author | hoelzl |
Wed, 24 Apr 2013 13:28:30 +0200 | |
changeset 51773 | 9328c6681f3c |
parent 51641 | cd05e9fcc63d |
child 51774 | 916271d52466 |
permissions | -rw-r--r-- |
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(* Title: HOL/Basic_Topology.thy |
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Author: Brian Huffman |
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Author: Johannes Hölzl |
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*) |
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header {* Topological Spaces *} |
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theory Topological_Spaces |
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imports Main Conditionally_Complete_Lattices |
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begin |
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subsection {* Topological space *} |
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class "open" = |
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fixes "open" :: "'a set \<Rightarrow> bool" |
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class topological_space = "open" + |
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assumes open_UNIV [simp, intro]: "open UNIV" |
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assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)" |
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assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union> K)" |
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begin |
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definition |
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closed :: "'a set \<Rightarrow> bool" where |
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"closed S \<longleftrightarrow> open (- S)" |
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lemma open_empty [intro, simp]: "open {}" |
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using open_Union [of "{}"] by simp |
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lemma open_Un [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)" |
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using open_Union [of "{S, T}"] by simp |
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lemma open_UN [intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)" |
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unfolding SUP_def by (rule open_Union) auto |
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lemma open_Inter [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)" |
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by (induct set: finite) auto |
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lemma open_INT [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)" |
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unfolding INF_def by (rule open_Inter) auto |
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lemma openI: |
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assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" |
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shows "open S" |
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proof - |
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have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto |
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moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms) |
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ultimately show "open S" by simp |
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qed |
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lemma closed_empty [intro, simp]: "closed {}" |
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unfolding closed_def by simp |
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lemma closed_Un [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)" |
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unfolding closed_def by auto |
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lemma closed_UNIV [intro, simp]: "closed UNIV" |
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unfolding closed_def by simp |
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lemma closed_Int [intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)" |
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unfolding closed_def by auto |
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lemma closed_INT [intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)" |
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unfolding closed_def by auto |
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lemma closed_Inter [intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter> K)" |
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unfolding closed_def uminus_Inf by auto |
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lemma closed_Union [intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)" |
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by (induct set: finite) auto |
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lemma closed_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)" |
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unfolding SUP_def by (rule closed_Union) auto |
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lemma open_closed: "open S \<longleftrightarrow> closed (- S)" |
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unfolding closed_def by simp |
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lemma closed_open: "closed S \<longleftrightarrow> open (- S)" |
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unfolding closed_def by simp |
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lemma open_Diff [intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)" |
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unfolding closed_open Diff_eq by (rule open_Int) |
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lemma closed_Diff [intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)" |
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unfolding open_closed Diff_eq by (rule closed_Int) |
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lemma open_Compl [intro]: "closed S \<Longrightarrow> open (- S)" |
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unfolding closed_open . |
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lemma closed_Compl [intro]: "open S \<Longrightarrow> closed (- S)" |
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unfolding open_closed . |
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end |
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subsection{* Hausdorff and other separation properties *} |
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class t0_space = topological_space + |
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assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)" |
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class t1_space = topological_space + |
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assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U" |
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instance t1_space \<subseteq> t0_space |
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proof qed (fast dest: t1_space) |
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lemma separation_t1: |
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fixes x y :: "'a::t1_space" |
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shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)" |
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using t1_space[of x y] by blast |
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lemma closed_singleton: |
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fixes a :: "'a::t1_space" |
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shows "closed {a}" |
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proof - |
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let ?T = "\<Union>{S. open S \<and> a \<notin> S}" |
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have "open ?T" by (simp add: open_Union) |
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also have "?T = - {a}" |
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by (simp add: set_eq_iff separation_t1, auto) |
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finally show "closed {a}" unfolding closed_def . |
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qed |
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lemma closed_insert [simp]: |
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fixes a :: "'a::t1_space" |
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assumes "closed S" shows "closed (insert a S)" |
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proof - |
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from closed_singleton assms |
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have "closed ({a} \<union> S)" by (rule closed_Un) |
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thus "closed (insert a S)" by simp |
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qed |
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lemma finite_imp_closed: |
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fixes S :: "'a::t1_space set" |
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shows "finite S \<Longrightarrow> closed S" |
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by (induct set: finite, simp_all) |
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text {* T2 spaces are also known as Hausdorff spaces. *} |
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class t2_space = topological_space + |
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assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" |
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instance t2_space \<subseteq> t1_space |
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proof qed (fast dest: hausdorff) |
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lemma separation_t2: |
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fixes x y :: "'a::t2_space" |
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shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})" |
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using hausdorff[of x y] by blast |
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lemma separation_t0: |
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fixes x y :: "'a::t0_space" |
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shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))" |
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using t0_space[of x y] by blast |
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text {* A perfect space is a topological space with no isolated points. *} |
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class perfect_space = topological_space + |
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assumes not_open_singleton: "\<not> open {x}" |
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subsection {* Generators for toplogies *} |
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inductive generate_topology for S where |
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UNIV: "generate_topology S UNIV" |
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| Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)" |
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| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)" |
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| Basis: "s \<in> S \<Longrightarrow> generate_topology S s" |
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hide_fact (open) UNIV Int UN Basis |
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lemma generate_topology_Union: |
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"(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)" |
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unfolding SUP_def by (intro generate_topology.UN) auto |
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lemma topological_space_generate_topology: |
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"class.topological_space (generate_topology S)" |
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by default (auto intro: generate_topology.intros) |
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subsection {* Order topologies *} |
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class order_topology = order + "open" + |
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assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))" |
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begin |
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subclass topological_space |
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unfolding open_generated_order |
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by (rule topological_space_generate_topology) |
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lemma open_greaterThan [simp]: "open {a <..}" |
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unfolding open_generated_order by (auto intro: generate_topology.Basis) |
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lemma open_lessThan [simp]: "open {..< a}" |
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unfolding open_generated_order by (auto intro: generate_topology.Basis) |
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lemma open_greaterThanLessThan [simp]: "open {a <..< b}" |
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unfolding greaterThanLessThan_eq by (simp add: open_Int) |
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end |
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class linorder_topology = linorder + order_topology |
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lemma closed_atMost [simp]: "closed {.. a::'a::linorder_topology}" |
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by (simp add: closed_open) |
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lemma closed_atLeast [simp]: "closed {a::'a::linorder_topology ..}" |
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by (simp add: closed_open) |
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lemma closed_atLeastAtMost [simp]: "closed {a::'a::linorder_topology .. b}" |
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proof - |
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have "{a .. b} = {a ..} \<inter> {.. b}" |
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by auto |
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then show ?thesis |
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by (simp add: closed_Int) |
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qed |
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lemma (in linorder) less_separate: |
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assumes "x < y" |
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shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}" |
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proof cases |
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assume "\<exists>z. x < z \<and> z < y" |
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then guess z .. |
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then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}" |
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by auto |
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then show ?thesis by blast |
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next |
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assume "\<not> (\<exists>z. x < z \<and> z < y)" |
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with `x < y` have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}" |
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by auto |
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then show ?thesis by blast |
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qed |
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instance linorder_topology \<subseteq> t2_space |
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proof |
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fix x y :: 'a |
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from less_separate[of x y] less_separate[of y x] |
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show "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" |
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by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+ |
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qed |
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lemma (in linorder_topology) open_right: |
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assumes "open S" "x \<in> S" and gt_ex: "x < y" shows "\<exists>b>x. {x ..< b} \<subseteq> S" |
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using assms unfolding open_generated_order |
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proof induction |
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case (Int A B) |
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then obtain a b where "a > x" "{x ..< a} \<subseteq> A" "b > x" "{x ..< b} \<subseteq> B" by auto |
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then show ?case by (auto intro!: exI[of _ "min a b"]) |
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next |
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case (Basis S) then show ?case by (fastforce intro: exI[of _ y] gt_ex) |
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qed blast+ |
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lemma (in linorder_topology) open_left: |
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assumes "open S" "x \<in> S" and lt_ex: "y < x" shows "\<exists>b<x. {b <.. x} \<subseteq> S" |
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using assms unfolding open_generated_order |
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proof induction |
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case (Int A B) |
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then obtain a b where "a < x" "{a <.. x} \<subseteq> A" "b < x" "{b <.. x} \<subseteq> B" by auto |
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then show ?case by (auto intro!: exI[of _ "max a b"]) |
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next |
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case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex) |
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qed blast+ |
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subsection {* Filters *} |
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text {* |
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This definition also allows non-proper filters. |
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*} |
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locale is_filter = |
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fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool" |
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assumes True: "F (\<lambda>x. True)" |
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assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)" |
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assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)" |
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typedef 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}" |
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proof |
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show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro) |
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qed |
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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)" |
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using Rep_filter [of F] by simp |
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lemma Abs_filter_inverse': |
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assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F" |
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using assms by (simp add: Abs_filter_inverse) |
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subsubsection {* Eventually *} |
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" |
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where "eventually P F \<longleftrightarrow> Rep_filter F P" |
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lemma eventually_Abs_filter: |
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assumes "is_filter F" shows "eventually P (Abs_filter F) = F P" |
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unfolding eventually_def using assms by (simp add: Abs_filter_inverse) |
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lemma filter_eq_iff: |
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shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')" |
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unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def .. |
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F" |
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unfolding eventually_def |
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by (rule is_filter.True [OF is_filter_Rep_filter]) |
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F" |
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proof - |
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assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext) |
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thus "eventually P F" by simp |
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qed |
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lemma eventually_mono: |
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"(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F" |
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unfolding eventually_def |
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by (rule is_filter.mono [OF is_filter_Rep_filter]) |
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lemma eventually_conj: |
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assumes P: "eventually (\<lambda>x. P x) F" |
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assumes Q: "eventually (\<lambda>x. Q x) F" |
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shows "eventually (\<lambda>x. P x \<and> Q x) F" |
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using assms unfolding eventually_def |
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by (rule is_filter.conj [OF is_filter_Rep_filter]) |
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lemma eventually_Ball_finite: |
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assumes "finite A" and "\<forall>y\<in>A. eventually (\<lambda>x. P x y) net" |
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shows "eventually (\<lambda>x. \<forall>y\<in>A. P x y) net" |
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using assms by (induct set: finite, simp, simp add: eventually_conj) |
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lemma eventually_all_finite: |
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fixes P :: "'a \<Rightarrow> 'b::finite \<Rightarrow> bool" |
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assumes "\<And>y. eventually (\<lambda>x. P x y) net" |
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shows "eventually (\<lambda>x. \<forall>y. P x y) net" |
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using eventually_Ball_finite [of UNIV P] assms by simp |
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lemma eventually_mp: |
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assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
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assumes "eventually (\<lambda>x. P x) F" |
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shows "eventually (\<lambda>x. Q x) F" |
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proof (rule eventually_mono) |
|
337 |
show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp |
|
338 |
show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F" |
|
339 |
using assms by (rule eventually_conj) |
|
340 |
qed |
|
341 |
||
342 |
lemma eventually_rev_mp: |
|
343 |
assumes "eventually (\<lambda>x. P x) F" |
|
344 |
assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
|
345 |
shows "eventually (\<lambda>x. Q x) F" |
|
346 |
using assms(2) assms(1) by (rule eventually_mp) |
|
347 |
||
348 |
lemma eventually_conj_iff: |
|
349 |
"eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F" |
|
350 |
by (auto intro: eventually_conj elim: eventually_rev_mp) |
|
351 |
||
352 |
lemma eventually_elim1: |
|
353 |
assumes "eventually (\<lambda>i. P i) F" |
|
354 |
assumes "\<And>i. P i \<Longrightarrow> Q i" |
|
355 |
shows "eventually (\<lambda>i. Q i) F" |
|
356 |
using assms by (auto elim!: eventually_rev_mp) |
|
357 |
||
358 |
lemma eventually_elim2: |
|
359 |
assumes "eventually (\<lambda>i. P i) F" |
|
360 |
assumes "eventually (\<lambda>i. Q i) F" |
|
361 |
assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i" |
|
362 |
shows "eventually (\<lambda>i. R i) F" |
|
363 |
using assms by (auto elim!: eventually_rev_mp) |
|
364 |
||
365 |
lemma eventually_subst: |
|
366 |
assumes "eventually (\<lambda>n. P n = Q n) F" |
|
367 |
shows "eventually P F = eventually Q F" (is "?L = ?R") |
|
368 |
proof - |
|
369 |
from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F" |
|
370 |
and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F" |
|
371 |
by (auto elim: eventually_elim1) |
|
372 |
then show ?thesis by (auto elim: eventually_elim2) |
|
373 |
qed |
|
374 |
||
375 |
ML {* |
|
376 |
fun eventually_elim_tac ctxt thms thm = |
|
377 |
let |
|
378 |
val thy = Proof_Context.theory_of ctxt |
|
379 |
val mp_thms = thms RL [@{thm eventually_rev_mp}] |
|
380 |
val raw_elim_thm = |
|
381 |
(@{thm allI} RS @{thm always_eventually}) |
|
382 |
|> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms |
|
383 |
|> fold (fn _ => fn thm => @{thm impI} RS thm) thms |
|
384 |
val cases_prop = prop_of (raw_elim_thm RS thm) |
|
385 |
val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])]) |
|
386 |
in |
|
387 |
CASES cases (rtac raw_elim_thm 1) thm |
|
388 |
end |
|
389 |
*} |
|
390 |
||
391 |
method_setup eventually_elim = {* |
|
392 |
Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt)) |
|
393 |
*} "elimination of eventually quantifiers" |
|
394 |
||
395 |
||
396 |
subsubsection {* Finer-than relation *} |
|
397 |
||
398 |
text {* @{term "F \<le> F'"} means that filter @{term F} is finer than |
|
399 |
filter @{term F'}. *} |
|
400 |
||
401 |
instantiation filter :: (type) complete_lattice |
|
402 |
begin |
|
403 |
||
404 |
definition le_filter_def: |
|
405 |
"F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)" |
|
406 |
||
407 |
definition |
|
408 |
"(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" |
|
409 |
||
410 |
definition |
|
411 |
"top = Abs_filter (\<lambda>P. \<forall>x. P x)" |
|
412 |
||
413 |
definition |
|
414 |
"bot = Abs_filter (\<lambda>P. True)" |
|
415 |
||
416 |
definition |
|
417 |
"sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')" |
|
418 |
||
419 |
definition |
|
420 |
"inf F F' = Abs_filter |
|
421 |
(\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" |
|
422 |
||
423 |
definition |
|
424 |
"Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)" |
|
425 |
||
426 |
definition |
|
427 |
"Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}" |
|
428 |
||
429 |
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)" |
|
430 |
unfolding top_filter_def |
|
431 |
by (rule eventually_Abs_filter, rule is_filter.intro, auto) |
|
432 |
||
433 |
lemma eventually_bot [simp]: "eventually P bot" |
|
434 |
unfolding bot_filter_def |
|
435 |
by (subst eventually_Abs_filter, rule is_filter.intro, auto) |
|
436 |
||
437 |
lemma eventually_sup: |
|
438 |
"eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'" |
|
439 |
unfolding sup_filter_def |
|
440 |
by (rule eventually_Abs_filter, rule is_filter.intro) |
|
441 |
(auto elim!: eventually_rev_mp) |
|
442 |
||
443 |
lemma eventually_inf: |
|
444 |
"eventually P (inf F F') \<longleftrightarrow> |
|
445 |
(\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))" |
|
446 |
unfolding inf_filter_def |
|
447 |
apply (rule eventually_Abs_filter, rule is_filter.intro) |
|
448 |
apply (fast intro: eventually_True) |
|
449 |
apply clarify |
|
450 |
apply (intro exI conjI) |
|
451 |
apply (erule (1) eventually_conj) |
|
452 |
apply (erule (1) eventually_conj) |
|
453 |
apply simp |
|
454 |
apply auto |
|
455 |
done |
|
456 |
||
457 |
lemma eventually_Sup: |
|
458 |
"eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)" |
|
459 |
unfolding Sup_filter_def |
|
460 |
apply (rule eventually_Abs_filter, rule is_filter.intro) |
|
461 |
apply (auto intro: eventually_conj elim!: eventually_rev_mp) |
|
462 |
done |
|
463 |
||
464 |
instance proof |
|
465 |
fix F F' F'' :: "'a filter" and S :: "'a filter set" |
|
466 |
{ show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F" |
|
467 |
by (rule less_filter_def) } |
|
468 |
{ show "F \<le> F" |
|
469 |
unfolding le_filter_def by simp } |
|
470 |
{ assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''" |
|
471 |
unfolding le_filter_def by simp } |
|
472 |
{ assume "F \<le> F'" and "F' \<le> F" thus "F = F'" |
|
473 |
unfolding le_filter_def filter_eq_iff by fast } |
|
474 |
{ show "F \<le> top" |
|
475 |
unfolding le_filter_def eventually_top by (simp add: always_eventually) } |
|
476 |
{ show "bot \<le> F" |
|
477 |
unfolding le_filter_def by simp } |
|
478 |
{ show "F \<le> sup F F'" and "F' \<le> sup F F'" |
|
479 |
unfolding le_filter_def eventually_sup by simp_all } |
|
480 |
{ assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''" |
|
481 |
unfolding le_filter_def eventually_sup by simp } |
|
482 |
{ show "inf F F' \<le> F" and "inf F F' \<le> F'" |
|
483 |
unfolding le_filter_def eventually_inf by (auto intro: eventually_True) } |
|
484 |
{ assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''" |
|
485 |
unfolding le_filter_def eventually_inf |
|
486 |
by (auto elim!: eventually_mono intro: eventually_conj) } |
|
487 |
{ assume "F \<in> S" thus "F \<le> Sup S" |
|
488 |
unfolding le_filter_def eventually_Sup by simp } |
|
489 |
{ assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'" |
|
490 |
unfolding le_filter_def eventually_Sup by simp } |
|
491 |
{ assume "F'' \<in> S" thus "Inf S \<le> F''" |
|
492 |
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } |
|
493 |
{ assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S" |
|
494 |
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp } |
|
495 |
qed |
|
496 |
||
497 |
end |
|
498 |
||
499 |
lemma filter_leD: |
|
500 |
"F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F" |
|
501 |
unfolding le_filter_def by simp |
|
502 |
||
503 |
lemma filter_leI: |
|
504 |
"(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'" |
|
505 |
unfolding le_filter_def by simp |
|
506 |
||
507 |
lemma eventually_False: |
|
508 |
"eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot" |
|
509 |
unfolding filter_eq_iff by (auto elim: eventually_rev_mp) |
|
510 |
||
511 |
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool" |
|
512 |
where "trivial_limit F \<equiv> F = bot" |
|
513 |
||
514 |
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F" |
|
515 |
by (rule eventually_False [symmetric]) |
|
516 |
||
517 |
lemma eventually_const: "\<not> trivial_limit net \<Longrightarrow> eventually (\<lambda>x. P) net \<longleftrightarrow> P" |
|
518 |
by (cases P) (simp_all add: eventually_False) |
|
519 |
||
520 |
||
521 |
subsubsection {* Map function for filters *} |
|
522 |
||
523 |
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter" |
|
524 |
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)" |
|
525 |
||
526 |
lemma eventually_filtermap: |
|
527 |
"eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F" |
|
528 |
unfolding filtermap_def |
|
529 |
apply (rule eventually_Abs_filter) |
|
530 |
apply (rule is_filter.intro) |
|
531 |
apply (auto elim!: eventually_rev_mp) |
|
532 |
done |
|
533 |
||
534 |
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F" |
|
535 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
536 |
||
537 |
lemma filtermap_filtermap: |
|
538 |
"filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F" |
|
539 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
540 |
||
541 |
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'" |
|
542 |
unfolding le_filter_def eventually_filtermap by simp |
|
543 |
||
544 |
lemma filtermap_bot [simp]: "filtermap f bot = bot" |
|
545 |
by (simp add: filter_eq_iff eventually_filtermap) |
|
546 |
||
547 |
lemma filtermap_sup: "filtermap f (sup F1 F2) = sup (filtermap f F1) (filtermap f F2)" |
|
548 |
by (auto simp: filter_eq_iff eventually_filtermap eventually_sup) |
|
549 |
||
550 |
subsubsection {* Order filters *} |
|
551 |
||
552 |
definition at_top :: "('a::order) filter" |
|
553 |
where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)" |
|
554 |
||
555 |
lemma eventually_at_top_linorder: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<forall>n\<ge>N. P n)" |
|
556 |
unfolding at_top_def |
|
557 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
|
558 |
fix P Q :: "'a \<Rightarrow> bool" |
|
559 |
assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n" |
|
560 |
then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto |
|
561 |
then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp |
|
562 |
then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" .. |
|
563 |
qed auto |
|
564 |
||
565 |
lemma eventually_ge_at_top: |
|
566 |
"eventually (\<lambda>x. (c::_::linorder) \<le> x) at_top" |
|
567 |
unfolding eventually_at_top_linorder by auto |
|
568 |
||
569 |
lemma eventually_at_top_dense: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::dense_linorder. \<forall>n>N. P n)" |
|
570 |
unfolding eventually_at_top_linorder |
|
571 |
proof safe |
|
572 |
fix N assume "\<forall>n\<ge>N. P n" then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N]) |
|
573 |
next |
|
574 |
fix N assume "\<forall>n>N. P n" |
|
575 |
moreover from gt_ex[of N] guess y .. |
|
576 |
ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y]) |
|
577 |
qed |
|
578 |
||
579 |
lemma eventually_gt_at_top: |
|
580 |
"eventually (\<lambda>x. (c::_::dense_linorder) < x) at_top" |
|
581 |
unfolding eventually_at_top_dense by auto |
|
582 |
||
583 |
definition at_bot :: "('a::order) filter" |
|
584 |
where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)" |
|
585 |
||
586 |
lemma eventually_at_bot_linorder: |
|
587 |
fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)" |
|
588 |
unfolding at_bot_def |
|
589 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
|
590 |
fix P Q :: "'a \<Rightarrow> bool" |
|
591 |
assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n" |
|
592 |
then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto |
|
593 |
then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp |
|
594 |
then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" .. |
|
595 |
qed auto |
|
596 |
||
597 |
lemma eventually_le_at_bot: |
|
598 |
"eventually (\<lambda>x. x \<le> (c::_::linorder)) at_bot" |
|
599 |
unfolding eventually_at_bot_linorder by auto |
|
600 |
||
601 |
lemma eventually_at_bot_dense: |
|
602 |
fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)" |
|
603 |
unfolding eventually_at_bot_linorder |
|
604 |
proof safe |
|
605 |
fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N]) |
|
606 |
next |
|
607 |
fix N assume "\<forall>n<N. P n" |
|
608 |
moreover from lt_ex[of N] guess y .. |
|
609 |
ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y]) |
|
610 |
qed |
|
611 |
||
612 |
lemma eventually_gt_at_bot: |
|
613 |
"eventually (\<lambda>x. x < (c::_::dense_linorder)) at_bot" |
|
614 |
unfolding eventually_at_bot_dense by auto |
|
615 |
||
616 |
subsection {* Sequentially *} |
|
617 |
||
618 |
abbreviation sequentially :: "nat filter" |
|
619 |
where "sequentially == at_top" |
|
620 |
||
621 |
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)" |
|
622 |
unfolding at_top_def by simp |
|
623 |
||
624 |
lemma eventually_sequentially: |
|
625 |
"eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)" |
|
626 |
by (rule eventually_at_top_linorder) |
|
627 |
||
628 |
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot" |
|
629 |
unfolding filter_eq_iff eventually_sequentially by auto |
|
630 |
||
631 |
lemmas trivial_limit_sequentially = sequentially_bot |
|
632 |
||
633 |
lemma eventually_False_sequentially [simp]: |
|
634 |
"\<not> eventually (\<lambda>n. False) sequentially" |
|
635 |
by (simp add: eventually_False) |
|
636 |
||
637 |
lemma le_sequentially: |
|
638 |
"F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)" |
|
639 |
unfolding le_filter_def eventually_sequentially |
|
640 |
by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp) |
|
641 |
||
642 |
lemma eventually_sequentiallyI: |
|
643 |
assumes "\<And>x. c \<le> x \<Longrightarrow> P x" |
|
644 |
shows "eventually P sequentially" |
|
645 |
using assms by (auto simp: eventually_sequentially) |
|
646 |
||
51474
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
647 |
lemma eventually_sequentially_seg: |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
648 |
"eventually (\<lambda>n. P (n + k)) sequentially \<longleftrightarrow> eventually P sequentially" |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
649 |
unfolding eventually_sequentially |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
650 |
apply safe |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
651 |
apply (rule_tac x="N + k" in exI) |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
652 |
apply rule |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
653 |
apply (erule_tac x="n - k" in allE) |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
654 |
apply auto [] |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
655 |
apply (rule_tac x=N in exI) |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
656 |
apply auto [] |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
657 |
done |
51471 | 658 |
|
659 |
subsubsection {* Standard filters *} |
|
660 |
||
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
661 |
definition principal :: "'a set \<Rightarrow> 'a filter" where |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
662 |
"principal S = Abs_filter (\<lambda>P. \<forall>x\<in>S. P x)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
663 |
|
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
664 |
lemma eventually_principal: "eventually P (principal S) \<longleftrightarrow> (\<forall>x\<in>S. P x)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
665 |
unfolding principal_def |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
666 |
by (rule eventually_Abs_filter, rule is_filter.intro) auto |
51471 | 667 |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
668 |
lemma eventually_inf_principal: "eventually P (inf F (principal s)) \<longleftrightarrow> eventually (\<lambda>x. x \<in> s \<longrightarrow> P x) F" |
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changeset
|
669 |
unfolding eventually_inf eventually_principal by (auto elim: eventually_elim1) |
cd05e9fcc63d
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parents:
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diff
changeset
|
670 |
|
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
671 |
lemma principal_UNIV[simp]: "principal UNIV = top" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
672 |
by (auto simp: filter_eq_iff eventually_principal) |
51471 | 673 |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
674 |
lemma principal_empty[simp]: "principal {} = bot" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
675 |
by (auto simp: filter_eq_iff eventually_principal) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
676 |
|
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
677 |
lemma principal_le_iff[iff]: "principal A \<le> principal B \<longleftrightarrow> A \<subseteq> B" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
678 |
by (auto simp: le_filter_def eventually_principal) |
51471 | 679 |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
680 |
lemma le_principal: "F \<le> principal A \<longleftrightarrow> eventually (\<lambda>x. x \<in> A) F" |
cd05e9fcc63d
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hoelzl
parents:
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diff
changeset
|
681 |
unfolding le_filter_def eventually_principal |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
682 |
apply safe |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
683 |
apply (erule_tac x="\<lambda>x. x \<in> A" in allE) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
684 |
apply (auto elim: eventually_elim1) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
685 |
done |
51471 | 686 |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
687 |
lemma principal_inject[iff]: "principal A = principal B \<longleftrightarrow> A = B" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
688 |
unfolding eq_iff by simp |
51471 | 689 |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
690 |
lemma sup_principal[simp]: "sup (principal A) (principal B) = principal (A \<union> B)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
691 |
unfolding filter_eq_iff eventually_sup eventually_principal by auto |
51471 | 692 |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
693 |
lemma inf_principal[simp]: "inf (principal A) (principal B) = principal (A \<inter> B)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
694 |
unfolding filter_eq_iff eventually_inf eventually_principal |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
695 |
by (auto intro: exI[of _ "\<lambda>x. x \<in> A"] exI[of _ "\<lambda>x. x \<in> B"]) |
51471 | 696 |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
697 |
lemma SUP_principal[simp]: "(SUP i : I. principal (A i)) = principal (\<Union>i\<in>I. A i)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
698 |
unfolding filter_eq_iff eventually_Sup SUP_def by (auto simp: eventually_principal) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
699 |
|
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
700 |
lemma filtermap_principal[simp]: "filtermap f (principal A) = principal (f ` A)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
701 |
unfolding filter_eq_iff eventually_filtermap eventually_principal by simp |
51471 | 702 |
|
703 |
subsubsection {* Topological filters *} |
|
704 |
||
705 |
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter" |
|
706 |
where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" |
|
707 |
||
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
708 |
definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
709 |
where "at a within s = inf (nhds a) (principal (s - {a}))" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
710 |
|
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
711 |
abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
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parents:
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diff
changeset
|
712 |
"at x \<equiv> at x within (CONST UNIV)" |
51471 | 713 |
|
51473 | 714 |
abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where |
51471 | 715 |
"at_right x \<equiv> at x within {x <..}" |
716 |
||
51473 | 717 |
abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where |
51471 | 718 |
"at_left x \<equiv> at x within {..< x}" |
719 |
||
51473 | 720 |
lemma (in topological_space) eventually_nhds: |
51471 | 721 |
"eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" |
722 |
unfolding nhds_def |
|
723 |
proof (rule eventually_Abs_filter, rule is_filter.intro) |
|
51473 | 724 |
have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp |
51471 | 725 |
thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" .. |
726 |
next |
|
727 |
fix P Q |
|
728 |
assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)" |
|
729 |
and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" |
|
730 |
then obtain S T where |
|
731 |
"open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)" |
|
732 |
"open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto |
|
733 |
hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)" |
|
734 |
by (simp add: open_Int) |
|
735 |
thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" .. |
|
736 |
qed auto |
|
737 |
||
738 |
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot" |
|
739 |
unfolding trivial_limit_def eventually_nhds by simp |
|
740 |
||
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
741 |
lemma eventually_at_filter: |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
742 |
"eventually P (at a within s) \<longleftrightarrow> eventually (\<lambda>x. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x) (nhds a)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
743 |
unfolding at_within_def eventually_inf_principal by (simp add: imp_conjL[symmetric] conj_commute) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
744 |
|
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
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diff
changeset
|
745 |
lemma at_le: "s \<subseteq> t \<Longrightarrow> at x within s \<le> at x within t" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
746 |
unfolding at_within_def by (intro inf_mono) auto |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
747 |
|
51471 | 748 |
lemma eventually_at_topological: |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
749 |
"eventually P (at a within s) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
750 |
unfolding eventually_nhds eventually_at_filter by simp |
51471 | 751 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
752 |
lemma at_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> at a within S = at a" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
753 |
unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I) |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
754 |
|
51471 | 755 |
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}" |
756 |
unfolding trivial_limit_def eventually_at_topological |
|
757 |
by (safe, case_tac "S = {a}", simp, fast, fast) |
|
758 |
||
759 |
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot" |
|
760 |
by (simp add: at_eq_bot_iff not_open_singleton) |
|
761 |
||
762 |
lemma eventually_at_right: |
|
763 |
fixes x :: "'a :: {no_top, linorder_topology}" |
|
764 |
shows "eventually P (at_right x) \<longleftrightarrow> (\<exists>b. x < b \<and> (\<forall>z. x < z \<and> z < b \<longrightarrow> P z))" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
765 |
unfolding eventually_at_topological |
51471 | 766 |
proof safe |
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
767 |
from gt_ex[of x] guess y .. |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
768 |
moreover fix S assume "open S" "x \<in> S" note open_right[OF this, of y] |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
769 |
moreover note gt_ex[of x] |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
770 |
moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {x<..} \<longrightarrow> P s" |
51471 | 771 |
ultimately show "\<exists>b>x. \<forall>z. x < z \<and> z < b \<longrightarrow> P z" |
772 |
by (auto simp: subset_eq Ball_def) |
|
773 |
next |
|
774 |
fix b assume "x < b" "\<forall>z. x < z \<and> z < b \<longrightarrow> P z" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
775 |
then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>xa\<in>S. xa \<noteq> x \<longrightarrow> xa \<in> {x<..} \<longrightarrow> P xa)" |
51471 | 776 |
by (intro exI[of _ "{..< b}"]) auto |
777 |
qed |
|
778 |
||
779 |
lemma eventually_at_left: |
|
780 |
fixes x :: "'a :: {no_bot, linorder_topology}" |
|
781 |
shows "eventually P (at_left x) \<longleftrightarrow> (\<exists>b. x > b \<and> (\<forall>z. b < z \<and> z < x \<longrightarrow> P z))" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
782 |
unfolding eventually_at_topological |
51471 | 783 |
proof safe |
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
784 |
from lt_ex[of x] guess y .. |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
785 |
moreover fix S assume "open S" "x \<in> S" note open_left[OF this, of y] |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
786 |
moreover assume "\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {..<x} \<longrightarrow> P s" |
51471 | 787 |
ultimately show "\<exists>b<x. \<forall>z. b < z \<and> z < x \<longrightarrow> P z" |
788 |
by (auto simp: subset_eq Ball_def) |
|
789 |
next |
|
790 |
fix b assume "b < x" "\<forall>z. b < z \<and> z < x \<longrightarrow> P z" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
791 |
then show "\<exists>S. open S \<and> x \<in> S \<and> (\<forall>s\<in>S. s \<noteq> x \<longrightarrow> s \<in> {..<x} \<longrightarrow> P s)" |
51471 | 792 |
by (intro exI[of _ "{b <..}"]) auto |
793 |
qed |
|
794 |
||
795 |
lemma trivial_limit_at_left_real [simp]: |
|
796 |
"\<not> trivial_limit (at_left (x::'a::{no_bot, dense_linorder, linorder_topology}))" |
|
797 |
unfolding trivial_limit_def eventually_at_left by (auto dest: dense) |
|
798 |
||
799 |
lemma trivial_limit_at_right_real [simp]: |
|
800 |
"\<not> trivial_limit (at_right (x::'a::{no_top, dense_linorder, linorder_topology}))" |
|
801 |
unfolding trivial_limit_def eventually_at_right by (auto dest: dense) |
|
802 |
||
803 |
lemma at_eq_sup_left_right: "at (x::'a::linorder_topology) = sup (at_left x) (at_right x)" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
804 |
by (auto simp: eventually_at_filter filter_eq_iff eventually_sup |
51471 | 805 |
elim: eventually_elim2 eventually_elim1) |
806 |
||
807 |
lemma eventually_at_split: |
|
808 |
"eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)" |
|
809 |
by (subst at_eq_sup_left_right) (simp add: eventually_sup) |
|
810 |
||
811 |
subsection {* Limits *} |
|
812 |
||
813 |
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where |
|
814 |
"filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2" |
|
815 |
||
816 |
syntax |
|
817 |
"_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10) |
|
818 |
||
819 |
translations |
|
820 |
"LIM x F1. f :> F2" == "CONST filterlim (%x. f) F2 F1" |
|
821 |
||
822 |
lemma filterlim_iff: |
|
823 |
"(LIM x F1. f x :> F2) \<longleftrightarrow> (\<forall>P. eventually P F2 \<longrightarrow> eventually (\<lambda>x. P (f x)) F1)" |
|
824 |
unfolding filterlim_def le_filter_def eventually_filtermap .. |
|
825 |
||
826 |
lemma filterlim_compose: |
|
827 |
"filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1" |
|
828 |
unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans) |
|
829 |
||
830 |
lemma filterlim_mono: |
|
831 |
"filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'" |
|
832 |
unfolding filterlim_def by (metis filtermap_mono order_trans) |
|
833 |
||
834 |
lemma filterlim_ident: "LIM x F. x :> F" |
|
835 |
by (simp add: filterlim_def filtermap_ident) |
|
836 |
||
837 |
lemma filterlim_cong: |
|
838 |
"F1 = F1' \<Longrightarrow> F2 = F2' \<Longrightarrow> eventually (\<lambda>x. f x = g x) F2 \<Longrightarrow> filterlim f F1 F2 = filterlim g F1' F2'" |
|
839 |
by (auto simp: filterlim_def le_filter_def eventually_filtermap elim: eventually_elim2) |
|
840 |
||
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
841 |
lemma filterlim_principal: |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
842 |
"(LIM x F. f x :> principal S) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> S) F)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
843 |
unfolding filterlim_def eventually_filtermap le_principal .. |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
844 |
|
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
845 |
lemma filterlim_inf: |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
846 |
"(LIM x F1. f x :> inf F2 F3) \<longleftrightarrow> ((LIM x F1. f x :> F2) \<and> (LIM x F1. f x :> F3))" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
847 |
unfolding filterlim_def by simp |
51471 | 848 |
|
849 |
lemma filterlim_filtermap: "filterlim f F1 (filtermap g F2) = filterlim (\<lambda>x. f (g x)) F1 F2" |
|
850 |
unfolding filterlim_def filtermap_filtermap .. |
|
851 |
||
852 |
lemma filterlim_sup: |
|
853 |
"filterlim f F F1 \<Longrightarrow> filterlim f F F2 \<Longrightarrow> filterlim f F (sup F1 F2)" |
|
854 |
unfolding filterlim_def filtermap_sup by auto |
|
855 |
||
856 |
lemma filterlim_Suc: "filterlim Suc sequentially sequentially" |
|
857 |
by (simp add: filterlim_iff eventually_sequentially) (metis le_Suc_eq) |
|
858 |
||
859 |
subsubsection {* Tendsto *} |
|
860 |
||
861 |
abbreviation (in topological_space) |
|
862 |
tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where |
|
863 |
"(f ---> l) F \<equiv> filterlim f (nhds l) F" |
|
864 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
865 |
definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
866 |
"Lim A f = (THE l. (f ---> l) A)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
867 |
|
51471 | 868 |
lemma tendsto_eq_rhs: "(f ---> x) F \<Longrightarrow> x = y \<Longrightarrow> (f ---> y) F" |
869 |
by simp |
|
870 |
||
871 |
ML {* |
|
872 |
||
873 |
structure Tendsto_Intros = Named_Thms |
|
874 |
( |
|
875 |
val name = @{binding tendsto_intros} |
|
876 |
val description = "introduction rules for tendsto" |
|
877 |
) |
|
878 |
||
879 |
*} |
|
880 |
||
881 |
setup {* |
|
882 |
Tendsto_Intros.setup #> |
|
883 |
Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros}, |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
884 |
map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])) o Tendsto_Intros.get o Context.proof_of); |
51471 | 885 |
*} |
886 |
||
51473 | 887 |
lemma (in topological_space) tendsto_def: |
888 |
"(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)" |
|
51471 | 889 |
unfolding filterlim_def |
890 |
proof safe |
|
891 |
fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l" |
|
892 |
then show "eventually (\<lambda>x. f x \<in> S) F" |
|
893 |
unfolding eventually_nhds eventually_filtermap le_filter_def |
|
894 |
by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp) |
|
895 |
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def) |
|
896 |
||
897 |
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F" |
|
898 |
unfolding tendsto_def le_filter_def by fast |
|
899 |
||
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
900 |
lemma tendsto_within_subset: "(f ---> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f ---> l) (at x within T)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
901 |
by (blast intro: tendsto_mono at_le) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
902 |
|
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
903 |
lemma filterlim_at: |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
904 |
"(LIM x F. f x :> at b within s) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f ---> b) F)" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
905 |
by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
906 |
|
51473 | 907 |
lemma (in topological_space) topological_tendstoI: |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
908 |
"(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f ---> l) F" |
51471 | 909 |
unfolding tendsto_def by auto |
910 |
||
51473 | 911 |
lemma (in topological_space) topological_tendstoD: |
51471 | 912 |
"(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F" |
913 |
unfolding tendsto_def by auto |
|
914 |
||
915 |
lemma order_tendstoI: |
|
916 |
fixes y :: "_ :: order_topology" |
|
917 |
assumes "\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F" |
|
918 |
assumes "\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F" |
|
919 |
shows "(f ---> y) F" |
|
920 |
proof (rule topological_tendstoI) |
|
921 |
fix S assume "open S" "y \<in> S" |
|
922 |
then show "eventually (\<lambda>x. f x \<in> S) F" |
|
923 |
unfolding open_generated_order |
|
924 |
proof induct |
|
925 |
case (UN K) |
|
926 |
then obtain k where "y \<in> k" "k \<in> K" by auto |
|
927 |
with UN(2)[of k] show ?case |
|
928 |
by (auto elim: eventually_elim1) |
|
929 |
qed (insert assms, auto elim: eventually_elim2) |
|
930 |
qed |
|
931 |
||
932 |
lemma order_tendstoD: |
|
933 |
fixes y :: "_ :: order_topology" |
|
934 |
assumes y: "(f ---> y) F" |
|
935 |
shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F" |
|
936 |
and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F" |
|
937 |
using topological_tendstoD[OF y, of "{..< a}"] topological_tendstoD[OF y, of "{a <..}"] by auto |
|
938 |
||
939 |
lemma order_tendsto_iff: |
|
940 |
fixes f :: "_ \<Rightarrow> 'a :: order_topology" |
|
941 |
shows "(f ---> x) F \<longleftrightarrow>(\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)" |
|
942 |
by (metis order_tendstoI order_tendstoD) |
|
943 |
||
944 |
lemma tendsto_bot [simp]: "(f ---> a) bot" |
|
945 |
unfolding tendsto_def by simp |
|
946 |
||
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
947 |
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a within s)" |
51471 | 948 |
unfolding tendsto_def eventually_at_topological by auto |
949 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
950 |
lemma (in topological_space) tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F" |
51471 | 951 |
by (simp add: tendsto_def) |
952 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
953 |
lemma (in t2_space) tendsto_unique: |
51471 | 954 |
assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F" |
955 |
shows "a = b" |
|
956 |
proof (rule ccontr) |
|
957 |
assume "a \<noteq> b" |
|
958 |
obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}" |
|
959 |
using hausdorff [OF `a \<noteq> b`] by fast |
|
960 |
have "eventually (\<lambda>x. f x \<in> U) F" |
|
961 |
using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD) |
|
962 |
moreover |
|
963 |
have "eventually (\<lambda>x. f x \<in> V) F" |
|
964 |
using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD) |
|
965 |
ultimately |
|
966 |
have "eventually (\<lambda>x. False) F" |
|
967 |
proof eventually_elim |
|
968 |
case (elim x) |
|
969 |
hence "f x \<in> U \<inter> V" by simp |
|
970 |
with `U \<inter> V = {}` show ?case by simp |
|
971 |
qed |
|
972 |
with `\<not> trivial_limit F` show "False" |
|
973 |
by (simp add: trivial_limit_def) |
|
974 |
qed |
|
975 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
976 |
lemma (in t2_space) tendsto_const_iff: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
977 |
assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) ---> b) F \<longleftrightarrow> a = b" |
51471 | 978 |
by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const]) |
979 |
||
980 |
lemma increasing_tendsto: |
|
981 |
fixes f :: "_ \<Rightarrow> 'a::order_topology" |
|
982 |
assumes bdd: "eventually (\<lambda>n. f n \<le> l) F" |
|
983 |
and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F" |
|
984 |
shows "(f ---> l) F" |
|
985 |
using assms by (intro order_tendstoI) (auto elim!: eventually_elim1) |
|
986 |
||
987 |
lemma decreasing_tendsto: |
|
988 |
fixes f :: "_ \<Rightarrow> 'a::order_topology" |
|
989 |
assumes bdd: "eventually (\<lambda>n. l \<le> f n) F" |
|
990 |
and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F" |
|
991 |
shows "(f ---> l) F" |
|
992 |
using assms by (intro order_tendstoI) (auto elim!: eventually_elim1) |
|
993 |
||
994 |
lemma tendsto_sandwich: |
|
995 |
fixes f g h :: "'a \<Rightarrow> 'b::order_topology" |
|
996 |
assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net" |
|
997 |
assumes lim: "(f ---> c) net" "(h ---> c) net" |
|
998 |
shows "(g ---> c) net" |
|
999 |
proof (rule order_tendstoI) |
|
1000 |
fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net" |
|
1001 |
using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2) |
|
1002 |
next |
|
1003 |
fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net" |
|
1004 |
using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2) |
|
1005 |
qed |
|
1006 |
||
1007 |
lemma tendsto_le: |
|
1008 |
fixes f g :: "'a \<Rightarrow> 'b::linorder_topology" |
|
1009 |
assumes F: "\<not> trivial_limit F" |
|
1010 |
assumes x: "(f ---> x) F" and y: "(g ---> y) F" |
|
1011 |
assumes ev: "eventually (\<lambda>x. g x \<le> f x) F" |
|
1012 |
shows "y \<le> x" |
|
1013 |
proof (rule ccontr) |
|
1014 |
assume "\<not> y \<le> x" |
|
1015 |
with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}" |
|
1016 |
by (auto simp: not_le) |
|
1017 |
then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F" |
|
1018 |
using x y by (auto intro: order_tendstoD) |
|
1019 |
with ev have "eventually (\<lambda>x. False) F" |
|
1020 |
by eventually_elim (insert xy, fastforce) |
|
1021 |
with F show False |
|
1022 |
by (simp add: eventually_False) |
|
1023 |
qed |
|
1024 |
||
1025 |
lemma tendsto_le_const: |
|
1026 |
fixes f :: "'a \<Rightarrow> 'b::linorder_topology" |
|
1027 |
assumes F: "\<not> trivial_limit F" |
|
1028 |
assumes x: "(f ---> x) F" and a: "eventually (\<lambda>x. a \<le> f x) F" |
|
1029 |
shows "a \<le> x" |
|
1030 |
using F x tendsto_const a by (rule tendsto_le) |
|
1031 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1032 |
subsubsection {* Rules about @{const Lim} *} |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1033 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1034 |
lemma (in t2_space) tendsto_Lim: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1035 |
"\<not>(trivial_limit net) \<Longrightarrow> (f ---> l) net \<Longrightarrow> Lim net f = l" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1036 |
unfolding Lim_def using tendsto_unique[of net f] by auto |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1037 |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1038 |
lemma Lim_ident_at: "\<not> trivial_limit (at x within s) \<Longrightarrow> Lim (at x within s) (\<lambda>x. x) = x" |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1039 |
by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1040 |
|
51471 | 1041 |
subsection {* Limits to @{const at_top} and @{const at_bot} *} |
1042 |
||
1043 |
lemma filterlim_at_top: |
|
1044 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" |
|
1045 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z \<le> f x) F)" |
|
1046 |
by (auto simp: filterlim_iff eventually_at_top_linorder elim!: eventually_elim1) |
|
1047 |
||
1048 |
lemma filterlim_at_top_dense: |
|
1049 |
fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" |
|
1050 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)" |
|
1051 |
by (metis eventually_elim1[of _ F] eventually_gt_at_top order_less_imp_le |
|
1052 |
filterlim_at_top[of f F] filterlim_iff[of f at_top F]) |
|
1053 |
||
1054 |
lemma filterlim_at_top_ge: |
|
1055 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" |
|
1056 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F)" |
|
1057 |
unfolding filterlim_at_top |
|
1058 |
proof safe |
|
1059 |
fix Z assume *: "\<forall>Z\<ge>c. eventually (\<lambda>x. Z \<le> f x) F" |
|
1060 |
with *[THEN spec, of "max Z c"] show "eventually (\<lambda>x. Z \<le> f x) F" |
|
1061 |
by (auto elim!: eventually_elim1) |
|
1062 |
qed simp |
|
1063 |
||
1064 |
lemma filterlim_at_top_at_top: |
|
1065 |
fixes f :: "'a::linorder \<Rightarrow> 'b::linorder" |
|
1066 |
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
1067 |
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" |
|
1068 |
assumes Q: "eventually Q at_top" |
|
1069 |
assumes P: "eventually P at_top" |
|
1070 |
shows "filterlim f at_top at_top" |
|
1071 |
proof - |
|
1072 |
from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" |
|
1073 |
unfolding eventually_at_top_linorder by auto |
|
1074 |
show ?thesis |
|
1075 |
proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) |
|
1076 |
fix z assume "x \<le> z" |
|
1077 |
with x have "P z" by auto |
|
1078 |
have "eventually (\<lambda>x. g z \<le> x) at_top" |
|
1079 |
by (rule eventually_ge_at_top) |
|
1080 |
with Q show "eventually (\<lambda>x. z \<le> f x) at_top" |
|
1081 |
by eventually_elim (metis mono bij `P z`) |
|
1082 |
qed |
|
1083 |
qed |
|
1084 |
||
1085 |
lemma filterlim_at_top_gt: |
|
1086 |
fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b" |
|
1087 |
shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z \<le> f x) F)" |
|
1088 |
by (metis filterlim_at_top order_less_le_trans gt_ex filterlim_at_top_ge) |
|
1089 |
||
1090 |
lemma filterlim_at_bot: |
|
1091 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" |
|
1092 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x \<le> Z) F)" |
|
1093 |
by (auto simp: filterlim_iff eventually_at_bot_linorder elim!: eventually_elim1) |
|
1094 |
||
1095 |
lemma filterlim_at_bot_le: |
|
1096 |
fixes f :: "'a \<Rightarrow> ('b::linorder)" and c :: "'b" |
|
1097 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F)" |
|
1098 |
unfolding filterlim_at_bot |
|
1099 |
proof safe |
|
1100 |
fix Z assume *: "\<forall>Z\<le>c. eventually (\<lambda>x. Z \<ge> f x) F" |
|
1101 |
with *[THEN spec, of "min Z c"] show "eventually (\<lambda>x. Z \<ge> f x) F" |
|
1102 |
by (auto elim!: eventually_elim1) |
|
1103 |
qed simp |
|
1104 |
||
1105 |
lemma filterlim_at_bot_lt: |
|
1106 |
fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b" |
|
1107 |
shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z \<ge> f x) F)" |
|
1108 |
by (metis filterlim_at_bot filterlim_at_bot_le lt_ex order_le_less_trans) |
|
1109 |
||
1110 |
lemma filterlim_at_bot_at_right: |
|
1111 |
fixes f :: "'a::{no_top, linorder_topology} \<Rightarrow> 'b::linorder" |
|
1112 |
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
1113 |
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" |
|
1114 |
assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b" |
|
1115 |
assumes P: "eventually P at_bot" |
|
1116 |
shows "filterlim f at_bot (at_right a)" |
|
1117 |
proof - |
|
1118 |
from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y" |
|
1119 |
unfolding eventually_at_bot_linorder by auto |
|
1120 |
show ?thesis |
|
1121 |
proof (intro filterlim_at_bot_le[THEN iffD2] allI impI) |
|
1122 |
fix z assume "z \<le> x" |
|
1123 |
with x have "P z" by auto |
|
1124 |
have "eventually (\<lambda>x. x \<le> g z) (at_right a)" |
|
1125 |
using bound[OF bij(2)[OF `P z`]] |
|
1126 |
unfolding eventually_at_right by (auto intro!: exI[of _ "g z"]) |
|
1127 |
with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)" |
|
1128 |
by eventually_elim (metis bij `P z` mono) |
|
1129 |
qed |
|
1130 |
qed |
|
1131 |
||
1132 |
lemma filterlim_at_top_at_left: |
|
1133 |
fixes f :: "'a::{no_bot, linorder_topology} \<Rightarrow> 'b::linorder" |
|
1134 |
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
1135 |
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" |
|
1136 |
assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a" |
|
1137 |
assumes P: "eventually P at_top" |
|
1138 |
shows "filterlim f at_top (at_left a)" |
|
1139 |
proof - |
|
1140 |
from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" |
|
1141 |
unfolding eventually_at_top_linorder by auto |
|
1142 |
show ?thesis |
|
1143 |
proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) |
|
1144 |
fix z assume "x \<le> z" |
|
1145 |
with x have "P z" by auto |
|
1146 |
have "eventually (\<lambda>x. g z \<le> x) (at_left a)" |
|
1147 |
using bound[OF bij(2)[OF `P z`]] |
|
1148 |
unfolding eventually_at_left by (auto intro!: exI[of _ "g z"]) |
|
1149 |
with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)" |
|
1150 |
by eventually_elim (metis bij `P z` mono) |
|
1151 |
qed |
|
1152 |
qed |
|
1153 |
||
1154 |
lemma filterlim_split_at: |
|
1155 |
"filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))" |
|
1156 |
by (subst at_eq_sup_left_right) (rule filterlim_sup) |
|
1157 |
||
1158 |
lemma filterlim_at_split: |
|
1159 |
"filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)" |
|
1160 |
by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup) |
|
1161 |
||
1162 |
||
1163 |
subsection {* Limits on sequences *} |
|
1164 |
||
1165 |
abbreviation (in topological_space) |
|
1166 |
LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool" |
|
1167 |
("((_)/ ----> (_))" [60, 60] 60) where |
|
1168 |
"X ----> L \<equiv> (X ---> L) sequentially" |
|
1169 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1170 |
abbreviation (in t2_space) lim :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" where |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1171 |
"lim X \<equiv> Lim sequentially X" |
51471 | 1172 |
|
1173 |
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where |
|
1174 |
"convergent X = (\<exists>L. X ----> L)" |
|
1175 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1176 |
lemma lim_def: "lim X = (THE L. X ----> L)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1177 |
unfolding Lim_def .. |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1178 |
|
51471 | 1179 |
subsubsection {* Monotone sequences and subsequences *} |
1180 |
||
1181 |
definition |
|
1182 |
monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where |
|
1183 |
--{*Definition of monotonicity. |
|
1184 |
The use of disjunction here complicates proofs considerably. |
|
1185 |
One alternative is to add a Boolean argument to indicate the direction. |
|
1186 |
Another is to develop the notions of increasing and decreasing first.*} |
|
1187 |
"monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))" |
|
1188 |
||
1189 |
definition |
|
1190 |
incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where |
|
1191 |
--{*Increasing sequence*} |
|
1192 |
"incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)" |
|
1193 |
||
1194 |
definition |
|
1195 |
decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where |
|
1196 |
--{*Decreasing sequence*} |
|
1197 |
"decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)" |
|
1198 |
||
1199 |
definition |
|
1200 |
subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where |
|
1201 |
--{*Definition of subsequence*} |
|
1202 |
"subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)" |
|
1203 |
||
1204 |
lemma incseq_mono: "mono f \<longleftrightarrow> incseq f" |
|
1205 |
unfolding mono_def incseq_def by auto |
|
1206 |
||
1207 |
lemma incseq_SucI: |
|
1208 |
"(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X" |
|
1209 |
using lift_Suc_mono_le[of X] |
|
1210 |
by (auto simp: incseq_def) |
|
1211 |
||
1212 |
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j" |
|
1213 |
by (auto simp: incseq_def) |
|
1214 |
||
1215 |
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)" |
|
1216 |
using incseqD[of A i "Suc i"] by auto |
|
1217 |
||
1218 |
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))" |
|
1219 |
by (auto intro: incseq_SucI dest: incseq_SucD) |
|
1220 |
||
1221 |
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)" |
|
1222 |
unfolding incseq_def by auto |
|
1223 |
||
1224 |
lemma decseq_SucI: |
|
1225 |
"(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X" |
|
1226 |
using order.lift_Suc_mono_le[OF dual_order, of X] |
|
1227 |
by (auto simp: decseq_def) |
|
1228 |
||
1229 |
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i" |
|
1230 |
by (auto simp: decseq_def) |
|
1231 |
||
1232 |
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i" |
|
1233 |
using decseqD[of A i "Suc i"] by auto |
|
1234 |
||
1235 |
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)" |
|
1236 |
by (auto intro: decseq_SucI dest: decseq_SucD) |
|
1237 |
||
1238 |
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)" |
|
1239 |
unfolding decseq_def by auto |
|
1240 |
||
1241 |
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X" |
|
1242 |
unfolding monoseq_def incseq_def decseq_def .. |
|
1243 |
||
1244 |
lemma monoseq_Suc: |
|
1245 |
"monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)" |
|
1246 |
unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff .. |
|
1247 |
||
1248 |
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X" |
|
1249 |
by (simp add: monoseq_def) |
|
1250 |
||
1251 |
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X" |
|
1252 |
by (simp add: monoseq_def) |
|
1253 |
||
1254 |
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X" |
|
1255 |
by (simp add: monoseq_Suc) |
|
1256 |
||
1257 |
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X" |
|
1258 |
by (simp add: monoseq_Suc) |
|
1259 |
||
1260 |
lemma monoseq_minus: |
|
1261 |
fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add" |
|
1262 |
assumes "monoseq a" |
|
1263 |
shows "monoseq (\<lambda> n. - a n)" |
|
1264 |
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n") |
|
1265 |
case True |
|
1266 |
hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto |
|
1267 |
thus ?thesis by (rule monoI2) |
|
1268 |
next |
|
1269 |
case False |
|
1270 |
hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto |
|
1271 |
thus ?thesis by (rule monoI1) |
|
1272 |
qed |
|
1273 |
||
1274 |
text{*Subsequence (alternative definition, (e.g. Hoskins)*} |
|
1275 |
||
1276 |
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))" |
|
1277 |
apply (simp add: subseq_def) |
|
1278 |
apply (auto dest!: less_imp_Suc_add) |
|
1279 |
apply (induct_tac k) |
|
1280 |
apply (auto intro: less_trans) |
|
1281 |
done |
|
1282 |
||
1283 |
text{* for any sequence, there is a monotonic subsequence *} |
|
1284 |
lemma seq_monosub: |
|
1285 |
fixes s :: "nat => 'a::linorder" |
|
1286 |
shows "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))" |
|
1287 |
proof cases |
|
1288 |
let "?P p n" = "p > n \<and> (\<forall>m\<ge>p. s m \<le> s p)" |
|
1289 |
assume *: "\<forall>n. \<exists>p. ?P p n" |
|
1290 |
def f \<equiv> "nat_rec (SOME p. ?P p 0) (\<lambda>_ n. SOME p. ?P p n)" |
|
1291 |
have f_0: "f 0 = (SOME p. ?P p 0)" unfolding f_def by simp |
|
1292 |
have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc .. |
|
1293 |
have P_0: "?P (f 0) 0" unfolding f_0 using *[rule_format] by (rule someI2_ex) auto |
|
1294 |
have P_Suc: "\<And>i. ?P (f (Suc i)) (f i)" unfolding f_Suc using *[rule_format] by (rule someI2_ex) auto |
|
1295 |
then have "subseq f" unfolding subseq_Suc_iff by auto |
|
1296 |
moreover have "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc |
|
1297 |
proof (intro disjI2 allI) |
|
1298 |
fix n show "s (f (Suc n)) \<le> s (f n)" |
|
1299 |
proof (cases n) |
|
1300 |
case 0 with P_Suc[of 0] P_0 show ?thesis by auto |
|
1301 |
next |
|
1302 |
case (Suc m) |
|
1303 |
from P_Suc[of n] Suc have "f (Suc m) \<le> f (Suc (Suc m))" by simp |
|
1304 |
with P_Suc Suc show ?thesis by simp |
|
1305 |
qed |
|
1306 |
qed |
|
1307 |
ultimately show ?thesis by auto |
|
1308 |
next |
|
1309 |
let "?P p m" = "m < p \<and> s m < s p" |
|
1310 |
assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))" |
|
1311 |
then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less) |
|
1312 |
def f \<equiv> "nat_rec (SOME p. ?P p (Suc N)) (\<lambda>_ n. SOME p. ?P p n)" |
|
1313 |
have f_0: "f 0 = (SOME p. ?P p (Suc N))" unfolding f_def by simp |
|
1314 |
have f_Suc: "\<And>i. f (Suc i) = (SOME p. ?P p (f i))" unfolding f_def nat_rec_Suc .. |
|
1315 |
have P_0: "?P (f 0) (Suc N)" |
|
1316 |
unfolding f_0 some_eq_ex[of "\<lambda>p. ?P p (Suc N)"] using N[of "Suc N"] by auto |
|
1317 |
{ fix i have "N < f i \<Longrightarrow> ?P (f (Suc i)) (f i)" |
|
1318 |
unfolding f_Suc some_eq_ex[of "\<lambda>p. ?P p (f i)"] using N[of "f i"] . } |
|
1319 |
note P' = this |
|
1320 |
{ fix i have "N < f i \<and> ?P (f (Suc i)) (f i)" |
|
1321 |
by (induct i) (insert P_0 P', auto) } |
|
1322 |
then have "subseq f" "monoseq (\<lambda>x. s (f x))" |
|
1323 |
unfolding subseq_Suc_iff monoseq_Suc by (auto simp: not_le intro: less_imp_le) |
|
1324 |
then show ?thesis by auto |
|
1325 |
qed |
|
1326 |
||
1327 |
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n" |
|
1328 |
proof(induct n) |
|
1329 |
case 0 thus ?case by simp |
|
1330 |
next |
|
1331 |
case (Suc n) |
|
1332 |
from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps |
|
1333 |
have "n < f (Suc n)" by arith |
|
1334 |
thus ?case by arith |
|
1335 |
qed |
|
1336 |
||
1337 |
lemma eventually_subseq: |
|
1338 |
"subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially" |
|
1339 |
unfolding eventually_sequentially by (metis seq_suble le_trans) |
|
1340 |
||
51473 | 1341 |
lemma not_eventually_sequentiallyD: |
1342 |
assumes P: "\<not> eventually P sequentially" |
|
1343 |
shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))" |
|
1344 |
proof - |
|
1345 |
from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m" |
|
1346 |
unfolding eventually_sequentially by (simp add: not_less) |
|
1347 |
then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)" |
|
1348 |
by (auto simp: choice_iff) |
|
1349 |
then show ?thesis |
|
1350 |
by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"] |
|
1351 |
simp: less_eq_Suc_le subseq_Suc_iff) |
|
1352 |
qed |
|
1353 |
||
51471 | 1354 |
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially" |
1355 |
unfolding filterlim_iff by (metis eventually_subseq) |
|
1356 |
||
1357 |
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)" |
|
1358 |
unfolding subseq_def by simp |
|
1359 |
||
1360 |
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n" |
|
1361 |
using assms by (auto simp: subseq_def) |
|
1362 |
||
1363 |
lemma incseq_imp_monoseq: "incseq X \<Longrightarrow> monoseq X" |
|
1364 |
by (simp add: incseq_def monoseq_def) |
|
1365 |
||
1366 |
lemma decseq_imp_monoseq: "decseq X \<Longrightarrow> monoseq X" |
|
1367 |
by (simp add: decseq_def monoseq_def) |
|
1368 |
||
1369 |
lemma decseq_eq_incseq: |
|
1370 |
fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" |
|
1371 |
by (simp add: decseq_def incseq_def) |
|
1372 |
||
1373 |
lemma INT_decseq_offset: |
|
1374 |
assumes "decseq F" |
|
1375 |
shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)" |
|
1376 |
proof safe |
|
1377 |
fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)" |
|
1378 |
show "x \<in> F i" |
|
1379 |
proof cases |
|
1380 |
from x have "x \<in> F n" by auto |
|
1381 |
also assume "i \<le> n" with `decseq F` have "F n \<subseteq> F i" |
|
1382 |
unfolding decseq_def by simp |
|
1383 |
finally show ?thesis . |
|
1384 |
qed (insert x, simp) |
|
1385 |
qed auto |
|
1386 |
||
1387 |
lemma LIMSEQ_const_iff: |
|
1388 |
fixes k l :: "'a::t2_space" |
|
1389 |
shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l" |
|
1390 |
using trivial_limit_sequentially by (rule tendsto_const_iff) |
|
1391 |
||
1392 |
lemma LIMSEQ_SUP: |
|
1393 |
"incseq X \<Longrightarrow> X ----> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})" |
|
1394 |
by (intro increasing_tendsto) |
|
1395 |
(auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans) |
|
1396 |
||
1397 |
lemma LIMSEQ_INF: |
|
1398 |
"decseq X \<Longrightarrow> X ----> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})" |
|
1399 |
by (intro decreasing_tendsto) |
|
1400 |
(auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans) |
|
1401 |
||
1402 |
lemma LIMSEQ_ignore_initial_segment: |
|
1403 |
"f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a" |
|
51474
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
1404 |
unfolding tendsto_def |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
1405 |
by (subst eventually_sequentially_seg[where k=k]) |
51471 | 1406 |
|
1407 |
lemma LIMSEQ_offset: |
|
1408 |
"(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a" |
|
51474
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
1409 |
unfolding tendsto_def |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
1410 |
by (subst (asm) eventually_sequentially_seg[where k=k]) |
51471 | 1411 |
|
1412 |
lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l" |
|
1413 |
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp) |
|
1414 |
||
1415 |
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l" |
|
1416 |
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp) |
|
1417 |
||
1418 |
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l" |
|
1419 |
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc) |
|
1420 |
||
1421 |
lemma LIMSEQ_unique: |
|
1422 |
fixes a b :: "'a::t2_space" |
|
1423 |
shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b" |
|
1424 |
using trivial_limit_sequentially by (rule tendsto_unique) |
|
1425 |
||
1426 |
lemma LIMSEQ_le_const: |
|
1427 |
"\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x" |
|
1428 |
using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially) |
|
1429 |
||
1430 |
lemma LIMSEQ_le: |
|
1431 |
"\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)" |
|
1432 |
using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially) |
|
1433 |
||
1434 |
lemma LIMSEQ_le_const2: |
|
1435 |
"\<lbrakk>X ----> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a" |
|
1436 |
by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) (auto simp: tendsto_const) |
|
1437 |
||
1438 |
lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)" |
|
1439 |
by (simp add: convergent_def) |
|
1440 |
||
1441 |
lemma convergentI: "(X ----> L) ==> convergent X" |
|
1442 |
by (auto simp add: convergent_def) |
|
1443 |
||
1444 |
lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)" |
|
1445 |
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def) |
|
1446 |
||
1447 |
lemma convergent_const: "convergent (\<lambda>n. c)" |
|
1448 |
by (rule convergentI, rule tendsto_const) |
|
1449 |
||
1450 |
lemma monoseq_le: |
|
1451 |
"monoseq a \<Longrightarrow> a ----> (x::'a::linorder_topology) \<Longrightarrow> |
|
1452 |
((\<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))" |
|
1453 |
by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff) |
|
1454 |
||
1455 |
lemma LIMSEQ_subseq_LIMSEQ: |
|
1456 |
"\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L" |
|
1457 |
unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq]) |
|
1458 |
||
1459 |
lemma convergent_subseq_convergent: |
|
1460 |
"\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)" |
|
1461 |
unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ) |
|
1462 |
||
1463 |
lemma limI: "X ----> L ==> lim X = L" |
|
1464 |
apply (simp add: lim_def) |
|
1465 |
apply (blast intro: LIMSEQ_unique) |
|
1466 |
done |
|
1467 |
||
1468 |
lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x" |
|
1469 |
using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff) |
|
1470 |
||
1471 |
subsubsection{*Increasing and Decreasing Series*} |
|
1472 |
||
1473 |
lemma incseq_le: "incseq X \<Longrightarrow> X ----> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)" |
|
1474 |
by (metis incseq_def LIMSEQ_le_const) |
|
1475 |
||
1476 |
lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n" |
|
1477 |
by (metis decseq_def LIMSEQ_le_const2) |
|
1478 |
||
51473 | 1479 |
subsection {* First countable topologies *} |
1480 |
||
1481 |
class first_countable_topology = topological_space + |
|
1482 |
assumes first_countable_basis: |
|
1483 |
"\<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))" |
|
1484 |
||
1485 |
lemma (in first_countable_topology) countable_basis_at_decseq: |
|
1486 |
obtains A :: "nat \<Rightarrow> 'a set" where |
|
1487 |
"\<And>i. open (A i)" "\<And>i. x \<in> (A i)" |
|
1488 |
"\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially" |
|
1489 |
proof atomize_elim |
|
1490 |
from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where |
|
1491 |
nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i" |
|
1492 |
and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S" by auto |
|
1493 |
def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i" |
|
1494 |
show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> |
|
1495 |
(\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)" |
|
1496 |
proof (safe intro!: exI[of _ F]) |
|
1497 |
fix i |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1498 |
show "open (F i)" using nhds(1) by (auto simp: F_def) |
51473 | 1499 |
show "x \<in> F i" using nhds(2) by (auto simp: F_def) |
1500 |
next |
|
1501 |
fix S assume "open S" "x \<in> S" |
|
1502 |
from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto |
|
1503 |
moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i" |
|
1504 |
by (auto simp: F_def) |
|
1505 |
ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially" |
|
1506 |
by (auto simp: eventually_sequentially) |
|
1507 |
qed |
|
1508 |
qed |
|
1509 |
||
1510 |
lemma (in first_countable_topology) countable_basis: |
|
1511 |
obtains A :: "nat \<Rightarrow> 'a set" where |
|
1512 |
"\<And>i. open (A i)" "\<And>i. x \<in> A i" |
|
1513 |
"\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x" |
|
1514 |
proof atomize_elim |
|
1515 |
from countable_basis_at_decseq[of x] guess A . note A = this |
|
1516 |
{ fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S" |
|
1517 |
with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially" |
|
1518 |
by (auto elim: eventually_elim1 simp: subset_eq) } |
|
1519 |
with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F ----> x)" |
|
1520 |
by (intro exI[of _ A]) (auto simp: tendsto_def) |
|
1521 |
qed |
|
1522 |
||
1523 |
lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within: |
|
1524 |
assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1525 |
shows "eventually P (inf (nhds a) (principal s))" |
51473 | 1526 |
proof (rule ccontr) |
1527 |
from countable_basis[of a] guess A . note A = this |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1528 |
assume "\<not> eventually P (inf (nhds a) (principal s))" |
51473 | 1529 |
with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1530 |
unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce |
51473 | 1531 |
then guess F .. |
1532 |
hence F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)" |
|
1533 |
by fast+ |
|
1534 |
with A have "F ----> a" by auto |
|
1535 |
hence "eventually (\<lambda>n. P (F n)) sequentially" |
|
1536 |
using assms F0 by simp |
|
1537 |
thus "False" by (simp add: F3) |
|
1538 |
qed |
|
1539 |
||
1540 |
lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially: |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1541 |
"eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> |
51473 | 1542 |
(\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)" |
1543 |
proof (safe intro!: sequentially_imp_eventually_nhds_within) |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1544 |
assume "eventually P (inf (nhds a) (principal s))" |
51473 | 1545 |
then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1546 |
by (auto simp: eventually_inf_principal eventually_nhds) |
51473 | 1547 |
moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a" |
1548 |
ultimately show "eventually (\<lambda>n. P (f n)) sequentially" |
|
1549 |
by (auto dest!: topological_tendstoD elim: eventually_elim1) |
|
1550 |
qed |
|
1551 |
||
1552 |
lemma (in first_countable_topology) eventually_nhds_iff_sequentially: |
|
1553 |
"eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)" |
|
1554 |
using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp |
|
1555 |
||
51471 | 1556 |
subsection {* Function limit at a point *} |
1557 |
||
1558 |
abbreviation |
|
1559 |
LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" |
|
1560 |
("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where |
|
1561 |
"f -- a --> L \<equiv> (f ---> L) (at a)" |
|
1562 |
||
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1563 |
lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l) (at a within S) \<longleftrightarrow> (f -- a --> l)" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1564 |
unfolding tendsto_def by (simp add: at_within_open[where S=S]) |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1565 |
|
51471 | 1566 |
lemma LIM_const_not_eq[tendsto_intros]: |
1567 |
fixes a :: "'a::perfect_space" |
|
1568 |
fixes k L :: "'b::t2_space" |
|
1569 |
shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) -- a --> L" |
|
1570 |
by (simp add: tendsto_const_iff) |
|
1571 |
||
1572 |
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0] |
|
1573 |
||
1574 |
lemma LIM_const_eq: |
|
1575 |
fixes a :: "'a::perfect_space" |
|
1576 |
fixes k L :: "'b::t2_space" |
|
1577 |
shows "(\<lambda>x. k) -- a --> L \<Longrightarrow> k = L" |
|
1578 |
by (simp add: tendsto_const_iff) |
|
1579 |
||
1580 |
lemma LIM_unique: |
|
1581 |
fixes a :: "'a::perfect_space" and L M :: "'b::t2_space" |
|
1582 |
shows "f -- a --> L \<Longrightarrow> f -- a --> M \<Longrightarrow> L = M" |
|
1583 |
using at_neq_bot by (rule tendsto_unique) |
|
1584 |
||
1585 |
text {* Limits are equal for functions equal except at limit point *} |
|
1586 |
||
1587 |
lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- a --> l)" |
|
1588 |
unfolding tendsto_def eventually_at_topological by simp |
|
1589 |
||
1590 |
lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f -- a --> l) \<longleftrightarrow> (g -- b --> m)" |
|
1591 |
by (simp add: LIM_equal) |
|
1592 |
||
1593 |
lemma LIM_cong_limit: "f -- x --> L \<Longrightarrow> K = L \<Longrightarrow> f -- x --> K" |
|
1594 |
by simp |
|
1595 |
||
1596 |
lemma tendsto_at_iff_tendsto_nhds: |
|
1597 |
"g -- l --> g l \<longleftrightarrow> (g ---> g l) (nhds l)" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1598 |
unfolding tendsto_def eventually_at_filter |
51471 | 1599 |
by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1) |
1600 |
||
1601 |
lemma tendsto_compose: |
|
1602 |
"g -- l --> g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F" |
|
1603 |
unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g]) |
|
1604 |
||
1605 |
lemma LIM_o: "\<lbrakk>g -- l --> g l; f -- a --> l\<rbrakk> \<Longrightarrow> (g \<circ> f) -- a --> g l" |
|
1606 |
unfolding o_def by (rule tendsto_compose) |
|
1607 |
||
1608 |
lemma tendsto_compose_eventually: |
|
1609 |
"g -- l --> m \<Longrightarrow> (f ---> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> m) F" |
|
1610 |
by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at) |
|
1611 |
||
1612 |
lemma LIM_compose_eventually: |
|
1613 |
assumes f: "f -- a --> b" |
|
1614 |
assumes g: "g -- b --> c" |
|
1615 |
assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)" |
|
1616 |
shows "(\<lambda>x. g (f x)) -- a --> c" |
|
1617 |
using g f inj by (rule tendsto_compose_eventually) |
|
1618 |
||
51473 | 1619 |
subsubsection {* Relation of LIM and LIMSEQ *} |
1620 |
||
1621 |
lemma (in first_countable_topology) sequentially_imp_eventually_within: |
|
1622 |
"(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> |
|
1623 |
eventually P (at a within s)" |
|
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1624 |
unfolding at_within_def |
51473 | 1625 |
by (intro sequentially_imp_eventually_nhds_within) auto |
1626 |
||
1627 |
lemma (in first_countable_topology) sequentially_imp_eventually_at: |
|
1628 |
"(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)" |
|
1629 |
using assms sequentially_imp_eventually_within [where s=UNIV] by simp |
|
1630 |
||
1631 |
lemma LIMSEQ_SEQ_conv1: |
|
1632 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" |
|
1633 |
assumes f: "f -- a --> l" |
|
1634 |
shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l" |
|
1635 |
using tendsto_compose_eventually [OF f, where F=sequentially] by simp |
|
1636 |
||
1637 |
lemma LIMSEQ_SEQ_conv2: |
|
1638 |
fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space" |
|
1639 |
assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l" |
|
1640 |
shows "f -- a --> l" |
|
1641 |
using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at) |
|
1642 |
||
1643 |
lemma LIMSEQ_SEQ_conv: |
|
1644 |
"(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) = |
|
1645 |
(X -- a --> (L::'b::topological_space))" |
|
1646 |
using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 .. |
|
1647 |
||
51471 | 1648 |
subsection {* Continuity *} |
1649 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1650 |
subsubsection {* Continuity on a set *} |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1651 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1652 |
definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1653 |
"continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f ---> f x) (at x within s))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1654 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1655 |
lemma continuous_on_cong [cong]: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1656 |
"s = t \<Longrightarrow> (\<And>x. x \<in> t \<Longrightarrow> f x = g x) \<Longrightarrow> continuous_on s f \<longleftrightarrow> continuous_on t g" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1657 |
unfolding continuous_on_def by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter) |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1658 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1659 |
lemma continuous_on_topological: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1660 |
"continuous_on s f \<longleftrightarrow> |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1661 |
(\<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)))" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1662 |
unfolding continuous_on_def tendsto_def eventually_at_topological by metis |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1663 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1664 |
lemma continuous_on_open_invariant: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1665 |
"continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1666 |
proof safe |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1667 |
fix B :: "'b set" assume "continuous_on s f" "open B" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1668 |
then have "\<forall>x\<in>f -` B \<inter> s. (\<exists>A. open A \<and> x \<in> A \<and> s \<inter> A \<subseteq> f -` B)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1669 |
by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1670 |
then guess A unfolding bchoice_iff .. |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1671 |
then show "\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1672 |
by (intro exI[of _ "\<Union>x\<in>f -` B \<inter> s. A x"]) auto |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1673 |
next |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1674 |
assume B: "\<forall>B. open B \<longrightarrow> (\<exists>A. open A \<and> A \<inter> s = f -` B \<inter> s)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1675 |
show "continuous_on s f" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1676 |
unfolding continuous_on_topological |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1677 |
proof safe |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1678 |
fix x B assume "x \<in> s" "open B" "f x \<in> B" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1679 |
with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1680 |
with `x \<in> s` `f x \<in> B` show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1681 |
by (intro exI[of _ A]) auto |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1682 |
qed |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1683 |
qed |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1684 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1685 |
lemma continuous_on_open_vimage: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1686 |
"open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. open B \<longrightarrow> open (f -` B \<inter> s))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1687 |
unfolding continuous_on_open_invariant |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1688 |
by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s]) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1689 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1690 |
lemma continuous_on_closed_invariant: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1691 |
"continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> (\<exists>A. closed A \<and> A \<inter> s = f -` B \<inter> s))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1692 |
proof - |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1693 |
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)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1694 |
by (metis double_compl) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1695 |
show ?thesis |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1696 |
unfolding continuous_on_open_invariant by (intro *) (auto simp: open_closed[symmetric]) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1697 |
qed |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1698 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1699 |
lemma continuous_on_closed_vimage: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1700 |
"closed s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>B. closed B \<longrightarrow> closed (f -` B \<inter> s))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1701 |
unfolding continuous_on_closed_invariant |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1702 |
by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s]) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1703 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1704 |
lemma continuous_on_open_Union: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1705 |
"(\<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" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1706 |
unfolding continuous_on_def by safe (metis open_Union at_within_open UnionI) |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1707 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1708 |
lemma continuous_on_open_UN: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1709 |
"(\<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" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1710 |
unfolding Union_image_eq[symmetric] by (rule continuous_on_open_Union) auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1711 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1712 |
lemma continuous_on_closed_Un: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1713 |
"closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1714 |
by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1715 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1716 |
lemma continuous_on_If: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1717 |
assumes closed: "closed s" "closed t" and cont: "continuous_on s f" "continuous_on t g" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1718 |
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" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1719 |
shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)" (is "continuous_on _ ?h") |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1720 |
proof- |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1721 |
from P have "\<forall>x\<in>s. f x = ?h x" "\<forall>x\<in>t. g x = ?h x" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1722 |
by auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1723 |
with cont have "continuous_on s ?h" "continuous_on t ?h" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1724 |
by simp_all |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1725 |
with closed show ?thesis |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1726 |
by (rule continuous_on_closed_Un) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1727 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1728 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1729 |
ML {* |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1730 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1731 |
structure Continuous_On_Intros = Named_Thms |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1732 |
( |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1733 |
val name = @{binding continuous_on_intros} |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1734 |
val description = "Structural introduction rules for setwise continuity" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1735 |
) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1736 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1737 |
*} |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1738 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1739 |
setup Continuous_On_Intros.setup |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1740 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1741 |
lemma continuous_on_id[continuous_on_intros]: "continuous_on s (\<lambda>x. x)" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1742 |
unfolding continuous_on_def by (fast intro: tendsto_ident_at) |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1743 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1744 |
lemma continuous_on_const[continuous_on_intros]: "continuous_on s (\<lambda>x. c)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1745 |
unfolding continuous_on_def by (auto intro: tendsto_const) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1746 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1747 |
lemma continuous_on_compose[continuous_on_intros]: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1748 |
"continuous_on s f \<Longrightarrow> continuous_on (f ` s) g \<Longrightarrow> continuous_on s (g o f)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1749 |
unfolding continuous_on_topological by simp metis |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1750 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1751 |
lemma continuous_on_compose2: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1752 |
"continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> t = f ` s \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1753 |
using continuous_on_compose[of s f g] by (simp add: comp_def) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1754 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1755 |
subsubsection {* Continuity at a point *} |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1756 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1757 |
definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1758 |
"continuous F f \<longleftrightarrow> (f ---> f (Lim F (\<lambda>x. x))) F" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1759 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1760 |
ML {* |
51471 | 1761 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1762 |
structure Continuous_Intros = Named_Thms |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1763 |
( |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1764 |
val name = @{binding continuous_intros} |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1765 |
val description = "Structural introduction rules for pointwise continuity" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1766 |
) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1767 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1768 |
*} |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1769 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1770 |
setup Continuous_Intros.setup |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1771 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1772 |
lemma continuous_bot[continuous_intros, simp]: "continuous bot f" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1773 |
unfolding continuous_def by auto |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1774 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1775 |
lemma continuous_trivial_limit: "trivial_limit net \<Longrightarrow> continuous net f" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1776 |
by simp |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1777 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1778 |
lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f ---> f x) (at x within s)" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1779 |
by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def) |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1780 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1781 |
lemma continuous_within_topological: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1782 |
"continuous (at x within s) f \<longleftrightarrow> |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1783 |
(\<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)))" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1784 |
unfolding continuous_within tendsto_def eventually_at_topological by metis |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1785 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1786 |
lemma continuous_within_compose[continuous_intros]: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1787 |
"continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow> |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1788 |
continuous (at x within s) (g o f)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1789 |
by (simp add: continuous_within_topological) metis |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1790 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1791 |
lemma continuous_within_compose2: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1792 |
"continuous (at x within s) f \<Longrightarrow> continuous (at (f x) within f ` s) g \<Longrightarrow> |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1793 |
continuous (at x within s) (\<lambda>x. g (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1794 |
using continuous_within_compose[of x s f g] by (simp add: comp_def) |
51471 | 1795 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1796 |
lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f -- x --> f x" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1797 |
using continuous_within[of x UNIV f] by simp |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1798 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1799 |
lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (\<lambda>x. x)" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1800 |
unfolding continuous_within by (rule tendsto_ident_at) |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1801 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1802 |
lemma continuous_const[continuous_intros, simp]: "continuous F (\<lambda>x. c)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1803 |
unfolding continuous_def by (rule tendsto_const) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1804 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1805 |
lemma continuous_on_eq_continuous_within: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1806 |
"continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. continuous (at x within s) f)" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1807 |
unfolding continuous_on_def continuous_within .. |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1808 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1809 |
abbreviation isCont :: "('a::t2_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1810 |
"isCont f a \<equiv> continuous (at a) f" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1811 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1812 |
lemma isCont_def: "isCont f a \<longleftrightarrow> f -- a --> f a" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1813 |
by (rule continuous_at) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1814 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1815 |
lemma continuous_at_within: "isCont f x \<Longrightarrow> continuous (at x within s) f" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1816 |
by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within) |
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1817 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1818 |
lemma continuous_on_eq_continuous_at: "open s \<Longrightarrow> continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. isCont f x)" |
51641
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
1819 |
by (simp add: continuous_on_def continuous_at at_within_open[of _ s]) |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1820 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1821 |
lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1822 |
unfolding continuous_on_def by (metis subset_eq tendsto_within_subset) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1823 |
|
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1824 |
lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1825 |
by (auto intro: continuous_at_within simp: continuous_on_eq_continuous_within) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1826 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1827 |
lemma isContI_continuous: "continuous (at x within UNIV) f \<Longrightarrow> isCont f x" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1828 |
by simp |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1829 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1830 |
lemma isCont_ident[continuous_intros, simp]: "isCont (\<lambda>x. x) a" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1831 |
using continuous_ident by (rule isContI_continuous) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1832 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1833 |
lemmas isCont_const = continuous_const |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1834 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1835 |
lemma isCont_o2: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (\<lambda>x. g (f x)) a" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1836 |
unfolding isCont_def by (rule tendsto_compose) |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1837 |
|
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1838 |
lemma isCont_o[continuous_intros]: "isCont f a \<Longrightarrow> isCont g (f a) \<Longrightarrow> isCont (g \<circ> f) a" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1839 |
unfolding o_def by (rule isCont_o2) |
51471 | 1840 |
|
1841 |
lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F" |
|
1842 |
unfolding isCont_def by (rule tendsto_compose) |
|
1843 |
||
51478
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1844 |
lemma continuous_within_compose3: |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1845 |
"isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))" |
270b21f3ae0a
move continuous and continuous_on to the HOL image; isCont is an abbreviation for continuous (at x) (isCont is now restricted to a T2 space)
hoelzl
parents:
51474
diff
changeset
|
1846 |
using continuous_within_compose2[of x s f g] by (simp add: continuous_at_within) |
51471 | 1847 |
|
51479
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1848 |
subsubsection{* Open-cover compactness *} |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1849 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1850 |
context topological_space |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1851 |
begin |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1852 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1853 |
definition compact :: "'a set \<Rightarrow> bool" where |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1854 |
compact_eq_heine_borel: -- "This name is used for backwards compatibility" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1855 |
"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))" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1856 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1857 |
lemma compactI: |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1858 |
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'" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1859 |
shows "compact s" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1860 |
unfolding compact_eq_heine_borel using assms by metis |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1861 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1862 |
lemma compact_empty[simp]: "compact {}" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1863 |
by (auto intro!: compactI) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1864 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1865 |
lemma compactE: |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1866 |
assumes "compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1867 |
obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1868 |
using assms unfolding compact_eq_heine_borel by metis |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1869 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1870 |
lemma compactE_image: |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1871 |
assumes "compact s" and "\<forall>t\<in>C. open (f t)" and "s \<subseteq> (\<Union>c\<in>C. f c)" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1872 |
obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> (\<Union>c\<in>C'. f c)" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1873 |
using assms unfolding ball_simps[symmetric] SUP_def |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1874 |
by (metis (lifting) finite_subset_image compact_eq_heine_borel[of s]) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1875 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1876 |
lemma compact_inter_closed [intro]: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1877 |
assumes "compact s" and "closed t" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1878 |
shows "compact (s \<inter> t)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1879 |
proof (rule compactI) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1880 |
fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1881 |
from C `closed t` have "\<forall>c\<in>C \<union> {-t}. open c" by auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1882 |
moreover from cover have "s \<subseteq> \<Union>(C \<union> {-t})" by auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1883 |
ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1884 |
using `compact s` unfolding compact_eq_heine_borel by auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1885 |
then guess D .. |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1886 |
then show "\<exists>D\<subseteq>C. finite D \<and> s \<inter> t \<subseteq> \<Union>D" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1887 |
by (intro exI[of _ "D - {-t}"]) auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1888 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1889 |
|
51471 | 1890 |
end |
1891 |
||
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1892 |
lemma (in t2_space) compact_imp_closed: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1893 |
assumes "compact s" shows "closed s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1894 |
unfolding closed_def |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1895 |
proof (rule openI) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1896 |
fix y assume "y \<in> - s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1897 |
let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1898 |
note `compact s` |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1899 |
moreover have "\<forall>u\<in>?C. open u" by simp |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1900 |
moreover have "s \<subseteq> \<Union>?C" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1901 |
proof |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1902 |
fix x assume "x \<in> s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1903 |
with `y \<in> - s` have "x \<noteq> y" by clarsimp |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1904 |
hence "\<exists>u v. open u \<and> open v \<and> x \<in> u \<and> y \<in> v \<and> u \<inter> v = {}" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1905 |
by (rule hausdorff) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1906 |
with `x \<in> s` show "x \<in> \<Union>?C" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1907 |
unfolding eventually_nhds by auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1908 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1909 |
ultimately obtain D where "D \<subseteq> ?C" and "finite D" and "s \<subseteq> \<Union>D" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1910 |
by (rule compactE) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1911 |
from `D \<subseteq> ?C` have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1912 |
with `finite D` have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1913 |
by (simp add: eventually_Ball_finite) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1914 |
with `s \<subseteq> \<Union>D` have "eventually (\<lambda>y. y \<notin> s) (nhds y)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1915 |
by (auto elim!: eventually_mono [rotated]) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1916 |
thus "\<exists>t. open t \<and> y \<in> t \<and> t \<subseteq> - s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1917 |
by (simp add: eventually_nhds subset_eq) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1918 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1919 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1920 |
lemma compact_continuous_image: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1921 |
assumes f: "continuous_on s f" and s: "compact s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1922 |
shows "compact (f ` s)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1923 |
proof (rule compactI) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1924 |
fix C assume "\<forall>c\<in>C. open c" and cover: "f`s \<subseteq> \<Union>C" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1925 |
with f have "\<forall>c\<in>C. \<exists>A. open A \<and> A \<inter> s = f -` c \<inter> s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1926 |
unfolding continuous_on_open_invariant by blast |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1927 |
then guess A unfolding bchoice_iff .. note A = this |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1928 |
with cover have "\<forall>c\<in>C. open (A c)" "s \<subseteq> (\<Union>c\<in>C. A c)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1929 |
by (fastforce simp add: subset_eq set_eq_iff)+ |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1930 |
from compactE_image[OF s this] obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> (\<Union>c\<in>D. A c)" . |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1931 |
with A show "\<exists>D \<subseteq> C. finite D \<and> f`s \<subseteq> \<Union>D" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1932 |
by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+ |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1933 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1934 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1935 |
lemma continuous_on_inv: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1936 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1937 |
assumes "continuous_on s f" "compact s" "\<forall>x\<in>s. g (f x) = x" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1938 |
shows "continuous_on (f ` s) g" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1939 |
unfolding continuous_on_topological |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1940 |
proof (clarsimp simp add: assms(3)) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1941 |
fix x :: 'a and B :: "'a set" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1942 |
assume "x \<in> s" and "open B" and "x \<in> B" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1943 |
have 1: "\<forall>x\<in>s. f x \<in> f ` (s - B) \<longleftrightarrow> x \<in> s - B" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1944 |
using assms(3) by (auto, metis) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1945 |
have "continuous_on (s - B) f" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1946 |
using `continuous_on s f` Diff_subset |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1947 |
by (rule continuous_on_subset) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1948 |
moreover have "compact (s - B)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1949 |
using `open B` and `compact s` |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1950 |
unfolding Diff_eq by (intro compact_inter_closed closed_Compl) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1951 |
ultimately have "compact (f ` (s - B))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1952 |
by (rule compact_continuous_image) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1953 |
hence "closed (f ` (s - B))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1954 |
by (rule compact_imp_closed) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1955 |
hence "open (- f ` (s - B))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1956 |
by (rule open_Compl) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1957 |
moreover have "f x \<in> - f ` (s - B)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1958 |
using `x \<in> s` and `x \<in> B` by (simp add: 1) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1959 |
moreover have "\<forall>y\<in>s. f y \<in> - f ` (s - B) \<longrightarrow> y \<in> B" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1960 |
by (simp add: 1) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1961 |
ultimately show "\<exists>A. open A \<and> f x \<in> A \<and> (\<forall>y\<in>s. f y \<in> A \<longrightarrow> y \<in> B)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1962 |
by fast |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1963 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1964 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1965 |
lemma continuous_on_inv_into: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1966 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1967 |
assumes s: "continuous_on s f" "compact s" and f: "inj_on f s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1968 |
shows "continuous_on (f ` s) (the_inv_into s f)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1969 |
by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f]) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1970 |
|
51479
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1971 |
lemma (in linorder_topology) compact_attains_sup: |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1972 |
assumes "compact S" "S \<noteq> {}" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1973 |
shows "\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1974 |
proof (rule classical) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1975 |
assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. t \<le> s)" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1976 |
then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. s < t s" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1977 |
by (metis not_le) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1978 |
then have "\<forall>s\<in>S. open {..< t s}" "S \<subseteq> (\<Union>s\<in>S. {..< t s})" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1979 |
by auto |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1980 |
with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1981 |
by (erule compactE_image) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1982 |
with `S \<noteq> {}` have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1983 |
by (auto intro!: Max_in) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1984 |
with C have "S \<subseteq> {..< Max (t`C)}" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1985 |
by (auto intro: less_le_trans simp: subset_eq) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1986 |
with t Max `C \<subseteq> S` show ?thesis |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1987 |
by fastforce |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1988 |
qed |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1989 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1990 |
lemma (in linorder_topology) compact_attains_inf: |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1991 |
assumes "compact S" "S \<noteq> {}" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1992 |
shows "\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1993 |
proof (rule classical) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1994 |
assume "\<not> (\<exists>s\<in>S. \<forall>t\<in>S. s \<le> t)" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1995 |
then obtain t where t: "\<forall>s\<in>S. t s \<in> S" and "\<forall>s\<in>S. t s < s" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1996 |
by (metis not_le) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1997 |
then have "\<forall>s\<in>S. open {t s <..}" "S \<subseteq> (\<Union>s\<in>S. {t s <..})" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1998 |
by auto |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
1999 |
with `compact S` obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2000 |
by (erule compactE_image) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2001 |
with `S \<noteq> {}` have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2002 |
by (auto intro!: Min_in) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2003 |
with C have "S \<subseteq> {Min (t`C) <..}" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2004 |
by (auto intro: le_less_trans simp: subset_eq) |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2005 |
with t Min `C \<subseteq> S` show ?thesis |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2006 |
by fastforce |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2007 |
qed |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2008 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2009 |
lemma continuous_attains_sup: |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2010 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2011 |
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)" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2012 |
using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2013 |
|
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2014 |
lemma continuous_attains_inf: |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2015 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::linorder_topology" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2016 |
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)" |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2017 |
using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2018 |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2019 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2020 |
subsection {* Connectedness *} |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2021 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2022 |
context topological_space |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2023 |
begin |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2024 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2025 |
definition "connected S \<longleftrightarrow> |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2026 |
\<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> {})" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2027 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2028 |
lemma connectedI: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2029 |
"(\<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) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2030 |
\<Longrightarrow> connected U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2031 |
by (auto simp: connected_def) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2032 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2033 |
lemma connected_empty[simp]: "connected {}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2034 |
by (auto intro!: connectedI) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2035 |
|
51479
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2036 |
end |
33db4b7189af
move compact to the HOL image; prove compactness of real closed intervals; show that continuous functions attain supremum and infimum on compact sets
hoelzl
parents:
51478
diff
changeset
|
2037 |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2038 |
lemma (in linorder_topology) connectedD_interval: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2039 |
assumes "connected U" and xy: "x \<in> U" "y \<in> U" and "x \<le> z" "z \<le> y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2040 |
shows "z \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2041 |
proof - |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2042 |
have eq: "{..<z} \<union> {z<..} = - {z}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2043 |
by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2044 |
{ assume "z \<notin> U" "x < z" "z < y" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2045 |
with xy have "\<not> connected U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2046 |
unfolding connected_def simp_thms |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2047 |
apply (rule_tac exI[of _ "{..< z}"]) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2048 |
apply (rule_tac exI[of _ "{z <..}"]) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2049 |
apply (auto simp add: eq) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2050 |
done } |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2051 |
with assms show "z \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2052 |
by (metis less_le) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2053 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2054 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2055 |
lemma connected_continuous_image: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2056 |
assumes *: "continuous_on s f" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2057 |
assumes "connected s" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2058 |
shows "connected (f ` s)" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2059 |
proof (rule connectedI) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2060 |
fix A B assume A: "open A" "A \<inter> f ` s \<noteq> {}" and B: "open B" "B \<inter> f ` s \<noteq> {}" and |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2061 |
AB: "A \<inter> B \<inter> f ` s = {}" "f ` s \<subseteq> A \<union> B" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2062 |
obtain A' where A': "open A'" "f -` A \<inter> s = A' \<inter> s" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2063 |
using * `open A` unfolding continuous_on_open_invariant by metis |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2064 |
obtain B' where B': "open B'" "f -` B \<inter> s = B' \<inter> s" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2065 |
using * `open B` unfolding continuous_on_open_invariant by metis |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2066 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2067 |
have "\<exists>A B. open A \<and> open B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2068 |
proof (rule exI[of _ A'], rule exI[of _ B'], intro conjI) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2069 |
have "s \<subseteq> (f -` A \<inter> s) \<union> (f -` B \<inter> s)" using AB by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2070 |
then show "s \<subseteq> A' \<union> B'" using A' B' by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2071 |
next |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2072 |
have "(f -` A \<inter> s) \<inter> (f -` B \<inter> s) = {}" using AB by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2073 |
then show "A' \<inter> B' \<inter> s = {}" using A' B' by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2074 |
qed (insert A' B' A B, auto) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2075 |
with `connected s` show False |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2076 |
unfolding connected_def by blast |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2077 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2078 |
|
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2079 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2080 |
section {* Connectedness *} |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2081 |
|
51773 | 2082 |
class linear_continuum_topology = linorder_topology + conditionally_complete_linorder + inner_dense_linorder |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2083 |
begin |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2084 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2085 |
lemma Inf_notin_open: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2086 |
assumes A: "open A" and bnd: "\<forall>a\<in>A. x < a" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2087 |
shows "Inf A \<notin> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2088 |
proof |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2089 |
assume "Inf A \<in> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2090 |
then obtain b where "b < Inf A" "{b <.. Inf A} \<subseteq> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2091 |
using open_left[of A "Inf A" x] assms by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2092 |
with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2093 |
by (auto simp: subset_eq) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2094 |
then show False |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2095 |
using cInf_lower[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2096 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2097 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2098 |
lemma Sup_notin_open: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2099 |
assumes A: "open A" and bnd: "\<forall>a\<in>A. a < x" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2100 |
shows "Sup A \<notin> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2101 |
proof |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2102 |
assume "Sup A \<in> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2103 |
then obtain b where "Sup A < b" "{Sup A ..< b} \<subseteq> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2104 |
using open_right[of A "Sup A" x] assms by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2105 |
with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2106 |
by (auto simp: subset_eq) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2107 |
then show False |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2108 |
using cSup_upper[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2109 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2110 |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2111 |
end |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2112 |
|
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2113 |
lemma connectedI_interval: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2114 |
fixes U :: "'a :: linear_continuum_topology set" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2115 |
assumes *: "\<And>x y z. x \<in> U \<Longrightarrow> y \<in> U \<Longrightarrow> x \<le> z \<Longrightarrow> z \<le> y \<Longrightarrow> z \<in> U" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2116 |
shows "connected U" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2117 |
proof (rule connectedI) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2118 |
{ fix A B assume "open A" "open B" "A \<inter> B \<inter> U = {}" "U \<subseteq> A \<union> B" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2119 |
fix x y assume "x < y" "x \<in> A" "y \<in> B" "x \<in> U" "y \<in> U" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2120 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2121 |
let ?z = "Inf (B \<inter> {x <..})" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2122 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2123 |
have "x \<le> ?z" "?z \<le> y" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2124 |
using `y \<in> B` `x < y` by (auto intro: cInf_lower[where z=x] cInf_greatest) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2125 |
with `x \<in> U` `y \<in> U` have "?z \<in> U" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2126 |
by (rule *) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2127 |
moreover have "?z \<notin> B \<inter> {x <..}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2128 |
using `open B` by (intro Inf_notin_open) auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2129 |
ultimately have "?z \<in> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2130 |
using `x \<le> ?z` `A \<inter> B \<inter> U = {}` `x \<in> A` `U \<subseteq> A \<union> B` by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2131 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2132 |
{ assume "?z < y" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2133 |
obtain a where "?z < a" "{?z ..< a} \<subseteq> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2134 |
using open_right[OF `open A` `?z \<in> A` `?z < y`] by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2135 |
moreover obtain b where "b \<in> B" "x < b" "b < min a y" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2136 |
using cInf_less_iff[of "B \<inter> {x <..}" x "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B` |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2137 |
by (auto intro: less_imp_le) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2138 |
moreover then have "?z \<le> b" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2139 |
by (intro cInf_lower[where z=x]) auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2140 |
moreover have "b \<in> U" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2141 |
using `x \<le> ?z` `?z \<le> b` `b < min a y` |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2142 |
by (intro *[OF `x \<in> U` `y \<in> U`]) (auto simp: less_imp_le) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2143 |
ultimately have "\<exists>b\<in>B. b \<in> A \<and> b \<in> U" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2144 |
by (intro bexI[of _ b]) auto } |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2145 |
then have False |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2146 |
using `?z \<le> y` `?z \<in> A` `y \<in> B` `y \<in> U` `A \<inter> B \<inter> U = {}` unfolding le_less by blast } |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2147 |
note not_disjoint = this |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2148 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2149 |
fix A B assume AB: "open A" "open B" "U \<subseteq> A \<union> B" "A \<inter> B \<inter> U = {}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2150 |
moreover assume "A \<inter> U \<noteq> {}" then obtain x where x: "x \<in> U" "x \<in> A" by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2151 |
moreover assume "B \<inter> U \<noteq> {}" then obtain y where y: "y \<in> U" "y \<in> B" by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2152 |
moreover note not_disjoint[of B A y x] not_disjoint[of A B x y] |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2153 |
ultimately show False by (cases x y rule: linorder_cases) auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2154 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2155 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2156 |
lemma connected_iff_interval: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2157 |
fixes U :: "'a :: linear_continuum_topology set" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2158 |
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)" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2159 |
by (auto intro: connectedI_interval dest: connectedD_interval) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2160 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2161 |
lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2162 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2163 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2164 |
lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2165 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2166 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2167 |
lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2168 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2169 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2170 |
lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2171 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2172 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2173 |
lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2174 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2175 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2176 |
lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2177 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2178 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2179 |
lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2180 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2181 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2182 |
lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2183 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2184 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2185 |
lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2186 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2187 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2188 |
lemma connected_contains_Ioo: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2189 |
fixes A :: "'a :: linorder_topology set" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2190 |
assumes A: "connected A" "a \<in> A" "b \<in> A" shows "{a <..< b} \<subseteq> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2191 |
using connectedD_interval[OF A] by (simp add: subset_eq Ball_def less_imp_le) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2192 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2193 |
subsection {* Intermediate Value Theorem *} |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2194 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2195 |
lemma IVT': |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2196 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2197 |
assumes y: "f a \<le> y" "y \<le> f b" "a \<le> b" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2198 |
assumes *: "continuous_on {a .. b} f" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2199 |
shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2200 |
proof - |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2201 |
have "connected {a..b}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2202 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2203 |
from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2204 |
show ?thesis |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2205 |
by (auto simp add: atLeastAtMost_def atLeast_def atMost_def) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2206 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2207 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2208 |
lemma IVT2': |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2209 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2210 |
assumes y: "f b \<le> y" "y \<le> f a" "a \<le> b" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2211 |
assumes *: "continuous_on {a .. b} f" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2212 |
shows "\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = y" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2213 |
proof - |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2214 |
have "connected {a..b}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2215 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2216 |
from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2217 |
show ?thesis |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2218 |
by (auto simp add: atLeastAtMost_def atLeast_def atMost_def) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2219 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2220 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2221 |
lemma IVT: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2222 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2223 |
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" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2224 |
by (rule IVT') (auto intro: continuous_at_imp_continuous_on) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2225 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2226 |
lemma IVT2: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2227 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2228 |
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" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2229 |
by (rule IVT2') (auto intro: continuous_at_imp_continuous_on) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2230 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2231 |
lemma continuous_inj_imp_mono: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2232 |
fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2233 |
assumes x: "a < x" "x < b" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2234 |
assumes cont: "continuous_on {a..b} f" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2235 |
assumes inj: "inj_on f {a..b}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2236 |
shows "(f a < f x \<and> f x < f b) \<or> (f b < f x \<and> f x < f a)" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2237 |
proof - |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2238 |
note I = inj_on_iff[OF inj] |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2239 |
{ assume "f x < f a" "f x < f b" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2240 |
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" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2241 |
using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2242 |
by (auto simp: continuous_on_subset[OF cont] less_imp_le) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2243 |
with x I have False by auto } |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2244 |
moreover |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2245 |
{ assume "f a < f x" "f b < f x" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2246 |
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" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2247 |
using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2248 |
by (auto simp: continuous_on_subset[OF cont] less_imp_le) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2249 |
with x I have False by auto } |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2250 |
ultimately show ?thesis |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2251 |
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) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2252 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2253 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2254 |
end |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2255 |