x \<in> F i" using nhds(2) by (auto simp: F_def)
hoelzl@51473
  1779
  next
hoelzl@51473
  1780
    fix S assume "open S" "x \<in> S"
hoelzl@51473
  1781
    from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto
hoelzl@51473
  1782
    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
hoelzl@51473
  1783
      by (auto simp: F_def)
hoelzl@51473
  1784
    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
hoelzl@51473
  1785
      by (auto simp: eventually_sequentially)
hoelzl@51473
  1786
  qed
hoelzl@51473
  1787
qed
hoelzl@51473
  1788
hoelzl@57448
  1789
lemma (in first_countable_topology) nhds_countable:
hoelzl@57448
  1790
  obtains X :: "nat \<Rightarrow> 'a set"
hoelzl@57448
  1791
  where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))"
hoelzl@57448
  1792
proof -
hoelzl@57448
  1793
  from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set"
hoelzl@57448
  1794
    where A: "\<And>n. x \<in> A n" "\<And>n. open (A n)" "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"
hoelzl@57448
  1795
    by metis
hoelzl@57448
  1796
  show thesis
hoelzl@57448
  1797
  proof
hoelzl@57448
  1798
    show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)"
hoelzl@57448
  1799
      by (auto simp: decseq_def)
hoelzl@57448
  1800
    show "\<And>n. x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)"
hoelzl@57448
  1801
      using A by auto
hoelzl@57448
  1802
    show "nhds x = (INF n. principal (\<Inter> i\<le>n. A i))"
hoelzl@57448
  1803
      using A unfolding nhds_def
hoelzl@57448
  1804
      apply (intro INF_eq)
hoelzl@57448
  1805
      apply simp_all
hoelzl@57448
  1806
      apply force
hoelzl@57448
  1807
      apply (intro exI[of _ "\<Inter> i\<le>n. A i" for n] conjI open_INT)
hoelzl@57448
  1808
      apply auto
hoelzl@57448
  1809
      done
hoelzl@57448
  1810
  qed
hoelzl@57448
  1811
qed
hoelzl@57448
  1812
hoelzl@51473
  1813
lemma (in first_countable_topology) countable_basis:
hoelzl@51473
  1814
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1815
    "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1816
    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
hoelzl@51473
  1817
proof atomize_elim
wenzelm@53381
  1818
  obtain A :: "nat \<Rightarrow> 'a set" where A:
wenzelm@53381
  1819
    "\<And>i. open (A i)"
wenzelm@53381
  1820
    "\<And>i. x \<in> A i"
wenzelm@53381
  1821
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
wenzelm@53381
  1822
    by (rule countable_basis_at_decseq) blast
wenzelm@53381
  1823
  {
wenzelm@53381
  1824
    fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
hoelzl@51473
  1825
    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
wenzelm@53381
  1826
      by (auto elim: eventually_elim1 simp: subset_eq)
wenzelm@53381
  1827
  }
hoelzl@51473
  1828
  with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F ----> x)"
hoelzl@51473
  1829
    by (intro exI[of _ A]) (auto simp: tendsto_def)
hoelzl@51473
  1830
qed
hoelzl@51473
  1831
hoelzl@51473
  1832
lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
hoelzl@51473
  1833
  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51641
  1834
  shows "eventually P (inf (nhds a) (principal s))"
hoelzl@51473
  1835
proof (rule ccontr)
wenzelm@53381
  1836
  obtain A :: "nat \<Rightarrow> 'a set" where A:
wenzelm@53381
  1837
    "\<And>i. open (A i)"
wenzelm@53381
  1838
    "\<And>i. a \<in> A i"
wenzelm@53381
  1839
    "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F ----> a"
wenzelm@53381
  1840
    by (rule countable_basis) blast
wenzelm@53381
  1841
  assume "\<not> ?thesis"
hoelzl@51473
  1842
  with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
hoelzl@51641
  1843
    unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce
wenzelm@53381
  1844
  then obtain F where F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
wenzelm@53381
  1845
    by blast
hoelzl@51473
  1846
  with A have "F ----> a" by auto
hoelzl@51473
  1847
  hence "eventually (\<lambda>n. P (F n)) sequentially"
hoelzl@51473
  1848
    using assms F0 by simp
hoelzl@51473
  1849
  thus "False" by (simp add: F3)
hoelzl@51473
  1850
qed
hoelzl@51473
  1851
hoelzl@51473
  1852
lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
hoelzl@51641
  1853
  "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> 
hoelzl@51473
  1854
    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1855
proof (safe intro!: sequentially_imp_eventually_nhds_within)
hoelzl@51641
  1856
  assume "eventually P (inf (nhds a) (principal s))" 
hoelzl@51473
  1857
  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
hoelzl@51641
  1858
    by (auto simp: eventually_inf_principal eventually_nhds)
hoelzl@51473
  1859
  moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
hoelzl@51473
  1860
  ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51473
  1861
    by (auto dest!: topological_tendstoD elim: eventually_elim1)
hoelzl@51473
  1862
qed
hoelzl@51473
  1863
hoelzl@51473
  1864
lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
hoelzl@51473
  1865
  "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1866
  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
hoelzl@51473
  1867
hoelzl@57447
  1868
lemma tendsto_at_iff_sequentially:
hoelzl@57447
  1869
  fixes f :: "'a :: first_countable_topology \<Rightarrow> _"
hoelzl@57447
  1870
  shows "(f ---> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X ----> x \<longrightarrow> ((f \<circ> X) ----> a))"
hoelzl@57447
  1871
  unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def
hoelzl@57447
  1872
  by metis
hoelzl@57447
  1873
hoelzl@51471
  1874
subsection {* Function limit at a point *}
hoelzl@51471
  1875
hoelzl@51471
  1876
abbreviation
hoelzl@51471
  1877
  LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
hoelzl@51471
  1878
        ("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
hoelzl@51471
  1879
  "f -- a --> L \<equiv> (f ---> L) (at a)"
hoelzl@51471
  1880
hoelzl@51481
  1881
lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l) (at a within S) \<longleftrightarrow> (f -- a --> l)"
hoelzl@51641
  1882
  unfolding tendsto_def by (simp add: at_within_open[where S=S])
hoelzl@51481
  1883
hoelzl@51471
  1884
lemma LIM_const_not_eq[tendsto_intros]:
hoelzl@51471
  1885
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1886
  fixes k L :: "'b::t2_space"
hoelzl@51471
  1887
  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L"
hoelzl@51471
  1888
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1889
hoelzl@51471
  1890
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
hoelzl@51471
  1891
hoelzl@51471
  1892
lemma LIM_const_eq:
hoelzl@51471
  1893
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1894
  fixes k L :: "'b::t2_space"
hoelzl@51471
  1895
  shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L"
hoelzl@51471
  1896
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1897
hoelzl@51471
  1898
lemma LIM_unique:
hoelzl@51471
  1899
  fixes a :: "'a::perfect_space" and L M :: "'b::t2_space"
hoelzl@51471
  1900
  shows "f -- a --> L \<Longrightarrow> f -- a --> M \<Longrightarrow> L = M"
hoelzl@51471
  1901
  using at_neq_bot by (rule tendsto_unique)
hoelzl@51471
  1902
hoelzl@51471
  1903
text {* Limits are equal for functions equal except at limit point *}
hoelzl@51471
  1904
hoelzl@51471
  1905
lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- a --> l)"
hoelzl@51471
  1906
  unfolding tendsto_def eventually_at_topological by simp
hoelzl@51471
  1907
hoelzl@51471
  1908
lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- b --> m)"
hoelzl@51471
  1909
  by (simp add: LIM_equal)
hoelzl@51471
  1910
hoelzl@51471
  1911
lemma LIM_cong_limit: "f -- x --> L \<Longrightarrow> K = L \<Longrightarrow> f -- x --> K"
hoelzl@51471
  1912
  by simp
hoelzl@51471
  1913
hoelzl@51471
  1914
lemma tendsto_at_iff_tendsto_nhds:
hoelzl@51471
  1915
  "g -- l --> g l \<longleftrightarrow> (g ---> g l) (nhds l)"
hoelzl@51641
  1916
  unfolding tendsto_def eventually_at_filter
hoelzl@51471
  1917
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@51471
  1918
hoelzl@51471
  1919
lemma tendsto_compose:
hoelzl@51471
  1920
  "g -- l --> g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@51471
  1921
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
hoelzl@51471
  1922
hoelzl@51471
  1923
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l"
hoelzl@51471
  1924
  unfolding o_def by (rule tendsto_compose)
hoelzl@51471
  1925
hoelzl@51471
  1926
lemma tendsto_compose_eventually:
hoelzl@51471
  1927
  "g -- l --> m \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F"
hoelzl@51471
  1928
  by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
hoelzl@51471
  1929
hoelzl@51471
  1930
lemma LIM_compose_eventually:
hoelzl@51471
  1931
  assumes f: "f -- a --> b"
hoelzl@51471
  1932
  assumes g: "g -- b --> c"
hoelzl@51471
  1933
  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
hoelzl@51471
  1934
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51471
  1935
  using g f inj by (rule tendsto_compose_eventually)
hoelzl@51471
  1936
hoelzl@57447
  1937
lemma tendsto_compose_filtermap: "((g \<circ> f) ---> T) F \<longleftrightarrow> (g ---> T) (filtermap f F)"
hoelzl@57447
  1938
  by (simp add: filterlim_def filtermap_filtermap comp_def)
hoelzl@57447
  1939
hoelzl@51473
  1940
subsubsection {* Relation of LIM and LIMSEQ *}
hoelzl@51473
  1941
hoelzl@51473
  1942
lemma (in first_countable_topology) sequentially_imp_eventually_within:
hoelzl@51473
  1943
  "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
hoelzl@51473
  1944
    eventually P (at a within s)"
hoelzl@51641
  1945
  unfolding at_within_def
hoelzl@51473
  1946
  by (intro sequentially_imp_eventually_nhds_within) auto
hoelzl@51473
  1947
hoelzl@51473
  1948
lemma (in first_countable_topology) sequentially_imp_eventually_at:
hoelzl@51473
  1949
  "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"
hoelzl@51473
  1950
  using assms sequentially_imp_eventually_within [where s=UNIV] by simp
hoelzl@51473
  1951
hoelzl@51473
  1952
lemma LIMSEQ_SEQ_conv1:
hoelzl@51473
  1953
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
hoelzl@51473
  1954
  assumes f: "f -- a --> l"
hoelzl@51473
  1955
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
hoelzl@51473
  1956
  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
hoelzl@51473
  1957
hoelzl@51473
  1958
lemma LIMSEQ_SEQ_conv2:
hoelzl@51473
  1959
  fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
hoelzl@51473
  1960
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
hoelzl@51473
  1961
  shows "f -- a --> l"
hoelzl@51473
  1962
  using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
hoelzl@51473
  1963
hoelzl@51473
  1964
lemma LIMSEQ_SEQ_conv:
hoelzl@51473
  1965
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
hoelzl@51473
  1966
   (X -- a --> (L::'b::topological_space))"
hoelzl@51473
  1967
  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
hoelzl@51473
  1968
hoelzl@57025
  1969
lemma sequentially_imp_eventually_at_left:
hoelzl@57025
  1970
  fixes a :: "'a :: {dense_linorder, linorder_topology, first_countable_topology}"
hoelzl@57025
  1971
  assumes b[simp]: "b < a"
hoelzl@57025
  1972
  assumes *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f ----> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@57025
  1973
  shows "eventually P (at_left a)"
hoelzl@57025
  1974
proof (safe intro!: sequentially_imp_eventually_within)
hoelzl@57447
  1975
  fix X assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X ----> a"
hoelzl@57025
  1976
  show "eventually (\<lambda>n. P (X n)) sequentially"
hoelzl@57025
  1977
  proof (rule ccontr)
hoelzl@57447
  1978
    assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
hoelzl@57447
  1979
    have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> b < X (s n)) \<and> (X (s n) \<le> X (s (Suc n)) \<and> Suc (s n) \<le> s (Suc n))"
hoelzl@57447
  1980
    proof (rule dependent_nat_choice)
hoelzl@57447
  1981
      have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially"
hoelzl@57447
  1982
        by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b])
hoelzl@57447
  1983
      then show "\<exists>x. \<not> P (X x) \<and> b < X x"
hoelzl@57447
  1984
        by (auto dest!: not_eventuallyD)
hoelzl@57447
  1985
    next
hoelzl@57447
  1986
      fix x n
hoelzl@57447
  1987
      have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> b < X n \<longrightarrow> X x < X n \<longrightarrow> P (X n)) sequentially"
hoelzl@57447
  1988
        using X by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
hoelzl@57447
  1989
      then show "\<exists>n. (\<not> P (X n) \<and> b < X n) \<and> (X x \<le> X n \<and> Suc x \<le> n)"
hoelzl@57447
  1990
        by (auto dest!: not_eventuallyD)
hoelzl@57025
  1991
    qed
hoelzl@57447
  1992
    then guess s ..
hoelzl@57447
  1993
    then have "\<And>n. b < X (s n)" "\<And>n. X (s n) < a" "incseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) ----> a" "\<And>n. \<not> P (X (s n))"
hoelzl@57447
  1994
      using X by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF `X ----> a`, unfolded comp_def])
hoelzl@57447
  1995
    from *[OF this(1,2,3,4)] this(5) show False by auto
hoelzl@57025
  1996
  qed
hoelzl@57025
  1997
qed
hoelzl@57025
  1998
hoelzl@57025
  1999
lemma tendsto_at_left_sequentially:
hoelzl@57025
  2000
  fixes a :: "_ :: {dense_linorder, linorder_topology, first_countable_topology}"
hoelzl@57025
  2001
  assumes "b < a"
hoelzl@57025
  2002
  assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S ----> a \<Longrightarrow> (\<lambda>n. X (S n)) ----> L"
hoelzl@57025
  2003
  shows "(X ---> L) (at_left a)"
hoelzl@57025
  2004
  using assms unfolding tendsto_def [where l=L]
hoelzl@57025
  2005
  by (simp add: sequentially_imp_eventually_at_left)
hoelzl@57025
  2006
hoelzl@57447
  2007
lemma sequentially_imp_eventually_at_right:
hoelzl@57447
  2008
  fixes a :: "'a :: {dense_linorder, linorder_topology, first_countable_topology}"
hoelzl@57447
  2009
  assumes b[simp]: "a < b"
hoelzl@57447
  2010
  assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f ----> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@57447
  2011
  shows "eventually P (at_right a)"
hoelzl@57447
  2012
proof (safe intro!: sequentially_imp_eventually_within)
hoelzl@57447
  2013
  fix X assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X ----> a"
hoelzl@57447
  2014
  show "eventually (\<lambda>n. P (X n)) sequentially"
hoelzl@57447
  2015
  proof (rule ccontr)
hoelzl@57447
  2016
    assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
hoelzl@57447
  2017
    have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> X (s n) < b) \<and> (X (s (Suc n)) \<le> X (s n) \<and> Suc (s n) \<le> s (Suc n))"
hoelzl@57447
  2018
    proof (rule dependent_nat_choice)
hoelzl@57447
  2019
      have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially"
hoelzl@57447
  2020
        by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b])
hoelzl@57447
  2021
      then show "\<exists>x. \<not> P (X x) \<and> X x < b"
hoelzl@57447
  2022
        by (auto dest!: not_eventuallyD)
hoelzl@57447
  2023
    next
hoelzl@57447
  2024
      fix x n
hoelzl@57447
  2025
      have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> X n < b \<longrightarrow> X n < X x \<longrightarrow> P (X n)) sequentially"
hoelzl@57447
  2026
        using X by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
hoelzl@57447
  2027
      then show "\<exists>n. (\<not> P (X n) \<and> X n < b) \<and> (X n \<le> X x \<and> Suc x \<le> n)"
hoelzl@57447
  2028
        by (auto dest!: not_eventuallyD)
hoelzl@57447
  2029
    qed
hoelzl@57447
  2030
    then guess s ..
hoelzl@57447
  2031
    then have "\<And>n. a < X (s n)" "\<And>n. X (s n) < b" "decseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) ----> a" "\<And>n. \<not> P (X (s n))"
hoelzl@57447
  2032
      using X by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF `X ----> a`, unfolded comp_def])
hoelzl@57447
  2033
    from *[OF this(1,2,3,4)] this(5) show False by auto
hoelzl@57447
  2034
  qed
hoelzl@57447
  2035
qed
hoelzl@57447
  2036
hoelzl@57447
  2037
lemma tendsto_at_right_sequentially:
hoelzl@57447
  2038
  fixes a :: "_ :: {dense_linorder, linorder_topology, first_countable_topology}"
hoelzl@57447
  2039
  assumes "a < b"
hoelzl@57447
  2040
  assumes *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S ----> a \<Longrightarrow> (\<lambda>n. X (S n)) ----> L"
hoelzl@57447
  2041
  shows "(X ---> L) (at_right a)"
hoelzl@57447
  2042
  using assms unfolding tendsto_def [where l=L]
hoelzl@57447
  2043
  by (simp add: sequentially_imp_eventually_at_right)
hoelzl@57447
  2044
hoelzl@51471
  2045
subsection {* Continuity *}
hoelzl@51471
  2046
hoelzl@51478
  2047
subsubsection {* Continuity on a set *}
hoelzl@51478
  2048
hoelzl@51478
  2049
definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where
hoelzl@51478
  2050
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))"
hoelzl@51478
  2051
hoelzl@51481
  2052
lemma continuous_on_cong [cong]:
hoelzl@51481
  2053
  "s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g"
hoelzl@51641
  2054
  unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
hoelzl@51481
  2055
hoelzl@51478
  2056
lemma continuous_on_topological:
hoelzl@51478
  2057
  "continuous_on s f \<longleftrightarrow>
hoelzl@51478
  2058
    (\<forall>x\<in>s. \<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
hoelzl@51641
  2059
  unfolding continuous_on_def tendsto_def eventually_at_topological by metis
hoelzl@51478
  2060
hoelzl@51478
  2061
lemma continuous_on_open_invariant:
hoelzl@51478
  2062
  "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  2063
proof safe
hoelzl@51478
  2064
  fix B :: "'b set" assume "continuous_on s f" "open B"
hoelzl@51478
  2065
  then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)"
hoelzl@51478
  2066
    by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
wenzelm@53381
  2067
  then obtain A where "\<forall>x\<in>f -` B \<inter> s. open (A x) \<and> x \<in> A x \<and> s \<inter> A x \<subseteq> f -` B"
wenzelm@53381
  2068
    unfolding bchoice_iff ..
hoelzl@51478
  2069
  then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"
hoelzl@51478
  2070
    by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto
hoelzl@51478
  2071
next
hoelzl@51478
  2072
  assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"
hoelzl@51478
  2073
  show "continuous_on s f"
hoelzl@51478
  2074
    unfolding continuous_on_topological
hoelzl@51478
  2075
  proof safe
hoelzl@51478
  2076
    fix x B assume "x \<in> s" "open B" "f x \<in> B"
hoelzl@51478
  2077
    with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto
hoelzl@51478
  2078
    with `x \<in> s` `f x \<in> B` show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
hoelzl@51478
  2079
      by (intro exI[of _ A]) auto
hoelzl@51478
  2080
  qed
hoelzl@51478
  2081
qed
hoelzl@51478
  2082
hoelzl@51481
  2083
lemma continuous_on_open_vimage:
hoelzl@51481
  2084
  "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
hoelzl@51481
  2085
  unfolding continuous_on_open_invariant
hoelzl@51481
  2086
  by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  2087
lp15@55734
  2088
corollary continuous_imp_open_vimage:
lp15@55734
  2089
  assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s"
lp15@55734
  2090
    shows "open (f -` B)"
lp15@55734
  2091
by (metis assms continuous_on_open_vimage le_iff_inf)
lp15@55734
  2092
hoelzl@56371
  2093
corollary open_vimage[continuous_intros]:
lp15@55775
  2094
  assumes "open s" and "continuous_on UNIV f"
lp15@55775
  2095
  shows "open (f -` s)"
lp15@55775
  2096
  using assms unfolding continuous_on_open_vimage [OF open_UNIV]
lp15@55775
  2097
  by simp
lp15@55775
  2098
hoelzl@51478
  2099
lemma continuous_on_closed_invariant:
hoelzl@51478
  2100
  "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  2101
proof -
hoelzl@51478
  2102
  have *: "\<And>P Q::'b set\<Rightarrow>bool. (\<And>A. P A \<longleftrightarrow> Q (- A)) \<Longrightarrow> (\<forall>A. P A) \<longleftrightarrow> (\<forall>A. Q A)"
hoelzl@51478
  2103
    by (metis double_compl)
hoelzl@51478
  2104
  show ?thesis
hoelzl@51478
  2105
    unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric])
hoelzl@51478
  2106
qed
hoelzl@51478
  2107
hoelzl@51481
  2108
lemma continuous_on_closed_vimage:
hoelzl@51481
  2109
  "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
hoelzl@51481
  2110
  unfolding continuous_on_closed_invariant
hoelzl@51481
  2111
  by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  2112
hoelzl@56371
  2113
corollary closed_vimage[continuous_intros]:
hoelzl@56371
  2114
  assumes "closed s" and "continuous_on UNIV f"
hoelzl@56371
  2115
  shows "closed (f -` s)"
hoelzl@56371
  2116
  using assms unfolding continuous_on_closed_vimage [OF closed_UNIV]
hoelzl@56371
  2117
  by simp
hoelzl@56371
  2118
hoelzl@51481
  2119
lemma continuous_on_open_Union:
hoelzl@51481
  2120
  "(\<And>s. s \<in> S \<Longrightarrow> open s) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on s f) \<Longrightarrow> continuous_on (\<Union>S) f"
hoelzl@51641
  2121
  unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI)
hoelzl@51481
  2122
hoelzl@51481
  2123
lemma continuous_on_open_UN:
hoelzl@51481
  2124
  "(\<And>s. s \<in> S \<Longrightarrow> open (A s)) \<Longrightarrow> (\<And>s. s \<in> S \<Longrightarrow> continuous_on (A s) f) \<Longrightarrow> continuous_on (\<Union>s\<in>S. A s) f"
hoelzl@51481
  2125
  unfolding Union_image_eq[symmetric] by (rule continuous_on_open_Union) auto
hoelzl@51481
  2126
hoelzl@51481
  2127
lemma continuous_on_closed_Un:
hoelzl@51481
  2128
  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
hoelzl@51481
  2129
  by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
hoelzl@51481
  2130
hoelzl@51481
  2131
lemma continuous_on_If:
hoelzl@51481
  2132
  assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g"
hoelzl@51481
  2133
    and P: "\<And>x. x \<in> s \<Longrightarrow> \<not> P x \<Longrightarrow> f x = g x" "\<And>x. x \<in> t \<Longrightarrow> P x \<Longrightarrow> f x = g x"
hoelzl@51481
  2134
  shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h")
hoelzl@51481
  2135
proof-
hoelzl@51481
  2136
  from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
hoelzl@51481
  2137
    by auto
hoelzl@51481
  2138
  with cont have "continuous_on s ?h" "continuous_on t ?h"
hoelzl@51481
  2139
    by simp_all
hoelzl@51481
  2140
  with closed show ?thesis
hoelzl@51481
  2141
    by (rule continuous_on_closed_Un)
hoelzl@51481
  2142
qed
hoelzl@51481
  2143
hoelzl@56371
  2144
lemma continuous_on_id[continuous_intros]: "continuous_on s (\<lambda>x. x)"
hoelzl@58729
  2145
  unfolding continuous_on_def by fast
hoelzl@51478
  2146
hoelzl@56371
  2147
lemma continuous_on_const[continuous_intros]: "continuous_on s (\<lambda>x. c)"
hoelzl@58729
  2148
  unfolding continuous_on_def by auto
hoelzl@51478
  2149
hoelzl@56371
  2150
lemma continuous_on_compose[continuous_intros]:
hoelzl@51478
  2151
  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
hoelzl@51478
  2152
  unfolding continuous_on_topological by simp metis
hoelzl@51478
  2153
hoelzl@51481
  2154
lemma continuous_on_compose2:
hoelzl@51481
  2155
  "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> t = f ` s \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
hoelzl@51481
  2156
  using continuous_on_compose[of s f g] by (simp add: comp_def)
hoelzl@51481
  2157
hoelzl@51478
  2158
subsubsection {* Continuity at a point *}
hoelzl@51478
  2159
hoelzl@51478
  2160
definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
hoelzl@51478
  2161
  "continuous F f \<longleftrightarrow> (f ---> f (Lim F (\<lambda>x. x))) F"
hoelzl@51478
  2162
hoelzl@51478
  2163
lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
hoelzl@51478
  2164
  unfolding continuous_def by auto
hoelzl@51478
  2165
hoelzl@51478
  2166
lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
hoelzl@51478
  2167
  by simp
hoelzl@51478
  2168
hoelzl@51478
  2169
lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f x) (at x within s)"
hoelzl@51641
  2170
  by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
hoelzl@51478
  2171
hoelzl@51478
  2172
lemma continuous_within_topological:
hoelzl@51478
  2173
  "continuous (at x within s) f \<longleftrightarrow>
hoelzl@51478
  2174
    (\<forall>B. open B \<longrightarrow> f x \<in> B \<longrightarrow> (\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)))"
hoelzl@51641
  2175
  unfolding continuous_within tendsto_def eventually_at_topological by metis
hoelzl@51478
  2176
hoelzl@51478
  2177
lemma continuous_within_compose[continuous_intros]:
hoelzl@51478
  2178
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  2179
  continuous (at x within s) (g o f)"
hoelzl@51478
  2180
  by (simp add: continuous_within_topological) metis
hoelzl@51478
  2181
hoelzl@51478
  2182
lemma continuous_within_compose2:
hoelzl@51478
  2183
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  2184
  continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  2185
  using continuous_within_compose[of x s f g] by (simp add: comp_def)
hoelzl@51471
  2186
hoelzl@51478
  2187
lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f -- x --> f x"
hoelzl@51478
  2188
  using continuous_within[of x UNIV f] by simp
hoelzl@51478
  2189
hoelzl@51478
  2190
lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"
hoelzl@51641
  2191
  unfolding continuous_within by (rule tendsto_ident_at)
hoelzl@51478
  2192
hoelzl@51478
  2193
lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"
hoelzl@51478
  2194
  unfolding continuous_def by (rule tendsto_const)
hoelzl@51478
  2195
hoelzl@51478
  2196
lemma continuous_on_eq_continuous_within:
hoelzl@51478
  2197
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
hoelzl@51478
  2198
  unfolding continuous_on_def continuous_within ..
hoelzl@51478
  2199
hoelzl@51478
  2200
abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where
hoelzl@51478
  2201
  "isCont f a \<equiv> continuous (at a) f"
hoelzl@51478
  2202
hoelzl@51478
  2203
lemma isCont_def: "isCont f a \<longleftrightarrow> f -- a --> f a"
hoelzl@51478
  2204
  by (rule continuous_at)
hoelzl@51478
  2205
hoelzl@51478
  2206
lemma continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"
hoelzl@51641
  2207
  by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)
hoelzl@51478
  2208
hoelzl@51481
  2209
lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"
hoelzl@51641
  2210
  by (simp add: continuous_on_def continuous_at at_within_open[of _ s])
hoelzl@51481
  2211
hoelzl@51481
  2212
lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"
hoelzl@51481
  2213
  unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)
hoelzl@51481
  2214
hoelzl@51478
  2215
lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"
hoelzl@51478
  2216
  by (auto intro: continuous_at_within simp: continuous_on_eq_continuous_within)
hoelzl@51478
  2217
hoelzl@51478
  2218
lemma isContI_continuous: "continuous (at x within UNIV) f \<Longrightarrow> isCont f x"
hoelzl@51478
  2219
  by simp
hoelzl@51478
  2220
hoelzl@51478
  2221
lemma isCont_ident[continuous_intros, simp]: "isCont (\<lambda>x. x) a"
hoelzl@51478
  2222
  using continuous_ident by (rule isContI_continuous)
hoelzl@51478
  2223
hoelzl@51478
  2224
lemmas isCont_const = continuous_const
hoelzl@51478
  2225
hoelzl@51478
  2226
lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
hoelzl@51478
  2227
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51478
  2228
hoelzl@51478
  2229
lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
hoelzl@51478
  2230
  unfolding o_def by (rule isCont_o2)
hoelzl@51471
  2231
hoelzl@51471
  2232
lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@51471
  2233
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51471
  2234
hoelzl@51478
  2235
lemma continuous_within_compose3:
hoelzl@51478
  2236
  "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  2237
  using continuous_within_compose2[of x s f g] by (simp add: continuous_at_within)
hoelzl@51471
  2238
hoelzl@57447
  2239
lemma filtermap_nhds_open_map:
hoelzl@57447
  2240
  assumes cont: "isCont f a" and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)"
hoelzl@57447
  2241
  shows "filtermap f (nhds a) = nhds (f a)"
hoelzl@57447
  2242
  unfolding filter_eq_iff
hoelzl@57447
  2243
proof safe
hoelzl@57447
  2244
  fix P assume "eventually P (filtermap f (nhds a))"
hoelzl@57447
  2245
  then guess S unfolding eventually_filtermap eventually_nhds ..
hoelzl@57447
  2246
  then show "eventually P (nhds (f a))"
hoelzl@57447
  2247
    unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map)
hoelzl@57447
  2248
qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)
hoelzl@57447
  2249
hoelzl@57447
  2250
lemma continuous_at_split: 
hoelzl@57447
  2251
  "continuous (at (x::'a::linorder_topology)) f = (continuous (at_left x) f \<and> continuous (at_right x) f)"
hoelzl@57447
  2252
  by (simp add: continuous_within filterlim_at_split)
hoelzl@57447
  2253
hoelzl@51479
  2254
subsubsection{* Open-cover compactness *}
hoelzl@51479
  2255
hoelzl@51479
  2256
context topological_space
hoelzl@51479
  2257
begin
hoelzl@51479
  2258
hoelzl@51479
  2259
definition compact :: "'a set \<Rightarrow> bool" where
hoelzl@51479
  2260
  compact_eq_heine_borel: -- "This name is used for backwards compatibility"
hoelzl@51479
  2261
    "compact S \<longleftrightarrow> (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union>C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
hoelzl@51479
  2262
hoelzl@51479
  2263
lemma compactI:
hoelzl@51479
  2264
  assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union> C \<Longrightarrow> \<exists>C'. C' \<subseteq> C \<and> finite C' \<and> s \<subseteq> \<Union> C'"
hoelzl@51479
  2265
  shows "compact s"
hoelzl@51479
  2266
  unfolding compact_eq_heine_borel using assms by metis
hoelzl@51479
  2267
hoelzl@51479
  2268
lemma compact_empty[simp]: "compact {}"
hoelzl@51479
  2269
  by (auto intro!: compactI)
hoelzl@51479
  2270
hoelzl@51479
  2271
lemma compactE:
hoelzl@51479
  2272
  assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
hoelzl@51479
  2273
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
hoelzl@51479
  2274
  using assms unfolding compact_eq_heine_borel by metis
hoelzl@51479
  2275
hoelzl@51479
  2276
lemma compactE_image:
hoelzl@51479
  2277
  assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
hoelzl@51479
  2278
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
hoelzl@51479
  2279
  using assms unfolding ball_simps[symmetric] SUP_def
hoelzl@51479
  2280
  by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
hoelzl@51479
  2281
hoelzl@51481
  2282
lemma compact_inter_closed [intro]:
hoelzl@51481
  2283
  assumes "compact s" and "closed t"
hoelzl@51481
  2284
  shows "compact (s \<inter> t)"
hoelzl@51481
  2285
proof (rule compactI)
hoelzl@51481
  2286
  fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
hoelzl@51481
  2287
  from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto
hoelzl@51481
  2288
  moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
hoelzl@51481
  2289
  ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
hoelzl@51481
  2290
    using `compact s` unfolding compact_eq_heine_borel by auto
wenzelm@53381
  2291
  then obtain D where "D \<subseteq> C \<union> {- t} \<and> finite D \<and> s \<subseteq> \<Union>D" ..
hoelzl@51481
  2292
  then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
hoelzl@51481
  2293
    by (intro exI[of _ "D - {-t}"]) auto
hoelzl@51481
  2294
qed
hoelzl@51481
  2295
hoelzl@54797
  2296
lemma inj_setminus: "inj_on uminus (A::'a set set)"
hoelzl@54797
  2297
  by (auto simp: inj_on_def)
hoelzl@54797
  2298
hoelzl@54797
  2299
lemma compact_fip:
hoelzl@54797
  2300
  "compact U \<longleftrightarrow>
hoelzl@54797
  2301
    (\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"
hoelzl@54797
  2302
  (is "_ \<longleftrightarrow> ?R")
hoelzl@54797
  2303
proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
hoelzl@54797
  2304
  fix A
hoelzl@54797
  2305
  assume "compact U"
hoelzl@54797
  2306
    and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
hoelzl@54797
  2307
    and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
hoelzl@54797
  2308
  from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"
hoelzl@54797
  2309
    by auto
hoelzl@54797
  2310
  with `compact U` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
hoelzl@54797
  2311
    unfolding compact_eq_heine_borel by (metis subset_image_iff)
hoelzl@54797
  2312
  with fi[THEN spec, of B] show False
hoelzl@54797
  2313
    by (auto dest: finite_imageD intro: inj_setminus)
hoelzl@54797
  2314
next
hoelzl@54797
  2315
  fix A
hoelzl@54797
  2316
  assume ?R
hoelzl@54797
  2317
  assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
hoelzl@54797
  2318
  then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
hoelzl@54797
  2319
    by auto
hoelzl@54797
  2320
  with `?R` obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"
hoelzl@54797
  2321
    by (metis subset_image_iff)
hoelzl@54797
  2322
  then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
hoelzl@54797
  2323
    by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
hoelzl@54797
  2324
qed
hoelzl@54797
  2325
hoelzl@54797
  2326
lemma compact_imp_fip:
hoelzl@54797
  2327
  "compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>
hoelzl@54797
  2328
    s \<inter> (\<Inter> f) \<noteq> {}"
hoelzl@54797
  2329
  unfolding compact_fip by auto
hoelzl@54797
  2330
hoelzl@54797
  2331
lemma compact_imp_fip_image:
haftmann@56166
  2332
  assumes "compact s"
haftmann@56166
  2333
    and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)"
haftmann@56166
  2334
    and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})"
haftmann@56166
  2335
  shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}"
haftmann@56166
  2336
proof -
haftmann@56166
  2337
  note `compact s`
haftmann@56166
  2338
  moreover from P have "\<forall>i \<in> f ` I. closed i" by blast
haftmann@56166
  2339
  moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})"
haftmann@56166
  2340
  proof (rule, rule, erule conjE)
haftmann@56166
  2341
    fix A :: "'a set set"