src/HOL/Topological_Spaces.thy
author eberlm
Mon Jan 04 17:45:36 2016 +0100 (2016-01-04)
changeset 62049 b0f941e207cf
parent 61976 3a27957ac658
child 62083 7582b39f51ed
permissions -rw-r--r--
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
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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 \<open>Topological Spaces\<close>
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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 \<open>Topological space\<close>
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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\<open>Hausdorff and other separation properties\<close>
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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 \<open>T2 spaces are also known as Hausdorff spaces.\<close>
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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 \<open>A perfect space is a topological space with no isolated points.\<close>
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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 \<open>Generators for toplogies\<close>
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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 standard (auto intro: generate_topology.intros)
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subsection \<open>Order topologies\<close>
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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 \<open>x < y\<close> 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)
hoelzl@51471
   294
  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
wenzelm@60758
   300
subsubsection \<open>Boolean is an order topology\<close>
hoelzl@59106
   301
wenzelm@60758
   302
text \<open>It also is a discrete topology, but don't have a type class for it (yet).\<close>
hoelzl@59106
   303
hoelzl@59106
   304
instantiation bool :: order_topology
hoelzl@59106
   305
begin
hoelzl@59106
   306
hoelzl@59106
   307
definition open_bool :: "bool set \<Rightarrow> bool" where
hoelzl@59106
   308
  "open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))"
hoelzl@59106
   309
hoelzl@59106
   310
instance
hoelzl@59106
   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
wenzelm@60758
   325
subsubsection \<open>Topological filters\<close>
hoelzl@51471
   326
hoelzl@51471
   327
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
hoelzl@57276
   328
  where "nhds a = (INF S:{S. open S \<and> a \<in> S}. principal S)"
hoelzl@51471
   329
hoelzl@51641
   330
definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60)
hoelzl@51641
   331
  where "at a within s = inf (nhds a) (principal (s - {a}))"
hoelzl@51641
   332
hoelzl@51641
   333
abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where
hoelzl@51641
   334
  "at x \<equiv> at x within (CONST UNIV)"
hoelzl@51471
   335
hoelzl@51473
   336
abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   337
  "at_right x \<equiv> at x within {x <..}"
hoelzl@51471
   338
hoelzl@51473
   339
abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
hoelzl@51471
   340
  "at_left x \<equiv> at x within {..< x}"
hoelzl@51471
   341
hoelzl@57448
   342
lemma (in topological_space) nhds_generated_topology:
hoelzl@57448
   343
  "open = generate_topology T \<Longrightarrow> nhds x = (INF S:{S\<in>T. x \<in> S}. principal S)"
hoelzl@57448
   344
  unfolding nhds_def
hoelzl@57448
   345
proof (safe intro!: antisym INF_greatest)
hoelzl@57448
   346
  fix S assume "generate_topology T S" "x \<in> S"
hoelzl@57448
   347
  then show "(INF S:{S \<in> T. x \<in> S}. principal S) \<le> principal S"
hoelzl@57448
   348
    by induction 
hoelzl@57448
   349
       (auto intro: INF_lower order_trans simp add: inf_principal[symmetric] simp del: inf_principal)
hoelzl@57448
   350
qed (auto intro!: INF_lower intro: generate_topology.intros)
hoelzl@57448
   351
hoelzl@51473
   352
lemma (in topological_space) eventually_nhds:
hoelzl@51471
   353
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
hoelzl@57276
   354
  unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal)
hoelzl@51471
   355
eberlm@61531
   356
lemma (in topological_space) eventually_nhds_in_open: 
eberlm@61531
   357
  "open s \<Longrightarrow> x \<in> s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
eberlm@61531
   358
  by (subst eventually_nhds) blast
eberlm@61531
   359
hoelzl@51471
   360
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
hoelzl@51471
   361
  unfolding trivial_limit_def eventually_nhds by simp
hoelzl@51471
   362
hoelzl@60182
   363
lemma (in t1_space) t1_space_nhds:
hoelzl@60182
   364
  "x \<noteq> y \<Longrightarrow> (\<forall>\<^sub>F x in nhds x. x \<noteq> y)"
hoelzl@60182
   365
  by (drule t1_space) (auto simp: eventually_nhds)
hoelzl@60182
   366
hoelzl@57448
   367
lemma at_within_eq: "at x within s = (INF S:{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))"
hoelzl@57448
   368
  unfolding nhds_def at_within_def by (subst INF_inf_const2[symmetric]) (auto simp add: Diff_Int_distrib)
hoelzl@57448
   369
hoelzl@51641
   370
lemma eventually_at_filter:
hoelzl@51641
   371
  "eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)"
hoelzl@51641
   372
  unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute)
hoelzl@51641
   373
hoelzl@51641
   374
lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t"
hoelzl@51641
   375
  unfolding at_within_def by (intro inf_mono) auto
hoelzl@51641
   376
hoelzl@51471
   377
lemma eventually_at_topological:
hoelzl@51641
   378
  "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
   379
  unfolding eventually_nhds eventually_at_filter by simp
hoelzl@51471
   380
hoelzl@51481
   381
lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a"
hoelzl@51641
   382
  unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)
hoelzl@51481
   383
lp15@61234
   384
lemma at_within_open_NO_MATCH:
lp15@61234
   385
  "a \<in> s \<Longrightarrow> open s \<Longrightarrow> NO_MATCH UNIV s \<Longrightarrow> at a within s = at a"
lp15@61234
   386
  by (simp only: at_within_open)
lp15@61234
   387
hoelzl@61245
   388
lemma at_within_nhd:
hoelzl@61245
   389
  assumes "x \<in> S" "open S" "T \<inter> S - {x} = U \<inter> S - {x}"
hoelzl@61245
   390
  shows "at x within T = at x within U"
hoelzl@61245
   391
  unfolding filter_eq_iff eventually_at_filter
hoelzl@61245
   392
proof (intro allI eventually_subst)
hoelzl@61245
   393
  have "eventually (\<lambda>x. x \<in> S) (nhds x)"
hoelzl@61245
   394
    using \<open>x \<in> S\<close> \<open>open S\<close> by (auto simp: eventually_nhds)
hoelzl@61245
   395
  then show "\<forall>\<^sub>F n in nhds x. (n \<noteq> x \<longrightarrow> n \<in> T \<longrightarrow> P n) = (n \<noteq> x \<longrightarrow> n \<in> U \<longrightarrow> P n)" for P  
hoelzl@61245
   396
    by eventually_elim (insert \<open>T \<inter> S - {x} = U \<inter> S - {x}\<close>, blast)
hoelzl@61245
   397
qed
hoelzl@61245
   398
huffman@53859
   399
lemma at_within_empty [simp]: "at a within {} = bot"
huffman@53859
   400
  unfolding at_within_def by simp
huffman@53859
   401
huffman@53860
   402
lemma at_within_union: "at x within (S \<union> T) = sup (at x within S) (at x within T)"
huffman@53860
   403
  unfolding filter_eq_iff eventually_sup eventually_at_filter
huffman@53860
   404
  by (auto elim!: eventually_rev_mp)
huffman@53860
   405
hoelzl@51471
   406
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
hoelzl@51471
   407
  unfolding trivial_limit_def eventually_at_topological
hoelzl@51471
   408
  by (safe, case_tac "S = {a}", simp, fast, fast)
hoelzl@51471
   409
hoelzl@51471
   410
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
hoelzl@51471
   411
  by (simp add: at_eq_bot_iff not_open_singleton)
hoelzl@51471
   412
hoelzl@57448
   413
lemma (in order_topology) nhds_order: "nhds x =
hoelzl@57448
   414
  inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})"
hoelzl@57448
   415
proof -
hoelzl@57448
   416
  have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} = 
hoelzl@57448
   417
      (\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}"
hoelzl@57448
   418
    by auto
hoelzl@57448
   419
  show ?thesis
hoelzl@57448
   420
    unfolding nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def ..
hoelzl@51471
   421
qed
hoelzl@51471
   422
hoelzl@57448
   423
lemma (in linorder_topology) at_within_order: "UNIV \<noteq> {x} \<Longrightarrow> 
hoelzl@57448
   424
  at x within s = inf (INF a:{x <..}. principal ({..< a} \<inter> s - {x}))
hoelzl@57448
   425
                      (INF a:{..< x}. principal ({a <..} \<inter> s - {x}))"
hoelzl@57448
   426
proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split[case_product case_split])
hoelzl@57448
   427
  assume "UNIV \<noteq> {x}" "{x<..} = {}" "{..< x} = {}"
hoelzl@57448
   428
  moreover have "UNIV = {..< x} \<union> {x} \<union> {x <..}"
hoelzl@57448
   429
    by auto
hoelzl@57448
   430
  ultimately show ?thesis
hoelzl@57448
   431
    by auto
hoelzl@57448
   432
qed (auto simp: at_within_def nhds_order Int_Diff inf_principal[symmetric] INF_inf_const2
hoelzl@57448
   433
                inf_sup_aci[where 'a="'a filter"]
hoelzl@57448
   434
          simp del: inf_principal)
hoelzl@57448
   435
hoelzl@57448
   436
lemma (in linorder_topology) at_left_eq:
hoelzl@57448
   437
  "y < x \<Longrightarrow> at_left x = (INF a:{..< x}. principal {a <..< x})"
hoelzl@57448
   438
  by (subst at_within_order)
hoelzl@57448
   439
     (auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant
hoelzl@57448
   440
           intro!: INF_lower2 inf_absorb2)
hoelzl@57448
   441
hoelzl@57448
   442
lemma (in linorder_topology) eventually_at_left:
hoelzl@57448
   443
  "y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)"
hoelzl@57448
   444
  unfolding at_left_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
hoelzl@57448
   445
hoelzl@57448
   446
lemma (in linorder_topology) at_right_eq:
hoelzl@57448
   447
  "x < y \<Longrightarrow> at_right x = (INF a:{x <..}. principal {x <..< a})"
hoelzl@57448
   448
  by (subst at_within_order)
hoelzl@57448
   449
     (auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute
hoelzl@57448
   450
           intro!: INF_lower2 inf_absorb1)
hoelzl@57448
   451
hoelzl@57448
   452
lemma (in linorder_topology) eventually_at_right:
hoelzl@57448
   453
  "x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)"
hoelzl@57448
   454
  unfolding at_right_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)
hoelzl@51471
   455
hoelzl@57275
   456
lemma trivial_limit_at_right_top: "at_right (top::_::{order_top, linorder_topology}) = bot"
hoelzl@57275
   457
  unfolding filter_eq_iff eventually_at_topological by auto
hoelzl@57275
   458
hoelzl@57275
   459
lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot, linorder_topology}) = bot"
hoelzl@57275
   460
  unfolding filter_eq_iff eventually_at_topological by auto
hoelzl@57275
   461
hoelzl@51471
   462
lemma trivial_limit_at_left_real [simp]:
hoelzl@57275
   463
  "\<not> trivial_limit (at_left (x::'a::{no_bot, dense_order, linorder_topology}))"
hoelzl@57275
   464
  using lt_ex[of x]
hoelzl@57275
   465
  by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense)
hoelzl@51471
   466
hoelzl@51471
   467
lemma trivial_limit_at_right_real [simp]:
hoelzl@57275
   468
  "\<not> trivial_limit (at_right (x::'a::{no_top, dense_order, linorder_topology}))"
hoelzl@57275
   469
  using gt_ex[of x]
hoelzl@57275
   470
  by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense)
hoelzl@51471
   471
hoelzl@51471
   472
lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)"
hoelzl@51641
   473
  by (auto simp: eventually_at_filter filter_eq_iff eventually_sup 
lp15@61810
   474
           elim: eventually_elim2 eventually_mono)
hoelzl@51471
   475
hoelzl@51471
   476
lemma eventually_at_split:
hoelzl@51471
   477
  "eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)"
hoelzl@51471
   478
  by (subst at_eq_sup_left_right) (simp add: eventually_sup)
hoelzl@51471
   479
wenzelm@60758
   480
subsubsection \<open>Tendsto\<close>
hoelzl@51471
   481
hoelzl@51471
   482
abbreviation (in topological_space)
wenzelm@61973
   483
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "\<longlongrightarrow>" 55) where
wenzelm@61973
   484
  "(f \<longlongrightarrow> l) F \<equiv> filterlim f (nhds l) F"
hoelzl@51471
   485
hoelzl@51478
   486
definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where
wenzelm@61973
   487
  "Lim A f = (THE l. (f \<longlongrightarrow> l) A)"
hoelzl@51478
   488
wenzelm@61973
   489
lemma tendsto_eq_rhs: "(f \<longlongrightarrow> x) F \<Longrightarrow> x = y \<Longrightarrow> (f \<longlongrightarrow> y) F"
hoelzl@51471
   490
  by simp
hoelzl@51471
   491
wenzelm@57953
   492
named_theorems tendsto_intros "introduction rules for tendsto"
wenzelm@60758
   493
setup \<open>
hoelzl@51471
   494
  Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros},
wenzelm@57953
   495
    fn context =>
wenzelm@57953
   496
      Named_Theorems.get (Context.proof_of context) @{named_theorems tendsto_intros}
wenzelm@57953
   497
      |> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])))
wenzelm@60758
   498
\<close>
hoelzl@51471
   499
hoelzl@51473
   500
lemma (in topological_space) tendsto_def:
wenzelm@61973
   501
   "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@57276
   502
   unfolding nhds_def filterlim_INF filterlim_principal by auto
hoelzl@51471
   503
eberlm@61531
   504
lemma tendsto_cong:
eberlm@61531
   505
  assumes "eventually (\<lambda>x. f x = g x) F"
wenzelm@61973
   506
  shows   "(f \<longlongrightarrow> c) F \<longleftrightarrow> (g \<longlongrightarrow> c) F"
eberlm@61531
   507
  by (rule filterlim_cong[OF refl refl assms])
eberlm@61531
   508
eberlm@61531
   509
wenzelm@61973
   510
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f \<longlongrightarrow> l) F' \<Longrightarrow> (f \<longlongrightarrow> l) F"
hoelzl@51471
   511
  unfolding tendsto_def le_filter_def by fast
hoelzl@51471
   512
wenzelm@61973
   513
lemma tendsto_within_subset: "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T)"
hoelzl@51641
   514
  by (blast intro: tendsto_mono at_le)
hoelzl@51641
   515
hoelzl@51641
   516
lemma filterlim_at:
wenzelm@61973
   517
  "(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 \<longlongrightarrow> b) F)"
hoelzl@51641
   518
  by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)
hoelzl@51641
   519
hoelzl@51473
   520
lemma (in topological_space) topological_tendstoI:
wenzelm@61973
   521
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f \<longlongrightarrow> l) F"
hoelzl@51471
   522
  unfolding tendsto_def by auto
hoelzl@51471
   523
hoelzl@51473
   524
lemma (in topological_space) topological_tendstoD:
wenzelm@61973
   525
  "(f \<longlongrightarrow> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
hoelzl@51471
   526
  unfolding tendsto_def by auto
hoelzl@51471
   527
hoelzl@57448
   528
lemma (in order_topology) order_tendsto_iff:
wenzelm@61973
   529
  "(f \<longlongrightarrow> 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
   530
  unfolding nhds_order filterlim_inf filterlim_INF filterlim_principal by auto
hoelzl@57448
   531
hoelzl@57448
   532
lemma (in order_topology) order_tendstoI:
hoelzl@57448
   533
  "(\<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>
wenzelm@61973
   534
    (f \<longlongrightarrow> y) F"
hoelzl@57448
   535
  unfolding order_tendsto_iff by auto
hoelzl@57448
   536
hoelzl@57448
   537
lemma (in order_topology) order_tendstoD:
wenzelm@61973
   538
  assumes "(f \<longlongrightarrow> y) F"
hoelzl@51471
   539
  shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F"
hoelzl@51471
   540
    and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F"
hoelzl@57448
   541
  using assms unfolding order_tendsto_iff by auto
hoelzl@51471
   542
wenzelm@61973
   543
lemma tendsto_bot [simp]: "(f \<longlongrightarrow> a) bot"
hoelzl@51471
   544
  unfolding tendsto_def by simp
hoelzl@51471
   545
hoelzl@57448
   546
lemma (in linorder_topology) tendsto_max:
wenzelm@61973
   547
  assumes X: "(X \<longlongrightarrow> x) net"
wenzelm@61973
   548
  assumes Y: "(Y \<longlongrightarrow> y) net"
wenzelm@61973
   549
  shows "((\<lambda>x. max (X x) (Y x)) \<longlongrightarrow> max x y) net"
hoelzl@56949
   550
proof (rule order_tendstoI)
hoelzl@56949
   551
  fix a assume "a < max x y"
hoelzl@56949
   552
  then show "eventually (\<lambda>x. a < max (X x) (Y x)) net"
hoelzl@56949
   553
    using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
lp15@61810
   554
    by (auto simp: less_max_iff_disj elim: eventually_mono)
hoelzl@56949
   555
next
hoelzl@56949
   556
  fix a assume "max x y < a"
hoelzl@56949
   557
  then show "eventually (\<lambda>x. max (X x) (Y x) < a) net"
hoelzl@56949
   558
    using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
hoelzl@56949
   559
    by (auto simp: eventually_conj_iff)
hoelzl@56949
   560
qed
hoelzl@56949
   561
hoelzl@57448
   562
lemma (in linorder_topology) tendsto_min:
wenzelm@61973
   563
  assumes X: "(X \<longlongrightarrow> x) net"
wenzelm@61973
   564
  assumes Y: "(Y \<longlongrightarrow> y) net"
wenzelm@61973
   565
  shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net"
hoelzl@56949
   566
proof (rule order_tendstoI)
hoelzl@56949
   567
  fix a assume "a < min x y"
hoelzl@56949
   568
  then show "eventually (\<lambda>x. a < min (X x) (Y x)) net"
hoelzl@56949
   569
    using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
hoelzl@56949
   570
    by (auto simp: eventually_conj_iff)
hoelzl@56949
   571
next
hoelzl@56949
   572
  fix a assume "min x y < a"
hoelzl@56949
   573
  then show "eventually (\<lambda>x. min (X x) (Y x) < a) net"
hoelzl@56949
   574
    using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
lp15@61810
   575
    by (auto simp: min_less_iff_disj elim: eventually_mono)
hoelzl@56949
   576
qed
hoelzl@56949
   577
wenzelm@61973
   578
lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) \<longlongrightarrow> a) (at a within s)"
hoelzl@51471
   579
  unfolding tendsto_def eventually_at_topological by auto
hoelzl@51471
   580
wenzelm@61973
   581
lemma (in topological_space) tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) \<longlongrightarrow> k) F"
hoelzl@51471
   582
  by (simp add: tendsto_def)
hoelzl@51471
   583
hoelzl@51478
   584
lemma (in t2_space) tendsto_unique:
wenzelm@61973
   585
  assumes "F \<noteq> bot" and "(f \<longlongrightarrow> a) F" and "(f \<longlongrightarrow> b) F"
hoelzl@51471
   586
  shows "a = b"
hoelzl@51471
   587
proof (rule ccontr)
hoelzl@51471
   588
  assume "a \<noteq> b"
hoelzl@51471
   589
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
wenzelm@60758
   590
    using hausdorff [OF \<open>a \<noteq> b\<close>] by fast
hoelzl@51471
   591
  have "eventually (\<lambda>x. f x \<in> U) F"
wenzelm@61973
   592
    using \<open>(f \<longlongrightarrow> a) F\<close> \<open>open U\<close> \<open>a \<in> U\<close> by (rule topological_tendstoD)
hoelzl@51471
   593
  moreover
hoelzl@51471
   594
  have "eventually (\<lambda>x. f x \<in> V) F"
wenzelm@61973
   595
    using \<open>(f \<longlongrightarrow> b) F\<close> \<open>open V\<close> \<open>b \<in> V\<close> by (rule topological_tendstoD)
hoelzl@51471
   596
  ultimately
hoelzl@51471
   597
  have "eventually (\<lambda>x. False) F"
hoelzl@51471
   598
  proof eventually_elim
hoelzl@51471
   599
    case (elim x)
hoelzl@51471
   600
    hence "f x \<in> U \<inter> V" by simp
wenzelm@60758
   601
    with \<open>U \<inter> V = {}\<close> show ?case by simp
hoelzl@51471
   602
  qed
wenzelm@60758
   603
  with \<open>\<not> trivial_limit F\<close> show "False"
hoelzl@51471
   604
    by (simp add: trivial_limit_def)
hoelzl@51471
   605
qed
hoelzl@51471
   606
hoelzl@51478
   607
lemma (in t2_space) tendsto_const_iff:
wenzelm@61973
   608
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) \<longlongrightarrow> b) F \<longleftrightarrow> a = b"
hoelzl@58729
   609
  by (auto intro!: tendsto_unique [OF assms tendsto_const])
hoelzl@51471
   610
hoelzl@51471
   611
lemma increasing_tendsto:
hoelzl@51471
   612
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
   613
  assumes bdd: "eventually (\<lambda>n. f n \<le> l) F"
hoelzl@51471
   614
      and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F"
wenzelm@61973
   615
  shows "(f \<longlongrightarrow> l) F"
lp15@61810
   616
  using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
hoelzl@51471
   617
hoelzl@51471
   618
lemma decreasing_tendsto:
hoelzl@51471
   619
  fixes f :: "_ \<Rightarrow> 'a::order_topology"
hoelzl@51471
   620
  assumes bdd: "eventually (\<lambda>n. l \<le> f n) F"
hoelzl@51471
   621
      and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F"
wenzelm@61973
   622
  shows "(f \<longlongrightarrow> l) F"
lp15@61810
   623
  using assms by (intro order_tendstoI) (auto elim!: eventually_mono)
hoelzl@51471
   624
hoelzl@51471
   625
lemma tendsto_sandwich:
hoelzl@51471
   626
  fixes f g h :: "'a \<Rightarrow> 'b::order_topology"
hoelzl@51471
   627
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
wenzelm@61973
   628
  assumes lim: "(f \<longlongrightarrow> c) net" "(h \<longlongrightarrow> c) net"
wenzelm@61973
   629
  shows "(g \<longlongrightarrow> c) net"
hoelzl@51471
   630
proof (rule order_tendstoI)
hoelzl@51471
   631
  fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net"
hoelzl@51471
   632
    using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
   633
next
hoelzl@51471
   634
  fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net"
hoelzl@51471
   635
    using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
hoelzl@51471
   636
qed
hoelzl@51471
   637
eberlm@61531
   638
lemma limit_frequently_eq:
eberlm@61531
   639
  assumes "F \<noteq> bot"
eberlm@61531
   640
  assumes "frequently (\<lambda>x. f x = c) F"
wenzelm@61973
   641
  assumes "(f \<longlongrightarrow> d) F"
eberlm@61531
   642
  shows   "d = (c :: 'a :: t1_space)"
eberlm@61531
   643
proof (rule ccontr)
eberlm@61531
   644
  assume "d \<noteq> c"
eberlm@61531
   645
  from t1_space[OF this] obtain U where "open U" "d \<in> U" "c \<notin> U" by blast
eberlm@61531
   646
  from this assms have "eventually (\<lambda>x. f x \<in> U) F" unfolding tendsto_def by blast
wenzelm@61799
   647
  hence "eventually (\<lambda>x. f x \<noteq> c) F" by eventually_elim (insert \<open>c \<notin> U\<close>, blast)
eberlm@61531
   648
  with assms(2) show False unfolding frequently_def by contradiction
eberlm@61531
   649
qed
eberlm@61531
   650
eberlm@61531
   651
lemma tendsto_imp_eventually_ne:
wenzelm@61973
   652
  assumes "F \<noteq> bot" "(f \<longlongrightarrow> c) F" "c \<noteq> (c' :: 'a :: t1_space)"
eberlm@61531
   653
  shows   "eventually (\<lambda>z. f z \<noteq> c') F"
eberlm@61531
   654
proof (rule ccontr)
eberlm@61531
   655
  assume "\<not>eventually (\<lambda>z. f z \<noteq> c') F"
eberlm@61531
   656
  hence "frequently (\<lambda>z. f z = c') F" by (simp add: frequently_def)
eberlm@61531
   657
  from limit_frequently_eq[OF assms(1) this assms(2)] and assms(3) show False by contradiction
eberlm@61531
   658
qed
eberlm@61531
   659
hoelzl@51471
   660
lemma tendsto_le:
hoelzl@51471
   661
  fixes f g :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
   662
  assumes F: "\<not> trivial_limit F"
wenzelm@61973
   663
  assumes x: "(f \<longlongrightarrow> x) F" and y: "(g \<longlongrightarrow> y) F"
hoelzl@51471
   664
  assumes ev: "eventually (\<lambda>x. g x \<le> f x) F"
hoelzl@51471
   665
  shows "y \<le> x"
hoelzl@51471
   666
proof (rule ccontr)
hoelzl@51471
   667
  assume "\<not> y \<le> x"
hoelzl@51471
   668
  with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}"
hoelzl@51471
   669
    by (auto simp: not_le)
hoelzl@51471
   670
  then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F"
hoelzl@51471
   671
    using x y by (auto intro: order_tendstoD)
hoelzl@51471
   672
  with ev have "eventually (\<lambda>x. False) F"
hoelzl@51471
   673
    by eventually_elim (insert xy, fastforce)
hoelzl@51471
   674
  with F show False
hoelzl@51471
   675
    by (simp add: eventually_False)
hoelzl@51471
   676
qed
hoelzl@51471
   677
hoelzl@51471
   678
lemma tendsto_le_const:
hoelzl@51471
   679
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
hoelzl@51471
   680
  assumes F: "\<not> trivial_limit F"
wenzelm@61973
   681
  assumes x: "(f \<longlongrightarrow> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F"
hoelzl@51471
   682
  shows "a \<le> x"
hoelzl@51471
   683
  using F x tendsto_const a by (rule tendsto_le)
hoelzl@51471
   684
lp15@56289
   685
lemma tendsto_ge_const:
lp15@56289
   686
  fixes f :: "'a \<Rightarrow> 'b::linorder_topology"
lp15@56289
   687
  assumes F: "\<not> trivial_limit F"
wenzelm@61973
   688
  assumes x: "(f \<longlongrightarrow> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F"
lp15@56289
   689
  shows "a \<ge> x"
lp15@56289
   690
  by (rule tendsto_le [OF F tendsto_const x a])
lp15@56289
   691
eberlm@61531
   692
eberlm@61531
   693
eberlm@61531
   694
wenzelm@60758
   695
subsubsection \<open>Rules about @{const Lim}\<close>
hoelzl@51478
   696
hoelzl@57276
   697
lemma tendsto_Lim:
wenzelm@61973
   698
  "\<not>(trivial_limit net) \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> Lim net f = l"
hoelzl@51478
   699
  unfolding Lim_def using tendsto_unique[of net f] by auto
hoelzl@51478
   700
hoelzl@51641
   701
lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x"
hoelzl@51478
   702
  by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto
hoelzl@51478
   703
hoelzl@51471
   704
lemma filterlim_at_bot_at_right:
hoelzl@57275
   705
  fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
hoelzl@51471
   706
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
   707
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
   708
  assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b"
hoelzl@51471
   709
  assumes P: "eventually P at_bot"
hoelzl@51471
   710
  shows "filterlim f at_bot (at_right a)"
hoelzl@51471
   711
proof -
hoelzl@51471
   712
  from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y"
hoelzl@51471
   713
    unfolding eventually_at_bot_linorder by auto
hoelzl@51471
   714
  show ?thesis
hoelzl@51471
   715
  proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
hoelzl@51471
   716
    fix z assume "z \<le> x"
hoelzl@51471
   717
    with x have "P z" by auto
hoelzl@51471
   718
    have "eventually (\<lambda>x. x \<le> g z) (at_right a)"
wenzelm@60758
   719
      using bound[OF bij(2)[OF \<open>P z\<close>]]
wenzelm@60758
   720
      unfolding eventually_at_right[OF bound[OF bij(2)[OF \<open>P z\<close>]]] by (auto intro!: exI[of _ "g z"])
hoelzl@51471
   721
    with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)"
wenzelm@60758
   722
      by eventually_elim (metis bij \<open>P z\<close> mono)
hoelzl@51471
   723
  qed
hoelzl@51471
   724
qed
hoelzl@51471
   725
hoelzl@51471
   726
lemma filterlim_at_top_at_left:
hoelzl@57275
   727
  fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder"
hoelzl@51471
   728
  assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@51471
   729
  assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)"
hoelzl@51471
   730
  assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a"
hoelzl@51471
   731
  assumes P: "eventually P at_top"
hoelzl@51471
   732
  shows "filterlim f at_top (at_left a)"
hoelzl@51471
   733
proof -
hoelzl@51471
   734
  from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y"
hoelzl@51471
   735
    unfolding eventually_at_top_linorder by auto
hoelzl@51471
   736
  show ?thesis
hoelzl@51471
   737
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
hoelzl@51471
   738
    fix z assume "x \<le> z"
hoelzl@51471
   739
    with x have "P z" by auto
hoelzl@51471
   740
    have "eventually (\<lambda>x. g z \<le> x) (at_left a)"
wenzelm@60758
   741
      using bound[OF bij(2)[OF \<open>P z\<close>]]
wenzelm@60758
   742
      unfolding eventually_at_left[OF bound[OF bij(2)[OF \<open>P z\<close>]]] by (auto intro!: exI[of _ "g z"])
hoelzl@51471
   743
    with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)"
wenzelm@60758
   744
      by eventually_elim (metis bij \<open>P z\<close> mono)
hoelzl@51471
   745
  qed
hoelzl@51471
   746
qed
hoelzl@51471
   747
hoelzl@51471
   748
lemma filterlim_split_at:
hoelzl@51471
   749
  "filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))"
hoelzl@51471
   750
  by (subst at_eq_sup_left_right) (rule filterlim_sup)
hoelzl@51471
   751
hoelzl@51471
   752
lemma filterlim_at_split:
hoelzl@51471
   753
  "filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)"
hoelzl@51471
   754
  by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)
hoelzl@51471
   755
hoelzl@57025
   756
lemma eventually_nhds_top:
hoelzl@57025
   757
  fixes P :: "'a :: {order_top, linorder_topology} \<Rightarrow> bool"
hoelzl@57025
   758
  assumes "(b::'a) < top"
hoelzl@57025
   759
  shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))"
hoelzl@57025
   760
  unfolding eventually_nhds
hoelzl@57025
   761
proof safe
hoelzl@57025
   762
  fix S :: "'a set" assume "open S" "top \<in> S"
wenzelm@60758
   763
  note open_left[OF this \<open>b < top\<close>]
hoelzl@57025
   764
  moreover assume "\<forall>s\<in>S. P s"
hoelzl@57025
   765
  ultimately show "\<exists>b<top. \<forall>z>b. P z"
hoelzl@57025
   766
    by (auto simp: subset_eq Ball_def)
hoelzl@57025
   767
next
hoelzl@57025
   768
  fix b assume "b < top" "\<forall>z>b. P z"
hoelzl@57025
   769
  then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)"
hoelzl@57025
   770
    by (intro exI[of _ "{b <..}"]) auto
hoelzl@57025
   771
qed
hoelzl@51471
   772
hoelzl@57447
   773
lemma tendsto_at_within_iff_tendsto_nhds:
wenzelm@61973
   774
  "(g \<longlongrightarrow> g l) (at l within S) \<longleftrightarrow> (g \<longlongrightarrow> g l) (inf (nhds l) (principal S))"
hoelzl@57447
   775
  unfolding tendsto_def eventually_at_filter eventually_inf_principal
lp15@61810
   776
  by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
hoelzl@57447
   777
wenzelm@60758
   778
subsection \<open>Limits on sequences\<close>
hoelzl@51471
   779
hoelzl@51471
   780
abbreviation (in topological_space)
hoelzl@51471
   781
  LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
wenzelm@61969
   782
    ("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60) where
wenzelm@61973
   783
  "X \<longlonglongrightarrow> L \<equiv> (X \<longlongrightarrow> L) sequentially"
hoelzl@51471
   784
hoelzl@51478
   785
abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where
hoelzl@51478
   786
  "lim X \<equiv> Lim sequentially X"
hoelzl@51471
   787
hoelzl@51471
   788
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where
wenzelm@61969
   789
  "convergent X = (\<exists>L. X \<longlonglongrightarrow> L)"
hoelzl@51471
   790
wenzelm@61969
   791
lemma lim_def: "lim X = (THE L. X \<longlonglongrightarrow> L)"
hoelzl@51478
   792
  unfolding Lim_def ..
hoelzl@51478
   793
wenzelm@60758
   794
subsubsection \<open>Monotone sequences and subsequences\<close>
hoelzl@51471
   795
hoelzl@51471
   796
definition
hoelzl@51471
   797
  monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
wenzelm@61799
   798
    \<comment>\<open>Definition of monotonicity.
hoelzl@51471
   799
        The use of disjunction here complicates proofs considerably.
hoelzl@51471
   800
        One alternative is to add a Boolean argument to indicate the direction.
wenzelm@60758
   801
        Another is to develop the notions of increasing and decreasing first.\<close>
hoelzl@56020
   802
  "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
   803
hoelzl@56020
   804
abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@56020
   805
  "incseq X \<equiv> mono X"
hoelzl@56020
   806
hoelzl@56020
   807
lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)"
hoelzl@56020
   808
  unfolding mono_def ..
hoelzl@56020
   809
hoelzl@56020
   810
abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where
hoelzl@56020
   811
  "decseq X \<equiv> antimono X"
hoelzl@56020
   812
hoelzl@56020
   813
lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
hoelzl@56020
   814
  unfolding antimono_def ..
hoelzl@51471
   815
hoelzl@51471
   816
definition
hoelzl@51471
   817
  subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where
wenzelm@61799
   818
    \<comment>\<open>Definition of subsequence\<close>
hoelzl@51471
   819
  "subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)"
hoelzl@51471
   820
hoelzl@51471
   821
lemma incseq_SucI:
hoelzl@51471
   822
  "(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X"
hoelzl@51471
   823
  using lift_Suc_mono_le[of X]
hoelzl@51471
   824
  by (auto simp: incseq_def)
hoelzl@51471
   825
hoelzl@51471
   826
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j"
hoelzl@51471
   827
  by (auto simp: incseq_def)
hoelzl@51471
   828
hoelzl@51471
   829
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)"
hoelzl@51471
   830
  using incseqD[of A i "Suc i"] by auto
hoelzl@51471
   831
hoelzl@51471
   832
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
hoelzl@51471
   833
  by (auto intro: incseq_SucI dest: incseq_SucD)
hoelzl@51471
   834
hoelzl@51471
   835
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)"
hoelzl@51471
   836
  unfolding incseq_def by auto
hoelzl@51471
   837
hoelzl@51471
   838
lemma decseq_SucI:
hoelzl@51471
   839
  "(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X"
hoelzl@51471
   840
  using order.lift_Suc_mono_le[OF dual_order, of X]
hoelzl@51471
   841
  by (auto simp: decseq_def)
hoelzl@51471
   842
hoelzl@51471
   843
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i"
hoelzl@51471
   844
  by (auto simp: decseq_def)
hoelzl@51471
   845
hoelzl@51471
   846
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i"
hoelzl@51471
   847
  using decseqD[of A i "Suc i"] by auto
hoelzl@51471
   848
hoelzl@51471
   849
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@51471
   850
  by (auto intro: decseq_SucI dest: decseq_SucD)
hoelzl@51471
   851
hoelzl@51471
   852
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)"
hoelzl@51471
   853
  unfolding decseq_def by auto
hoelzl@51471
   854
hoelzl@51471
   855
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X"
hoelzl@51471
   856
  unfolding monoseq_def incseq_def decseq_def ..
hoelzl@51471
   857
hoelzl@51471
   858
lemma monoseq_Suc:
hoelzl@51471
   859
  "monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)"
hoelzl@51471
   860
  unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..
hoelzl@51471
   861
hoelzl@51471
   862
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
hoelzl@51471
   863
by (simp add: monoseq_def)
hoelzl@51471
   864
hoelzl@51471
   865
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
hoelzl@51471
   866
by (simp add: monoseq_def)
hoelzl@51471
   867
hoelzl@51471
   868
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
hoelzl@51471
   869
by (simp add: monoseq_Suc)
hoelzl@51471
   870
hoelzl@51471
   871
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
hoelzl@51471
   872
by (simp add: monoseq_Suc)
hoelzl@51471
   873
hoelzl@51471
   874
lemma monoseq_minus:
hoelzl@51471
   875
  fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add"
hoelzl@51471
   876
  assumes "monoseq a"
hoelzl@51471
   877
  shows "monoseq (\<lambda> n. - a n)"
hoelzl@51471
   878
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
hoelzl@51471
   879
  case True
hoelzl@51471
   880
  hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
hoelzl@51471
   881
  thus ?thesis by (rule monoI2)
hoelzl@51471
   882
next
hoelzl@51471
   883
  case False
wenzelm@60758
   884
  hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using \<open>monoseq a\<close>[unfolded monoseq_def] by auto
hoelzl@51471
   885
  thus ?thesis by (rule monoI1)
hoelzl@51471
   886
qed
hoelzl@51471
   887
wenzelm@60758
   888
text\<open>Subsequence (alternative definition, (e.g. Hoskins)\<close>
hoelzl@51471
   889
hoelzl@51471
   890
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
hoelzl@51471
   891
apply (simp add: subseq_def)
hoelzl@51471
   892
apply (auto dest!: less_imp_Suc_add)
hoelzl@51471
   893
apply (induct_tac k)
hoelzl@51471
   894
apply (auto intro: less_trans)
hoelzl@51471
   895
done
hoelzl@51471
   896
wenzelm@60758
   897
text\<open>for any sequence, there is a monotonic subsequence\<close>
hoelzl@51471
   898
lemma seq_monosub:
hoelzl@51471
   899
  fixes s :: "nat => 'a::linorder"
hoelzl@57448
   900
  shows "\<exists>f. subseq f \<and> monoseq (\<lambda>n. (s (f n)))"
hoelzl@51471
   901
proof cases
hoelzl@57448
   902
  assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p"
hoelzl@57448
   903
  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
   904
    by (intro dependent_nat_choice) (auto simp: conj_commute)
hoelzl@57448
   905
  then obtain f where "subseq f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)"
hoelzl@57448
   906
    by (auto simp: subseq_Suc_iff)
hoelzl@57448
   907
  moreover 
hoelzl@57448
   908
  then have "incseq f"
hoelzl@57448
   909
    unfolding subseq_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le)
hoelzl@57448
   910
  then have "monoseq (\<lambda>n. s (f n))"
hoelzl@57448
   911
    by (auto simp add: incseq_def intro!: mono monoI2)
hoelzl@57448
   912
  ultimately show ?thesis
hoelzl@57448
   913
    by auto
hoelzl@51471
   914
next
hoelzl@51471
   915
  assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))"
hoelzl@51471
   916
  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
   917
  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
   918
  proof (intro dependent_nat_choice)
hoelzl@57448
   919
    fix x assume "N < x" with N[of x] show "\<exists>y>N. x < y \<and> s x \<le> s y"
hoelzl@57448
   920
      by (auto intro: less_trans)
hoelzl@57448
   921
  qed auto
hoelzl@57448
   922
  then show ?thesis
hoelzl@57448
   923
    by (auto simp: monoseq_iff incseq_Suc_iff subseq_Suc_iff)
hoelzl@51471
   924
qed
hoelzl@51471
   925
hoelzl@51471
   926
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
hoelzl@51471
   927
proof(induct n)
hoelzl@51471
   928
  case 0 thus ?case by simp
hoelzl@51471
   929
next
hoelzl@51471
   930
  case (Suc n)
hoelzl@51471
   931
  from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
hoelzl@51471
   932
  have "n < f (Suc n)" by arith
hoelzl@51471
   933
  thus ?case by arith
hoelzl@51471
   934
qed
hoelzl@51471
   935
hoelzl@51471
   936
lemma eventually_subseq:
hoelzl@51471
   937
  "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
hoelzl@51471
   938
  unfolding eventually_sequentially by (metis seq_suble le_trans)
hoelzl@51471
   939
hoelzl@51473
   940
lemma not_eventually_sequentiallyD:
hoelzl@51473
   941
  assumes P: "\<not> eventually P sequentially"
hoelzl@51473
   942
  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
hoelzl@51473
   943
proof -
hoelzl@51473
   944
  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
hoelzl@51473
   945
    unfolding eventually_sequentially by (simp add: not_less)
hoelzl@51473
   946
  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
hoelzl@51473
   947
    by (auto simp: choice_iff)
hoelzl@51473
   948
  then show ?thesis
hoelzl@51473
   949
    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
hoelzl@51473
   950
             simp: less_eq_Suc_le subseq_Suc_iff)
hoelzl@51473
   951
qed
hoelzl@51473
   952
hoelzl@51471
   953
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
hoelzl@51471
   954
  unfolding filterlim_iff by (metis eventually_subseq)
hoelzl@51471
   955
hoelzl@51471
   956
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)"
hoelzl@51471
   957
  unfolding subseq_def by simp
hoelzl@51471
   958
hoelzl@51471
   959
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n"
hoelzl@51471
   960
  using assms by (auto simp: subseq_def)
hoelzl@51471
   961
eberlm@61531
   962
lemma subseq_imp_inj_on: "subseq g \<Longrightarrow> inj_on g A"
eberlm@61531
   963
proof (rule inj_onI)
eberlm@61531
   964
  assume g: "subseq g"
eberlm@61531
   965
  fix x y assume "g x = g y"
eberlm@61531
   966
  with subseq_mono[OF g, of x y] subseq_mono[OF g, of y x] show "x = y" 
eberlm@61531
   967
    by (cases x y rule: linorder_cases) simp_all
eberlm@61531
   968
qed
eberlm@61531
   969
eberlm@61531
   970
lemma subseq_strict_mono: "subseq g \<Longrightarrow> strict_mono g"
eberlm@61531
   971
  by (intro strict_monoI subseq_mono[of g])
eberlm@61531
   972
hoelzl@51471
   973
lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
hoelzl@51471
   974
  by (simp add: incseq_def monoseq_def)
hoelzl@51471
   975
hoelzl@51471
   976
lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
hoelzl@51471
   977
  by (simp add: decseq_def monoseq_def)
hoelzl@51471
   978
hoelzl@51471
   979
lemma decseq_eq_incseq:
hoelzl@51471
   980
  fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" 
hoelzl@51471
   981
  by (simp add: decseq_def incseq_def)
hoelzl@51471
   982
hoelzl@51471
   983
lemma INT_decseq_offset:
hoelzl@51471
   984
  assumes "decseq F"
hoelzl@51471
   985
  shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
   986
proof safe
hoelzl@51471
   987
  fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)"
hoelzl@51471
   988
  show "x \<in> F i"
hoelzl@51471
   989
  proof cases
hoelzl@51471
   990
    from x have "x \<in> F n" by auto
wenzelm@60758
   991
    also assume "i \<le> n" with \<open>decseq F\<close> have "F n \<subseteq> F i"
hoelzl@51471
   992
      unfolding decseq_def by simp
hoelzl@51471
   993
    finally show ?thesis .
hoelzl@51471
   994
  qed (insert x, simp)
hoelzl@51471
   995
qed auto
hoelzl@51471
   996
hoelzl@51471
   997
lemma LIMSEQ_const_iff:
hoelzl@51471
   998
  fixes k l :: "'a::t2_space"
wenzelm@61969
   999
  shows "(\<lambda>n. k) \<longlonglongrightarrow> l \<longleftrightarrow> k = l"
hoelzl@51471
  1000
  using trivial_limit_sequentially by (rule tendsto_const_iff)
hoelzl@51471
  1001
hoelzl@51471
  1002
lemma LIMSEQ_SUP:
wenzelm@61969
  1003
  "incseq X \<Longrightarrow> X \<longlonglongrightarrow> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1004
  by (intro increasing_tendsto)
hoelzl@51471
  1005
     (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)
hoelzl@51471
  1006
hoelzl@51471
  1007
lemma LIMSEQ_INF:
wenzelm@61969
  1008
  "decseq X \<Longrightarrow> X \<longlonglongrightarrow> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})"
hoelzl@51471
  1009
  by (intro decreasing_tendsto)
hoelzl@51471
  1010
     (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)
hoelzl@51471
  1011
hoelzl@51471
  1012
lemma LIMSEQ_ignore_initial_segment:
wenzelm@61969
  1013
  "f \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (n + k)) \<longlonglongrightarrow> a"
hoelzl@51474
  1014
  unfolding tendsto_def
hoelzl@51474
  1015
  by (subst eventually_sequentially_seg[where k=k])
hoelzl@51471
  1016
hoelzl@51471
  1017
lemma LIMSEQ_offset:
wenzelm@61969
  1018
  "(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a \<Longrightarrow> f \<longlonglongrightarrow> a"
hoelzl@51474
  1019
  unfolding tendsto_def
hoelzl@51474
  1020
  by (subst (asm) eventually_sequentially_seg[where k=k])
hoelzl@51471
  1021
wenzelm@61969
  1022
lemma LIMSEQ_Suc: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l"
hoelzl@51471
  1023
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
hoelzl@51471
  1024
wenzelm@61969
  1025
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l \<Longrightarrow> f \<longlonglongrightarrow> l"
hoelzl@51471
  1026
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
hoelzl@51471
  1027
wenzelm@61969
  1028
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l = f \<longlonglongrightarrow> l"
hoelzl@51471
  1029
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
hoelzl@51471
  1030
hoelzl@51471
  1031
lemma LIMSEQ_unique:
hoelzl@51471
  1032
  fixes a b :: "'a::t2_space"
wenzelm@61969
  1033
  shows "\<lbrakk>X \<longlonglongrightarrow> a; X \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> a = b"
hoelzl@51471
  1034
  using trivial_limit_sequentially by (rule tendsto_unique)
hoelzl@51471
  1035
hoelzl@51471
  1036
lemma LIMSEQ_le_const:
wenzelm@61969
  1037
  "\<lbrakk>X \<longlonglongrightarrow> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
hoelzl@51471
  1038
  using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially)
hoelzl@51471
  1039
hoelzl@51471
  1040
lemma LIMSEQ_le:
wenzelm@61969
  1041
  "\<lbrakk>X \<longlonglongrightarrow> x; Y \<longlonglongrightarrow> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)"
hoelzl@51471
  1042
  using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)
hoelzl@51471
  1043
hoelzl@51471
  1044
lemma LIMSEQ_le_const2:
wenzelm@61969
  1045
  "\<lbrakk>X \<longlonglongrightarrow> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
hoelzl@58729
  1046
  by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) auto
hoelzl@51471
  1047
wenzelm@61969
  1048
lemma convergentD: "convergent X ==> \<exists>L. (X \<longlonglongrightarrow> L)"
hoelzl@51471
  1049
by (simp add: convergent_def)
hoelzl@51471
  1050
wenzelm@61969
  1051
lemma convergentI: "(X \<longlonglongrightarrow> L) ==> convergent X"
hoelzl@51471
  1052
by (auto simp add: convergent_def)
hoelzl@51471
  1053
wenzelm@61969
  1054
lemma convergent_LIMSEQ_iff: "convergent X = (X \<longlonglongrightarrow> lim X)"
hoelzl@51471
  1055
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
hoelzl@51471
  1056
hoelzl@51471
  1057
lemma convergent_const: "convergent (\<lambda>n. c)"
hoelzl@51471
  1058
  by (rule convergentI, rule tendsto_const)
hoelzl@51471
  1059
hoelzl@51471
  1060
lemma monoseq_le:
wenzelm@61969
  1061
  "monoseq a \<Longrightarrow> a \<longlonglongrightarrow> (x::'a::linorder_topology) \<Longrightarrow>
hoelzl@51471
  1062
    ((\<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
  1063
  by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)
hoelzl@51471
  1064
hoelzl@51471
  1065
lemma LIMSEQ_subseq_LIMSEQ:
wenzelm@61969
  1066
  "\<lbrakk> X \<longlonglongrightarrow> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) \<longlonglongrightarrow> L"
hoelzl@51471
  1067
  unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq])
hoelzl@51471
  1068
hoelzl@51471
  1069
lemma convergent_subseq_convergent:
hoelzl@51471
  1070
  "\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)"
hoelzl@51471
  1071
  unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ)
hoelzl@51471
  1072
wenzelm@61969
  1073
lemma limI: "X \<longlonglongrightarrow> L ==> lim X = L"
hoelzl@57276
  1074
  by (rule tendsto_Lim) (rule trivial_limit_sequentially)
hoelzl@51471
  1075
hoelzl@51471
  1076
lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x"
hoelzl@51471
  1077
  using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)
hoelzl@51471
  1078
wenzelm@60758
  1079
subsubsection\<open>Increasing and Decreasing Series\<close>
hoelzl@51471
  1080
wenzelm@61969
  1081
lemma incseq_le: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)"
hoelzl@51471
  1082
  by (metis incseq_def LIMSEQ_le_const)
hoelzl@51471
  1083
wenzelm@61969
  1084
lemma decseq_le: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
hoelzl@51471
  1085
  by (metis decseq_def LIMSEQ_le_const2)
hoelzl@51471
  1086
wenzelm@60758
  1087
subsection \<open>First countable topologies\<close>
hoelzl@51473
  1088
hoelzl@51473
  1089
class first_countable_topology = topological_space +
hoelzl@51473
  1090
  assumes first_countable_basis:
hoelzl@51473
  1091
    "\<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
  1092
hoelzl@51473
  1093
lemma (in first_countable_topology) countable_basis_at_decseq:
hoelzl@51473
  1094
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1095
    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
hoelzl@51473
  1096
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
hoelzl@51473
  1097
proof atomize_elim
hoelzl@51473
  1098
  from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1099
    nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
hoelzl@51473
  1100
    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto
hoelzl@51473
  1101
  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i"
hoelzl@51473
  1102
  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
hoelzl@51473
  1103
      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
hoelzl@51473
  1104
  proof (safe intro!: exI[of _ F])
hoelzl@51473
  1105
    fix i
hoelzl@51480
  1106
    show "open (F i)" using nhds(1) by (auto simp: F_def)
hoelzl@51473
  1107
    show "x \<in> F i" using nhds(2) by (auto simp: F_def)
hoelzl@51473
  1108
  next
hoelzl@51473
  1109
    fix S assume "open S" "x \<in> S"
hoelzl@51473
  1110
    from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto
hoelzl@51473
  1111
    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
hoelzl@51473
  1112
      by (auto simp: F_def)
hoelzl@51473
  1113
    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
hoelzl@51473
  1114
      by (auto simp: eventually_sequentially)
hoelzl@51473
  1115
  qed
hoelzl@51473
  1116
qed
hoelzl@51473
  1117
hoelzl@57448
  1118
lemma (in first_countable_topology) nhds_countable:
hoelzl@57448
  1119
  obtains X :: "nat \<Rightarrow> 'a set"
hoelzl@57448
  1120
  where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))"
hoelzl@57448
  1121
proof -
hoelzl@57448
  1122
  from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set"
hoelzl@57448
  1123
    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
  1124
    by metis
hoelzl@57448
  1125
  show thesis
hoelzl@57448
  1126
  proof
hoelzl@57448
  1127
    show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)"
hoelzl@57448
  1128
      by (auto simp: decseq_def)
hoelzl@57448
  1129
    show "\<And>n. x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)"
hoelzl@57448
  1130
      using A by auto
wenzelm@60585
  1131
    show "nhds x = (INF n. principal (\<Inter>i\<le>n. A i))"
hoelzl@57448
  1132
      using A unfolding nhds_def
hoelzl@57448
  1133
      apply (intro INF_eq)
hoelzl@57448
  1134
      apply simp_all
hoelzl@57448
  1135
      apply force
wenzelm@60585
  1136
      apply (intro exI[of _ "\<Inter>i\<le>n. A i" for n] conjI open_INT)
hoelzl@57448
  1137
      apply auto
hoelzl@57448
  1138
      done
hoelzl@57448
  1139
  qed
hoelzl@57448
  1140
qed
hoelzl@57448
  1141
hoelzl@51473
  1142
lemma (in first_countable_topology) countable_basis:
hoelzl@51473
  1143
  obtains A :: "nat \<Rightarrow> 'a set" where
hoelzl@51473
  1144
    "\<And>i. open (A i)" "\<And>i. x \<in> A i"
wenzelm@61969
  1145
    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x"
hoelzl@51473
  1146
proof atomize_elim
wenzelm@53381
  1147
  obtain A :: "nat \<Rightarrow> 'a set" where A:
wenzelm@53381
  1148
    "\<And>i. open (A i)"
wenzelm@53381
  1149
    "\<And>i. x \<in> A i"
wenzelm@53381
  1150
    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
wenzelm@53381
  1151
    by (rule countable_basis_at_decseq) blast
wenzelm@53381
  1152
  {
wenzelm@53381
  1153
    fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
hoelzl@51473
  1154
    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
lp15@61810
  1155
      by (auto elim: eventually_mono simp: subset_eq)
wenzelm@53381
  1156
  }
wenzelm@61969
  1157
  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 \<longlonglongrightarrow> x)"
hoelzl@51473
  1158
    by (intro exI[of _ A]) (auto simp: tendsto_def)
hoelzl@51473
  1159
qed
hoelzl@51473
  1160
hoelzl@51473
  1161
lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
wenzelm@61969
  1162
  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@51641
  1163
  shows "eventually P (inf (nhds a) (principal s))"
hoelzl@51473
  1164
proof (rule ccontr)
wenzelm@53381
  1165
  obtain A :: "nat \<Rightarrow> 'a set" where A:
wenzelm@53381
  1166
    "\<And>i. open (A i)"
wenzelm@53381
  1167
    "\<And>i. a \<in> A i"
wenzelm@61969
  1168
    "\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> a"
wenzelm@53381
  1169
    by (rule countable_basis) blast
wenzelm@53381
  1170
  assume "\<not> ?thesis"
hoelzl@51473
  1171
  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
  1172
    unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce
wenzelm@53381
  1173
  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
  1174
    by blast
wenzelm@61969
  1175
  with A have "F \<longlonglongrightarrow> a" by auto
hoelzl@51473
  1176
  hence "eventually (\<lambda>n. P (F n)) sequentially"
hoelzl@51473
  1177
    using assms F0 by simp
hoelzl@51473
  1178
  thus "False" by (simp add: F3)
hoelzl@51473
  1179
qed
hoelzl@51473
  1180
hoelzl@51473
  1181
lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
hoelzl@51641
  1182
  "eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> 
wenzelm@61969
  1183
    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1184
proof (safe intro!: sequentially_imp_eventually_nhds_within)
hoelzl@51641
  1185
  assume "eventually P (inf (nhds a) (principal s))" 
hoelzl@51473
  1186
  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
hoelzl@51641
  1187
    by (auto simp: eventually_inf_principal eventually_nhds)
wenzelm@61969
  1188
  moreover fix f assume "\<forall>n. f n \<in> s" "f \<longlonglongrightarrow> a"
hoelzl@51473
  1189
  ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
lp15@61810
  1190
    by (auto dest!: topological_tendstoD elim: eventually_mono)
hoelzl@51473
  1191
qed
hoelzl@51473
  1192
hoelzl@51473
  1193
lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
wenzelm@61969
  1194
  "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
hoelzl@51473
  1195
  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
hoelzl@51473
  1196
hoelzl@57447
  1197
lemma tendsto_at_iff_sequentially:
hoelzl@57447
  1198
  fixes f :: "'a :: first_countable_topology \<Rightarrow> _"
wenzelm@61973
  1199
  shows "(f \<longlongrightarrow> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> ((f \<circ> X) \<longlonglongrightarrow> a))"
hoelzl@57447
  1200
  unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def
hoelzl@57447
  1201
  by metis
hoelzl@57447
  1202
wenzelm@60758
  1203
subsection \<open>Function limit at a point\<close>
hoelzl@51471
  1204
hoelzl@51471
  1205
abbreviation
hoelzl@51471
  1206
  LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
wenzelm@61976
  1207
        ("((_)/ \<midarrow>(_)/\<rightarrow> (_))" [60, 0, 60] 60) where
wenzelm@61976
  1208
  "f \<midarrow>a\<rightarrow> L \<equiv> (f \<longlongrightarrow> L) (at a)"
hoelzl@51471
  1209
wenzelm@61976
  1210
lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<midarrow>a\<rightarrow> l)"
hoelzl@51641
  1211
  unfolding tendsto_def by (simp add: at_within_open[where S=S])
hoelzl@51481
  1212
hoelzl@51471
  1213
lemma LIM_const_not_eq[tendsto_intros]:
hoelzl@51471
  1214
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1215
  fixes k L :: "'b::t2_space"
wenzelm@61976
  1216
  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> L"
hoelzl@51471
  1217
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1218
hoelzl@51471
  1219
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]
hoelzl@51471
  1220
hoelzl@51471
  1221
lemma LIM_const_eq:
hoelzl@51471
  1222
  fixes a :: "'a::perfect_space"
hoelzl@51471
  1223
  fixes k L :: "'b::t2_space"
wenzelm@61976
  1224
  shows "(\<lambda>x. k) \<midarrow>a\<rightarrow> L \<Longrightarrow> k = L"
hoelzl@51471
  1225
  by (simp add: tendsto_const_iff)
hoelzl@51471
  1226
hoelzl@51471
  1227
lemma LIM_unique:
hoelzl@51471
  1228
  fixes a :: "'a::perfect_space" and L M :: "'b::t2_space"
wenzelm@61976
  1229
  shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> M \<Longrightarrow> L = M"
hoelzl@51471
  1230
  using at_neq_bot by (rule tendsto_unique)
hoelzl@51471
  1231
wenzelm@60758
  1232
text \<open>Limits are equal for functions equal except at limit point\<close>
hoelzl@51471
  1233
wenzelm@61976
  1234
lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>a\<rightarrow> l)"
hoelzl@51471
  1235
  unfolding tendsto_def eventually_at_topological by simp
hoelzl@51471
  1236
wenzelm@61976
  1237
lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>b\<rightarrow> m)"
hoelzl@51471
  1238
  by (simp add: LIM_equal)
hoelzl@51471
  1239
wenzelm@61976
  1240
lemma LIM_cong_limit: "f \<midarrow>x\<rightarrow> L \<Longrightarrow> K = L \<Longrightarrow> f \<midarrow>x\<rightarrow> K"
hoelzl@51471
  1241
  by simp
hoelzl@51471
  1242
hoelzl@51471
  1243
lemma tendsto_at_iff_tendsto_nhds:
wenzelm@61976
  1244
  "g \<midarrow>l\<rightarrow> g l \<longleftrightarrow> (g \<longlongrightarrow> g l) (nhds l)"
hoelzl@51641
  1245
  unfolding tendsto_def eventually_at_filter
lp15@61810
  1246
  by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)
hoelzl@51471
  1247
hoelzl@51471
  1248
lemma tendsto_compose:
wenzelm@61976
  1249
  "g \<midarrow>l\<rightarrow> g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
hoelzl@51471
  1250
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
hoelzl@51471
  1251
wenzelm@61976
  1252
lemma LIM_o: "\<lbrakk>g \<midarrow>l\<rightarrow> g l; f \<midarrow>a\<rightarrow> l\<rbrakk> \<Longrightarrow> (g \<circ> f) \<midarrow>a\<rightarrow> g l"
hoelzl@51471
  1253
  unfolding o_def by (rule tendsto_compose)
hoelzl@51471
  1254
hoelzl@51471
  1255
lemma tendsto_compose_eventually:
wenzelm@61976
  1256
  "g \<midarrow>l\<rightarrow> m \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> m) F"
hoelzl@51471
  1257
  by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)
hoelzl@51471
  1258
hoelzl@51471
  1259
lemma LIM_compose_eventually:
wenzelm@61976
  1260
  assumes f: "f \<midarrow>a\<rightarrow> b"
wenzelm@61976
  1261
  assumes g: "g \<midarrow>b\<rightarrow> c"
hoelzl@51471
  1262
  assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)"
wenzelm@61976
  1263
  shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c"
hoelzl@51471
  1264
  using g f inj by (rule tendsto_compose_eventually)
hoelzl@51471
  1265
wenzelm@61973
  1266
lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)"
hoelzl@57447
  1267
  by (simp add: filterlim_def filtermap_filtermap comp_def)
hoelzl@57447
  1268
wenzelm@60758
  1269
subsubsection \<open>Relation of LIM and LIMSEQ\<close>
hoelzl@51473
  1270
hoelzl@51473
  1271
lemma (in first_countable_topology) sequentially_imp_eventually_within:
wenzelm@61969
  1272
  "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
hoelzl@51473
  1273
    eventually P (at a within s)"
hoelzl@51641
  1274
  unfolding at_within_def
hoelzl@51473
  1275
  by (intro sequentially_imp_eventually_nhds_within) auto
hoelzl@51473
  1276
hoelzl@51473
  1277
lemma (in first_countable_topology) sequentially_imp_eventually_at:
wenzelm@61969
  1278
  "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"
hoelzl@51473
  1279
  using assms sequentially_imp_eventually_within [where s=UNIV] by simp
hoelzl@51473
  1280
hoelzl@51473
  1281
lemma LIMSEQ_SEQ_conv1:
hoelzl@51473
  1282
  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
wenzelm@61976
  1283
  assumes f: "f \<midarrow>a\<rightarrow> l"
wenzelm@61969
  1284
  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
hoelzl@51473
  1285
  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
hoelzl@51473
  1286
hoelzl@51473
  1287
lemma LIMSEQ_SEQ_conv2:
hoelzl@51473
  1288
  fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
wenzelm@61969
  1289
  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l"
wenzelm@61976
  1290
  shows "f \<midarrow>a\<rightarrow> l"
hoelzl@51473
  1291
  using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
hoelzl@51473
  1292
hoelzl@51473
  1293
lemma LIMSEQ_SEQ_conv:
wenzelm@61969
  1294
  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L) =
wenzelm@61976
  1295
   (X \<midarrow>a\<rightarrow> (L::'b::topological_space))"
hoelzl@51473
  1296
  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
hoelzl@51473
  1297
hoelzl@57025
  1298
lemma sequentially_imp_eventually_at_left:
hoelzl@60172
  1299
  fixes a :: "'a :: {linorder_topology, first_countable_topology}"
hoelzl@57025
  1300
  assumes b[simp]: "b < a"
wenzelm@61969
  1301
  assumes *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@57025
  1302
  shows "eventually P (at_left a)"
hoelzl@57025
  1303
proof (safe intro!: sequentially_imp_eventually_within)
wenzelm@61969
  1304
  fix X assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
hoelzl@57025
  1305
  show "eventually (\<lambda>n. P (X n)) sequentially"
hoelzl@57025
  1306
  proof (rule ccontr)
hoelzl@57447
  1307
    assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
hoelzl@57447
  1308
    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
  1309
    proof (rule dependent_nat_choice)
hoelzl@57447
  1310
      have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially"
hoelzl@57447
  1311
        by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b])
hoelzl@57447
  1312
      then show "\<exists>x. \<not> P (X x) \<and> b < X x"
hoelzl@57447
  1313
        by (auto dest!: not_eventuallyD)
hoelzl@57447
  1314
    next
hoelzl@57447
  1315
      fix x n
hoelzl@57447
  1316
      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
  1317
        using X by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
hoelzl@57447
  1318
      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
  1319
        by (auto dest!: not_eventuallyD)
hoelzl@57025
  1320
    qed
hoelzl@57447
  1321
    then guess s ..
wenzelm@61969
  1322
    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)) \<longlonglongrightarrow> a" "\<And>n. \<not> P (X (s n))"
wenzelm@61969
  1323
      using X by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
hoelzl@57447
  1324
    from *[OF this(1,2,3,4)] this(5) show False by auto
hoelzl@57025
  1325
  qed
hoelzl@57025
  1326
qed
hoelzl@57025
  1327
hoelzl@57025
  1328
lemma tendsto_at_left_sequentially:
hoelzl@60172
  1329
  fixes a :: "_ :: {linorder_topology, first_countable_topology}"
hoelzl@57025
  1330
  assumes "b < a"
wenzelm@61969
  1331
  assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
wenzelm@61973
  1332
  shows "(X \<longlongrightarrow> L) (at_left a)"
hoelzl@57025
  1333
  using assms unfolding tendsto_def [where l=L]
hoelzl@57025
  1334
  by (simp add: sequentially_imp_eventually_at_left)
hoelzl@57025
  1335
hoelzl@57447
  1336
lemma sequentially_imp_eventually_at_right:
hoelzl@60172
  1337
  fixes a :: "'a :: {linorder_topology, first_countable_topology}"
hoelzl@57447
  1338
  assumes b[simp]: "a < b"
wenzelm@61969
  1339
  assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
hoelzl@57447
  1340
  shows "eventually P (at_right a)"
hoelzl@57447
  1341
proof (safe intro!: sequentially_imp_eventually_within)
wenzelm@61969
  1342
  fix X assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a"
hoelzl@57447
  1343
  show "eventually (\<lambda>n. P (X n)) sequentially"
hoelzl@57447
  1344
  proof (rule ccontr)
hoelzl@57447
  1345
    assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially"
hoelzl@57447
  1346
    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
  1347
    proof (rule dependent_nat_choice)
hoelzl@57447
  1348
      have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially"
hoelzl@57447
  1349
        by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b])
hoelzl@57447
  1350
      then show "\<exists>x. \<not> P (X x) \<and> X x < b"
hoelzl@57447
  1351
        by (auto dest!: not_eventuallyD)
hoelzl@57447
  1352
    next
hoelzl@57447
  1353
      fix x n
hoelzl@57447
  1354
      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
  1355
        using X by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
hoelzl@57447
  1356
      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
  1357
        by (auto dest!: not_eventuallyD)
hoelzl@57447
  1358
    qed
hoelzl@57447
  1359
    then guess s ..
wenzelm@61969
  1360
    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)) \<longlonglongrightarrow> a" "\<And>n. \<not> P (X (s n))"
wenzelm@61969
  1361
      using X by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def])
hoelzl@57447
  1362
    from *[OF this(1,2,3,4)] this(5) show False by auto
hoelzl@57447
  1363
  qed
hoelzl@57447
  1364
qed
hoelzl@57447
  1365
hoelzl@57447
  1366
lemma tendsto_at_right_sequentially:
hoelzl@60172
  1367
  fixes a :: "_ :: {linorder_topology, first_countable_topology}"
hoelzl@57447
  1368
  assumes "a < b"
wenzelm@61969
  1369
  assumes *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
wenzelm@61973
  1370
  shows "(X \<longlongrightarrow> L) (at_right a)"
hoelzl@57447
  1371
  using assms unfolding tendsto_def [where l=L]
hoelzl@57447
  1372
  by (simp add: sequentially_imp_eventually_at_right)
hoelzl@57447
  1373
wenzelm@60758
  1374
subsection \<open>Continuity\<close>
hoelzl@51471
  1375
wenzelm@60758
  1376
subsubsection \<open>Continuity on a set\<close>
hoelzl@51478
  1377
hoelzl@51478
  1378
definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where
wenzelm@61973
  1379
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))"
hoelzl@51478
  1380
hoelzl@51481
  1381
lemma continuous_on_cong [cong]:
hoelzl@51481
  1382
  "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
  1383
  unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)
hoelzl@51481
  1384
hoelzl@51478
  1385
lemma continuous_on_topological:
hoelzl@51478
  1386
  "continuous_on s f \<longleftrightarrow>
hoelzl@51478
  1387
    (\<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
  1388
  unfolding continuous_on_def tendsto_def eventually_at_topological by metis
hoelzl@51478
  1389
hoelzl@51478
  1390
lemma continuous_on_open_invariant:
hoelzl@51478
  1391
  "continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  1392
proof safe
hoelzl@51478
  1393
  fix B :: "'b set" assume "continuous_on s f" "open B"
hoelzl@51478
  1394
  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
  1395
    by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
wenzelm@53381
  1396
  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
  1397
    unfolding bchoice_iff ..
hoelzl@51478
  1398
  then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s"
hoelzl@51478
  1399
    by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto
hoelzl@51478
  1400
next
hoelzl@51478
  1401
  assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)"
hoelzl@51478
  1402
  show "continuous_on s f"
hoelzl@51478
  1403
    unfolding continuous_on_topological
hoelzl@51478
  1404
  proof safe
hoelzl@51478
  1405
    fix x B assume "x \<in> s" "open B" "f x \<in> B"
hoelzl@51478
  1406
    with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto
wenzelm@60758
  1407
    with \<open>x \<in> s\<close> \<open>f x \<in> B\<close> show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)"
hoelzl@51478
  1408
      by (intro exI[of _ A]) auto
hoelzl@51478
  1409
  qed
hoelzl@51478
  1410
qed
hoelzl@51478
  1411
hoelzl@51481
  1412
lemma continuous_on_open_vimage:
hoelzl@51481
  1413
  "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))"
hoelzl@51481
  1414
  unfolding continuous_on_open_invariant
hoelzl@51481
  1415
  by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  1416
lp15@55734
  1417
corollary continuous_imp_open_vimage:
lp15@55734
  1418
  assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s"
lp15@55734
  1419
    shows "open (f -` B)"
lp15@55734
  1420
by (metis assms continuous_on_open_vimage le_iff_inf)
lp15@55734
  1421
hoelzl@56371
  1422
corollary open_vimage[continuous_intros]:
lp15@55775
  1423
  assumes "open s" and "continuous_on UNIV f"
lp15@55775
  1424
  shows "open (f -` s)"
lp15@55775
  1425
  using assms unfolding continuous_on_open_vimage [OF open_UNIV]
lp15@55775
  1426
  by simp
lp15@55775
  1427
hoelzl@51478
  1428
lemma continuous_on_closed_invariant:
hoelzl@51478
  1429
  "continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))"
hoelzl@51478
  1430
proof -
hoelzl@51478
  1431
  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
  1432
    by (metis double_compl)
hoelzl@51478
  1433
  show ?thesis
hoelzl@51478
  1434
    unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric])
hoelzl@51478
  1435
qed
hoelzl@51478
  1436
hoelzl@51481
  1437
lemma continuous_on_closed_vimage:
hoelzl@51481
  1438
  "closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))"
hoelzl@51481
  1439
  unfolding continuous_on_closed_invariant
hoelzl@51481
  1440
  by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])
hoelzl@51481
  1441
lp15@61426
  1442
corollary closed_vimage_Int[continuous_intros]:
lp15@61426
  1443
  assumes "closed s" and "continuous_on t f" and t: "closed t"
lp15@61426
  1444
  shows "closed (f -` s \<inter> t)"
lp15@61426
  1445
  using assms unfolding continuous_on_closed_vimage [OF t]  by simp
lp15@61426
  1446
hoelzl@56371
  1447
corollary closed_vimage[continuous_intros]:
hoelzl@56371
  1448
  assumes "closed s" and "continuous_on UNIV f"
hoelzl@56371
  1449
  shows "closed (f -` s)"
lp15@61426
  1450
  using closed_vimage_Int [OF assms] by simp
hoelzl@56371
  1451
lp15@61907
  1452
lemma continuous_on_empty: "continuous_on {} f"
lp15@61907
  1453
  by (simp add: continuous_on_def)
lp15@61907
  1454
lp15@61907
  1455
lemma continuous_on_sing: "continuous_on {x} f"
lp15@61907
  1456
  by (simp add: continuous_on_def at_within_def)
lp15@61907
  1457
hoelzl@51481
  1458
lemma continuous_on_open_Union:
hoelzl@51481
  1459
  "(\<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
  1460
  unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI)
hoelzl@51481
  1461
hoelzl@51481
  1462
lemma continuous_on_open_UN:
hoelzl@51481
  1463
  "(\<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
  1464
  unfolding Union_image_eq[symmetric] by (rule continuous_on_open_Union) auto
hoelzl@51481
  1465
paulson@61204
  1466
lemma continuous_on_open_Un:
paulson@61204
  1467
  "open s \<Longrightarrow> open t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
paulson@61204
  1468
  using continuous_on_open_Union [of "{s,t}"] by auto
paulson@61204
  1469
hoelzl@51481
  1470
lemma continuous_on_closed_Un:
hoelzl@51481
  1471
  "closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
hoelzl@51481
  1472
  by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)
hoelzl@51481
  1473
hoelzl@51481
  1474
lemma continuous_on_If:
hoelzl@51481
  1475
  assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g"
hoelzl@51481
  1476
    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
  1477
  shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h")
hoelzl@51481
  1478
proof-
hoelzl@51481
  1479
  from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x"
hoelzl@51481
  1480
    by auto
hoelzl@51481
  1481
  with cont have "continuous_on s ?h" "continuous_on t ?h"
hoelzl@51481
  1482
    by simp_all
hoelzl@51481
  1483
  with closed show ?thesis
hoelzl@51481
  1484
    by (rule continuous_on_closed_Un)
hoelzl@51481
  1485
qed
hoelzl@51481
  1486
hoelzl@56371
  1487
lemma continuous_on_id[continuous_intros]: "continuous_on s (\<lambda>x. x)"
hoelzl@58729
  1488
  unfolding continuous_on_def by fast
hoelzl@51478
  1489
hoelzl@56371
  1490
lemma continuous_on_const[continuous_intros]: "continuous_on s (\<lambda>x. c)"
hoelzl@58729
  1491
  unfolding continuous_on_def by auto
hoelzl@51478
  1492
lp15@61738
  1493
lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f"
lp15@61738
  1494
  unfolding continuous_on_def by (metis subset_eq tendsto_within_subset)
lp15@61738
  1495
hoelzl@56371
  1496
lemma continuous_on_compose[continuous_intros]:
hoelzl@51478
  1497
  "continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)"
hoelzl@51478
  1498
  unfolding continuous_on_topological by simp metis
hoelzl@51478
  1499
hoelzl@51481
  1500
lemma continuous_on_compose2:
lp15@61738
  1501
  "continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))"
lp15@61738
  1502
  using continuous_on_compose[of s f g] continuous_on_subset by (force simp add: comp_def)
hoelzl@51481
  1503
hoelzl@60720
  1504
lemma continuous_on_generate_topology:
hoelzl@60720
  1505
  assumes *: "open = generate_topology X"
hoelzl@60720
  1506
  assumes **: "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
hoelzl@60720
  1507
  shows "continuous_on A f"
hoelzl@60720
  1508
  unfolding continuous_on_open_invariant
hoelzl@60720
  1509
proof safe
hoelzl@60720
  1510
  fix B :: "'a set" assume "open B" then show "\<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A"
hoelzl@60720
  1511
    unfolding *
hoelzl@60720
  1512
  proof induction
hoelzl@60720
  1513
    case (UN K)
hoelzl@60720
  1514
    then obtain C where "\<And>k. k \<in> K \<Longrightarrow> open (C k)" "\<And>k. k \<in> K \<Longrightarrow> C k \<inter> A = f -` k \<inter> A"
hoelzl@60720
  1515
      by metis
hoelzl@60720
  1516
    then show ?case
hoelzl@60720
  1517
      by (intro exI[of _ "\<Union>k\<in>K. C k"]) blast
hoelzl@60720
  1518
  qed (auto intro: **)
hoelzl@60720
  1519
qed
hoelzl@60720
  1520
hoelzl@60720
  1521
lemma continuous_onI_mono:
hoelzl@60720
  1522
  fixes f :: "'a::linorder_topology \<Rightarrow> 'b::{dense_order, linorder_topology}"
hoelzl@60720
  1523
  assumes "open (f`A)"
hoelzl@60720
  1524
  assumes mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
hoelzl@60720
  1525
  shows "continuous_on A f"
hoelzl@60720
  1526
proof (rule continuous_on_generate_topology[OF open_generated_order], safe)
hoelzl@60720
  1527
  have monoD: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x < f y \<Longrightarrow> x < y"
hoelzl@60720
  1528
    by (auto simp: not_le[symmetric] mono)
hoelzl@60720
  1529
hoelzl@60720
  1530
  { fix a b assume "a \<in> A" "f a < b"
hoelzl@60720
  1531
    moreover
hoelzl@60720
  1532
    with open_right[OF \<open>open (f`A)\<close>, of "f a" b] obtain y where "f a < y" "{f a ..< y} \<subseteq> f`A"
hoelzl@60720
  1533
      by auto
hoelzl@60720
  1534
    moreover then obtain z where "f a < z" "z < min b y"
hoelzl@60720
  1535
      using dense[of "f a" "min b y"] \<open>f a < y\<close> \<open>f a < b\<close> by auto
hoelzl@60720
  1536
    moreover then obtain c where "z = f c" "c \<in> A"
hoelzl@60720
  1537
      using \<open>{f a ..< y} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
hoelzl@60720
  1538
    ultimately have "\<exists>x. x \<in> A \<and> f x < b \<and> a < x"
hoelzl@60720
  1539
      by (auto intro!: exI[of _ c] simp: monoD) }
hoelzl@60720
  1540
  then show "\<exists>C. open C \<and> C \<inter> A = f -` {..<b} \<inter> A" for b
hoelzl@60720
  1541
    by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. f x < b}. {..< x})"])
hoelzl@60720
  1542
       (auto intro: le_less_trans[OF mono] less_imp_le)
hoelzl@60720
  1543
hoelzl@60720
  1544
  { fix a b assume "a \<in> A" "b < f a"
hoelzl@60720
  1545
    moreover
hoelzl@60720
  1546
    with open_left[OF \<open>open (f`A)\<close>, of "f a" b] obtain y where "y < f a" "{y <.. f a} \<subseteq> f`A"
hoelzl@60720
  1547
      by auto
hoelzl@60720
  1548
    moreover then obtain z where "max b y < z" "z < f a"
hoelzl@60720
  1549
      using dense[of "max b y" "f a"] \<open>y < f a\<close> \<open>b < f a\<close> by auto
hoelzl@60720
  1550
    moreover then obtain c where "z = f c" "c \<in> A"
hoelzl@60720
  1551
      using \<open>{y <.. f a} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le)
hoelzl@60720
  1552
    ultimately have "\<exists>x. x \<in> A \<and> b < f x \<and> x < a"
hoelzl@60720
  1553
      by (auto intro!: exI[of _ c] simp: monoD) }
hoelzl@60720
  1554
  then show "\<exists>C. open C \<and> C \<inter> A = f -` {b <..} \<inter> A" for b
hoelzl@60720
  1555
    by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. b < f x}. {x <..})"])
hoelzl@60720
  1556
       (auto intro: less_le_trans[OF _ mono] less_imp_le)
hoelzl@60720
  1557
qed
hoelzl@60720
  1558
wenzelm@60758
  1559
subsubsection \<open>Continuity at a point\<close>
hoelzl@51478
  1560
hoelzl@51478
  1561
definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
wenzelm@61973
  1562
  "continuous F f \<longleftrightarrow> (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F"
hoelzl@51478
  1563
hoelzl@51478
  1564
lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
hoelzl@51478
  1565
  unfolding continuous_def by auto
hoelzl@51478
  1566
hoelzl@51478
  1567
lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f"
hoelzl@51478
  1568
  by simp
hoelzl@51478
  1569
wenzelm@61973
  1570
lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f \<longlongrightarrow> f x) (at x within s)"
hoelzl@51641
  1571
  by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)
hoelzl@51478
  1572
hoelzl@51478
  1573
lemma continuous_within_topological:
hoelzl@51478
  1574
  "continuous (at x within s) f \<longleftrightarrow>
hoelzl@51478
  1575
    (\<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
  1576
  unfolding continuous_within tendsto_def eventually_at_topological by metis
hoelzl@51478
  1577
hoelzl@51478
  1578
lemma continuous_within_compose[continuous_intros]:
hoelzl@51478
  1579
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  1580
  continuous (at x within s) (g o f)"
hoelzl@51478
  1581
  by (simp add: continuous_within_topological) metis
hoelzl@51478
  1582
hoelzl@51478
  1583
lemma continuous_within_compose2:
hoelzl@51478
  1584
  "continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow>
hoelzl@51478
  1585
  continuous (at x within s) (\<lambda>x. g (f x))"
hoelzl@51478
  1586
  using continuous_within_compose[of x s f g] by (simp add: comp_def)
hoelzl@51471
  1587
wenzelm@61976
  1588
lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f \<midarrow>x\<rightarrow> f x"
hoelzl@51478
  1589
  using continuous_within[of x UNIV f] by simp
hoelzl@51478
  1590
hoelzl@51478
  1591
lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)"
hoelzl@51641
  1592
  unfolding continuous_within by (rule tendsto_ident_at)
hoelzl@51478
  1593
hoelzl@51478
  1594
lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)"
hoelzl@51478
  1595
  unfolding continuous_def by (rule tendsto_const)
hoelzl@51478
  1596
hoelzl@51478
  1597
lemma continuous_on_eq_continuous_within:
hoelzl@51478
  1598
  "continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)"
hoelzl@51478
  1599
  unfolding continuous_on_def continuous_within ..
hoelzl@51478
  1600
hoelzl@51478
  1601
abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where
hoelzl@51478
  1602
  "isCont f a \<equiv> continuous (at a) f"
hoelzl@51478
  1603
wenzelm@61976
  1604
lemma isCont_def: "isCont f a \<longleftrightarrow> f \<midarrow>a\<rightarrow> f a"
hoelzl@51478
  1605
  by (rule continuous_at)
hoelzl@51478
  1606
paulson@60762
  1607
lemma continuous_at_imp_continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f"
hoelzl@51641
  1608
  by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)
hoelzl@51478
  1609
hoelzl@51481
  1610
lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)"
hoelzl@51641
  1611
  by (simp add: continuous_on_def continuous_at at_within_open[of _ s])
hoelzl@51481
  1612
hoelzl@51478
  1613
lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f"
paulson@60762
  1614
  by (auto intro: continuous_at_imp_continuous_at_within simp: continuous_on_eq_continuous_within)
hoelzl@51478
  1615
hoelzl@51478
  1616
lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a"
hoelzl@51478
  1617
  unfolding isCont_def by (rule tendsto_compose)
hoelzl@51478
  1618
hoelzl@51478
  1619
lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a"
hoelzl@51478
  1620
  unfolding o_def by (rule isCont_o2)
hoelzl@51471
  1621
wenzelm@61973
  1622
lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F"
hoelzl@51471
  1623
  unfolding isCont_def by (rule tendsto_compose)
eberlm@62049
  1624
  
eberlm@62049
  1625
lemma continuous_on_tendsto_compose:
eberlm@62049
  1626
  assumes f_cont: "continuous_on s f"
eberlm@62049
  1627
  assumes g: "(g \<longlongrightarrow> l) F"
eberlm@62049
  1628
  assumes l: "l \<in> s"
eberlm@62049
  1629
  assumes ev: "\<forall>\<^sub>F x in F. g x \<in> s"
eberlm@62049
  1630
  shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F"
eberlm@62049
  1631
proof -
eberlm@62049
  1632
  from f_cont l have f: "(f \<longlongrightarrow> f l) (at l within s)"
eberlm@62049
  1633
    by (simp add: continuous_on_def)
eberlm@62049
  1634
  have i: "((\<lambda>x. if g x = l then f l else f (g x)) \<longlongrightarrow> f l) F"
eberlm@62049
  1635
    by (rule filterlim_If)
eberlm@62049
  1636
       (auto intro!: filterlim_compose[OF f] eventually_conj tendsto_mono[OF _ g]
eberlm@62049
  1637
             simp: filterlim_at eventually_inf_principal eventually_mono[OF ev])
eberlm@62049
  1638
  show ?thesis
eberlm@62049
  1639
    by (rule filterlim_cong[THEN iffD1[OF _ i]]) auto
eberlm@62049
  1640
qed
hoelzl@51471
  1641
hoelzl@51478
  1642
lemma continuous_within_compose3:
hoelzl@51478
  1643
  "isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))"
paulson@60762
  1644
  using continuous_within_compose2[of x s f g] by (simp add: continuous_at_imp_continuous_at_within)
hoelzl@51471
  1645
hoelzl@57447
  1646
lemma filtermap_nhds_open_map:
hoelzl@57447
  1647
  assumes cont: "isCont f a" and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)"
hoelzl@57447
  1648
  shows "filtermap f (nhds a) = nhds (f a)"
hoelzl@57447
  1649
  unfolding filter_eq_iff
hoelzl@57447
  1650
proof safe
hoelzl@57447
  1651
  fix P assume "eventually P (filtermap f (nhds a))"
hoelzl@57447
  1652
  then guess S unfolding eventually_filtermap eventually_nhds ..
hoelzl@57447
  1653
  then show "eventually P (nhds (f a))"
hoelzl@57447
  1654
    unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map)
hoelzl@57447
  1655
qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)
hoelzl@57447
  1656
hoelzl@57447
  1657
lemma continuous_at_split: 
hoelzl@57447
  1658
  "continuous (at (x::'a::linorder_topology)) f = (continuous (at_left x) f \<and> continuous (at_right x) f)"
hoelzl@57447
  1659
  by (simp add: continuous_within filterlim_at_split)
hoelzl@57447
  1660
hoelzl@61245
  1661
subsubsection \<open>Open-cover compactness\<close>
hoelzl@51479
  1662
hoelzl@51479
  1663
context topological_space
hoelzl@51479
  1664
begin
hoelzl@51479
  1665
hoelzl@51479
  1666
definition compact :: "'a set \<Rightarrow> bool" where
wenzelm@61799
  1667
  compact_eq_heine_borel: \<comment> "This name is used for backwards compatibility"
hoelzl@51479
  1668
    "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
  1669
hoelzl@51479
  1670
lemma compactI:
wenzelm@60585
  1671
  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
  1672
  shows "compact s"
hoelzl@51479
  1673
  unfolding compact_eq_heine_borel using assms by metis
hoelzl@51479
  1674
hoelzl@51479
  1675
lemma compact_empty[simp]: "compact {}"
hoelzl@51479
  1676
  by (auto intro!: compactI)
hoelzl@51479
  1677
hoelzl@51479
  1678
lemma compactE:
hoelzl@51479
  1679
  assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C"
hoelzl@51479
  1680
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
hoelzl@51479
  1681
  using assms unfolding compact_eq_heine_borel by metis
hoelzl@51479
  1682
hoelzl@51479
  1683
lemma compactE_image:
hoelzl@51479
  1684
  assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)"
hoelzl@51479
  1685
  obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)"
hoelzl@51479
  1686
  using assms unfolding ball_simps[symmetric] SUP_def
hoelzl@51479
  1687
  by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s])
hoelzl@51479
  1688
hoelzl@51481
  1689
lemma compact_inter_closed [intro]:
hoelzl@51481
  1690
  assumes "compact s" and "closed t"
hoelzl@51481
  1691
  shows "compact (s \<inter> t)"
hoelzl@51481
  1692
proof (rule compactI)
hoelzl@51481
  1693
  fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C"
wenzelm@60758
  1694
  from C \<open>closed t\<close> have "\<forall>c\<in>C \<union> {-t}. open c" by auto
hoelzl@51481
  1695
  moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto
hoelzl@51481
  1696
  ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D"
wenzelm@60758
  1697
    using \<open>compact s\<close> unfolding compact_eq_heine_borel by auto
wenzelm@53381
  1698
  then obtain D where "D \<subseteq> C \<union> {- t} \<and> finite D \<and> s \<subseteq> \<Union>D" ..
hoelzl@51481
  1699
  then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D"
hoelzl@51481
  1700
    by (intro exI[of _ "D - {-t}"]) auto
hoelzl@51481
  1701
qed
hoelzl@51481
  1702
hoelzl@54797
  1703
lemma inj_setminus: "inj_on uminus (A::'a set set)"
hoelzl@54797
  1704
  by (auto simp: inj_on_def)
hoelzl@54797
  1705
hoelzl@54797
  1706
lemma compact_fip:
hoelzl@54797
  1707
  "compact U \<longleftrightarrow>
hoelzl@54797
  1708
    (\<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
  1709
  (is "_ \<longleftrightarrow> ?R")
hoelzl@54797
  1710
proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
hoelzl@54797
  1711
  fix A
hoelzl@54797
  1712
  assume "compact U"
hoelzl@54797
  1713
    and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
hoelzl@54797
  1714
    and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
hoelzl@54797
  1715
  from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)"
hoelzl@54797
  1716
    by auto
wenzelm@60758
  1717
  with \<open>compact U\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)"
hoelzl@54797
  1718
    unfolding compact_eq_heine_borel by (metis subset_image_iff)
hoelzl@54797
  1719
  with fi[THEN spec, of B] show False
hoelzl@54797
  1720
    by (auto dest: finite_imageD intro: inj_setminus)
hoelzl@54797
  1721
next
hoelzl@54797
  1722
  fix A
hoelzl@54797
  1723
  assume ?R
hoelzl@54797
  1724
  assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
hoelzl@54797
  1725
  then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a"
hoelzl@54797
  1726
    by auto
wenzelm@60758
  1727
  with \<open>?R\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}"
hoelzl@54797
  1728
    by (metis subset_image_iff)
hoelzl@54797
  1729
  then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
hoelzl@54797
  1730
    by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
hoelzl@54797
  1731
qed
hoelzl@54797
  1732
hoelzl@54797
  1733
lemma compact_imp_fip:
wenzelm@60585
  1734
  "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>
wenzelm@60585
  1735
    s \<inter> (\<Inter>f) \<noteq> {}"
hoelzl@54797
  1736
  unfolding compact_fip by auto
hoelzl@54797
  1737
hoelzl@54797
  1738
lemma compact_imp_fip_image:
haftmann@56166
  1739
  assumes "compact s"
haftmann@56166
  1740
    and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)"
haftmann@56166
  1741
    and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})"
haftmann@56166
  1742
  shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}"
haftmann@56166
  1743
proof -
wenzelm@60758
  1744
  note \<open>compact s\<close>
haftmann@56166
  1745
  moreover from P have "\<forall>i \<in> f ` I. closed i" by blast
haftmann@56166
  1746
  moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})"
haftmann@56166
  1747
  proof (rule, rule, erule conjE)
haftmann@56166
  1748
    fix A :: "'a set set"
haftmann@56166
  1749
    assume "finite A"
haftmann@56166
  1750
    moreover assume "A \<subseteq> f ` I"
haftmann@56166
  1751
    ultimately obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B"
haftmann@56166
  1752
      using finite_subset_image [of A f I] by blast
haftmann@56166
  1753
    with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}" by simp
haftmann@56166
  1754
  qed
haftmann@56166
  1755
  ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}" by (rule compact_imp_fip)
haftmann@56166
  1756
  then show ?thesis by simp
haftmann@56166
  1757
qed
hoelzl@54797
  1758
hoelzl@51471
  1759
end
hoelzl@51471
  1760
hoelzl@51481
  1761
lemma (in t2_space) compact_imp_closed:
hoelzl@51481
  1762
  assumes "compact s" shows "closed s"
hoelzl@51481
  1763
unfolding closed_def
hoelzl@51481
  1764
proof (rule openI)
hoelzl@51481
  1765
  fix y assume "y \<in> - s"
hoelzl@51481
  1766
  let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}"
wenzelm@60758
  1767
  note \<open>compact s\<close>
hoelzl@51481
  1768
  moreover have "\<forall>u\<in>?C. open u" by simp
hoelzl@51481
  1769
  moreover have "s \<subseteq> \<Union>?C"
hoelzl@51481
  1770
  proof
hoelzl@51481
  1771
    fix x assume "x \<in> s"
wenzelm@60758
  1772
    with \<open>y \<in> - s\<close> have "x \<noteq> y" by clarsimp
hoelzl@51481
  1773
    hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}"
hoelzl@51481
  1774
      by (rule hausdorff)
wenzelm@60758
  1775
    with \<open>x \<in> s\<close> show "x \<in> \<Union>?C"
hoelzl@51481
  1776
      unfolding eventually_nhds by auto
hoelzl@51481
  1777
  qed
hoelzl@51481
  1778
  ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D"
hoelzl@51481
  1779
    by (rule compactE)
wenzelm@60758
  1780
  from \<open>D \<subseteq> ?C\<close> have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto
wenzelm@60758
  1781
  with \<open>finite D\<close> have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)"
hoelzl@60040
  1782
    by (simp add: eventually_ball_finite)
wenzelm@60758
  1783
  with \<open>s \<subseteq> \<Union>D\<close> have "eventually (\<lambda>y. y \<notin> s) (nhds y)"
lp15@61810
  1784
    by (auto elim!: eventually_mono)
hoelzl@51481
  1785
  thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s"
hoelzl@51481
  1786
    by (simp add: eventually_nhds subset_eq)
hoelzl@51481
  1787
qed
hoelzl@51481
  1788
hoelzl@51481
  1789
lemma compact_continuous_image:
hoelzl@51481
  1790
  assumes f: "continuous_on s f" and s: "compact s"
hoelzl@51481
  1791
  shows "compact (f ` s)"
hoelzl@51481
  1792
proof (rule compactI)
hoelzl@51481
  1793
  fix C assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C"
hoelzl@51481
  1794
  with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s"
hoelzl@51481
  1795
    unfolding continuous_on_open_invariant by blast
wenzelm@53381
  1796
  then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s"
wenzelm@53381
  1797
    unfolding bchoice_iff ..
hoelzl@51481
  1798
  with cover have "\<forall>c\<in>C. open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)"
hoelzl@51481
  1799
    by (fastforce simp add: subset_eq set_eq_iff)+
hoelzl@51481
  1800
  from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" .
hoelzl@51481
  1801
  with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D"
hoelzl@51481
  1802
    by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+
hoelzl@51481
  1803
qed
hoelzl@51481
  1804
hoelzl@51481
  1805
lemma continuous_on_inv:
hoelzl@51481
  1806
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
hoelzl@51481
  1807
  assumes "continuous_on s f"  "compact s"  "\<forall>x\<in>s. g (f x) = x"
hoelzl@51481
  1808
  shows "continuous_on (f ` s) g"
hoelzl@51481
  1809
unfolding continuous_on_topological
hoelzl@51481
  1810
proof (clarsimp simp add: assms(3))
hoelzl@51481
  1811
  fix x :: 'a and B :: "'a set"
hoelzl@51481
  1812
  assume "x \<in> s" and "open B" and "x \<in> B"
hoelzl@51481
  1813
  have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B"
hoelzl@51481
  1814
    using assms(3) by (auto, metis)
hoelzl@51481
  1815
  have "continuous_on (s - B) f"
wenzelm@60758
  1816
    using \<open>continuous_on s f\<close> Diff_subset
hoelzl@51481
  1817
    by (rule continuous_on_subset)
hoelzl@51481
  1818
  moreover have "compact (s - B)"
wenzelm@60758
  1819
    using \<open>open B\<close> and \<open>compact s\<close>
hoelzl@51481
  1820
    unfolding Diff_eq by (intro compact_inter_closed closed_Compl)
hoelzl@51481
  1821
  ultimately have "compact (f ` (s - B))"
hoelzl@51481
  1822
    by (rule compact_continuous_image)
hoelzl@51481
  1823
  hence "closed (f ` (s - B))"
hoelzl@51481
  1824
    by (rule compact_imp_closed)
hoelzl@51481
  1825
  hence "open (- f ` (s - B))"
hoelzl@51481
  1826
    by (rule open_Compl)
hoelzl@51481
  1827
  moreover have "f x \<in> - f ` (s - B)"
wenzelm@60758
  1828
    using \<open>x \<in> s\<close> and \<open>x \<in> B\<close> by (simp add: 1)
hoelzl@51481
  1829
  moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B"
hoelzl@51481
  1830
    by (simp add: 1)
hoelzl@51481
  1831
  ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)"
hoelzl@51481
  1832
    by fast
hoelzl@51481
  1833
qed
hoelzl@51481
  1834
hoelzl@51481
  1835
lemma continuous_on_inv_into:
hoelzl@51481
  1836
  fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
hoelzl@51481
  1837
  assumes s: "continuous_on s f" "compact s" and f: "inj_on f s"
hoelzl@51481
  1838
  shows "continuous_on (f ` s) (the_inv_into s f)"
hoelzl@51481
  1839
  by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])
hoelzl@51481
  1840
hoelzl@51479
  1841
lemma (in linorder_topology) compact_attains_sup:
hoelzl@51479
  1842
  assumes "compact S" "S \<noteq> {}"
hoelzl@51479
  1843
  shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s"
hoelzl@51479
  1844
proof (rule classical)
hoelzl@51479
  1845
  assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)"
hoelzl@51479
  1846
  then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s"
hoelzl@51479
  1847
    by (metis not_le)
hoelzl@51479
  1848
  then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})"
hoelzl@51479
  1849
    by auto
wenzelm@60758
  1850
  with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})"
hoelzl@51479
  1851
    by (erule compactE_image)
wenzelm@60758
  1852
  with \<open>S \<noteq> {}\<close> have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)"
hoelzl@51479
  1853
    by (auto intro!: Max_in)
hoelzl@51479
  1854
  with C have "S \<subseteq> {..< Max (t`C)}"
hoelzl@51479
  1855
    by (auto intro: less_le_trans simp: subset_eq)
wenzelm@60758
  1856
  with t Max \<open>C \<subseteq> S\<close> show ?thesis
hoelzl@51479
  1857
    by fastforce
hoelzl@51479
  1858
qed
hoelzl@51479
  1859
hoelzl@51479
  1860
lemma (in linorder_topology) compact_attains_inf:
hoelzl@51479
  1861
  assumes "compact S" "S \<noteq> {}"
hoelzl@51479
  1862
  shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t"
hoelzl@51479
  1863
proof (rule classical)
hoelzl@51479
  1864
  assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)"
hoelzl@51479
  1865
  then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s"
hoelzl@51479
  1866
    by (metis not_le)
hoelzl@51479
  1867
  then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})"
hoelzl@51479
  1868
    by auto
wenzelm@60758
  1869
  with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})"
hoelzl@51479
  1870
    by (erule compactE_image)
wenzelm@60758
  1871
  with \<open>S \<noteq> {}\<close> have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s"
hoelzl@51479
  1872
    by (auto intro!: Min_in)
hoelzl@51479
  1873
  with C have "S \<subseteq> {Min (t`C) <..}"
hoelzl@51479
  1874
    by (auto intro: le_less_trans simp: subset_eq)
wenzelm@60758
  1875
  with t Min \<open>C \<subseteq> S\<close> show ?thesis
hoelzl@51479
  1876
    by fastforce
hoelzl@51479
  1877
qed
hoelzl@51479
  1878
hoelzl@51479
  1879
lemma continuous_attains_sup:
hoelzl@51479
  1880
  fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
hoelzl@51479
  1881
  shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s.  f y \<le> f x)"
hoelzl@51479
  1882
  using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto
hoelzl@51479
  1883
hoelzl@51479
  1884
lemma continuous_attains_inf:
hoelzl@51479
  1885
  fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology"
hoelzl@51479
  1886
  shows "compact s \<Longrightarrow> s \<noteq> {} \<Longrightarrow> continuous_on s f \<Longrightarrow> (\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y)"
hoelzl@51479
  1887
  using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto
hoelzl@51479
  1888
wenzelm@60758
  1889
subsection \<open>Connectedness\<close>
hoelzl@51480
  1890
hoelzl@51480
  1891
context topological_space
hoelzl@51480
  1892
begin
hoelzl@51480
  1893
hoelzl@51480
  1894
definition "connected S \<longleftrightarrow>
hoelzl@51480
  1895
  \<not> (\<exists>A B. open A \<and> open B \<and> S \<subseteq> A \<union> B \<and> A \<inter> B \<inter> S = {} \<and> A \<inter> S \<noteq> {} \<and> B \<inter> S \<noteq> {})"
hoelzl@51480
  1896
hoelzl@51480
  1897
lemma connectedI:
hoelzl@51480
  1898
  "(\<And>A B. open A \<Longrightarrow> open B \<Longrightarrow> A \<inter> U \<noteq> {} \<Longrightarrow> B \<inter> U \<noteq> {} \<Longrightarrow> A \<inter> B \<inter> U = {} \<Longrightarrow> U \<subseteq> A \<union> B \<Longrightarrow> False)
hoelzl@51480
  1899
  \<Longrightarrow> connected U"
hoelzl@51480
  1900
  by (auto simp: connected_def)
hoelzl@51480
  1901
lp15@61306
  1902
lemma connected_empty [simp]: "connected {}"
lp15@61306
  1903
  by (auto intro!: connectedI)
lp15@61306
  1904
lp15@61306
  1905
lemma connected_sing [simp]: "connected {x}"
hoelzl@51480
  1906
  by (auto intro!: connectedI)
hoelzl@51480
  1907
hoelzl@56329
  1908
lemma connectedD:
hoelzl@56329
  1909
  "connected A \<Longrightarrow> open U \<Longrightarrow> open V \<Longrightarrow> U \<inter> V \<inter> A = {} \<Longrightarrow> A \<subseteq> U \<union> V \<Longrightarrow> U \<inter> A = {} \<or> V \<inter> A = {}" 
hoelzl@56329
  1910
  by (auto simp: connected_def)
hoelzl@56329
  1911
hoelzl@51479
  1912
end
hoelzl@51479
  1913
lp15@61306
  1914
lemma connected_closed:
lp15@61306
  1915
    "connected s \<longleftrightarrow>
lp15@61306
  1916
     ~ (\<exists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})"
lp15@61306
  1917
apply (simp add: connected_def del: ex_simps, safe)
lp15@61306
  1918
apply (drule_tac x="-A" in spec)
lp15@61306
  1919
apply (drule_tac x="-B" in spec)
lp15@61306
  1920
apply (fastforce simp add: closed_def [symmetric])
lp15@61306
  1921
apply (drule_tac x="-A" in spec)
lp15@61306
  1922
apply (drule_tac x="-B" in spec)
lp15@61306
  1923
apply (fastforce simp add: open_closed [symmetric])
lp15@61306
  1924
done
lp15@61306
  1925
lp15@61306
  1926
lp15@61306
  1927
lemma connected_Union:
lp15@61306
  1928
  assumes cs: "\<And>s. s \<in> S \<Longrightarrow> connected s" and ne: "\<Inter>S \<noteq> {}"
lp15@61306
  1929
    shows "connected(\<Union>S)"
lp15@61306
  1930
proof (rule connectedI)
lp15@61306
  1931
  fix A B
lp15@61306
  1932
  assume A: "open A" and B: "open B" and Alap: "A \<inter> \<Union>S \<noteq> {}" and Blap: "B \<inter> \<Union>S \<noteq> {}"
lp15@61306
  1933
     and disj: "A \<inter> B \<inter> \<Union>S = {}" and cover: "\<Union>S \<subseteq> A \<union> B"
lp15@61306
  1934
  have disjs:"\<And>s. s \<in> S \<Longrightarrow> A \<inter> B \<inter> s = {}"
lp15@61306
  1935
    using disj by auto
lp15@61306
  1936
  obtain sa where sa: "sa \<in> S" "A \<inter> sa \<noteq> {}"
lp15@61306
  1937
    using Alap by auto
lp15@61306
  1938
  obtain sb where sb: "sb \<in> S" "B \<inter> sb \<noteq> {}"
lp15@61306
  1939
    using Blap by auto
lp15@61306
  1940
  obtain x where x: "\<And>s. s \<in> S \<Longrightarrow> x \<in> s"
lp15@61306
  1941
    using ne by auto
lp15@61306
  1942
  then have "x \<in> \<Union>S"
wenzelm@61342
  1943
    using \<open>sa \<in> S\<close> by blast
lp15@61306
  1944
  then have "x \<in> A \<or> x \<in> B"
lp15@61306
  1945
    using cover by auto
lp15@61306
  1946
  then show False
lp15@61306
  1947
    using cs [unfolded connected_def]
lp15@61306
  1948
    by (metis A B IntI Sup_upper sa sb disjs x cover empty_iff subset_trans)
lp15@61306
  1949
qed
lp15@61306
  1950
lp15@61306
  1951
lemma connected_Un: "\<lbrakk>connected s; connected t; s \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> connected (s \<union> t)"
lp15@61306
  1952
  using connected_Union [of "{s,t}"] by auto
lp15@61306
  1953
lp15@61306
  1954
lemma connected_diff_open_from_closed:
lp15@61306
  1955
  assumes st: "s \<subseteq> t" and tu: "t \<subseteq> u" and s: "open s"
lp15@61306
  1956
      and t: "closed t" and u: "connected u" and ts: "connected (t - s)"
lp15@61306
  1957
  shows "connected(u - s)"
lp15@61306
  1958
proof (rule connectedI)
lp15@61306
  1959
  fix A B
lp15@61306
  1960
  assume AB: "open A" "open B" "A \<inter> (u - s) \<noteq> {}" "B \<inter> (u - s) \<noteq> {}"
lp15@61306
  1961
     and disj: "A \<inter> B \<inter> (u - s) = {}" and cover: "u - s \<subseteq> A \<union> B"
lp15@61306
  1962
  then consider "A \<inter> (t - s) = {}" | "B \<inter> (t - s) = {}"
lp15@61306
  1963
    using st ts tu connectedD [of "t-s" "A" "B"]
lp15@61306
  1964
    by auto
lp15@61306
  1965
  then show False
lp15@61306
  1966
  proof cases
lp15@61306
  1967
    case 1
lp15@61306
  1968
    then have "(A - t) \<inter> (B \<union> s) \<inter> u = {}"
lp15@61306
  1969
      using disj st by auto
lp15@61306
  1970
    moreover have  "u \<subseteq> (A - t) \<union> (B \<union> s)" using 1 cover by auto
lp15@61306
  1971
    ultimately show False
lp15@61306
  1972
      using connectedD [of u "A - t" "B \<union> s"] AB s t 1 u
lp15@61306
  1973
      by auto
lp15@61306
  1974
  next
lp15@61306
  1975
    case 2
lp15@61306
  1976
    then have "(A \<union> s) \<inter> (B - t) \<inter> u = {}"
lp15@61306
  1977
      using disj st
lp15@61306
  1978
      by auto
lp15@61306
  1979
    moreover have "u \<subseteq> (A \<union> s) \<union> (B - t)" using 2 cover by auto
lp15@61306
  1980
    ultimately show False
lp15@61306
  1981
      using connectedD [of u "A \<union> s" "B - t"] AB s t 2 u
lp15@61306
  1982
      by auto
lp15@61306
  1983
  qed
lp15@61306
  1984
qed
lp15@61306
  1985
hoelzl@59106
  1986
lemma connected_iff_const:
hoelzl@59106
  1987
  fixes S :: "'a::topological_space set"
hoelzl@59106
  1988
  shows "connected S \<longleftrightarrow> (\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c))"
hoelzl@59106
  1989
proof safe
hoelzl@59106
  1990
  fix P :: "'a \<Rightarrow> bool" assume "connected S" "continuous_on S P"
hoelzl@59106
  1991
  then have "\<And>b. \<exists>A. open A \<and> A \<inter> S = P -` {b} \<inter> S"
hoelzl@59106
  1992
    unfolding continuous_on_open_invariant by simp
hoelzl@59106
  1993
  from this[of True] this[of False]
hoelzl@59106
  1994
  obtain t f where "open t" "open f" and *: "f \<inter> S = P -` {False} \<inter> S" "t \<inter> S = P -` {True} \<inter> S"
hoelzl@59106
  1995
    by auto
hoelzl@59106
  1996
  then have "t \<inter> S = {} \<or> f \<inter> S = {}"
wenzelm@60758
  1997
    by (intro connectedD[OF \<open>connected S\<close>])  auto
hoelzl@59106
  1998
  then show "\<exists>c. \<forall>s\<in>S. P s = c"
hoelzl@59106
  1999
  proof (rule disjE)
hoelzl@59106
  2000
    assume "t \<inter> S = {}" then show ?thesis
hoelzl@59106
  2001
      unfolding * by (intro exI[of _ False]) auto
hoelzl@59106
  2002
  next
hoelzl@59106
  2003
    assume "f \<inter> S = {}" then show ?thesis
hoelzl@59106
  2004
      unfolding * by (intro exI[of _ True]) auto
hoelzl@59106
  2005
  qed
hoelzl@59106
  2006
next
hoelzl@59106
  2007
  assume P: "\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c)"
hoelzl@59106
  2008
  show "connected S"
hoelzl@59106
  2009
  proof (rule connectedI)
hoelzl@59106
  2010
    fix A B assume *: "open A" "open B" "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B"
hoelzl@59106
  2011
    have "continuous_on S (\<lambda>x. x \<in> A)"
hoelzl@59106
  2012
      unfolding continuous_on_open_invariant
hoelzl@59106
  2013
    proof safe
hoelzl@59106
  2014
      fix C :: "bool set"
hoelzl@59106
  2015
      have "C = UNIV \<or> C = {True} \<or> C = {False} \<or> C = {}"
hoelzl@59106
  2016
        using subset_UNIV[of C] unfolding UNIV_bool by auto
hoelzl@59106
  2017
      with * show "\<exists>T. open T \<and> T \<inter> S = (\<lambda>x. x \<in> A) -` C \<inter> S"
hoelzl@59106
  2018
        by (intro exI[of _ "(if True \<in> C then A else {}) \<union> (if False \<in> C then B else {})"]) auto
hoelzl@59106
  2019
    qed
hoelzl@59106
  2020
    from P[rule_format, OF this] obtain c where "\<And>s. s \<in> S \<Longrightarrow> (s \<in> A) = c" by blast
hoelzl@59106
  2021
    with * show False
hoelzl@59106
  2022
      by (cases c) auto
hoelzl@59106
  2023
  qed
hoelzl@59106
  2024
qed
hoelzl@59106
  2025
hoelzl@59106
  2026
lemma connectedD_const:
hoelzl@59106
  2027
  fixes P :: "'a::topological_space \<Rightarrow> bool"
hoelzl@59106
  2028
  shows "connected S \<Longrightarrow> continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c"
hoelzl@59106
  2029
  unfolding connected_iff_const by auto
hoelzl@59106
  2030
hoelzl@59106
  2031
lemma connectedI_const:
hoelzl@59106
  2032
  "(\<And>P::'a::topological_space \<Rightarrow> bool. continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c) \<Longrightarrow> connected S"
hoelzl@59106
  2033
  unfolding connected_iff_const by auto
hoelzl@59106
  2034
hoelzl@56329
  2035
lemma connected_local_const:
hoelzl@56329
  2036
  assumes "connected A" "a \<in> A" "b \<in> A"
hoelzl@56329
  2037
  assumes *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)"
hoelzl@56329
  2038
  shows "f a = f b"
hoelzl@56329
  2039
proof -
hoelzl@56329
  2040
  obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)"
hoelzl@56329
  2041
    "\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x"
hoelzl@56329
  2042
    using * unfolding eventually_at_topological by metis
hoelzl@56329
  2043
hoelzl@56329
  2044
  let ?P = "\<Union>b\<in>{b\<in>A. f a = f b}. S b" and ?N = "\<Union>b\<in>{b\<in>A. f a \<noteq> f b}. S b"
hoelzl@56329
  2045
  have "?P \<inter> A = {} \<or> ?N \<inter> A = {}"
wenzelm@60758
  2046
    using \<open>connected A\<close> S \<open>a\<in>A\<close>
hoelzl@56329
  2047
    by (intro connectedD) (auto, metis)
hoelzl@56329
  2048
  then show "f a = f b"
hoelzl@56329
  2049
  proof
hoelzl@56329
  2050
    assume "?N \<inter> A = {}"
hoelzl@56329
  2051
    then have "\<forall>x\<in>A. f a = f x"
hoelzl@56329
  2052
      using S(1) by auto
wenzelm@60758
  2053
    with \<open>b\<in>A\<close> show ?thesis by auto
hoelzl@56329
  2054
  next
hoelzl@56329
  2055
    assume "?P \<inter> A = {}" then show ?thesis
wenzelm@60758
  2056
      using \<open>a \<in> A\<close> S(1)[of a] by auto
hoelzl@56329
  2057
  qed
hoelzl@56329
  2058
qed
hoelzl@56329
  2059
hoelzl@51480
  2060
lemma (in linorder_topology) connectedD_interval:
hoelzl@51480
  2061
  assumes "connected U" and xy: "x \<in> U" "y \<in> U" and "x \<le> z" "z \<le> y"
hoelzl@51480
  2062
  shows "z \<in> U"
hoelzl@51480
  2063
proof -
hoelzl@51480
  2064
  have eq: "{..<z} \<union> {z<..} = - {z}"
hoelzl@51480
  2065
    by auto
hoelzl@51480
  2066
  { assume "z \<notin> U" "x < z" "z < y"
hoelzl@51480
  2067
    with xy have "\<not> connected U"
hoelzl@51480
  2068
      unfolding connected_def simp_thms
hoelzl@51480
  2069
      apply (rule_tac exI[of _ "{..< z}"])
hoelzl@51480
  2070
      apply (rule_tac exI[of _ "{z <..}"])
hoelzl@51480
  2071
      apply (auto simp add: eq)
hoelzl@51480
  2072
      done }
hoelzl@51480
  2073
  with assms show "z \<in> U"
hoelzl@51480
  2074
    by (metis less_le)
hoelzl@51480
  2075
qed
hoelzl@51480
  2076
hoelzl@51480
  2077
lemma connected_continuous_image:
hoelzl@51480
  2078
  assumes *: "continuous_on s f"
hoelzl@51480
  2079
  assumes "connected s"
hoelzl@51480
  2080
  shows "connected (f ` s)"
hoelzl@59106
  2081
proof (rule connectedI_const)
hoelzl@59106
  2082
  fix P :: "'b \<Rightarrow> bool" assume "continuous_on (f ` s) P"
hoelzl@59106
  2083
  then have "continuous_on s (P \<circ> f)"
hoelzl@59106
  2084
    by (rule continuous_on_compose[OF *])
wenzelm@60758
  2085
  from connectedD_const[OF \<open>connected s\<close> this] show "\<exists>c. \<forall>s\<in>f ` s. P s = c"
hoelzl@59106
  2086
    by auto
hoelzl@51480
  2087
qed
hoelzl@51480
  2088
lp15@61306
  2089
lp15@61306
  2090
section \<open>Linear Continuum Topologies\<close>
hoelzl@51518
  2091
hoelzl@51775
  2092
class linear_continuum_topology = linorder_topology + linear_continuum
hoelzl@51518
  2093
begin
hoelzl@51518
  2094
hoelzl@51518
  2095
lemma Inf_notin_open:
hoelzl@51518
  2096
  assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a"
hoelzl@51518
  2097
  shows "Inf A \<notin> A"
hoelzl@51518
  2098
proof
hoelzl@51518
  2099
  assume "Inf A \<in> A"
hoelzl@51518
  2100
  then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A"
hoelzl@51518
  2101
    using open_left[of A "Inf A" x] assms by auto
hoelzl@51518
  2102
  with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
hoelzl@51518
  2103
    by (auto simp: subset_eq)
hoelzl@51518
  2104
  then show False
wenzelm@60758
  2105
    using cInf_lower[OF \<open>c \<in> A\<close>] bnd by (metis not_le less_imp_le bdd_belowI)
hoelzl@51518
  2106
qed
hoelzl@51518
  2107
hoelzl@51518
  2108
lemma Sup_notin_open:
hoelzl@51518
  2109
  assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x"
hoelzl@51518
  2110
  shows "Sup A \<notin> A"
hoelzl@51518
  2111
proof
hoelzl@51518
  2112
  assume "Sup A \<in> A"
hoelzl@51518
  2113
  then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A"
hoelzl@51518
  2114
    using open_right[of A "Sup A" x] assms by auto
hoelzl@51518
  2115
  with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
hoelzl@51518
  2116
    by (auto simp: subset_eq)
hoelzl@51518
  2117
  then show False
wenzelm@60758
  2118
    using cSup_upper[OF \<open>c \<in> A\<close>] bnd by (metis less_imp_le not_le bdd_aboveI)
hoelzl@51518
  2119
qed
hoelzl@51518
  2120
hoelzl@51480
  2121
end
hoelzl@51480
  2122
hoelzl@51775
  2123
instance linear_continuum_topology \<subseteq> perfect_space
hoelzl@51775
  2124
proof
hoelzl@51775
  2125
  fix x :: 'a
wenzelm@53381
  2126
  obtain y where "x < y \<or> y < x"
wenzelm@53381
  2127
    using ex_gt_or_lt [of x] ..
hoelzl@51775
  2128
  with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y]
hoelzl@51775
  2129
  show "\<not> open {x}"
hoelzl@51775
  2130
    by auto
hoelzl@51775
  2131
qed
hoelzl@51775
  2132
hoelzl@51518
  2133
lemma connectedI_interval:
hoelzl@51775
  2134
  fixes U :: "'a :: linear_continuum_topology set"
hoelzl@51518
  2135
  assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U"
hoelzl@51518
  2136
  shows "connected U"
hoelzl@51518
  2137
proof (rule connectedI)
hoelzl@51518
  2138
  { fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B"
hoelzl@51518
  2139
    fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U"
hoelzl@51518
  2140
hoelzl@51518
  2141
    let ?z = "Inf (B \<inter> {x <..})"
hoelzl@51518
  2142
hoelzl@51518
  2143
    have "x \<le> ?z" "?z \<le> y"
wenzelm@60758
  2144
      using \<open>y \<in> B\<close> \<open>x < y\<close> by (auto intro: cInf_lower cInf_greatest)
wenzelm@60758
  2145
    with \<open>x \<in> U\<close> \<open>y \<in> U\<close> have "?z \<in> U"
hoelzl@51518
  2146
      by (rule *)
hoelzl@51518
  2147
    moreover have "?z \<notin> B \<inter> {x <..}"
wenzelm@60758
  2148
      using \<open>open B\<close> by (intro Inf_notin_open) auto
hoelzl@51518
  2149
    ultimately have "?z \<in> A"
wenzelm@60758
  2150
      using \<open>x \<le> ?z\<close> \<open>A \<inter> B \<inter> U = {}\<close> \<open>x \<in> A\<close> \<open>U \<subseteq> A \<union> B\<close> by auto
hoelzl@51518
  2151
hoelzl@51518
  2152
    { assume "?z < y"
hoelzl@51518
  2153
      obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
wenzelm@60758
  2154
        using open_right[OF \<open>open A\<close> \<open>?z \<in> A\<close> \<open>?z < y\<close>] by auto
hoelzl@51518
  2155
      moreover obtain b where "b \<in> B" "x < b" "b < min a y"
wenzelm@60758
  2156
        using cInf_less_iff[of "B \<inter> {x <..}" "min a y"] \<open>?z < a\<close> \<open>?z < y\<close> \<open>x < y\<close> \<open>y \<in> B\<close>
hoelzl@51518
  2157
        by (auto intro: less_imp_le)
wenzelm@53374
  2158
      moreover have "?z \<le> b"
wenzelm@60758
  2159
        using \<open>b \<in> B\<close> \<open>x < b\<close>
hoelzl@54258
  2160
        by (intro cInf_lower) auto
hoelzl@51518
  2161
      moreover have "b \<in> U"
wenzelm@60758
  2162
        using \<open>x \<le> ?z\<close> \<open>?z \<le> b\<close> \<open>b < min a y\<close>
wenzelm@60758
  2163
        by (intro *[OF \<open>x \<in> U\<close> \<open>y \<in> U\<close>]) (auto simp: less_imp_le)
hoelzl@51518
  2164
      ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U"
hoelzl@51518
  2165
        by (intro bexI[of _ b]) auto }
hoelzl@51518
  2166
    then have False
wenzelm@60758
  2167
      using \<open>?z \<le> y\<close> \<open>?z \<in> A\<close> \<open>y \<in> B\<close> \<open>y \<in> U\<close> \<open>A \<inter> B \<inter> U = {}\<close> unfolding le_less by blast }
hoelzl@51518
  2168
  note not_disjoint = this
hoelzl@51518
  2169
hoelzl@51518
  2170
  fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}"
hoelzl@51518
  2171
  moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto
hoelzl@51518
  2172
  moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto
hoelzl@51518
  2173
  moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
hoelzl@51518
  2174
  ultimately show False by (cases x y rule: linorder_cases) auto
hoelzl@51518
  2175
qed
hoelzl@51518
  2176
hoelzl@51518
  2177
lemma connected_iff_interval:
hoelzl@51775
  2178
  fixes U :: "'a :: linear_continuum_topology set"
hoelzl@51518
  2179
  shows "connected U \<longleftrightarrow> (\<forall>x\<in>U. \<forall>y\<in>U. \<forall>z. x \<le> z \<longrightarrow> z \<le> y \<longrightarrow> z \<in> U)"
hoelzl@51518
  2180
  by (auto intro: connectedI_interval dest: connectedD_interval)
hoelzl@51518
  2181
hoelzl@51775
  2182
lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
hoelzl@51518
  2183
  unfolding connected_iff_interval by auto
hoelzl@51518
  2184
hoelzl@51775
  2185
lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}"
hoelzl@51518
  2186
  unfolding connected_iff_interval by auto
hoelzl@51518
  2187
hoelzl@51775
  2188
lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}"
hoelzl@51518
  2189
  unfolding connected_iff_interval by auto
hoelzl@51518
  2190
hoelzl@51775
  2191
lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}"
hoelzl@51518
  2192
  unfolding connected_iff_interval by auto
hoelzl@51518
  2193
hoelzl@51775
  2194
lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}"
hoelzl@51518
  2195
  unfolding connected_iff_interval by auto
hoelzl@51518
  2196
hoelzl@51775
  2197
lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}"
hoelzl@51518
  2198
  unfolding connected_iff_interval by auto
hoelzl@51518
  2199
hoelzl@51775
  2200
lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}"
hoelzl@51518
  2201
  unfolding connected_iff_interval by auto
hoelzl@51518
  2202
hoelzl@51775
  2203
lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}"
hoelzl@51518
  2204
  unfolding connected_iff_interval by auto
hoelzl@51518
  2205
hoelzl@51775
  2206
lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}"
hoelzl@51518
  2207
  unfolding connected_iff_interval by auto
hoelzl@51518
  2208
hoelzl@51518
  2209
lemma connected_contains_Ioo: 
hoelzl@51518
  2210
  fixes A :: "'a :: linorder_topology set"
hoelzl@51518
  2211
  assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A"
hoelzl@51518
  2212
  using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le)
hoelzl@51518
  2213
eberlm@61531
  2214
lemma connected_contains_Icc:
eberlm@61531
  2215
  assumes "connected (A :: ('a :: {linorder_topology}) set)" "a \<in> A" "b \<in> A"
eberlm@61531
  2216
  shows   "{a..b} \<subseteq> A"
eberlm@61531
  2217
proof
eberlm@61531
  2218
  fix x assume "x \<in> {a..b}"
eberlm@61531
  2219
  hence "x = a \<or> x = b \<or> x \<in> {a<..<b}" by auto
eberlm@61531
  2220
  thus "x \<in> A" using assms connected_contains_Ioo[of A a b] by auto
eberlm@61531
  2221
qed
eberlm@61531
  2222
wenzelm@60758
  2223
subsection \<open>Intermediate Value Theorem\<close>
hoelzl@51518
  2224
hoelzl@51518
  2225
lemma IVT':
hoelzl@51775
  2226
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2227
  assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b"
hoelzl@51518
  2228
  assumes *: "continuous_on {a .. b} f"
hoelzl@51518
  2229
  shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2230
proof -
hoelzl@51518
  2231
  have "connected {a..b}"
hoelzl@51518
  2232
    unfolding connected_iff_interval by auto
hoelzl@51518
  2233
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y
hoelzl@51518
  2234
  show ?thesis
hoelzl@51518
  2235
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
hoelzl@51518
  2236
qed
hoelzl@51518
  2237
hoelzl@51518
  2238
lemma IVT2':
hoelzl@51775
  2239
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2240
  assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b"
hoelzl@51518
  2241
  assumes *: "continuous_on {a .. b} f"
hoelzl@51518
  2242
  shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2243
proof -
hoelzl@51518
  2244
  have "connected {a..b}"
hoelzl@51518
  2245
    unfolding connected_iff_interval by auto
hoelzl@51518
  2246
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y
hoelzl@51518
  2247
  show ?thesis
hoelzl@51518
  2248
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
hoelzl@51518
  2249
qed
hoelzl@51518
  2250
hoelzl@51518
  2251
lemma IVT:
hoelzl@51775
  2252
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2253
  shows "f a \<le> y \<Longrightarrow> y \<le> f b \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2254
  by (rule IVT') (auto intro: continuous_at_imp_continuous_on)
hoelzl@51518
  2255
hoelzl@51518
  2256
lemma IVT2:
hoelzl@51775
  2257
  fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2258
  shows "f b \<le> y \<Longrightarrow> y \<le> f a \<Longrightarrow> a \<le> b \<Longrightarrow> (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x) \<Longrightarrow> \<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y"
hoelzl@51518
  2259
  by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)
hoelzl@51518
  2260
hoelzl@51518
  2261
lemma continuous_inj_imp_mono:
hoelzl@51775
  2262
  fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology"
hoelzl@51518
  2263
  assumes x: "a < x" "x < b"
hoelzl@51518
  2264
  assumes cont: "continuous_on {a..b} f"
hoelzl@51518
  2265
  assumes inj: "inj_on f {a..b}"
hoelzl@51518
  2266
  shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)"
hoelzl@51518
  2267
proof -
lp15@61520
  2268
  note I = inj_on_eq_iff[OF inj]
hoelzl@51518
  2269
  { assume "f x < f a" "f x < f b"
hoelzl@51518
  2270
    then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f x < f s"
hoelzl@51518
  2271
      using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x
hoelzl@51518
  2272
      by (auto simp: continuous_on_subset[OF cont] less_imp_le)
hoelzl@51518
  2273
    with x I have False by auto }
hoelzl@51518
  2274
  moreover
hoelzl@51518
  2275
  { assume "f a < f x" "f b < f x"
hoelzl@51518
  2276
    then obtain s t where "x \<le> s" "s \<le> b" "a \<le> t" "t \<le> x" "f s = f t" "f s < f x"
hoelzl@51518
  2277
      using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x
hoelzl@51518
  2278
      by (auto simp: continuous_on_subset[OF cont] less_imp_le)
hoelzl@51518
  2279
    with x I have False by auto }
hoelzl@51518
  2280
  ultimately show ?thesis
hoelzl@51518
  2281
    using I[of a x] I[of x b] x less_trans[OF x] by (auto simp add: le_less less_imp_neq neq_iff)
hoelzl@51518
  2282
qed
hoelzl@51518
  2283
hoelzl@59452
  2284
lemma continuous_at_Sup_mono:
hoelzl@59452
  2285
  fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}"
hoelzl@59452
  2286
  assumes "mono f"
hoelzl@59452
  2287
  assumes cont: "continuous (at_left (Sup S)) f"
hoelzl@59452
  2288
  assumes S: "S \<noteq> {}" "bdd_above S"
hoelzl@59452
  2289
  shows "f (Sup S) = (SUP s:S. f s)"
hoelzl@59452
  2290
proof (rule antisym)
wenzelm@61973
  2291
  have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))"
hoelzl@59452
  2292
    using cont unfolding continuous_within .
hoelzl@59452
  2293
hoelzl@59452
  2294
  show "f (Sup S) \<le> (SUP s:S. f s)"
hoelzl@59452
  2295
  proof cases
hoelzl@59452
  2296
    assume "Sup S \<in> S" then show ?thesis
wenzelm@60758
  2297
      by (rule cSUP_upper) (auto intro: bdd_above_image_mono S \<open>mono f\<close>)
hoelzl@59452
  2298
  next
hoelzl@59452
  2299
    assume "Sup S \<notin> S"
wenzelm@60758
  2300
    from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S"
hoelzl@59452
  2301
      by auto
wenzelm@60758
  2302
    with \<open>Sup S \<notin> S\<close> S have "s < Sup S"
hoelzl@59452
  2303
      unfolding less_le by (blast intro: cSup_upper)
hoelzl@59452
  2304
    show ?thesis
hoelzl@59452
  2305
    proof (rule ccontr)
hoelzl@59452
  2306
      assume "\<not> ?thesis"
hoelzl@59452
  2307
      with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "b < Sup S"
hoelzl@59452
  2308
        and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> (SUP s:S. f s) < f y"
wenzelm@60758
  2309
        by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>])
wenzelm@60758
  2310
      with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c"
hoelzl@59452
  2311
        using less_cSupD[of S b] by auto
wenzelm@60758
  2312
      with \<open>Sup S \<notin> S\<close> S have "c < Sup S"
hoelzl@59452
  2313
        unfolding less_le by (blast intro: cSup_upper)
wenzelm@60758
  2314
      from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cSUP_upper[OF \<open>c \<in> S\<close>