author | hoelzl |
Wed, 06 Jan 2016 12:18:53 +0100 | |
changeset 62083 | 7582b39f51ed |
parent 62049 | b0f941e207cf |
child 62101 | 26c0a70f78a3 |
permissions | -rw-r--r-- |
52265 | 1 |
(* Title: HOL/Topological_Spaces.thy |
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Author: Brian Huffman |
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Author: Johannes Hölzl |
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*) |
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section \<open>Topological Spaces\<close> |
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theory Topological_Spaces |
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imports Main Conditionally_Complete_Lattices |
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begin |
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named_theorems continuous_intros "structural introduction rules for continuity" |
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subsection \<open>Topological space\<close> |
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class "open" = |
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fixes "open" :: "'a set \<Rightarrow> bool" |
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class topological_space = "open" + |
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assumes open_UNIV [simp, intro]: "open UNIV" |
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assumes open_Int [intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<inter> T)" |
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assumes open_Union [intro]: "\<forall>S\<in>K. open S \<Longrightarrow> open (\<Union>K)" |
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begin |
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definition |
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closed :: "'a set \<Rightarrow> bool" where |
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"closed S \<longleftrightarrow> open (- S)" |
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lemma open_empty [continuous_intros, intro, simp]: "open {}" |
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using open_Union [of "{}"] by simp |
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lemma open_Un [continuous_intros, intro]: "open S \<Longrightarrow> open T \<Longrightarrow> open (S \<union> T)" |
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using open_Union [of "{S, T}"] by simp |
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lemma open_UN [continuous_intros, intro]: "\<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Union>x\<in>A. B x)" |
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using open_Union [of "B ` A"] by simp |
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lemma open_Inter [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. open T \<Longrightarrow> open (\<Inter>S)" |
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by (induct set: finite) auto |
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lemma open_INT [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. open (B x) \<Longrightarrow> open (\<Inter>x\<in>A. B x)" |
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using open_Inter [of "B ` A"] by simp |
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lemma openI: |
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assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" |
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shows "open S" |
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proof - |
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have "open (\<Union>{T. open T \<and> T \<subseteq> S})" by auto |
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moreover have "\<Union>{T. open T \<and> T \<subseteq> S} = S" by (auto dest!: assms) |
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ultimately show "open S" by simp |
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qed |
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lemma closed_empty [continuous_intros, intro, simp]: "closed {}" |
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unfolding closed_def by simp |
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lemma closed_Un [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<union> T)" |
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unfolding closed_def by auto |
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lemma closed_UNIV [continuous_intros, intro, simp]: "closed UNIV" |
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unfolding closed_def by simp |
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lemma closed_Int [continuous_intros, intro]: "closed S \<Longrightarrow> closed T \<Longrightarrow> closed (S \<inter> T)" |
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unfolding closed_def by auto |
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lemma closed_INT [continuous_intros, intro]: "\<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Inter>x\<in>A. B x)" |
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unfolding closed_def by auto |
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lemma closed_Inter [continuous_intros, intro]: "\<forall>S\<in>K. closed S \<Longrightarrow> closed (\<Inter>K)" |
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unfolding closed_def uminus_Inf by auto |
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lemma closed_Union [continuous_intros, intro]: "finite S \<Longrightarrow> \<forall>T\<in>S. closed T \<Longrightarrow> closed (\<Union>S)" |
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by (induct set: finite) auto |
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lemma closed_UN [continuous_intros, intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. closed (B x) \<Longrightarrow> closed (\<Union>x\<in>A. B x)" |
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using closed_Union [of "B ` A"] by simp |
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lemma open_closed: "open S \<longleftrightarrow> closed (- S)" |
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unfolding closed_def by simp |
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lemma closed_open: "closed S \<longleftrightarrow> open (- S)" |
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unfolding closed_def by simp |
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lemma open_Diff [continuous_intros, intro]: "open S \<Longrightarrow> closed T \<Longrightarrow> open (S - T)" |
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unfolding closed_open Diff_eq by (rule open_Int) |
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lemma closed_Diff [continuous_intros, intro]: "closed S \<Longrightarrow> open T \<Longrightarrow> closed (S - T)" |
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unfolding open_closed Diff_eq by (rule closed_Int) |
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lemma open_Compl [continuous_intros, intro]: "closed S \<Longrightarrow> open (- S)" |
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unfolding closed_open . |
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lemma closed_Compl [continuous_intros, intro]: "open S \<Longrightarrow> closed (- S)" |
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unfolding open_closed . |
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lemma open_Collect_neg: "closed {x. P x} \<Longrightarrow> open {x. \<not> P x}" |
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unfolding Collect_neg_eq by (rule open_Compl) |
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lemma open_Collect_conj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<and> Q x}" |
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using open_Int[OF assms] by (simp add: Int_def) |
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lemma open_Collect_disj: assumes "open {x. P x}" "open {x. Q x}" shows "open {x. P x \<or> Q x}" |
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using open_Un[OF assms] by (simp add: Un_def) |
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lemma open_Collect_ex: "(\<And>i. open {x. P i x}) \<Longrightarrow> open {x. \<exists>i. P i x}" |
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using open_UN[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_ex_eq by simp |
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lemma open_Collect_imp: "closed {x. P x} \<Longrightarrow> open {x. Q x} \<Longrightarrow> open {x. P x \<longrightarrow> Q x}" |
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unfolding imp_conv_disj by (intro open_Collect_disj open_Collect_neg) |
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lemma open_Collect_const: "open {x. P}" |
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by (cases P) auto |
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lemma closed_Collect_neg: "open {x. P x} \<Longrightarrow> closed {x. \<not> P x}" |
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unfolding Collect_neg_eq by (rule closed_Compl) |
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lemma closed_Collect_conj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<and> Q x}" |
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using closed_Int[OF assms] by (simp add: Int_def) |
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lemma closed_Collect_disj: assumes "closed {x. P x}" "closed {x. Q x}" shows "closed {x. P x \<or> Q x}" |
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using closed_Un[OF assms] by (simp add: Un_def) |
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lemma closed_Collect_all: "(\<And>i. closed {x. P i x}) \<Longrightarrow> closed {x. \<forall>i. P i x}" |
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using closed_INT[of UNIV "\<lambda>i. {x. P i x}"] unfolding Collect_all_eq by simp |
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lemma closed_Collect_imp: "open {x. P x} \<Longrightarrow> closed {x. Q x} \<Longrightarrow> closed {x. P x \<longrightarrow> Q x}" |
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unfolding imp_conv_disj by (intro closed_Collect_disj closed_Collect_neg) |
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lemma closed_Collect_const: "closed {x. P}" |
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by (cases P) auto |
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end |
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subsection\<open>Hausdorff and other separation properties\<close> |
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class t0_space = topological_space + |
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assumes t0_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> \<not> (x \<in> U \<longleftrightarrow> y \<in> U)" |
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class t1_space = topological_space + |
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assumes t1_space: "x \<noteq> y \<Longrightarrow> \<exists>U. open U \<and> x \<in> U \<and> y \<notin> U" |
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instance t1_space \<subseteq> t0_space |
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proof qed (fast dest: t1_space) |
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lemma separation_t1: |
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fixes x y :: "'a::t1_space" |
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shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> x \<in> U \<and> y \<notin> U)" |
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using t1_space[of x y] by blast |
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lemma closed_singleton: |
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fixes a :: "'a::t1_space" |
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shows "closed {a}" |
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proof - |
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let ?T = "\<Union>{S. open S \<and> a \<notin> S}" |
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have "open ?T" by (simp add: open_Union) |
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also have "?T = - {a}" |
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by (simp add: set_eq_iff separation_t1, auto) |
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finally show "closed {a}" unfolding closed_def . |
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qed |
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lemma closed_insert [continuous_intros, simp]: |
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fixes a :: "'a::t1_space" |
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assumes "closed S" shows "closed (insert a S)" |
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proof - |
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from closed_singleton assms |
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have "closed ({a} \<union> S)" by (rule closed_Un) |
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thus "closed (insert a S)" by simp |
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qed |
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lemma finite_imp_closed: |
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fixes S :: "'a::t1_space set" |
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shows "finite S \<Longrightarrow> closed S" |
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by (induct set: finite, simp_all) |
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text \<open>T2 spaces are also known as Hausdorff spaces.\<close> |
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class t2_space = topological_space + |
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assumes hausdorff: "x \<noteq> y \<Longrightarrow> \<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {}" |
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instance t2_space \<subseteq> t1_space |
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proof qed (fast dest: hausdorff) |
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lemma separation_t2: |
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fixes x y :: "'a::t2_space" |
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shows "x \<noteq> y \<longleftrightarrow> (\<exists>U V. open U \<and> open V \<and> x \<in> U \<and> y \<in> V \<and> U \<inter> V = {})" |
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using hausdorff[of x y] by blast |
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lemma separation_t0: |
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fixes x y :: "'a::t0_space" |
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shows "x \<noteq> y \<longleftrightarrow> (\<exists>U. open U \<and> ~(x\<in>U \<longleftrightarrow> y\<in>U))" |
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using t0_space[of x y] by blast |
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text \<open>A perfect space is a topological space with no isolated points.\<close> |
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class perfect_space = topological_space + |
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assumes not_open_singleton: "\<not> open {x}" |
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subsection \<open>Generators for toplogies\<close> |
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inductive generate_topology for S where |
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UNIV: "generate_topology S UNIV" |
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| Int: "generate_topology S a \<Longrightarrow> generate_topology S b \<Longrightarrow> generate_topology S (a \<inter> b)" |
|
204 |
| UN: "(\<And>k. k \<in> K \<Longrightarrow> generate_topology S k) \<Longrightarrow> generate_topology S (\<Union>K)" |
|
205 |
| Basis: "s \<in> S \<Longrightarrow> generate_topology S s" |
|
206 |
||
207 |
hide_fact (open) UNIV Int UN Basis |
|
208 |
||
209 |
lemma generate_topology_Union: |
|
210 |
"(\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)) \<Longrightarrow> generate_topology S (\<Union>k\<in>I. K k)" |
|
56166 | 211 |
using generate_topology.UN [of "K ` I"] by auto |
51471 | 212 |
|
213 |
lemma topological_space_generate_topology: |
|
214 |
"class.topological_space (generate_topology S)" |
|
61169 | 215 |
by standard (auto intro: generate_topology.intros) |
51471 | 216 |
|
60758 | 217 |
subsection \<open>Order topologies\<close> |
51471 | 218 |
|
219 |
class order_topology = order + "open" + |
|
220 |
assumes open_generated_order: "open = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))" |
|
221 |
begin |
|
222 |
||
223 |
subclass topological_space |
|
224 |
unfolding open_generated_order |
|
225 |
by (rule topological_space_generate_topology) |
|
226 |
||
56371
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diff
changeset
|
227 |
lemma open_greaterThan [continuous_intros, simp]: "open {a <..}" |
51471 | 228 |
unfolding open_generated_order by (auto intro: generate_topology.Basis) |
229 |
||
56371
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changeset
|
230 |
lemma open_lessThan [continuous_intros, simp]: "open {..< a}" |
51471 | 231 |
unfolding open_generated_order by (auto intro: generate_topology.Basis) |
232 |
||
56371
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diff
changeset
|
233 |
lemma open_greaterThanLessThan [continuous_intros, simp]: "open {a <..< b}" |
51471 | 234 |
unfolding greaterThanLessThan_eq by (simp add: open_Int) |
235 |
||
236 |
end |
|
237 |
||
238 |
class linorder_topology = linorder + order_topology |
|
239 |
||
56371
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56329
diff
changeset
|
240 |
lemma closed_atMost [continuous_intros, simp]: "closed {.. a::'a::linorder_topology}" |
51471 | 241 |
by (simp add: closed_open) |
242 |
||
56371
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extend continuous_intros; remove continuous_on_intros and isCont_intros
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56329
diff
changeset
|
243 |
lemma closed_atLeast [continuous_intros, simp]: "closed {a::'a::linorder_topology ..}" |
51471 | 244 |
by (simp add: closed_open) |
245 |
||
56371
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extend continuous_intros; remove continuous_on_intros and isCont_intros
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parents:
56329
diff
changeset
|
246 |
lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a::'a::linorder_topology .. b}" |
51471 | 247 |
proof - |
248 |
have "{a .. b} = {a ..} \<inter> {.. b}" |
|
249 |
by auto |
|
250 |
then show ?thesis |
|
251 |
by (simp add: closed_Int) |
|
252 |
qed |
|
253 |
||
254 |
lemma (in linorder) less_separate: |
|
255 |
assumes "x < y" |
|
256 |
shows "\<exists>a b. x \<in> {..< a} \<and> y \<in> {b <..} \<and> {..< a} \<inter> {b <..} = {}" |
|
53381 | 257 |
proof (cases "\<exists>z. x < z \<and> z < y") |
258 |
case True |
|
259 |
then obtain z where "x < z \<and> z < y" .. |
|
51471 | 260 |
then have "x \<in> {..< z} \<and> y \<in> {z <..} \<and> {z <..} \<inter> {..< z} = {}" |
261 |
by auto |
|
262 |
then show ?thesis by blast |
|
263 |
next |
|
53381 | 264 |
case False |
60758 | 265 |
with \<open>x < y\<close> have "x \<in> {..< y} \<and> y \<in> {x <..} \<and> {x <..} \<inter> {..< y} = {}" |
51471 | 266 |
by auto |
267 |
then show ?thesis by blast |
|
268 |
qed |
|
269 |
||
270 |
instance linorder_topology \<subseteq> t2_space |
|
271 |
proof |
|
272 |
fix x y :: 'a |
|
273 |
from less_separate[of x y] less_separate[of y x] |
|
274 |
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 = {}" |
|
275 |
by (elim neqE) (metis open_lessThan open_greaterThan Int_commute)+ |
|
276 |
qed |
|
277 |
||
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
278 |
lemma (in linorder_topology) open_right: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
279 |
assumes "open S" "x \<in> S" and gt_ex: "x < y" shows "\<exists>b>x. {x ..< b} \<subseteq> S" |
51471 | 280 |
using assms unfolding open_generated_order |
281 |
proof induction |
|
282 |
case (Int A B) |
|
283 |
then obtain a b where "a > x" "{x ..< a} \<subseteq> A" "b > x" "{x ..< b} \<subseteq> B" by auto |
|
284 |
then show ?case by (auto intro!: exI[of _ "min a b"]) |
|
285 |
next |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
286 |
case (Basis S) then show ?case by (fastforce intro: exI[of _ y] gt_ex) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
287 |
qed blast+ |
51471 | 288 |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
289 |
lemma (in linorder_topology) open_left: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
290 |
assumes "open S" "x \<in> S" and lt_ex: "y < x" shows "\<exists>b<x. {b <.. x} \<subseteq> S" |
51471 | 291 |
using assms unfolding open_generated_order |
292 |
proof induction |
|
293 |
case (Int A B) |
|
294 |
then obtain a b where "a < x" "{a <.. x} \<subseteq> A" "b < x" "{b <.. x} \<subseteq> B" by auto |
|
295 |
then show ?case by (auto intro!: exI[of _ "max a b"]) |
|
296 |
next |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
297 |
case (Basis S) then show ?case by (fastforce intro: exI[of _ y] lt_ex) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
298 |
qed blast+ |
51471 | 299 |
|
60758 | 300 |
subsubsection \<open>Boolean is an order topology\<close> |
59106 | 301 |
|
60758 | 302 |
text \<open>It also is a discrete topology, but don't have a type class for it (yet).\<close> |
59106 | 303 |
|
304 |
instantiation bool :: order_topology |
|
305 |
begin |
|
306 |
||
307 |
definition open_bool :: "bool set \<Rightarrow> bool" where |
|
308 |
"open_bool = generate_topology (range (\<lambda>a. {..< a}) \<union> range (\<lambda>a. {a <..}))" |
|
309 |
||
310 |
instance |
|
311 |
proof qed (rule open_bool_def) |
|
312 |
||
313 |
end |
|
314 |
||
315 |
lemma open_bool[simp, intro!]: "open (A::bool set)" |
|
316 |
proof - |
|
317 |
have *: "{False <..} = {True}" "{..< True} = {False}" |
|
318 |
by auto |
|
319 |
have "A = UNIV \<or> A = {} \<or> A = {False <..} \<or> A = {..< True}" |
|
320 |
using subset_UNIV[of A] unfolding UNIV_bool * by auto |
|
321 |
then show "open A" |
|
322 |
by auto |
|
323 |
qed |
|
324 |
||
60758 | 325 |
subsubsection \<open>Topological filters\<close> |
51471 | 326 |
|
327 |
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter" |
|
57276 | 328 |
where "nhds a = (INF S:{S. open S \<and> a \<in> S}. principal S)" |
51471 | 329 |
|
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
|
330 |
definition (in topological_space) at_within :: "'a \<Rightarrow> 'a set \<Rightarrow> 'a filter" ("at (_) within (_)" [1000, 60] 60) |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
331 |
where "at a within s = inf (nhds a) (principal (s - {a}))" |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
332 |
|
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
|
333 |
abbreviation (in topological_space) at :: "'a \<Rightarrow> 'a filter" ("at") where |
cd05e9fcc63d
remove the within-filter, replace "at" by "at _ within UNIV" (This allows to remove a couple of redundant lemmas)
hoelzl
parents:
51518
diff
changeset
|
334 |
"at x \<equiv> at x within (CONST UNIV)" |
51471 | 335 |
|
51473 | 336 |
abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where |
51471 | 337 |
"at_right x \<equiv> at x within {x <..}" |
338 |
||
51473 | 339 |
abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where |
51471 | 340 |
"at_left x \<equiv> at x within {..< x}" |
341 |
||
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
342 |
lemma (in topological_space) nhds_generated_topology: |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
343 |
"open = generate_topology T \<Longrightarrow> nhds x = (INF S:{S\<in>T. x \<in> S}. principal S)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
344 |
unfolding nhds_def |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
345 |
proof (safe intro!: antisym INF_greatest) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
346 |
fix S assume "generate_topology T S" "x \<in> S" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
347 |
then show "(INF S:{S \<in> T. x \<in> S}. principal S) \<le> principal S" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
348 |
by induction |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
349 |
(auto intro: INF_lower order_trans simp add: inf_principal[symmetric] simp del: inf_principal) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
350 |
qed (auto intro!: INF_lower intro: generate_topology.intros) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
351 |
|
51473 | 352 |
lemma (in topological_space) eventually_nhds: |
51471 | 353 |
"eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))" |
57276 | 354 |
unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal) |
51471 | 355 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
356 |
lemma (in topological_space) eventually_nhds_in_open: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
357 |
"open s \<Longrightarrow> x \<in> s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
358 |
by (subst eventually_nhds) blast |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
359 |
|
51471 | 360 |
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot" |
361 |
unfolding trivial_limit_def eventually_nhds by simp |
|
362 |
||
60182
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60172
diff
changeset
|
363 |
lemma (in t1_space) t1_space_nhds: |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60172
diff
changeset
|
364 |
"x \<noteq> y \<Longrightarrow> (\<forall>\<^sub>F x in nhds x. x \<noteq> y)" |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60172
diff
changeset
|
365 |
by (drule t1_space) (auto simp: eventually_nhds) |
e1ea5a6379c9
generalized tends over powr; added DERIV rule for powr
hoelzl
parents:
60172
diff
changeset
|
366 |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
367 |
lemma at_within_eq: "at x within s = (INF S:{S. open S \<and> x \<in> S}. principal (S \<inter> s - {x}))" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
368 |
unfolding nhds_def at_within_def by (subst INF_inf_const2[symmetric]) (auto simp add: Diff_Int_distrib) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
369 |
|
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
|
370 |
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
|
371 |
"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
|
372 |
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
|
373 |
|
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
|
374 |
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
|
375 |
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
|
376 |
|
51471 | 377 |
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
|
378 |
"eventually P (at a within s) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> x \<in> s \<longrightarrow> P x))" |
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
|
379 |
unfolding eventually_nhds eventually_at_filter by simp |
51471 | 380 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
381 |
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
|
382 |
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
|
383 |
|
61234 | 384 |
lemma at_within_open_NO_MATCH: |
385 |
"a \<in> s \<Longrightarrow> open s \<Longrightarrow> NO_MATCH UNIV s \<Longrightarrow> at a within s = at a" |
|
386 |
by (simp only: at_within_open) |
|
387 |
||
61245 | 388 |
lemma at_within_nhd: |
389 |
assumes "x \<in> S" "open S" "T \<inter> S - {x} = U \<inter> S - {x}" |
|
390 |
shows "at x within T = at x within U" |
|
391 |
unfolding filter_eq_iff eventually_at_filter |
|
392 |
proof (intro allI eventually_subst) |
|
393 |
have "eventually (\<lambda>x. x \<in> S) (nhds x)" |
|
394 |
using \<open>x \<in> S\<close> \<open>open S\<close> by (auto simp: eventually_nhds) |
|
395 |
then show "\<forall>\<^sub>F n in nhds x. (n \<noteq> x \<longrightarrow> n \<in> T \<longrightarrow> P n) = (n \<noteq> x \<longrightarrow> n \<in> U \<longrightarrow> P n)" for P |
|
396 |
by eventually_elim (insert \<open>T \<inter> S - {x} = U \<inter> S - {x}\<close>, blast) |
|
397 |
qed |
|
398 |
||
53859 | 399 |
lemma at_within_empty [simp]: "at a within {} = bot" |
400 |
unfolding at_within_def by simp |
|
401 |
||
53860 | 402 |
lemma at_within_union: "at x within (S \<union> T) = sup (at x within S) (at x within T)" |
403 |
unfolding filter_eq_iff eventually_sup eventually_at_filter |
|
404 |
by (auto elim!: eventually_rev_mp) |
|
405 |
||
51471 | 406 |
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}" |
407 |
unfolding trivial_limit_def eventually_at_topological |
|
408 |
by (safe, case_tac "S = {a}", simp, fast, fast) |
|
409 |
||
410 |
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot" |
|
411 |
by (simp add: at_eq_bot_iff not_open_singleton) |
|
412 |
||
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
413 |
lemma (in order_topology) nhds_order: "nhds x = |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
414 |
inf (INF a:{x <..}. principal {..< a}) (INF a:{..< x}. principal {a <..})" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
415 |
proof - |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
416 |
have 1: "{S \<in> range lessThan \<union> range greaterThan. x \<in> S} = |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
417 |
(\<lambda>a. {..< a}) ` {x <..} \<union> (\<lambda>a. {a <..}) ` {..< x}" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
418 |
by auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
419 |
show ?thesis |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
420 |
unfolding nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def .. |
51471 | 421 |
qed |
422 |
||
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
423 |
lemma (in linorder_topology) at_within_order: "UNIV \<noteq> {x} \<Longrightarrow> |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
424 |
at x within s = inf (INF a:{x <..}. principal ({..< a} \<inter> s - {x})) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
425 |
(INF a:{..< x}. principal ({a <..} \<inter> s - {x}))" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
426 |
proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split[case_product case_split]) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
427 |
assume "UNIV \<noteq> {x}" "{x<..} = {}" "{..< x} = {}" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
428 |
moreover have "UNIV = {..< x} \<union> {x} \<union> {x <..}" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
429 |
by auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
430 |
ultimately show ?thesis |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
431 |
by auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
432 |
qed (auto simp: at_within_def nhds_order Int_Diff inf_principal[symmetric] INF_inf_const2 |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
433 |
inf_sup_aci[where 'a="'a filter"] |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
434 |
simp del: inf_principal) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
435 |
|
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
436 |
lemma (in linorder_topology) at_left_eq: |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
437 |
"y < x \<Longrightarrow> at_left x = (INF a:{..< x}. principal {a <..< x})" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
438 |
by (subst at_within_order) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
439 |
(auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
440 |
intro!: INF_lower2 inf_absorb2) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
441 |
|
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
442 |
lemma (in linorder_topology) eventually_at_left: |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
443 |
"y < x \<Longrightarrow> eventually P (at_left x) \<longleftrightarrow> (\<exists>b<x. \<forall>y>b. y < x \<longrightarrow> P y)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
444 |
unfolding at_left_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
445 |
|
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
446 |
lemma (in linorder_topology) at_right_eq: |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
447 |
"x < y \<Longrightarrow> at_right x = (INF a:{x <..}. principal {x <..< a})" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
448 |
by (subst at_within_order) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
449 |
(auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
450 |
intro!: INF_lower2 inf_absorb1) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
451 |
|
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
452 |
lemma (in linorder_topology) eventually_at_right: |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
453 |
"x < y \<Longrightarrow> eventually P (at_right x) \<longleftrightarrow> (\<exists>b>x. \<forall>y>x. y < b \<longrightarrow> P y)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
454 |
unfolding at_right_eq by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def) |
51471 | 455 |
|
62083 | 456 |
lemma eventually_at_right_less: "\<forall>\<^sub>F y in at_right (x::'a::{linorder_topology, no_top}). x < y" |
457 |
using gt_ex[of x] eventually_at_right[of x] by auto |
|
458 |
||
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
459 |
lemma trivial_limit_at_right_top: "at_right (top::_::{order_top, linorder_topology}) = bot" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
460 |
unfolding filter_eq_iff eventually_at_topological by auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
461 |
|
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
462 |
lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot, linorder_topology}) = bot" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
463 |
unfolding filter_eq_iff eventually_at_topological by auto |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
464 |
|
51471 | 465 |
lemma trivial_limit_at_left_real [simp]: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
466 |
"\<not> trivial_limit (at_left (x::'a::{no_bot, dense_order, linorder_topology}))" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
467 |
using lt_ex[of x] |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
468 |
by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense) |
51471 | 469 |
|
470 |
lemma trivial_limit_at_right_real [simp]: |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
471 |
"\<not> trivial_limit (at_right (x::'a::{no_top, dense_order, linorder_topology}))" |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
472 |
using gt_ex[of x] |
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
473 |
by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense) |
51471 | 474 |
|
475 |
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
|
476 |
by (auto simp: eventually_at_filter filter_eq_iff eventually_sup |
61810 | 477 |
elim: eventually_elim2 eventually_mono) |
51471 | 478 |
|
479 |
lemma eventually_at_split: |
|
480 |
"eventually P (at (x::'a::linorder_topology)) \<longleftrightarrow> eventually P (at_left x) \<and> eventually P (at_right x)" |
|
481 |
by (subst at_eq_sup_left_right) (simp add: eventually_sup) |
|
482 |
||
60758 | 483 |
subsubsection \<open>Tendsto\<close> |
51471 | 484 |
|
485 |
abbreviation (in topological_space) |
|
61973 | 486 |
tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "\<longlongrightarrow>" 55) where |
487 |
"(f \<longlongrightarrow> l) F \<equiv> filterlim f (nhds l) F" |
|
51471 | 488 |
|
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
|
489 |
definition (in t2_space) Lim :: "'f filter \<Rightarrow> ('f \<Rightarrow> 'a) \<Rightarrow> 'a" where |
61973 | 490 |
"Lim A f = (THE l. (f \<longlongrightarrow> l) A)" |
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
|
491 |
|
61973 | 492 |
lemma tendsto_eq_rhs: "(f \<longlongrightarrow> x) F \<Longrightarrow> x = y \<Longrightarrow> (f \<longlongrightarrow> y) F" |
51471 | 493 |
by simp |
494 |
||
57953 | 495 |
named_theorems tendsto_intros "introduction rules for tendsto" |
60758 | 496 |
setup \<open> |
51471 | 497 |
Global_Theory.add_thms_dynamic (@{binding tendsto_eq_intros}, |
57953 | 498 |
fn context => |
499 |
Named_Theorems.get (Context.proof_of context) @{named_theorems tendsto_intros} |
|
500 |
|> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm]))) |
|
60758 | 501 |
\<close> |
51471 | 502 |
|
51473 | 503 |
lemma (in topological_space) tendsto_def: |
61973 | 504 |
"(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)" |
57276 | 505 |
unfolding nhds_def filterlim_INF filterlim_principal by auto |
51471 | 506 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
507 |
lemma tendsto_cong: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
508 |
assumes "eventually (\<lambda>x. f x = g x) F" |
61973 | 509 |
shows "(f \<longlongrightarrow> c) F \<longleftrightarrow> (g \<longlongrightarrow> c) F" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
510 |
by (rule filterlim_cong[OF refl refl assms]) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
511 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
512 |
|
61973 | 513 |
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f \<longlongrightarrow> l) F' \<Longrightarrow> (f \<longlongrightarrow> l) F" |
51471 | 514 |
unfolding tendsto_def le_filter_def by fast |
515 |
||
61973 | 516 |
lemma tendsto_within_subset: "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> T \<subseteq> S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T)" |
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
|
517 |
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
|
518 |
|
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
|
519 |
lemma filterlim_at: |
61973 | 520 |
"(LIM x F. f x :> at b within s) \<longleftrightarrow> (eventually (\<lambda>x. f x \<in> s \<and> f x \<noteq> b) F \<and> (f \<longlongrightarrow> b) F)" |
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
|
521 |
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
|
522 |
|
51473 | 523 |
lemma (in topological_space) topological_tendstoI: |
61973 | 524 |
"(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F) \<Longrightarrow> (f \<longlongrightarrow> l) F" |
51471 | 525 |
unfolding tendsto_def by auto |
526 |
||
51473 | 527 |
lemma (in topological_space) topological_tendstoD: |
61973 | 528 |
"(f \<longlongrightarrow> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F" |
51471 | 529 |
unfolding tendsto_def by auto |
530 |
||
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
531 |
lemma (in order_topology) order_tendsto_iff: |
61973 | 532 |
"(f \<longlongrightarrow> x) F \<longleftrightarrow> (\<forall>l<x. eventually (\<lambda>x. l < f x) F) \<and> (\<forall>u>x. eventually (\<lambda>x. f x < u) F)" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
533 |
unfolding nhds_order filterlim_inf filterlim_INF filterlim_principal by auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
534 |
|
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
535 |
lemma (in order_topology) order_tendstoI: |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
536 |
"(\<And>a. a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F) \<Longrightarrow> (\<And>a. y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F) \<Longrightarrow> |
61973 | 537 |
(f \<longlongrightarrow> y) F" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
538 |
unfolding order_tendsto_iff by auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
539 |
|
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
540 |
lemma (in order_topology) order_tendstoD: |
61973 | 541 |
assumes "(f \<longlongrightarrow> y) F" |
51471 | 542 |
shows "a < y \<Longrightarrow> eventually (\<lambda>x. a < f x) F" |
543 |
and "y < a \<Longrightarrow> eventually (\<lambda>x. f x < a) F" |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
544 |
using assms unfolding order_tendsto_iff by auto |
51471 | 545 |
|
61973 | 546 |
lemma tendsto_bot [simp]: "(f \<longlongrightarrow> a) bot" |
51471 | 547 |
unfolding tendsto_def by simp |
548 |
||
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
549 |
lemma (in linorder_topology) tendsto_max: |
61973 | 550 |
assumes X: "(X \<longlongrightarrow> x) net" |
551 |
assumes Y: "(Y \<longlongrightarrow> y) net" |
|
552 |
shows "((\<lambda>x. max (X x) (Y x)) \<longlongrightarrow> max x y) net" |
|
56949 | 553 |
proof (rule order_tendstoI) |
554 |
fix a assume "a < max x y" |
|
555 |
then show "eventually (\<lambda>x. a < max (X x) (Y x)) net" |
|
556 |
using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a] |
|
61810 | 557 |
by (auto simp: less_max_iff_disj elim: eventually_mono) |
56949 | 558 |
next |
559 |
fix a assume "max x y < a" |
|
560 |
then show "eventually (\<lambda>x. max (X x) (Y x) < a) net" |
|
561 |
using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a] |
|
562 |
by (auto simp: eventually_conj_iff) |
|
563 |
qed |
|
564 |
||
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
565 |
lemma (in linorder_topology) tendsto_min: |
61973 | 566 |
assumes X: "(X \<longlongrightarrow> x) net" |
567 |
assumes Y: "(Y \<longlongrightarrow> y) net" |
|
568 |
shows "((\<lambda>x. min (X x) (Y x)) \<longlongrightarrow> min x y) net" |
|
56949 | 569 |
proof (rule order_tendstoI) |
570 |
fix a assume "a < min x y" |
|
571 |
then show "eventually (\<lambda>x. a < min (X x) (Y x)) net" |
|
572 |
using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a] |
|
573 |
by (auto simp: eventually_conj_iff) |
|
574 |
next |
|
575 |
fix a assume "min x y < a" |
|
576 |
then show "eventually (\<lambda>x. min (X x) (Y x) < a) net" |
|
577 |
using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a] |
|
61810 | 578 |
by (auto simp: min_less_iff_disj elim: eventually_mono) |
56949 | 579 |
qed |
580 |
||
61973 | 581 |
lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((\<lambda>x. x) \<longlongrightarrow> a) (at a within s)" |
51471 | 582 |
unfolding tendsto_def eventually_at_topological by auto |
583 |
||
61973 | 584 |
lemma (in topological_space) tendsto_const [tendsto_intros, simp, intro]: "((\<lambda>x. k) \<longlongrightarrow> k) F" |
51471 | 585 |
by (simp add: tendsto_def) |
586 |
||
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
|
587 |
lemma (in t2_space) tendsto_unique: |
61973 | 588 |
assumes "F \<noteq> bot" and "(f \<longlongrightarrow> a) F" and "(f \<longlongrightarrow> b) F" |
51471 | 589 |
shows "a = b" |
590 |
proof (rule ccontr) |
|
591 |
assume "a \<noteq> b" |
|
592 |
obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}" |
|
60758 | 593 |
using hausdorff [OF \<open>a \<noteq> b\<close>] by fast |
51471 | 594 |
have "eventually (\<lambda>x. f x \<in> U) F" |
61973 | 595 |
using \<open>(f \<longlongrightarrow> a) F\<close> \<open>open U\<close> \<open>a \<in> U\<close> by (rule topological_tendstoD) |
51471 | 596 |
moreover |
597 |
have "eventually (\<lambda>x. f x \<in> V) F" |
|
61973 | 598 |
using \<open>(f \<longlongrightarrow> b) F\<close> \<open>open V\<close> \<open>b \<in> V\<close> by (rule topological_tendstoD) |
51471 | 599 |
ultimately |
600 |
have "eventually (\<lambda>x. False) F" |
|
601 |
proof eventually_elim |
|
602 |
case (elim x) |
|
603 |
hence "f x \<in> U \<inter> V" by simp |
|
60758 | 604 |
with \<open>U \<inter> V = {}\<close> show ?case by simp |
51471 | 605 |
qed |
60758 | 606 |
with \<open>\<not> trivial_limit F\<close> show "False" |
51471 | 607 |
by (simp add: trivial_limit_def) |
608 |
qed |
|
609 |
||
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
|
610 |
lemma (in t2_space) tendsto_const_iff: |
61973 | 611 |
assumes "\<not> trivial_limit F" shows "((\<lambda>x. a :: 'a) \<longlongrightarrow> b) F \<longleftrightarrow> a = b" |
58729
e8ecc79aee43
add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents:
57953
diff
changeset
|
612 |
by (auto intro!: tendsto_unique [OF assms tendsto_const]) |
51471 | 613 |
|
614 |
lemma increasing_tendsto: |
|
615 |
fixes f :: "_ \<Rightarrow> 'a::order_topology" |
|
616 |
assumes bdd: "eventually (\<lambda>n. f n \<le> l) F" |
|
617 |
and en: "\<And>x. x < l \<Longrightarrow> eventually (\<lambda>n. x < f n) F" |
|
61973 | 618 |
shows "(f \<longlongrightarrow> l) F" |
61810 | 619 |
using assms by (intro order_tendstoI) (auto elim!: eventually_mono) |
51471 | 620 |
|
621 |
lemma decreasing_tendsto: |
|
622 |
fixes f :: "_ \<Rightarrow> 'a::order_topology" |
|
623 |
assumes bdd: "eventually (\<lambda>n. l \<le> f n) F" |
|
624 |
and en: "\<And>x. l < x \<Longrightarrow> eventually (\<lambda>n. f n < x) F" |
|
61973 | 625 |
shows "(f \<longlongrightarrow> l) F" |
61810 | 626 |
using assms by (intro order_tendstoI) (auto elim!: eventually_mono) |
51471 | 627 |
|
628 |
lemma tendsto_sandwich: |
|
629 |
fixes f g h :: "'a \<Rightarrow> 'b::order_topology" |
|
630 |
assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net" |
|
61973 | 631 |
assumes lim: "(f \<longlongrightarrow> c) net" "(h \<longlongrightarrow> c) net" |
632 |
shows "(g \<longlongrightarrow> c) net" |
|
51471 | 633 |
proof (rule order_tendstoI) |
634 |
fix a show "a < c \<Longrightarrow> eventually (\<lambda>x. a < g x) net" |
|
635 |
using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2) |
|
636 |
next |
|
637 |
fix a show "c < a \<Longrightarrow> eventually (\<lambda>x. g x < a) net" |
|
638 |
using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2) |
|
639 |
qed |
|
640 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
641 |
lemma limit_frequently_eq: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
642 |
assumes "F \<noteq> bot" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
643 |
assumes "frequently (\<lambda>x. f x = c) F" |
61973 | 644 |
assumes "(f \<longlongrightarrow> d) F" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
645 |
shows "d = (c :: 'a :: t1_space)" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
646 |
proof (rule ccontr) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
647 |
assume "d \<noteq> c" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
648 |
from t1_space[OF this] obtain U where "open U" "d \<in> U" "c \<notin> U" by blast |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
649 |
from this assms have "eventually (\<lambda>x. f x \<in> U) F" unfolding tendsto_def by blast |
61799 | 650 |
hence "eventually (\<lambda>x. f x \<noteq> c) F" by eventually_elim (insert \<open>c \<notin> U\<close>, blast) |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
651 |
with assms(2) show False unfolding frequently_def by contradiction |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
652 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
653 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
654 |
lemma tendsto_imp_eventually_ne: |
61973 | 655 |
assumes "F \<noteq> bot" "(f \<longlongrightarrow> c) F" "c \<noteq> (c' :: 'a :: t1_space)" |
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
656 |
shows "eventually (\<lambda>z. f z \<noteq> c') F" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
657 |
proof (rule ccontr) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
658 |
assume "\<not>eventually (\<lambda>z. f z \<noteq> c') F" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
659 |
hence "frequently (\<lambda>z. f z = c') F" by (simp add: frequently_def) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
660 |
from limit_frequently_eq[OF assms(1) this assms(2)] and assms(3) show False by contradiction |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
661 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
662 |
|
51471 | 663 |
lemma tendsto_le: |
664 |
fixes f g :: "'a \<Rightarrow> 'b::linorder_topology" |
|
665 |
assumes F: "\<not> trivial_limit F" |
|
61973 | 666 |
assumes x: "(f \<longlongrightarrow> x) F" and y: "(g \<longlongrightarrow> y) F" |
51471 | 667 |
assumes ev: "eventually (\<lambda>x. g x \<le> f x) F" |
668 |
shows "y \<le> x" |
|
669 |
proof (rule ccontr) |
|
670 |
assume "\<not> y \<le> x" |
|
671 |
with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a} \<inter> {b<..} = {}" |
|
672 |
by (auto simp: not_le) |
|
673 |
then have "eventually (\<lambda>x. f x < a) F" "eventually (\<lambda>x. b < g x) F" |
|
674 |
using x y by (auto intro: order_tendstoD) |
|
675 |
with ev have "eventually (\<lambda>x. False) F" |
|
676 |
by eventually_elim (insert xy, fastforce) |
|
677 |
with F show False |
|
678 |
by (simp add: eventually_False) |
|
679 |
qed |
|
680 |
||
681 |
lemma tendsto_le_const: |
|
682 |
fixes f :: "'a \<Rightarrow> 'b::linorder_topology" |
|
683 |
assumes F: "\<not> trivial_limit F" |
|
61973 | 684 |
assumes x: "(f \<longlongrightarrow> x) F" and a: "eventually (\<lambda>i. a \<le> f i) F" |
51471 | 685 |
shows "a \<le> x" |
686 |
using F x tendsto_const a by (rule tendsto_le) |
|
687 |
||
56289 | 688 |
lemma tendsto_ge_const: |
689 |
fixes f :: "'a \<Rightarrow> 'b::linorder_topology" |
|
690 |
assumes F: "\<not> trivial_limit F" |
|
61973 | 691 |
assumes x: "(f \<longlongrightarrow> x) F" and a: "eventually (\<lambda>i. a \<ge> f i) F" |
56289 | 692 |
shows "a \<ge> x" |
693 |
by (rule tendsto_le [OF F tendsto_const x a]) |
|
694 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
695 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
696 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
697 |
|
60758 | 698 |
subsubsection \<open>Rules about @{const Lim}\<close> |
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
|
699 |
|
57276 | 700 |
lemma tendsto_Lim: |
61973 | 701 |
"\<not>(trivial_limit net) \<Longrightarrow> (f \<longlongrightarrow> l) net \<Longrightarrow> Lim net f = l" |
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
|
702 |
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
|
703 |
|
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
|
704 |
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
|
705 |
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
|
706 |
|
51471 | 707 |
lemma filterlim_at_bot_at_right: |
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
708 |
fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder" |
51471 | 709 |
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
710 |
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" |
|
711 |
assumes Q: "eventually Q (at_right a)" and bound: "\<And>b. Q b \<Longrightarrow> a < b" |
|
712 |
assumes P: "eventually P at_bot" |
|
713 |
shows "filterlim f at_bot (at_right a)" |
|
714 |
proof - |
|
715 |
from P obtain x where x: "\<And>y. y \<le> x \<Longrightarrow> P y" |
|
716 |
unfolding eventually_at_bot_linorder by auto |
|
717 |
show ?thesis |
|
718 |
proof (intro filterlim_at_bot_le[THEN iffD2] allI impI) |
|
719 |
fix z assume "z \<le> x" |
|
720 |
with x have "P z" by auto |
|
721 |
have "eventually (\<lambda>x. x \<le> g z) (at_right a)" |
|
60758 | 722 |
using bound[OF bij(2)[OF \<open>P z\<close>]] |
723 |
unfolding eventually_at_right[OF bound[OF bij(2)[OF \<open>P z\<close>]]] by (auto intro!: exI[of _ "g z"]) |
|
51471 | 724 |
with Q show "eventually (\<lambda>x. f x \<le> z) (at_right a)" |
60758 | 725 |
by eventually_elim (metis bij \<open>P z\<close> mono) |
51471 | 726 |
qed |
727 |
qed |
|
728 |
||
729 |
lemma filterlim_at_top_at_left: |
|
57275
0ddb5b755cdc
moved lemmas from the proof of the Central Limit Theorem by Jeremy Avigad and Luke Serafin
hoelzl
parents:
57025
diff
changeset
|
730 |
fixes f :: "'a::linorder_topology \<Rightarrow> 'b::linorder" |
51471 | 731 |
assumes mono: "\<And>x y. Q x \<Longrightarrow> Q y \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
732 |
assumes bij: "\<And>x. P x \<Longrightarrow> f (g x) = x" "\<And>x. P x \<Longrightarrow> Q (g x)" |
|
733 |
assumes Q: "eventually Q (at_left a)" and bound: "\<And>b. Q b \<Longrightarrow> b < a" |
|
734 |
assumes P: "eventually P at_top" |
|
735 |
shows "filterlim f at_top (at_left a)" |
|
736 |
proof - |
|
737 |
from P obtain x where x: "\<And>y. x \<le> y \<Longrightarrow> P y" |
|
738 |
unfolding eventually_at_top_linorder by auto |
|
739 |
show ?thesis |
|
740 |
proof (intro filterlim_at_top_ge[THEN iffD2] allI impI) |
|
741 |
fix z assume "x \<le> z" |
|
742 |
with x have "P z" by auto |
|
743 |
have "eventually (\<lambda>x. g z \<le> x) (at_left a)" |
|
60758 | 744 |
using bound[OF bij(2)[OF \<open>P z\<close>]] |
745 |
unfolding eventually_at_left[OF bound[OF bij(2)[OF \<open>P z\<close>]]] by (auto intro!: exI[of _ "g z"]) |
|
51471 | 746 |
with Q show "eventually (\<lambda>x. z \<le> f x) (at_left a)" |
60758 | 747 |
by eventually_elim (metis bij \<open>P z\<close> mono) |
51471 | 748 |
qed |
749 |
qed |
|
750 |
||
751 |
lemma filterlim_split_at: |
|
752 |
"filterlim f F (at_left x) \<Longrightarrow> filterlim f F (at_right x) \<Longrightarrow> filterlim f F (at (x::'a::linorder_topology))" |
|
753 |
by (subst at_eq_sup_left_right) (rule filterlim_sup) |
|
754 |
||
755 |
lemma filterlim_at_split: |
|
756 |
"filterlim f F (at (x::'a::linorder_topology)) \<longleftrightarrow> filterlim f F (at_left x) \<and> filterlim f F (at_right x)" |
|
757 |
by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup) |
|
758 |
||
57025 | 759 |
lemma eventually_nhds_top: |
760 |
fixes P :: "'a :: {order_top, linorder_topology} \<Rightarrow> bool" |
|
761 |
assumes "(b::'a) < top" |
|
762 |
shows "eventually P (nhds top) \<longleftrightarrow> (\<exists>b<top. (\<forall>z. b < z \<longrightarrow> P z))" |
|
763 |
unfolding eventually_nhds |
|
764 |
proof safe |
|
765 |
fix S :: "'a set" assume "open S" "top \<in> S" |
|
60758 | 766 |
note open_left[OF this \<open>b < top\<close>] |
57025 | 767 |
moreover assume "\<forall>s\<in>S. P s" |
768 |
ultimately show "\<exists>b<top. \<forall>z>b. P z" |
|
769 |
by (auto simp: subset_eq Ball_def) |
|
770 |
next |
|
771 |
fix b assume "b < top" "\<forall>z>b. P z" |
|
772 |
then show "\<exists>S. open S \<and> top \<in> S \<and> (\<forall>xa\<in>S. P xa)" |
|
773 |
by (intro exI[of _ "{b <..}"]) auto |
|
774 |
qed |
|
51471 | 775 |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
776 |
lemma tendsto_at_within_iff_tendsto_nhds: |
61973 | 777 |
"(g \<longlongrightarrow> g l) (at l within S) \<longleftrightarrow> (g \<longlongrightarrow> g l) (inf (nhds l) (principal S))" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
778 |
unfolding tendsto_def eventually_at_filter eventually_inf_principal |
61810 | 779 |
by (intro ext all_cong imp_cong) (auto elim!: eventually_mono) |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
780 |
|
60758 | 781 |
subsection \<open>Limits on sequences\<close> |
51471 | 782 |
|
783 |
abbreviation (in topological_space) |
|
784 |
LIMSEQ :: "[nat \<Rightarrow> 'a, 'a] \<Rightarrow> bool" |
|
61969 | 785 |
("((_)/ \<longlonglongrightarrow> (_))" [60, 60] 60) where |
61973 | 786 |
"X \<longlonglongrightarrow> L \<equiv> (X \<longlongrightarrow> L) sequentially" |
51471 | 787 |
|
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
|
788 |
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
|
789 |
"lim X \<equiv> Lim sequentially X" |
51471 | 790 |
|
791 |
definition (in topological_space) convergent :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" where |
|
61969 | 792 |
"convergent X = (\<exists>L. X \<longlonglongrightarrow> L)" |
51471 | 793 |
|
61969 | 794 |
lemma lim_def: "lim X = (THE L. X \<longlonglongrightarrow> L)" |
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
|
795 |
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
|
796 |
|
60758 | 797 |
subsubsection \<open>Monotone sequences and subsequences\<close> |
51471 | 798 |
|
799 |
definition |
|
800 |
monoseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where |
|
61799 | 801 |
\<comment>\<open>Definition of monotonicity. |
51471 | 802 |
The use of disjunction here complicates proofs considerably. |
803 |
One alternative is to add a Boolean argument to indicate the direction. |
|
60758 | 804 |
Another is to develop the notions of increasing and decreasing first.\<close> |
56020
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
805 |
"monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) \<or> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
806 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
807 |
abbreviation incseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
808 |
"incseq X \<equiv> mono X" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
809 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
810 |
lemma incseq_def: "incseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<ge> X m)" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
811 |
unfolding mono_def .. |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
812 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
813 |
abbreviation decseq :: "(nat \<Rightarrow> 'a::order) \<Rightarrow> bool" where |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
814 |
"decseq X \<equiv> antimono X" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
815 |
|
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
816 |
lemma decseq_def: "decseq X \<longleftrightarrow> (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)" |
f92479477c52
introduced antimono; incseq, decseq are now abbreviations for mono and antimono; renamed Library/Continuity to Library/Order_Continuity; removed up_cont; renamed down_cont to down_continuity and generalized to complete_lattices
hoelzl
parents:
55945
diff
changeset
|
817 |
unfolding antimono_def .. |
51471 | 818 |
|
819 |
definition |
|
820 |
subseq :: "(nat \<Rightarrow> nat) \<Rightarrow> bool" where |
|
61799 | 821 |
\<comment>\<open>Definition of subsequence\<close> |
51471 | 822 |
"subseq f \<longleftrightarrow> (\<forall>m. \<forall>n>m. f m < f n)" |
823 |
||
824 |
lemma incseq_SucI: |
|
825 |
"(\<And>n. X n \<le> X (Suc n)) \<Longrightarrow> incseq X" |
|
826 |
using lift_Suc_mono_le[of X] |
|
827 |
by (auto simp: incseq_def) |
|
828 |
||
829 |
lemma incseqD: "\<And>i j. incseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f i \<le> f j" |
|
830 |
by (auto simp: incseq_def) |
|
831 |
||
832 |
lemma incseq_SucD: "incseq A \<Longrightarrow> A i \<le> A (Suc i)" |
|
833 |
using incseqD[of A i "Suc i"] by auto |
|
834 |
||
835 |
lemma incseq_Suc_iff: "incseq f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))" |
|
836 |
by (auto intro: incseq_SucI dest: incseq_SucD) |
|
837 |
||
838 |
lemma incseq_const[simp, intro]: "incseq (\<lambda>x. k)" |
|
839 |
unfolding incseq_def by auto |
|
840 |
||
841 |
lemma decseq_SucI: |
|
842 |
"(\<And>n. X (Suc n) \<le> X n) \<Longrightarrow> decseq X" |
|
843 |
using order.lift_Suc_mono_le[OF dual_order, of X] |
|
844 |
by (auto simp: decseq_def) |
|
845 |
||
846 |
lemma decseqD: "\<And>i j. decseq f \<Longrightarrow> i \<le> j \<Longrightarrow> f j \<le> f i" |
|
847 |
by (auto simp: decseq_def) |
|
848 |
||
849 |
lemma decseq_SucD: "decseq A \<Longrightarrow> A (Suc i) \<le> A i" |
|
850 |
using decseqD[of A i "Suc i"] by auto |
|
851 |
||
852 |
lemma decseq_Suc_iff: "decseq f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)" |
|
853 |
by (auto intro: decseq_SucI dest: decseq_SucD) |
|
854 |
||
855 |
lemma decseq_const[simp, intro]: "decseq (\<lambda>x. k)" |
|
856 |
unfolding decseq_def by auto |
|
857 |
||
858 |
lemma monoseq_iff: "monoseq X \<longleftrightarrow> incseq X \<or> decseq X" |
|
859 |
unfolding monoseq_def incseq_def decseq_def .. |
|
860 |
||
861 |
lemma monoseq_Suc: |
|
862 |
"monoseq X \<longleftrightarrow> (\<forall>n. X n \<le> X (Suc n)) \<or> (\<forall>n. X (Suc n) \<le> X n)" |
|
863 |
unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff .. |
|
864 |
||
865 |
lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X" |
|
866 |
by (simp add: monoseq_def) |
|
867 |
||
868 |
lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X" |
|
869 |
by (simp add: monoseq_def) |
|
870 |
||
871 |
lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X" |
|
872 |
by (simp add: monoseq_Suc) |
|
873 |
||
874 |
lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X" |
|
875 |
by (simp add: monoseq_Suc) |
|
876 |
||
877 |
lemma monoseq_minus: |
|
878 |
fixes a :: "nat \<Rightarrow> 'a::ordered_ab_group_add" |
|
879 |
assumes "monoseq a" |
|
880 |
shows "monoseq (\<lambda> n. - a n)" |
|
881 |
proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n") |
|
882 |
case True |
|
883 |
hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto |
|
884 |
thus ?thesis by (rule monoI2) |
|
885 |
next |
|
886 |
case False |
|
60758 | 887 |
hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using \<open>monoseq a\<close>[unfolded monoseq_def] by auto |
51471 | 888 |
thus ?thesis by (rule monoI1) |
889 |
qed |
|
890 |
||
60758 | 891 |
text\<open>Subsequence (alternative definition, (e.g. Hoskins)\<close> |
51471 | 892 |
|
893 |
lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))" |
|
894 |
apply (simp add: subseq_def) |
|
895 |
apply (auto dest!: less_imp_Suc_add) |
|
896 |
apply (induct_tac k) |
|
897 |
apply (auto intro: less_trans) |
|
898 |
done |
|
899 |
||
60758 | 900 |
text\<open>for any sequence, there is a monotonic subsequence\<close> |
51471 | 901 |
lemma seq_monosub: |
902 |
fixes s :: "nat => 'a::linorder" |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
903 |
shows "\<exists>f. subseq f \<and> monoseq (\<lambda>n. (s (f n)))" |
51471 | 904 |
proof cases |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
905 |
assume "\<forall>n. \<exists>p>n. \<forall>m\<ge>p. s m \<le> s p" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
906 |
then have "\<exists>f. \<forall>n. (\<forall>m\<ge>f n. s m \<le> s (f n)) \<and> f n < f (Suc n)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
907 |
by (intro dependent_nat_choice) (auto simp: conj_commute) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
908 |
then obtain f where "subseq f" and mono: "\<And>n m. f n \<le> m \<Longrightarrow> s m \<le> s (f n)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
909 |
by (auto simp: subseq_Suc_iff) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
910 |
moreover |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
911 |
then have "incseq f" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
912 |
unfolding subseq_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
913 |
then have "monoseq (\<lambda>n. s (f n))" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
914 |
by (auto simp add: incseq_def intro!: mono monoI2) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
915 |
ultimately show ?thesis |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
916 |
by auto |
51471 | 917 |
next |
918 |
assume "\<not> (\<forall>n. \<exists>p>n. (\<forall>m\<ge>p. s m \<le> s p))" |
|
919 |
then obtain N where N: "\<And>p. p > N \<Longrightarrow> \<exists>m>p. s p < s m" by (force simp: not_le le_less) |
|
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
920 |
have "\<exists>f. \<forall>n. N < f n \<and> f n < f (Suc n) \<and> s (f n) \<le> s (f (Suc n))" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
921 |
proof (intro dependent_nat_choice) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
922 |
fix x assume "N < x" with N[of x] show "\<exists>y>N. x < y \<and> s x \<le> s y" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
923 |
by (auto intro: less_trans) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
924 |
qed auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
925 |
then show ?thesis |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
926 |
by (auto simp: monoseq_iff incseq_Suc_iff subseq_Suc_iff) |
51471 | 927 |
qed |
928 |
||
929 |
lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n" |
|
930 |
proof(induct n) |
|
931 |
case 0 thus ?case by simp |
|
932 |
next |
|
933 |
case (Suc n) |
|
934 |
from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps |
|
935 |
have "n < f (Suc n)" by arith |
|
936 |
thus ?case by arith |
|
937 |
qed |
|
938 |
||
939 |
lemma eventually_subseq: |
|
940 |
"subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially" |
|
941 |
unfolding eventually_sequentially by (metis seq_suble le_trans) |
|
942 |
||
51473 | 943 |
lemma not_eventually_sequentiallyD: |
944 |
assumes P: "\<not> eventually P sequentially" |
|
945 |
shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))" |
|
946 |
proof - |
|
947 |
from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m" |
|
948 |
unfolding eventually_sequentially by (simp add: not_less) |
|
949 |
then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)" |
|
950 |
by (auto simp: choice_iff) |
|
951 |
then show ?thesis |
|
952 |
by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"] |
|
953 |
simp: less_eq_Suc_le subseq_Suc_iff) |
|
954 |
qed |
|
955 |
||
51471 | 956 |
lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially" |
957 |
unfolding filterlim_iff by (metis eventually_subseq) |
|
958 |
||
959 |
lemma subseq_o: "subseq r \<Longrightarrow> subseq s \<Longrightarrow> subseq (r \<circ> s)" |
|
960 |
unfolding subseq_def by simp |
|
961 |
||
962 |
lemma subseq_mono: assumes "subseq r" "m < n" shows "r m < r n" |
|
963 |
using assms by (auto simp: subseq_def) |
|
964 |
||
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
965 |
lemma subseq_imp_inj_on: "subseq g \<Longrightarrow> inj_on g A" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
966 |
proof (rule inj_onI) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
967 |
assume g: "subseq g" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
968 |
fix x y assume "g x = g y" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
969 |
with subseq_mono[OF g, of x y] subseq_mono[OF g, of y x] show "x = y" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
970 |
by (cases x y rule: linorder_cases) simp_all |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
971 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
972 |
|
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
973 |
lemma subseq_strict_mono: "subseq g \<Longrightarrow> strict_mono g" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
974 |
by (intro strict_monoI subseq_mono[of g]) |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
975 |
|
51471 | 976 |
lemma incseq_imp_monoseq: "incseq X \<Longrightarrow> monoseq X" |
977 |
by (simp add: incseq_def monoseq_def) |
|
978 |
||
979 |
lemma decseq_imp_monoseq: "decseq X \<Longrightarrow> monoseq X" |
|
980 |
by (simp add: decseq_def monoseq_def) |
|
981 |
||
982 |
lemma decseq_eq_incseq: |
|
983 |
fixes X :: "nat \<Rightarrow> 'a::ordered_ab_group_add" shows "decseq X = incseq (\<lambda>n. - X n)" |
|
984 |
by (simp add: decseq_def incseq_def) |
|
985 |
||
986 |
lemma INT_decseq_offset: |
|
987 |
assumes "decseq F" |
|
988 |
shows "(\<Inter>i. F i) = (\<Inter>i\<in>{n..}. F i)" |
|
989 |
proof safe |
|
990 |
fix x i assume x: "x \<in> (\<Inter>i\<in>{n..}. F i)" |
|
991 |
show "x \<in> F i" |
|
992 |
proof cases |
|
993 |
from x have "x \<in> F n" by auto |
|
60758 | 994 |
also assume "i \<le> n" with \<open>decseq F\<close> have "F n \<subseteq> F i" |
51471 | 995 |
unfolding decseq_def by simp |
996 |
finally show ?thesis . |
|
997 |
qed (insert x, simp) |
|
998 |
qed auto |
|
999 |
||
1000 |
lemma LIMSEQ_const_iff: |
|
1001 |
fixes k l :: "'a::t2_space" |
|
61969 | 1002 |
shows "(\<lambda>n. k) \<longlonglongrightarrow> l \<longleftrightarrow> k = l" |
51471 | 1003 |
using trivial_limit_sequentially by (rule tendsto_const_iff) |
1004 |
||
1005 |
lemma LIMSEQ_SUP: |
|
61969 | 1006 |
"incseq X \<Longrightarrow> X \<longlonglongrightarrow> (SUP i. X i :: 'a :: {complete_linorder, linorder_topology})" |
51471 | 1007 |
by (intro increasing_tendsto) |
1008 |
(auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans) |
|
1009 |
||
1010 |
lemma LIMSEQ_INF: |
|
61969 | 1011 |
"decseq X \<Longrightarrow> X \<longlonglongrightarrow> (INF i. X i :: 'a :: {complete_linorder, linorder_topology})" |
51471 | 1012 |
by (intro decreasing_tendsto) |
1013 |
(auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans) |
|
1014 |
||
1015 |
lemma LIMSEQ_ignore_initial_segment: |
|
61969 | 1016 |
"f \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (n + k)) \<longlonglongrightarrow> a" |
51474
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
1017 |
unfolding tendsto_def |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
1018 |
by (subst eventually_sequentially_seg[where k=k]) |
51471 | 1019 |
|
1020 |
lemma LIMSEQ_offset: |
|
61969 | 1021 |
"(\<lambda>n. f (n + k)) \<longlonglongrightarrow> a \<Longrightarrow> f \<longlonglongrightarrow> a" |
51474
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
1022 |
unfolding tendsto_def |
1e9e68247ad1
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
hoelzl
parents:
51473
diff
changeset
|
1023 |
by (subst (asm) eventually_sequentially_seg[where k=k]) |
51471 | 1024 |
|
61969 | 1025 |
lemma LIMSEQ_Suc: "f \<longlonglongrightarrow> l \<Longrightarrow> (\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l" |
51471 | 1026 |
by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp) |
1027 |
||
61969 | 1028 |
lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l \<Longrightarrow> f \<longlonglongrightarrow> l" |
51471 | 1029 |
by (rule_tac k="Suc 0" in LIMSEQ_offset, simp) |
1030 |
||
61969 | 1031 |
lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) \<longlonglongrightarrow> l = f \<longlonglongrightarrow> l" |
51471 | 1032 |
by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc) |
1033 |
||
1034 |
lemma LIMSEQ_unique: |
|
1035 |
fixes a b :: "'a::t2_space" |
|
61969 | 1036 |
shows "\<lbrakk>X \<longlonglongrightarrow> a; X \<longlonglongrightarrow> b\<rbrakk> \<Longrightarrow> a = b" |
51471 | 1037 |
using trivial_limit_sequentially by (rule tendsto_unique) |
1038 |
||
1039 |
lemma LIMSEQ_le_const: |
|
61969 | 1040 |
"\<lbrakk>X \<longlonglongrightarrow> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x" |
51471 | 1041 |
using tendsto_le_const[of sequentially X x a] by (simp add: eventually_sequentially) |
1042 |
||
1043 |
lemma LIMSEQ_le: |
|
61969 | 1044 |
"\<lbrakk>X \<longlonglongrightarrow> x; Y \<longlonglongrightarrow> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::'a::linorder_topology)" |
51471 | 1045 |
using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially) |
1046 |
||
1047 |
lemma LIMSEQ_le_const2: |
|
61969 | 1048 |
"\<lbrakk>X \<longlonglongrightarrow> (x::'a::linorder_topology); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a" |
58729
e8ecc79aee43
add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents:
57953
diff
changeset
|
1049 |
by (rule LIMSEQ_le[of X x "\<lambda>n. a"]) auto |
51471 | 1050 |
|
61969 | 1051 |
lemma convergentD: "convergent X ==> \<exists>L. (X \<longlonglongrightarrow> L)" |
51471 | 1052 |
by (simp add: convergent_def) |
1053 |
||
61969 | 1054 |
lemma convergentI: "(X \<longlonglongrightarrow> L) ==> convergent X" |
51471 | 1055 |
by (auto simp add: convergent_def) |
1056 |
||
61969 | 1057 |
lemma convergent_LIMSEQ_iff: "convergent X = (X \<longlonglongrightarrow> lim X)" |
51471 | 1058 |
by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def) |
1059 |
||
1060 |
lemma convergent_const: "convergent (\<lambda>n. c)" |
|
1061 |
by (rule convergentI, rule tendsto_const) |
|
1062 |
||
1063 |
lemma monoseq_le: |
|
61969 | 1064 |
"monoseq a \<Longrightarrow> a \<longlonglongrightarrow> (x::'a::linorder_topology) \<Longrightarrow> |
51471 | 1065 |
((\<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))" |
1066 |
by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff) |
|
1067 |
||
1068 |
lemma LIMSEQ_subseq_LIMSEQ: |
|
61969 | 1069 |
"\<lbrakk> X \<longlonglongrightarrow> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) \<longlonglongrightarrow> L" |
51471 | 1070 |
unfolding comp_def by (rule filterlim_compose[of X, OF _ filterlim_subseq]) |
1071 |
||
1072 |
lemma convergent_subseq_convergent: |
|
1073 |
"\<lbrakk>convergent X; subseq f\<rbrakk> \<Longrightarrow> convergent (X o f)" |
|
1074 |
unfolding convergent_def by (auto intro: LIMSEQ_subseq_LIMSEQ) |
|
1075 |
||
61969 | 1076 |
lemma limI: "X \<longlonglongrightarrow> L ==> lim X = L" |
57276 | 1077 |
by (rule tendsto_Lim) (rule trivial_limit_sequentially) |
51471 | 1078 |
|
1079 |
lemma lim_le: "convergent f \<Longrightarrow> (\<And>n. f n \<le> (x::'a::linorder_topology)) \<Longrightarrow> lim f \<le> x" |
|
1080 |
using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff) |
|
1081 |
||
60758 | 1082 |
subsubsection\<open>Increasing and Decreasing Series\<close> |
51471 | 1083 |
|
61969 | 1084 |
lemma incseq_le: "incseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> X n \<le> (L::'a::linorder_topology)" |
51471 | 1085 |
by (metis incseq_def LIMSEQ_le_const) |
1086 |
||
61969 | 1087 |
lemma decseq_le: "decseq X \<Longrightarrow> X \<longlonglongrightarrow> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n" |
51471 | 1088 |
by (metis decseq_def LIMSEQ_le_const2) |
1089 |
||
60758 | 1090 |
subsection \<open>First countable topologies\<close> |
51473 | 1091 |
|
1092 |
class first_countable_topology = topological_space + |
|
1093 |
assumes first_countable_basis: |
|
1094 |
"\<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))" |
|
1095 |
||
1096 |
lemma (in first_countable_topology) countable_basis_at_decseq: |
|
1097 |
obtains A :: "nat \<Rightarrow> 'a set" where |
|
1098 |
"\<And>i. open (A i)" "\<And>i. x \<in> (A i)" |
|
1099 |
"\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially" |
|
1100 |
proof atomize_elim |
|
1101 |
from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where |
|
1102 |
nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i" |
|
1103 |
and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S" by auto |
|
1104 |
def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i" |
|
1105 |
show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> |
|
1106 |
(\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)" |
|
1107 |
proof (safe intro!: exI[of _ F]) |
|
1108 |
fix i |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1109 |
show "open (F i)" using nhds(1) by (auto simp: F_def) |
51473 | 1110 |
show "x \<in> F i" using nhds(2) by (auto simp: F_def) |
1111 |
next |
|
1112 |
fix S assume "open S" "x \<in> S" |
|
1113 |
from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto |
|
1114 |
moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i" |
|
1115 |
by (auto simp: F_def) |
|
1116 |
ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially" |
|
1117 |
by (auto simp: eventually_sequentially) |
|
1118 |
qed |
|
1119 |
qed |
|
1120 |
||
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1121 |
lemma (in first_countable_topology) nhds_countable: |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1122 |
obtains X :: "nat \<Rightarrow> 'a set" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1123 |
where "decseq X" "\<And>n. open (X n)" "\<And>n. x \<in> X n" "nhds x = (INF n. principal (X n))" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1124 |
proof - |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1125 |
from first_countable_basis obtain A :: "nat \<Rightarrow> 'a set" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1126 |
where A: "\<And>n. x \<in> A n" "\<And>n. open (A n)" "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1127 |
by metis |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1128 |
show thesis |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1129 |
proof |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1130 |
show "decseq (\<lambda>n. \<Inter>i\<le>n. A i)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1131 |
by (auto simp: decseq_def) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1132 |
show "\<And>n. x \<in> (\<Inter>i\<le>n. A i)" "\<And>n. open (\<Inter>i\<le>n. A i)" |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1133 |
using A by auto |
60585 | 1134 |
show "nhds x = (INF n. principal (\<Inter>i\<le>n. A i))" |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1135 |
using A unfolding nhds_def |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1136 |
apply (intro INF_eq) |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1137 |
apply simp_all |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1138 |
apply force |
60585 | 1139 |
apply (intro exI[of _ "\<Inter>i\<le>n. A i" for n] conjI open_INT) |
57448
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1140 |
apply auto |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1141 |
done |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1142 |
qed |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1143 |
qed |
159e45728ceb
more equalities of topological filters; strengthen dependent_nat_choice; tuned a couple of proofs
hoelzl
parents:
57447
diff
changeset
|
1144 |
|
51473 | 1145 |
lemma (in first_countable_topology) countable_basis: |
1146 |
obtains A :: "nat \<Rightarrow> 'a set" where |
|
1147 |
"\<And>i. open (A i)" "\<And>i. x \<in> A i" |
|
61969 | 1148 |
"\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F \<longlonglongrightarrow> x" |
51473 | 1149 |
proof atomize_elim |
53381 | 1150 |
obtain A :: "nat \<Rightarrow> 'a set" where A: |
1151 |
"\<And>i. open (A i)" |
|
1152 |
"\<And>i. x \<in> A i" |
|
1153 |
"\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially" |
|
1154 |
by (rule countable_basis_at_decseq) blast |
|
1155 |
{ |
|
1156 |
fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S" |
|
51473 | 1157 |
with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially" |
61810 | 1158 |
by (auto elim: eventually_mono simp: subset_eq) |
53381 | 1159 |
} |
61969 | 1160 |
with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F \<longlonglongrightarrow> x)" |
51473 | 1161 |
by (intro exI[of _ A]) (auto simp: tendsto_def) |
1162 |
qed |
|
1163 |
||
1164 |
lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within: |
|
61969 | 1165 |
assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> 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
|
1166 |
shows "eventually P (inf (nhds a) (principal s))" |
51473 | 1167 |
proof (rule ccontr) |
53381 | 1168 |
obtain A :: "nat \<Rightarrow> 'a set" where A: |
1169 |
"\<And>i. open (A i)" |
|
1170 |
"\<And>i. a \<in> A i" |
|
61969 | 1171 |
"\<And>F. \<forall>n. F n \<in> A n \<Longrightarrow> F \<longlonglongrightarrow> a" |
53381 | 1172 |
by (rule countable_basis) blast |
1173 |
assume "\<not> ?thesis" |
|
51473 | 1174 |
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
|
1175 |
unfolding eventually_inf_principal eventually_nhds by (intro choice) fastforce |
53381 | 1176 |
then obtain F where F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)" |
1177 |
by blast |
|
61969 | 1178 |
with A have "F \<longlonglongrightarrow> a" by auto |
51473 | 1179 |
hence "eventually (\<lambda>n. P (F n)) sequentially" |
1180 |
using assms F0 by simp |
|
1181 |
thus "False" by (simp add: F3) |
|
1182 |
qed |
|
1183 |
||
1184 |
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
|
1185 |
"eventually P (inf (nhds a) (principal s)) \<longleftrightarrow> |
61969 | 1186 |
(\<forall>f. (\<forall>n. f n \<in> s) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)" |
51473 | 1187 |
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
|
1188 |
assume "eventually P (inf (nhds a) (principal s))" |
51473 | 1189 |
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
|
1190 |
by (auto simp: eventually_inf_principal eventually_nhds) |
61969 | 1191 |
moreover fix f assume "\<forall>n. f n \<in> s" "f \<longlonglongrightarrow> a" |
51473 | 1192 |
ultimately show "eventually (\<lambda>n. P (f n)) sequentially" |
61810 | 1193 |
by (auto dest!: topological_tendstoD elim: eventually_mono) |
51473 | 1194 |
qed |
1195 |
||
1196 |
lemma (in first_countable_topology) eventually_nhds_iff_sequentially: |
|
61969 | 1197 |
"eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)" |
51473 | 1198 |
using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp |
1199 |
||
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1200 |
lemma tendsto_at_iff_sequentially: |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1201 |
fixes f :: "'a :: first_countable_topology \<Rightarrow> _" |
61973 | 1202 |
shows "(f \<longlongrightarrow> a) (at x within s) \<longleftrightarrow> (\<forall>X. (\<forall>i. X i \<in> s - {x}) \<longrightarrow> X \<longlonglongrightarrow> x \<longrightarrow> ((f \<circ> X) \<longlonglongrightarrow> a))" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1203 |
unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap at_within_def eventually_nhds_within_iff_sequentially comp_def |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1204 |
by metis |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1205 |
|
60758 | 1206 |
subsection \<open>Function limit at a point\<close> |
51471 | 1207 |
|
1208 |
abbreviation |
|
1209 |
LIM :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" |
|
61976 | 1210 |
("((_)/ \<midarrow>(_)/\<rightarrow> (_))" [60, 0, 60] 60) where |
1211 |
"f \<midarrow>a\<rightarrow> L \<equiv> (f \<longlongrightarrow> L) (at a)" |
|
51471 | 1212 |
|
61976 | 1213 |
lemma tendsto_within_open: "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow> (f \<midarrow>a\<rightarrow> 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
|
1214 |
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
|
1215 |
|
51471 | 1216 |
lemma LIM_const_not_eq[tendsto_intros]: |
1217 |
fixes a :: "'a::perfect_space" |
|
1218 |
fixes k L :: "'b::t2_space" |
|
61976 | 1219 |
shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow> L" |
51471 | 1220 |
by (simp add: tendsto_const_iff) |
1221 |
||
1222 |
lemmas LIM_not_zero = LIM_const_not_eq [where L = 0] |
|
1223 |
||
1224 |
lemma LIM_const_eq: |
|
1225 |
fixes a :: "'a::perfect_space" |
|
1226 |
fixes k L :: "'b::t2_space" |
|
61976 | 1227 |
shows "(\<lambda>x. k) \<midarrow>a\<rightarrow> L \<Longrightarrow> k = L" |
51471 | 1228 |
by (simp add: tendsto_const_iff) |
1229 |
||
1230 |
lemma LIM_unique: |
|
1231 |
fixes a :: "'a::perfect_space" and L M :: "'b::t2_space" |
|
61976 | 1232 |
shows "f \<midarrow>a\<rightarrow> L \<Longrightarrow> f \<midarrow>a\<rightarrow> M \<Longrightarrow> L = M" |
51471 | 1233 |
using at_neq_bot by (rule tendsto_unique) |
1234 |
||
60758 | 1235 |
text \<open>Limits are equal for functions equal except at limit point\<close> |
51471 | 1236 |
|
61976 | 1237 |
lemma LIM_equal: "\<forall>x. x \<noteq> a --> (f x = g x) \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>a\<rightarrow> l)" |
51471 | 1238 |
unfolding tendsto_def eventually_at_topological by simp |
1239 |
||
61976 | 1240 |
lemma LIM_cong: "a = b \<Longrightarrow> (\<And>x. x \<noteq> b \<Longrightarrow> f x = g x) \<Longrightarrow> l = m \<Longrightarrow> (f \<midarrow>a\<rightarrow> l) \<longleftrightarrow> (g \<midarrow>b\<rightarrow> m)" |
51471 | 1241 |
by (simp add: LIM_equal) |
1242 |
||
61976 | 1243 |
lemma LIM_cong_limit: "f \<midarrow>x\<rightarrow> L \<Longrightarrow> K = L \<Longrightarrow> f \<midarrow>x\<rightarrow> K" |
51471 | 1244 |
by simp |
1245 |
||
1246 |
lemma tendsto_at_iff_tendsto_nhds: |
|
61976 | 1247 |
"g \<midarrow>l\<rightarrow> g l \<longleftrightarrow> (g \<longlongrightarrow> 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
|
1248 |
unfolding tendsto_def eventually_at_filter |
61810 | 1249 |
by (intro ext all_cong imp_cong) (auto elim!: eventually_mono) |
51471 | 1250 |
|
1251 |
lemma tendsto_compose: |
|
61976 | 1252 |
"g \<midarrow>l\<rightarrow> g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F" |
51471 | 1253 |
unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g]) |
1254 |
||
61976 | 1255 |
lemma LIM_o: "\<lbrakk>g \<midarrow>l\<rightarrow> g l; f \<midarrow>a\<rightarrow> l\<rbrakk> \<Longrightarrow> (g \<circ> f) \<midarrow>a\<rightarrow> g l" |
51471 | 1256 |
unfolding o_def by (rule tendsto_compose) |
1257 |
||
1258 |
lemma tendsto_compose_eventually: |
|
61976 | 1259 |
"g \<midarrow>l\<rightarrow> m \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> eventually (\<lambda>x. f x \<noteq> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> m) F" |
51471 | 1260 |
by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at) |
1261 |
||
1262 |
lemma LIM_compose_eventually: |
|
61976 | 1263 |
assumes f: "f \<midarrow>a\<rightarrow> b" |
1264 |
assumes g: "g \<midarrow>b\<rightarrow> c" |
|
51471 | 1265 |
assumes inj: "eventually (\<lambda>x. f x \<noteq> b) (at a)" |
61976 | 1266 |
shows "(\<lambda>x. g (f x)) \<midarrow>a\<rightarrow> c" |
51471 | 1267 |
using g f inj by (rule tendsto_compose_eventually) |
1268 |
||
61973 | 1269 |
lemma tendsto_compose_filtermap: "((g \<circ> f) \<longlongrightarrow> T) F \<longleftrightarrow> (g \<longlongrightarrow> T) (filtermap f F)" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1270 |
by (simp add: filterlim_def filtermap_filtermap comp_def) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1271 |
|
60758 | 1272 |
subsubsection \<open>Relation of LIM and LIMSEQ\<close> |
51473 | 1273 |
|
1274 |
lemma (in first_countable_topology) sequentially_imp_eventually_within: |
|
61969 | 1275 |
"(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> |
51473 | 1276 |
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
|
1277 |
unfolding at_within_def |
51473 | 1278 |
by (intro sequentially_imp_eventually_nhds_within) auto |
1279 |
||
1280 |
lemma (in first_countable_topology) sequentially_imp_eventually_at: |
|
61969 | 1281 |
"(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f \<longlonglongrightarrow> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)" |
51473 | 1282 |
using assms sequentially_imp_eventually_within [where s=UNIV] by simp |
1283 |
||
1284 |
lemma LIMSEQ_SEQ_conv1: |
|
1285 |
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space" |
|
61976 | 1286 |
assumes f: "f \<midarrow>a\<rightarrow> l" |
61969 | 1287 |
shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l" |
51473 | 1288 |
using tendsto_compose_eventually [OF f, where F=sequentially] by simp |
1289 |
||
1290 |
lemma LIMSEQ_SEQ_conv2: |
|
1291 |
fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space" |
|
61969 | 1292 |
assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. f (S n)) \<longlonglongrightarrow> l" |
61976 | 1293 |
shows "f \<midarrow>a\<rightarrow> l" |
51473 | 1294 |
using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at) |
1295 |
||
1296 |
lemma LIMSEQ_SEQ_conv: |
|
61969 | 1297 |
"(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S \<longlonglongrightarrow> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L) = |
61976 | 1298 |
(X \<midarrow>a\<rightarrow> (L::'b::topological_space))" |
51473 | 1299 |
using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 .. |
1300 |
||
57025 | 1301 |
lemma sequentially_imp_eventually_at_left: |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60150
diff
changeset
|
1302 |
fixes a :: "'a :: {linorder_topology, first_countable_topology}" |
57025 | 1303 |
assumes b[simp]: "b < a" |
61969 | 1304 |
assumes *: "\<And>f. (\<And>n. b < f n) \<Longrightarrow> (\<And>n. f n < a) \<Longrightarrow> incseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially" |
57025 | 1305 |
shows "eventually P (at_left a)" |
1306 |
proof (safe intro!: sequentially_imp_eventually_within) |
|
61969 | 1307 |
fix X assume X: "\<forall>n. X n \<in> {..< a} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a" |
57025 | 1308 |
show "eventually (\<lambda>n. P (X n)) sequentially" |
1309 |
proof (rule ccontr) |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1310 |
assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1311 |
have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> b < X (s n)) \<and> (X (s n) \<le> X (s (Suc n)) \<and> Suc (s n) \<le> s (Suc n))" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1312 |
proof (rule dependent_nat_choice) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1313 |
have "\<not> eventually (\<lambda>n. b < X n \<longrightarrow> P (X n)) sequentially" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1314 |
by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b]) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1315 |
then show "\<exists>x. \<not> P (X x) \<and> b < X x" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1316 |
by (auto dest!: not_eventuallyD) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1317 |
next |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1318 |
fix x n |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1319 |
have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> b < X n \<longrightarrow> X x < X n \<longrightarrow> P (X n)) sequentially" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1320 |
using X by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1321 |
then show "\<exists>n. (\<not> P (X n) \<and> b < X n) \<and> (X x \<le> X n \<and> Suc x \<le> n)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1322 |
by (auto dest!: not_eventuallyD) |
57025 | 1323 |
qed |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1324 |
then guess s .. |
61969 | 1325 |
then have "\<And>n. b < X (s n)" "\<And>n. X (s n) < a" "incseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a" "\<And>n. \<not> P (X (s n))" |
1326 |
using X by (auto simp: subseq_Suc_iff Suc_le_eq incseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def]) |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1327 |
from *[OF this(1,2,3,4)] this(5) show False by auto |
57025 | 1328 |
qed |
1329 |
qed |
|
1330 |
||
1331 |
lemma tendsto_at_left_sequentially: |
|
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60150
diff
changeset
|
1332 |
fixes a :: "_ :: {linorder_topology, first_countable_topology}" |
57025 | 1333 |
assumes "b < a" |
61969 | 1334 |
assumes *: "\<And>S. (\<And>n. S n < a) \<Longrightarrow> (\<And>n. b < S n) \<Longrightarrow> incseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" |
61973 | 1335 |
shows "(X \<longlongrightarrow> L) (at_left a)" |
57025 | 1336 |
using assms unfolding tendsto_def [where l=L] |
1337 |
by (simp add: sequentially_imp_eventually_at_left) |
|
1338 |
||
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1339 |
lemma sequentially_imp_eventually_at_right: |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60150
diff
changeset
|
1340 |
fixes a :: "'a :: {linorder_topology, first_countable_topology}" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1341 |
assumes b[simp]: "a < b" |
61969 | 1342 |
assumes *: "\<And>f. (\<And>n. a < f n) \<Longrightarrow> (\<And>n. f n < b) \<Longrightarrow> decseq f \<Longrightarrow> f \<longlonglongrightarrow> a \<Longrightarrow> eventually (\<lambda>n. P (f n)) sequentially" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1343 |
shows "eventually P (at_right a)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1344 |
proof (safe intro!: sequentially_imp_eventually_within) |
61969 | 1345 |
fix X assume X: "\<forall>n. X n \<in> {a <..} \<and> X n \<noteq> a" "X \<longlonglongrightarrow> a" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1346 |
show "eventually (\<lambda>n. P (X n)) sequentially" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1347 |
proof (rule ccontr) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1348 |
assume neg: "\<not> eventually (\<lambda>n. P (X n)) sequentially" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1349 |
have "\<exists>s. \<forall>n. (\<not> P (X (s n)) \<and> X (s n) < b) \<and> (X (s (Suc n)) \<le> X (s n) \<and> Suc (s n) \<le> s (Suc n))" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1350 |
proof (rule dependent_nat_choice) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1351 |
have "\<not> eventually (\<lambda>n. X n < b \<longrightarrow> P (X n)) sequentially" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1352 |
by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b]) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1353 |
then show "\<exists>x. \<not> P (X x) \<and> X x < b" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1354 |
by (auto dest!: not_eventuallyD) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1355 |
next |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1356 |
fix x n |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1357 |
have "\<not> eventually (\<lambda>n. Suc x \<le> n \<longrightarrow> X n < b \<longrightarrow> X n < X x \<longrightarrow> P (X n)) sequentially" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1358 |
using X by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1359 |
then show "\<exists>n. (\<not> P (X n) \<and> X n < b) \<and> (X n \<le> X x \<and> Suc x \<le> n)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1360 |
by (auto dest!: not_eventuallyD) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1361 |
qed |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1362 |
then guess s .. |
61969 | 1363 |
then have "\<And>n. a < X (s n)" "\<And>n. X (s n) < b" "decseq (\<lambda>n. X (s n))" "(\<lambda>n. X (s n)) \<longlonglongrightarrow> a" "\<And>n. \<not> P (X (s n))" |
1364 |
using X by (auto simp: subseq_Suc_iff Suc_le_eq decseq_Suc_iff intro!: LIMSEQ_subseq_LIMSEQ[OF \<open>X \<longlonglongrightarrow> a\<close>, unfolded comp_def]) |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1365 |
from *[OF this(1,2,3,4)] this(5) show False by auto |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1366 |
qed |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1367 |
qed |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1368 |
|
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1369 |
lemma tendsto_at_right_sequentially: |
60172
423273355b55
rename continuous and down_continuous in Order_Continuity to sup_/inf_continuous; relate them with topological continuity
hoelzl
parents:
60150
diff
changeset
|
1370 |
fixes a :: "_ :: {linorder_topology, first_countable_topology}" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1371 |
assumes "a < b" |
61969 | 1372 |
assumes *: "\<And>S. (\<And>n. a < S n) \<Longrightarrow> (\<And>n. S n < b) \<Longrightarrow> decseq S \<Longrightarrow> S \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L" |
61973 | 1373 |
shows "(X \<longlongrightarrow> L) (at_right a)" |
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1374 |
using assms unfolding tendsto_def [where l=L] |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1375 |
by (simp add: sequentially_imp_eventually_at_right) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1376 |
|
60758 | 1377 |
subsection \<open>Continuity\<close> |
51471 | 1378 |
|
60758 | 1379 |
subsubsection \<open>Continuity on a set\<close> |
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
|
1380 |
|
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
|
1381 |
definition continuous_on :: "'a set \<Rightarrow> ('a :: topological_space \<Rightarrow> 'b :: topological_space) \<Rightarrow> bool" where |
61973 | 1382 |
"continuous_on s f \<longleftrightarrow> (\<forall>x\<in>s. (f \<longlongrightarrow> f x) (at x within s))" |
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
|
1383 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1384 |
lemma continuous_on_cong [cong]: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1385 |
"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
|
1386 |
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
|
1387 |
|
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
|
1388 |
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
|
1389 |
"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
|
1390 |
(\<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
|
1391 |
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
|
1392 |
|
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
|
1393 |
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
|
1394 |
"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
|
1395 |
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
|
1396 |
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
|
1397 |
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
|
1398 |
by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL) |
53381 | 1399 |
then obtain A where "\<forall>x\<in>f -` B \<inter> s. open (A x) \<and> x \<in> A x \<and> s \<inter> A x \<subseteq> f -` B" |
1400 |
unfolding bchoice_iff .. |
|
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
|
1401 |
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
|
1402 |
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
|
1403 |
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
|
1404 |
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
|
1405 |
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
|
1406 |
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
|
1407 |
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
|
1408 |
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
|
1409 |
with B obtain A where A: "open A" "A \<inter> s = f -` B \<inter> s" by auto |
60758 | 1410 |
with \<open>x \<in> s\<close> \<open>f x \<in> B\<close> show "\<exists>A. open A \<and> x \<in> A \<and> (\<forall>y\<in>s. y \<in> A \<longrightarrow> f y \<in> B)" |
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
|
1411 |
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
|
1412 |
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
|
1413 |
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
|
1414 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1415 |
lemma continuous_on_open_vimage: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1416 |
"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
|
1417 |
unfolding continuous_on_open_invariant |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1418 |
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
|
1419 |
|
55734 | 1420 |
corollary continuous_imp_open_vimage: |
1421 |
assumes "continuous_on s f" "open s" "open B" "f -` B \<subseteq> s" |
|
1422 |
shows "open (f -` B)" |
|
1423 |
by (metis assms continuous_on_open_vimage le_iff_inf) |
|
1424 |
||
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56329
diff
changeset
|
1425 |
corollary open_vimage[continuous_intros]: |
55775 | 1426 |
assumes "open s" and "continuous_on UNIV f" |
1427 |
shows "open (f -` s)" |
|
1428 |
using assms unfolding continuous_on_open_vimage [OF open_UNIV] |
|
1429 |
by simp |
|
1430 |
||
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
|
1431 |
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
|
1432 |
"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
|
1433 |
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
|
1434 |
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
|
1435 |
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
|
1436 |
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
|
1437 |
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
|
1438 |
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
|
1439 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1440 |
lemma continuous_on_closed_vimage: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1441 |
"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
|
1442 |
unfolding continuous_on_closed_invariant |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1443 |
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
|
1444 |
|
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61342
diff
changeset
|
1445 |
corollary closed_vimage_Int[continuous_intros]: |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61342
diff
changeset
|
1446 |
assumes "closed s" and "continuous_on t f" and t: "closed t" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61342
diff
changeset
|
1447 |
shows "closed (f -` s \<inter> t)" |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61342
diff
changeset
|
1448 |
using assms unfolding continuous_on_closed_vimage [OF t] by simp |
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61342
diff
changeset
|
1449 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56329
diff
changeset
|
1450 |
corollary closed_vimage[continuous_intros]: |
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56329
diff
changeset
|
1451 |
assumes "closed s" and "continuous_on UNIV f" |
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56329
diff
changeset
|
1452 |
shows "closed (f -` s)" |
61426
d53db136e8fd
new material on path_component_sets, inside, outside, etc. And more default simprules
paulson <lp15@cam.ac.uk>
parents:
61342
diff
changeset
|
1453 |
using closed_vimage_Int [OF assms] by simp |
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56329
diff
changeset
|
1454 |
|
61907
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61810
diff
changeset
|
1455 |
lemma continuous_on_empty: "continuous_on {} f" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61810
diff
changeset
|
1456 |
by (simp add: continuous_on_def) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61810
diff
changeset
|
1457 |
|
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61810
diff
changeset
|
1458 |
lemma continuous_on_sing: "continuous_on {x} f" |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61810
diff
changeset
|
1459 |
by (simp add: continuous_on_def at_within_def) |
f0c894ab18c9
Liouville theorem, Fundamental Theorem of Algebra, etc.
paulson <lp15@cam.ac.uk>
parents:
61810
diff
changeset
|
1460 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1461 |
lemma continuous_on_open_Union: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1462 |
"(\<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
|
1463 |
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
|
1464 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1465 |
lemma continuous_on_open_UN: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1466 |
"(\<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
|
1467 |
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
|
1468 |
|
61204 | 1469 |
lemma continuous_on_open_Un: |
1470 |
"open s \<Longrightarrow> open t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f" |
|
1471 |
using continuous_on_open_Union [of "{s,t}"] by auto |
|
1472 |
||
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1473 |
lemma continuous_on_closed_Un: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1474 |
"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
|
1475 |
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
|
1476 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1477 |
lemma continuous_on_If: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1478 |
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
|
1479 |
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
|
1480 |
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
|
1481 |
proof- |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1482 |
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
|
1483 |
by auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1484 |
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
|
1485 |
by simp_all |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1486 |
with closed show ?thesis |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1487 |
by (rule continuous_on_closed_Un) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1488 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1489 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56329
diff
changeset
|
1490 |
lemma continuous_on_id[continuous_intros]: "continuous_on s (\<lambda>x. x)" |
58729
e8ecc79aee43
add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents:
57953
diff
changeset
|
1491 |
unfolding continuous_on_def by fast |
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
|
1492 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56329
diff
changeset
|
1493 |
lemma continuous_on_const[continuous_intros]: "continuous_on s (\<lambda>x. c)" |
58729
e8ecc79aee43
add tendsto_const and tendsto_ident_at as simp and intro rules
hoelzl
parents:
57953
diff
changeset
|
1494 |
unfolding continuous_on_def by auto |
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
|
1495 |
|
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1496 |
lemma continuous_on_subset: "continuous_on s f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous_on t f" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1497 |
unfolding continuous_on_def by (metis subset_eq tendsto_within_subset) |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1498 |
|
56371
fb9ae0727548
extend continuous_intros; remove continuous_on_intros and isCont_intros
hoelzl
parents:
56329
diff
changeset
|
1499 |
lemma continuous_on_compose[continuous_intros]: |
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
|
1500 |
"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
|
1501 |
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
|
1502 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1503 |
lemma continuous_on_compose2: |
61738
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1504 |
"continuous_on t g \<Longrightarrow> continuous_on s f \<Longrightarrow> f ` s \<subseteq> t \<Longrightarrow> continuous_on s (\<lambda>x. g (f x))" |
c4f6031f1310
New material about paths, winding numbers, etc. Added lemmas to divide_const_simps. Misc tuning.
paulson <lp15@cam.ac.uk>
parents:
61531
diff
changeset
|
1505 |
using continuous_on_compose[of s f g] continuous_on_subset by (force simp add: comp_def) |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1506 |
|
60720 | 1507 |
lemma continuous_on_generate_topology: |
1508 |
assumes *: "open = generate_topology X" |
|
1509 |
assumes **: "\<And>B. B \<in> X \<Longrightarrow> \<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A" |
|
1510 |
shows "continuous_on A f" |
|
1511 |
unfolding continuous_on_open_invariant |
|
1512 |
proof safe |
|
1513 |
fix B :: "'a set" assume "open B" then show "\<exists>C. open C \<and> C \<inter> A = f -` B \<inter> A" |
|
1514 |
unfolding * |
|
1515 |
proof induction |
|
1516 |
case (UN K) |
|
1517 |
then obtain C where "\<And>k. k \<in> K \<Longrightarrow> open (C k)" "\<And>k. k \<in> K \<Longrightarrow> C k \<inter> A = f -` k \<inter> A" |
|
1518 |
by metis |
|
1519 |
then show ?case |
|
1520 |
by (intro exI[of _ "\<Union>k\<in>K. C k"]) blast |
|
1521 |
qed (auto intro: **) |
|
1522 |
qed |
|
1523 |
||
1524 |
lemma continuous_onI_mono: |
|
1525 |
fixes f :: "'a::linorder_topology \<Rightarrow> 'b::{dense_order, linorder_topology}" |
|
1526 |
assumes "open (f`A)" |
|
1527 |
assumes mono: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y" |
|
1528 |
shows "continuous_on A f" |
|
1529 |
proof (rule continuous_on_generate_topology[OF open_generated_order], safe) |
|
1530 |
have monoD: "\<And>x y. x \<in> A \<Longrightarrow> y \<in> A \<Longrightarrow> f x < f y \<Longrightarrow> x < y" |
|
1531 |
by (auto simp: not_le[symmetric] mono) |
|
1532 |
||
1533 |
{ fix a b assume "a \<in> A" "f a < b" |
|
1534 |
moreover |
|
1535 |
with open_right[OF \<open>open (f`A)\<close>, of "f a" b] obtain y where "f a < y" "{f a ..< y} \<subseteq> f`A" |
|
1536 |
by auto |
|
1537 |
moreover then obtain z where "f a < z" "z < min b y" |
|
1538 |
using dense[of "f a" "min b y"] \<open>f a < y\<close> \<open>f a < b\<close> by auto |
|
1539 |
moreover then obtain c where "z = f c" "c \<in> A" |
|
1540 |
using \<open>{f a ..< y} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le) |
|
1541 |
ultimately have "\<exists>x. x \<in> A \<and> f x < b \<and> a < x" |
|
1542 |
by (auto intro!: exI[of _ c] simp: monoD) } |
|
1543 |
then show "\<exists>C. open C \<and> C \<inter> A = f -` {..<b} \<inter> A" for b |
|
1544 |
by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. f x < b}. {..< x})"]) |
|
1545 |
(auto intro: le_less_trans[OF mono] less_imp_le) |
|
1546 |
||
1547 |
{ fix a b assume "a \<in> A" "b < f a" |
|
1548 |
moreover |
|
1549 |
with open_left[OF \<open>open (f`A)\<close>, of "f a" b] obtain y where "y < f a" "{y <.. f a} \<subseteq> f`A" |
|
1550 |
by auto |
|
1551 |
moreover then obtain z where "max b y < z" "z < f a" |
|
1552 |
using dense[of "max b y" "f a"] \<open>y < f a\<close> \<open>b < f a\<close> by auto |
|
1553 |
moreover then obtain c where "z = f c" "c \<in> A" |
|
1554 |
using \<open>{y <.. f a} \<subseteq> f`A\<close>[THEN subsetD, of z] by (auto simp: less_imp_le) |
|
1555 |
ultimately have "\<exists>x. x \<in> A \<and> b < f x \<and> x < a" |
|
1556 |
by (auto intro!: exI[of _ c] simp: monoD) } |
|
1557 |
then show "\<exists>C. open C \<and> C \<inter> A = f -` {b <..} \<inter> A" for b |
|
1558 |
by (intro exI[of _ "(\<Union>x\<in>{x\<in>A. b < f x}. {x <..})"]) |
|
1559 |
(auto intro: less_le_trans[OF _ mono] less_imp_le) |
|
1560 |
qed |
|
1561 |
||
60758 | 1562 |
subsubsection \<open>Continuity at a point\<close> |
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
|
1563 |
|
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
|
1564 |
definition continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where |
61973 | 1565 |
"continuous F f \<longleftrightarrow> (f \<longlongrightarrow> f (Lim F (\<lambda>x. x))) F" |
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
|
1566 |
|
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
|
1567 |
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
|
1568 |
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
|
1569 |
|
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
|
1570 |
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
|
1571 |
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
|
1572 |
|
61973 | 1573 |
lemma continuous_within: "continuous (at x within s) f \<longleftrightarrow> (f \<longlongrightarrow> 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
|
1574 |
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
|
1575 |
|
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
|
1576 |
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
|
1577 |
"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
|
1578 |
(\<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
|
1579 |
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
|
1580 |
|
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
|
1581 |
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
|
1582 |
"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
|
1583 |
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
|
1584 |
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
|
1585 |
|
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
|
1586 |
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
|
1587 |
"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
|
1588 |
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
|
1589 |
using continuous_within_compose[of x s f g] by (simp add: comp_def) |
51471 | 1590 |
|
61976 | 1591 |
lemma continuous_at: "continuous (at x) f \<longleftrightarrow> f \<midarrow>x\<rightarrow> f 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
|
1592 |
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
|
1593 |
|
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
|
1594 |
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
|
1595 |
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
|
1596 |
|
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
|
1597 |
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
|
1598 |
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
|
1599 |
|
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
|
1600 |
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
|
1601 |
"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
|
1602 |
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
|
1603 |
|
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
|
1604 |
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
|
1605 |
"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
|
1606 |
|
61976 | 1607 |
lemma isCont_def: "isCont f a \<longleftrightarrow> f \<midarrow>a\<rightarrow> f a" |
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
|
1608 |
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
|
1609 |
|
60762 | 1610 |
lemma continuous_at_imp_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
|
1611 |
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
|
1612 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1613 |
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
|
1614 |
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
|
1615 |
|
62083 | 1616 |
lemma continuous_within_open: "a \<in> A \<Longrightarrow> open A \<Longrightarrow> continuous (at a within A) f \<longleftrightarrow> isCont f a" |
1617 |
by (simp add: at_within_open_NO_MATCH) |
|
1618 |
||
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
|
1619 |
lemma continuous_at_imp_continuous_on: "\<forall>x\<in>s. isCont f x \<Longrightarrow> continuous_on s f" |
60762 | 1620 |
by (auto intro: continuous_at_imp_continuous_at_within simp: continuous_on_eq_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
|
1621 |
|
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
|
1622 |
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
|
1623 |
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
|
1624 |
|
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
|
1625 |
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
|
1626 |
unfolding o_def by (rule isCont_o2) |
51471 | 1627 |
|
61973 | 1628 |
lemma isCont_tendsto_compose: "isCont g l \<Longrightarrow> (f \<longlongrightarrow> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) \<longlongrightarrow> g l) F" |
51471 | 1629 |
unfolding isCont_def by (rule tendsto_compose) |
62049
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1630 |
|
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1631 |
lemma continuous_on_tendsto_compose: |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1632 |
assumes f_cont: "continuous_on s f" |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1633 |
assumes g: "(g \<longlongrightarrow> l) F" |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1634 |
assumes l: "l \<in> s" |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1635 |
assumes ev: "\<forall>\<^sub>F x in F. g x \<in> s" |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1636 |
shows "((\<lambda>x. f (g x)) \<longlongrightarrow> f l) F" |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1637 |
proof - |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1638 |
from f_cont l have f: "(f \<longlongrightarrow> f l) (at l within s)" |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1639 |
by (simp add: continuous_on_def) |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1640 |
have i: "((\<lambda>x. if g x = l then f l else f (g x)) \<longlongrightarrow> f l) F" |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1641 |
by (rule filterlim_If) |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1642 |
(auto intro!: filterlim_compose[OF f] eventually_conj tendsto_mono[OF _ g] |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1643 |
simp: filterlim_at eventually_inf_principal eventually_mono[OF ev]) |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1644 |
show ?thesis |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1645 |
by (rule filterlim_cong[THEN iffD1[OF _ i]]) auto |
b0f941e207cf
Added lots of material on infinite sums, convergence radii, harmonic numbers, Gamma function
eberlm
parents:
61976
diff
changeset
|
1646 |
qed |
51471 | 1647 |
|
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
|
1648 |
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
|
1649 |
"isCont g (f x) \<Longrightarrow> continuous (at x within s) f \<Longrightarrow> continuous (at x within s) (\<lambda>x. g (f x))" |
60762 | 1650 |
using continuous_within_compose2[of x s f g] by (simp add: continuous_at_imp_continuous_at_within) |
51471 | 1651 |
|
57447
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1652 |
lemma filtermap_nhds_open_map: |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1653 |
assumes cont: "isCont f a" and open_map: "\<And>S. open S \<Longrightarrow> open (f`S)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1654 |
shows "filtermap f (nhds a) = nhds (f a)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1655 |
unfolding filter_eq_iff |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1656 |
proof safe |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1657 |
fix P assume "eventually P (filtermap f (nhds a))" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1658 |
then guess S unfolding eventually_filtermap eventually_nhds .. |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1659 |
then show "eventually P (nhds (f a))" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1660 |
unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1661 |
qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1662 |
|
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1663 |
lemma continuous_at_split: |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1664 |
"continuous (at (x::'a::linorder_topology)) f = (continuous (at_left x) f \<and> continuous (at_right x) f)" |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1665 |
by (simp add: continuous_within filterlim_at_split) |
87429bdecad5
import more stuff from the CLT proof; base the lborel measure on interval_measure; remove lebesgue measure
hoelzl
parents:
57276
diff
changeset
|
1666 |
|
61245 | 1667 |
subsubsection \<open>Open-cover compactness\<close> |
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
|
1668 |
|
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
|
1669 |
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
|
1670 |
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
|
1671 |
|
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
|
1672 |
definition compact :: "'a set \<Rightarrow> bool" where |
61799 | 1673 |
compact_eq_heine_borel: \<comment> "This name is used for backwards compatibility" |
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
|
1674 |
"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
|
1675 |
|
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
|
1676 |
lemma compactI: |
60585 | 1677 |
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'" |
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
|
1678 |
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
|
1679 |
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
|
1680 |
|
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
|
1681 |
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
|
1682 |
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
|
1683 |
|
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
|
1684 |
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
|
1685 |
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
|
1686 |
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
|
1687 |
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
|
1688 |
|
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
|
1689 |
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
|
1690 |
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
|
1691 |
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
|
1692 |
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
|
1693 |
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
|
1694 |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1695 |
lemma compact_inter_closed [intro]: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1696 |
assumes "compact s" and "closed t" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1697 |
shows "compact (s \<inter> t)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1698 |
proof (rule compactI) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1699 |
fix C assume C: "\<forall>c\<in>C. open c" and cover: "s \<inter> t \<subseteq> \<Union>C" |
60758 | 1700 |
from C \<open>closed t\<close> have "\<forall>c\<in>C \<union> {-t}. open c" by auto |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1701 |
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
|
1702 |
ultimately have "\<exists>D\<subseteq>C \<union> {-t}. finite D \<and> s \<subseteq> \<Union>D" |
60758 | 1703 |
using \<open>compact s\<close> unfolding compact_eq_heine_borel by auto |
53381 | 1704 |
then obtain D where "D \<subseteq> C \<union> {- t} \<and> finite D \<and> s \<subseteq> \<Union>D" .. |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1705 |
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
|
1706 |
by (intro exI[of _ "D - {-t}"]) auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1707 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1708 |
|
54797
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1709 |
lemma inj_setminus: "inj_on uminus (A::'a set set)" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1710 |
by (auto simp: inj_on_def) |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1711 |
|
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1712 |
lemma compact_fip: |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1713 |
"compact U \<longleftrightarrow> |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1714 |
(\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1715 |
(is "_ \<longleftrightarrow> ?R") |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1716 |
proof (safe intro!: compact_eq_heine_borel[THEN iffD2]) |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1717 |
fix A |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1718 |
assume "compact U" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1719 |
and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1720 |
and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1721 |
from A have "(\<forall>a\<in>uminus`A. open a) \<and> U \<subseteq> \<Union>(uminus`A)" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1722 |
by auto |
60758 | 1723 |
with \<open>compact U\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<subseteq> \<Union>(uminus`B)" |
54797
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1724 |
unfolding compact_eq_heine_borel by (metis subset_image_iff) |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1725 |
with fi[THEN spec, of B] show False |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1726 |
by (auto dest: finite_imageD intro: inj_setminus) |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1727 |
next |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1728 |
fix A |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1729 |
assume ?R |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1730 |
assume "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1731 |
then have "U \<inter> \<Inter>(uminus`A) = {}" "\<forall>a\<in>uminus`A. closed a" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1732 |
by auto |
60758 | 1733 |
with \<open>?R\<close> obtain B where "B \<subseteq> A" "finite (uminus`B)" "U \<inter> \<Inter>(uminus`B) = {}" |
54797
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1734 |
by (metis subset_image_iff) |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1735 |
then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T" |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1736 |
by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD) |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1737 |
qed |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1738 |
|
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1739 |
lemma compact_imp_fip: |
60585 | 1740 |
"compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter>f') \<noteq> {}) \<Longrightarrow> |
1741 |
s \<inter> (\<Inter>f) \<noteq> {}" |
|
54797
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1742 |
unfolding compact_fip by auto |
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1743 |
|
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1744 |
lemma compact_imp_fip_image: |
56166 | 1745 |
assumes "compact s" |
1746 |
and P: "\<And>i. i \<in> I \<Longrightarrow> closed (f i)" |
|
1747 |
and Q: "\<And>I'. finite I' \<Longrightarrow> I' \<subseteq> I \<Longrightarrow> (s \<inter> (\<Inter>i\<in>I'. f i) \<noteq> {})" |
|
1748 |
shows "s \<inter> (\<Inter>i\<in>I. f i) \<noteq> {}" |
|
1749 |
proof - |
|
60758 | 1750 |
note \<open>compact s\<close> |
56166 | 1751 |
moreover from P have "\<forall>i \<in> f ` I. closed i" by blast |
1752 |
moreover have "\<forall>A. finite A \<and> A \<subseteq> f ` I \<longrightarrow> (s \<inter> (\<Inter>A) \<noteq> {})" |
|
1753 |
proof (rule, rule, erule conjE) |
|
1754 |
fix A :: "'a set set" |
|
1755 |
assume "finite A" |
|
1756 |
moreover assume "A \<subseteq> f ` I" |
|
1757 |
ultimately obtain B where "B \<subseteq> I" and "finite B" and "A = f ` B" |
|
1758 |
using finite_subset_image [of A f I] by blast |
|
1759 |
with Q [of B] show "s \<inter> \<Inter>A \<noteq> {}" by simp |
|
1760 |
qed |
|
1761 |
ultimately have "s \<inter> (\<Inter>(f ` I)) \<noteq> {}" by (rule compact_imp_fip) |
|
1762 |
then show ?thesis by simp |
|
1763 |
qed |
|
54797
be020ec8560c
modernized ContNotDenum: use Set_Interval, and finite intersection property to show the nested interval property
hoelzl
parents:
54258
diff
changeset
|
1764 |
|
51471 | 1765 |
end |
1766 |
||
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1767 |
lemma (in t2_space) compact_imp_closed: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1768 |
assumes "compact s" shows "closed s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1769 |
unfolding closed_def |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1770 |
proof (rule openI) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1771 |
fix y assume "y \<in> - s" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1772 |
let ?C = "\<Union>x\<in>s. {u. open u \<and> x \<in> u \<and> eventually (\<lambda>y. y \<notin> u) (nhds y)}" |
60758 | 1773 |
note \<open>compact s\<close> |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1774 |
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
|
1775 |
moreover have "s \<subseteq> \<Union>?C" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1776 |
proof |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1777 |
fix x assume "x \<in> s" |
60758 | 1778 |
with \<open>y \<in> - s\<close> have "x \<noteq> y" by clarsimp |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1779 |
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
|
1780 |
by (rule hausdorff) |
60758 | 1781 |
with \<open>x \<in> s\<close> show "x \<in> \<Union>?C" |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1782 |
unfolding eventually_nhds by auto |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1783 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1784 |
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
|
1785 |
by (rule compactE) |
60758 | 1786 |
from \<open>D \<subseteq> ?C\<close> have "\<forall>x\<in>D. eventually (\<lambda>y. y \<notin> x) (nhds y)" by auto |
1787 |
with \<open>finite D\<close> have "eventually (\<lambda>y. y \<notin> \<Union>D) (nhds y)" |
|
60040
1fa1023b13b9
move MOST and INFM in Infinite_Set to Filter; change them to abbreviations over the cofinite filter
hoelzl
parents:
60036
diff
changeset
|
1788 |
by (simp add: eventually_ball_finite) |
60758 | 1789 |
with \<open>s \<subseteq> \<Union>D\<close> have "eventually (\<lambda>y. y \<notin> s) (nhds y)" |
61810 | 1790 |
by (auto elim!: eventually_mono) |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1791 |
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
|
1792 |
by (simp add: eventually_nhds subset_eq) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1793 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1794 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1795 |
lemma compact_continuous_image: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1796 |
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
|
1797 |
shows "compact (f ` s)" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1798 |
proof (rule compactI) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1799 |
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
|
1800 |
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
|
1801 |
unfolding continuous_on_open_invariant by blast |
53381 | 1802 |
then obtain A where A: "\<forall>c\<in>C. open (A c) \<and> A c \<inter> s = f -` c \<inter> s" |
1803 |
unfolding bchoice_iff .. |
|
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1804 |
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
|
1805 |
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
|
1806 |
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
|
1807 |
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
|
1808 |
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
|
1809 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1810 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1811 |
lemma continuous_on_inv: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1812 |
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
|
1813 |
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
|
1814 |
shows "continuous_on (f ` s) g" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1815 |
unfolding continuous_on_topological |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1816 |
proof (clarsimp simp add: assms(3)) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1817 |
fix x :: 'a and B :: "'a set" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1818 |
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
|
1819 |
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
|
1820 |
using assms(3) by (auto, metis) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1821 |
have "continuous_on (s - B) f" |
60758 | 1822 |
using \<open>continuous_on s f\<close> Diff_subset |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1823 |
by (rule continuous_on_subset) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1824 |
moreover have "compact (s - B)" |
60758 | 1825 |
using \<open>open B\<close> and \<open>compact s\<close> |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1826 |
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
|
1827 |
ultimately have "compact (f ` (s - B))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1828 |
by (rule compact_continuous_image) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1829 |
hence "closed (f ` (s - B))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1830 |
by (rule compact_imp_closed) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1831 |
hence "open (- f ` (s - B))" |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1832 |
by (rule open_Compl) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1833 |
moreover have "f x \<in> - f ` (s - B)" |
60758 | 1834 |
using \<open>x \<in> s\<close> and \<open>x \<in> B\<close> by (simp add: 1) |
51481
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1835 |
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
|
1836 |
by (simp add: 1) |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1837 |
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
|
1838 |
by fast |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1839 |
qed |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1840 |
|
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1841 |
lemma continuous_on_inv_into: |
ef949192e5d6
move continuous_on_inv to HOL image (simplifies isCont_inverse_function)
hoelzl
parents:
51480
diff
changeset
|
1842 |
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
|
1843 |
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
|
1844 |
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
|
1845 |
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
|
1846 |
|
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
|
1847 |
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
|
1848 |
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
|
1849 |
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
|
1850 |
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
|
1851 |
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
|
1852 |
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
|
1853 |
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
|
1854 |
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
|
1855 |
by auto |
60758 | 1856 |
with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {..< t s})" |
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
|
1857 |
by (erule compactE_image) |
60758 | 1858 |
with \<open>S \<noteq> {}\<close> have Max: "Max (t`C) \<in> t`C" and "\<forall>s\<in>t`C. s \<le> Max (t`C)" |
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
|
1859 |
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
|
1860 |
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
|
1861 |
by (auto intro: less_le_trans simp: subset_eq) |
60758 | 1862 |
with t Max \<open>C \<subseteq> S\<close> show ?thesis |
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
|
1863 |
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
|
1864 |
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
|
1865 |
|
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 |
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
|
1867 |
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
|
1868 |
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
|
1869 |
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
|
1870 |
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
|
1871 |
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
|
1872 |
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
|
1873 |
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
|
1874 |
by auto |
60758 | 1875 |
with \<open>compact S\<close> obtain C where "C \<subseteq> S" "finite C" and C: "S \<subseteq> (\<Union>s\<in>C. {t s <..})" |
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
|
1876 |
by (erule compactE_image) |
60758 | 1877 |
with \<open>S \<noteq> {}\<close> have Min: "Min (t`C) \<in> t`C" and "\<forall>s\<in>t`C. Min (t`C) \<le> s" |
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
|
1878 |
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
|
1879 |
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
|
1880 |
by (auto intro: le_less_trans simp: subset_eq) |
60758 | 1881 |
with t Min \<open>C \<subseteq> S\<close> show ?thesis |
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
|
1882 |
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
|
1883 |
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
|
1884 |
|
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
|
1885 |
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
|
1886 |
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
|
1887 |
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
|
1888 |
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
|
1889 |
|
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
|
1890 |
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
|
1891 |
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
|
1892 |
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
|
1893 |
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
|
1894 |
|
60758 | 1895 |
subsection \<open>Connectedness\<close> |
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1896 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1897 |
context topological_space |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1898 |
begin |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1899 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1900 |
definition "connected S \<longleftrightarrow> |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1901 |
\<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
|
1902 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1903 |
lemma connectedI: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1904 |
"(\<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
|
1905 |
\<Longrightarrow> connected U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1906 |
by (auto simp: connected_def) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1907 |
|
61306
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1908 |
lemma connected_empty [simp]: "connected {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1909 |
by (auto intro!: connectedI) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1910 |
|
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1911 |
lemma connected_sing [simp]: "connected {x}" |
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1912 |
by (auto intro!: connectedI) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
1913 |
|
56329 | 1914 |
lemma connectedD: |
1915 |
"connected A \<Longrightarrow> open U \<Longrightarrow> open V \<Longrightarrow> U \<inter> V \<inter> A = {} \<Longrightarrow> A \<subseteq> U \<union> V \<Longrightarrow> U \<inter> A = {} \<or> V \<inter> A = {}" |
|
1916 |
by (auto simp: connected_def) |
|
1917 |
||
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
|
1918 |
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
|
1919 |
|
61306
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1920 |
lemma connected_closed: |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1921 |
"connected s \<longleftrightarrow> |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1922 |
~ (\<exists>A B. closed A \<and> closed B \<and> s \<subseteq> A \<union> B \<and> A \<inter> B \<inter> s = {} \<and> A \<inter> s \<noteq> {} \<and> B \<inter> s \<noteq> {})" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1923 |
apply (simp add: connected_def del: ex_simps, safe) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1924 |
apply (drule_tac x="-A" in spec) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1925 |
apply (drule_tac x="-B" in spec) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1926 |
apply (fastforce simp add: closed_def [symmetric]) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1927 |
apply (drule_tac x="-A" in spec) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1928 |
apply (drule_tac x="-B" in spec) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1929 |
apply (fastforce simp add: open_closed [symmetric]) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1930 |
done |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1931 |
|
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1932 |
|
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1933 |
lemma connected_Union: |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1934 |
assumes cs: "\<And>s. s \<in> S \<Longrightarrow> connected s" and ne: "\<Inter>S \<noteq> {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1935 |
shows "connected(\<Union>S)" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1936 |
proof (rule connectedI) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1937 |
fix A B |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1938 |
assume A: "open A" and B: "open B" and Alap: "A \<inter> \<Union>S \<noteq> {}" and Blap: "B \<inter> \<Union>S \<noteq> {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1939 |
and disj: "A \<inter> B \<inter> \<Union>S = {}" and cover: "\<Union>S \<subseteq> A \<union> B" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1940 |
have disjs:"\<And>s. s \<in> S \<Longrightarrow> A \<inter> B \<inter> s = {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1941 |
using disj by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1942 |
obtain sa where sa: "sa \<in> S" "A \<inter> sa \<noteq> {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1943 |
using Alap by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1944 |
obtain sb where sb: "sb \<in> S" "B \<inter> sb \<noteq> {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1945 |
using Blap by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1946 |
obtain x where x: "\<And>s. s \<in> S \<Longrightarrow> x \<in> s" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1947 |
using ne by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1948 |
then have "x \<in> \<Union>S" |
61342 | 1949 |
using \<open>sa \<in> S\<close> by blast |
61306
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1950 |
then have "x \<in> A \<or> x \<in> B" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1951 |
using cover by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1952 |
then show False |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1953 |
using cs [unfolded connected_def] |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1954 |
by (metis A B IntI Sup_upper sa sb disjs x cover empty_iff subset_trans) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1955 |
qed |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1956 |
|
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1957 |
lemma connected_Un: "\<lbrakk>connected s; connected t; s \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> connected (s \<union> t)" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1958 |
using connected_Union [of "{s,t}"] by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1959 |
|
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1960 |
lemma connected_diff_open_from_closed: |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1961 |
assumes st: "s \<subseteq> t" and tu: "t \<subseteq> u" and s: "open s" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1962 |
and t: "closed t" and u: "connected u" and ts: "connected (t - s)" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1963 |
shows "connected(u - s)" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1964 |
proof (rule connectedI) |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1965 |
fix A B |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1966 |
assume AB: "open A" "open B" "A \<inter> (u - s) \<noteq> {}" "B \<inter> (u - s) \<noteq> {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1967 |
and disj: "A \<inter> B \<inter> (u - s) = {}" and cover: "u - s \<subseteq> A \<union> B" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1968 |
then consider "A \<inter> (t - s) = {}" | "B \<inter> (t - s) = {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1969 |
using st ts tu connectedD [of "t-s" "A" "B"] |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1970 |
by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1971 |
then show False |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1972 |
proof cases |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1973 |
case 1 |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1974 |
then have "(A - t) \<inter> (B \<union> s) \<inter> u = {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1975 |
using disj st by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1976 |
moreover have "u \<subseteq> (A - t) \<union> (B \<union> s)" using 1 cover by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1977 |
ultimately show False |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1978 |
using connectedD [of u "A - t" "B \<union> s"] AB s t 1 u |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1979 |
by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1980 |
next |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1981 |
case 2 |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1982 |
then have "(A \<union> s) \<inter> (B - t) \<inter> u = {}" |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1983 |
using disj st |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1984 |
by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1985 |
moreover have "u \<subseteq> (A \<union> s) \<union> (B - t)" using 2 cover by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1986 |
ultimately show False |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1987 |
using connectedD [of u "A \<union> s" "B - t"] AB s t 2 u |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1988 |
by auto |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1989 |
qed |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1990 |
qed |
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
1991 |
|
59106 | 1992 |
lemma connected_iff_const: |
1993 |
fixes S :: "'a::topological_space set" |
|
1994 |
shows "connected S \<longleftrightarrow> (\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c))" |
|
1995 |
proof safe |
|
1996 |
fix P :: "'a \<Rightarrow> bool" assume "connected S" "continuous_on S P" |
|
1997 |
then have "\<And>b. \<exists>A. open A \<and> A \<inter> S = P -` {b} \<inter> S" |
|
1998 |
unfolding continuous_on_open_invariant by simp |
|
1999 |
from this[of True] this[of False] |
|
2000 |
obtain t f where "open t" "open f" and *: "f \<inter> S = P -` {False} \<inter> S" "t \<inter> S = P -` {True} \<inter> S" |
|
2001 |
by auto |
|
2002 |
then have "t \<inter> S = {} \<or> f \<inter> S = {}" |
|
60758 | 2003 |
by (intro connectedD[OF \<open>connected S\<close>]) auto |
59106 | 2004 |
then show "\<exists>c. \<forall>s\<in>S. P s = c" |
2005 |
proof (rule disjE) |
|
2006 |
assume "t \<inter> S = {}" then show ?thesis |
|
2007 |
unfolding * by (intro exI[of _ False]) auto |
|
2008 |
next |
|
2009 |
assume "f \<inter> S = {}" then show ?thesis |
|
2010 |
unfolding * by (intro exI[of _ True]) auto |
|
2011 |
qed |
|
2012 |
next |
|
2013 |
assume P: "\<forall>P::'a \<Rightarrow> bool. continuous_on S P \<longrightarrow> (\<exists>c. \<forall>s\<in>S. P s = c)" |
|
2014 |
show "connected S" |
|
2015 |
proof (rule connectedI) |
|
2016 |
fix A B assume *: "open A" "open B" "A \<inter> S \<noteq> {}" "B \<inter> S \<noteq> {}" "A \<inter> B \<inter> S = {}" "S \<subseteq> A \<union> B" |
|
2017 |
have "continuous_on S (\<lambda>x. x \<in> A)" |
|
2018 |
unfolding continuous_on_open_invariant |
|
2019 |
proof safe |
|
2020 |
fix C :: "bool set" |
|
2021 |
have "C = UNIV \<or> C = {True} \<or> C = {False} \<or> C = {}" |
|
2022 |
using subset_UNIV[of C] unfolding UNIV_bool by auto |
|
2023 |
with * show "\<exists>T. open T \<and> T \<inter> S = (\<lambda>x. x \<in> A) -` C \<inter> S" |
|
2024 |
by (intro exI[of _ "(if True \<in> C then A else {}) \<union> (if False \<in> C then B else {})"]) auto |
|
2025 |
qed |
|
2026 |
from P[rule_format, OF this] obtain c where "\<And>s. s \<in> S \<Longrightarrow> (s \<in> A) = c" by blast |
|
2027 |
with * show False |
|
2028 |
by (cases c) auto |
|
2029 |
qed |
|
2030 |
qed |
|
2031 |
||
2032 |
lemma connectedD_const: |
|
2033 |
fixes P :: "'a::topological_space \<Rightarrow> bool" |
|
2034 |
shows "connected S \<Longrightarrow> continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c" |
|
2035 |
unfolding connected_iff_const by auto |
|
2036 |
||
2037 |
lemma connectedI_const: |
|
2038 |
"(\<And>P::'a::topological_space \<Rightarrow> bool. continuous_on S P \<Longrightarrow> \<exists>c. \<forall>s\<in>S. P s = c) \<Longrightarrow> connected S" |
|
2039 |
unfolding connected_iff_const by auto |
|
2040 |
||
56329 | 2041 |
lemma connected_local_const: |
2042 |
assumes "connected A" "a \<in> A" "b \<in> A" |
|
2043 |
assumes *: "\<forall>a\<in>A. eventually (\<lambda>b. f a = f b) (at a within A)" |
|
2044 |
shows "f a = f b" |
|
2045 |
proof - |
|
2046 |
obtain S where S: "\<And>a. a \<in> A \<Longrightarrow> a \<in> S a" "\<And>a. a \<in> A \<Longrightarrow> open (S a)" |
|
2047 |
"\<And>a x. a \<in> A \<Longrightarrow> x \<in> S a \<Longrightarrow> x \<in> A \<Longrightarrow> f a = f x" |
|
2048 |
using * unfolding eventually_at_topological by metis |
|
2049 |
||
2050 |
let ?P = "\<Union>b\<in>{b\<in>A. f a = f b}. S b" and ?N = "\<Union>b\<in>{b\<in>A. f a \<noteq> f b}. S b" |
|
2051 |
have "?P \<inter> A = {} \<or> ?N \<inter> A = {}" |
|
60758 | 2052 |
using \<open>connected A\<close> S \<open>a\<in>A\<close> |
56329 | 2053 |
by (intro connectedD) (auto, metis) |
2054 |
then show "f a = f b" |
|
2055 |
proof |
|
2056 |
assume "?N \<inter> A = {}" |
|
2057 |
then have "\<forall>x\<in>A. f a = f x" |
|
2058 |
using S(1) by auto |
|
60758 | 2059 |
with \<open>b\<in>A\<close> show ?thesis by auto |
56329 | 2060 |
next |
2061 |
assume "?P \<inter> A = {}" then show ?thesis |
|
60758 | 2062 |
using \<open>a \<in> A\<close> S(1)[of a] by auto |
56329 | 2063 |
qed |
2064 |
qed |
|
2065 |
||
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2066 |
lemma (in linorder_topology) connectedD_interval: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2067 |
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
|
2068 |
shows "z \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2069 |
proof - |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2070 |
have eq: "{..<z} \<union> {z<..} = - {z}" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2071 |
by auto |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2072 |
{ 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
|
2073 |
with xy have "\<not> connected U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2074 |
unfolding connected_def simp_thms |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2075 |
apply (rule_tac exI[of _ "{..< z}"]) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2076 |
apply (rule_tac exI[of _ "{z <..}"]) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2077 |
apply (auto simp add: eq) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2078 |
done } |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2079 |
with assms show "z \<in> U" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2080 |
by (metis less_le) |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2081 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2082 |
|
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2083 |
lemma connected_continuous_image: |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2084 |
assumes *: "continuous_on s f" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2085 |
assumes "connected s" |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2086 |
shows "connected (f ` s)" |
59106 | 2087 |
proof (rule connectedI_const) |
2088 |
fix P :: "'b \<Rightarrow> bool" assume "continuous_on (f ` s) P" |
|
2089 |
then have "continuous_on s (P \<circ> f)" |
|
2090 |
by (rule continuous_on_compose[OF *]) |
|
60758 | 2091 |
from connectedD_const[OF \<open>connected s\<close> this] show "\<exists>c. \<forall>s\<in>f ` s. P s = c" |
59106 | 2092 |
by auto |
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2093 |
qed |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2094 |
|
61306
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
2095 |
|
9dd394c866fc
New theorems about connected sets. And pairwise moved to Set.thy.
paulson <lp15@cam.ac.uk>
parents:
61245
diff
changeset
|
2096 |
section \<open>Linear Continuum Topologies\<close> |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2097 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2098 |
class linear_continuum_topology = linorder_topology + linear_continuum |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2099 |
begin |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2100 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2101 |
lemma Inf_notin_open: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2102 |
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
|
2103 |
shows "Inf A \<notin> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2104 |
proof |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2105 |
assume "Inf A \<in> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2106 |
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
|
2107 |
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
|
2108 |
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
|
2109 |
by (auto simp: subset_eq) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2110 |
then show False |
60758 | 2111 |
using cInf_lower[OF \<open>c \<in> A\<close>] bnd by (metis not_le less_imp_le bdd_belowI) |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2112 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2113 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2114 |
lemma Sup_notin_open: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2115 |
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
|
2116 |
shows "Sup A \<notin> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2117 |
proof |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2118 |
assume "Sup A \<in> A" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2119 |
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
|
2120 |
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
|
2121 |
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
|
2122 |
by (auto simp: subset_eq) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2123 |
then show False |
60758 | 2124 |
using cSup_upper[OF \<open>c \<in> A\<close>] bnd by (metis less_imp_le not_le bdd_aboveI) |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2125 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2126 |
|
51480
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2127 |
end |
3793c3a11378
move connected to HOL image; used to show intermediate value theorem
hoelzl
parents:
51479
diff
changeset
|
2128 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2129 |
instance linear_continuum_topology \<subseteq> perfect_space |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2130 |
proof |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2131 |
fix x :: 'a |
53381 | 2132 |
obtain y where "x < y \<or> y < x" |
2133 |
using ex_gt_or_lt [of x] .. |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2134 |
with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y] |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2135 |
show "\<not> open {x}" |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2136 |
by auto |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2137 |
qed |
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2138 |
|
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2139 |
lemma connectedI_interval: |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2140 |
fixes U :: "'a :: linear_continuum_topology set" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2141 |
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
|
2142 |
shows "connected U" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2143 |
proof (rule connectedI) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2144 |
{ 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
|
2145 |
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
|
2146 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2147 |
let ?z = "Inf (B \<inter> {x <..})" |
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 |
have "x \<le> ?z" "?z \<le> y" |
60758 | 2150 |
using \<open>y \<in> B\<close> \<open>x < y\<close> by (auto intro: cInf_lower cInf_greatest) |
2151 |
with \<open>x \<in> U\<close> \<open>y \<in> U\<close> have "?z \<in> U" |
|
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2152 |
by (rule *) |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2153 |
moreover have "?z \<notin> B \<inter> {x <..}" |
60758 | 2154 |
using \<open>open B\<close> by (intro Inf_notin_open) auto |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2155 |
ultimately have "?z \<in> A" |
60758 | 2156 |
using \<open>x \<le> ?z\<close> \<open>A \<inter> B \<inter> U = {}\<close> \<open>x \<in> A\<close> \<open>U \<subseteq> A \<union> B\<close> by auto |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2157 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2158 |
{ assume "?z < y" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2159 |
obtain a where "?z < a" "{?z ..< a} \<subseteq> A" |
60758 | 2160 |
using open_right[OF \<open>open A\<close> \<open>?z \<in> A\<close> \<open>?z < y\<close>] by auto |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2161 |
moreover obtain b where "b \<in> B" "x < b" "b < min a y" |
60758 | 2162 |
using cInf_less_iff[of "B \<inter> {x <..}" "min a y"] \<open>?z < a\<close> \<open>?z < y\<close> \<open>x < y\<close> \<open>y \<in> B\<close> |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2163 |
by (auto intro: less_imp_le) |
53374
a14d2a854c02
tuned proofs -- clarified flow of facts wrt. calculation;
wenzelm
parents:
53215
diff
changeset
|
2164 |
moreover have "?z \<le> b" |
60758 | 2165 |
using \<open>b \<in> B\<close> \<open>x < b\<close> |
54258
adfc759263ab
use bdd_above and bdd_below for conditionally complete lattices
hoelzl
parents:
53946
diff
changeset
|
2166 |
by (intro cInf_lower) auto |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2167 |
moreover have "b \<in> U" |
60758 | 2168 |
using \<open>x \<le> ?z\<close> \<open>?z \<le> b\<close> \<open>b < min a y\<close> |
2169 |
by (intro *[OF \<open>x \<in> U\<close> \<open>y \<in> U\<close>]) (auto simp: less_imp_le) |
|
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2170 |
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
|
2171 |
by (intro bexI[of _ b]) auto } |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2172 |
then have False |
60758 | 2173 |
using \<open>?z \<le> y\<close> \<open>?z \<in> A\<close> \<open>y \<in> B\<close> \<open>y \<in> U\<close> \<open>A \<inter> B \<inter> U = {}\<close> unfolding le_less by blast } |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2174 |
note not_disjoint = this |
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 |
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
|
2177 |
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
|
2178 |
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
|
2179 |
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
|
2180 |
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
|
2181 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2182 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2183 |
lemma connected_iff_interval: |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2184 |
fixes U :: "'a :: linear_continuum_topology set" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2185 |
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
|
2186 |
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
|
2187 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2188 |
lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2189 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2190 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2191 |
lemma connected_Ioi[simp]: "connected {a::'a::linear_continuum_topology <..}" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2192 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2193 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2194 |
lemma connected_Ici[simp]: "connected {a::'a::linear_continuum_topology ..}" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2195 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2196 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2197 |
lemma connected_Iio[simp]: "connected {..< a::'a::linear_continuum_topology}" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2198 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2199 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2200 |
lemma connected_Iic[simp]: "connected {.. a::'a::linear_continuum_topology}" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2201 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2202 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2203 |
lemma connected_Ioo[simp]: "connected {a <..< b::'a::linear_continuum_topology}" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2204 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2205 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2206 |
lemma connected_Ioc[simp]: "connected {a <.. b::'a::linear_continuum_topology}" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2207 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2208 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2209 |
lemma connected_Ico[simp]: "connected {a ..< b::'a::linear_continuum_topology}" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2210 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2211 |
|
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2212 |
lemma connected_Icc[simp]: "connected {a .. b::'a::linear_continuum_topology}" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2213 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2214 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2215 |
lemma connected_contains_Ioo: |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2216 |
fixes A :: "'a :: linorder_topology set" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2217 |
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
|
2218 |
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
|
2219 |
|
61531
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2220 |
lemma connected_contains_Icc: |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2221 |
assumes "connected (A :: ('a :: {linorder_topology}) set)" "a \<in> A" "b \<in> A" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2222 |
shows "{a..b} \<subseteq> A" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2223 |
proof |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2224 |
fix x assume "x \<in> {a..b}" |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2225 |
hence "x = a \<or> x = b \<or> x \<in> {a<..<b}" by auto |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2226 |
thus "x \<in> A" using assms connected_contains_Ioo[of A a b] by auto |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2227 |
qed |
ab2e862263e7
Rounding function, uniform limits, cotangent, binomial identities
eberlm
parents:
61520
diff
changeset
|
2228 |
|
60758 | 2229 |
subsection \<open>Intermediate Value Theorem\<close> |
51518
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 IVT': |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2232 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2233 |
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
|
2234 |
assumes *: "continuous_on {a .. b} f" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2235 |
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
|
2236 |
proof - |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2237 |
have "connected {a..b}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2238 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2239 |
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
|
2240 |
show ?thesis |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2241 |
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
|
2242 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2243 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2244 |
lemma IVT2': |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2245 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2246 |
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
|
2247 |
assumes *: "continuous_on {a .. b} f" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2248 |
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
|
2249 |
proof - |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2250 |
have "connected {a..b}" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2251 |
unfolding connected_iff_interval by auto |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2252 |
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
|
2253 |
show ?thesis |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2254 |
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
|
2255 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2256 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2257 |
lemma IVT: |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2258 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2259 |
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
|
2260 |
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
|
2261 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2262 |
lemma IVT2: |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2263 |
fixes f :: "'a :: linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2264 |
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
|
2265 |
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
|
2266 |
|
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2267 |
lemma continuous_inj_imp_mono: |
51775
408d937c9486
revert #916271d52466; add non-topological linear_continuum type class; show linear_continuum_topology is a perfect_space
hoelzl
parents:
51774
diff
changeset
|
2268 |
fixes f :: "'a::linear_continuum_topology \<Rightarrow> 'b :: linorder_topology" |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2269 |
assumes x: "a < x" "x < b" |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2270 |
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
|
2271 |
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
|
2272 |
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
|
2273 |
proof - |
61520
8f85bb443d33
Cauchy's integral formula, required lemmas, and a bit of reorganisation
paulson <lp15@cam.ac.uk>
parents:
61426
diff
changeset
|
2274 |
note I = inj_on_eq_iff[OF inj] |
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2275 |
{ 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
|
2276 |
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
|
2277 |
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
|
2278 |
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
|
2279 |
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
|
2280 |
moreover |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2281 |
{ 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
|
2282 |
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
|
2283 |
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
|
2284 |
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
|
2285 |
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
|
2286 |
ultimately show ?thesis |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2287 |
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
|
2288 |
qed |
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2289 |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2290 |
lemma continuous_at_Sup_mono: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2291 |
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2292 |
assumes "mono f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2293 |
assumes cont: "continuous (at_left (Sup S)) f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2294 |
assumes S: "S \<noteq> {}" "bdd_above S" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2295 |
shows "f (Sup S) = (SUP s:S. f s)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2296 |
proof (rule antisym) |
61973 | 2297 |
have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2298 |
using cont unfolding continuous_within . |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2299 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2300 |
show "f (Sup S) \<le> (SUP s:S. f s)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2301 |
proof cases |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2302 |
assume "Sup S \<in> S" then show ?thesis |
60758 | 2303 |
by (rule cSUP_upper) (auto intro: bdd_above_image_mono S \<open>mono f\<close>) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2304 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2305 |
assume "Sup S \<notin> S" |
60758 | 2306 |
from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2307 |
by auto |
60758 | 2308 |
with \<open>Sup S \<notin> S\<close> S have "s < Sup S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2309 |
unfolding less_le by (blast intro: cSup_upper) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2310 |
show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2311 |
proof (rule ccontr) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2312 |
assume "\<not> ?thesis" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2313 |
with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "b < Sup S" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2314 |
and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> (SUP s:S. f s) < f y" |
60758 | 2315 |
by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>]) |
2316 |
with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2317 |
using less_cSupD[of S b] by auto |
60758 | 2318 |
with \<open>Sup S \<notin> S\<close> S have "c < Sup S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2319 |
unfolding less_le by (blast intro: cSup_upper) |
60758 | 2320 |
from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cSUP_upper[OF \<open>c \<in> S\<close> bdd_above_image_mono[of f]] |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2321 |
show False |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2322 |
by (auto simp: assms) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2323 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2324 |
qed |
60758 | 2325 |
qed (intro cSUP_least \<open>mono f\<close>[THEN monoD] cSup_upper S) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2326 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2327 |
lemma continuous_at_Sup_antimono: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2328 |
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2329 |
assumes "antimono f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2330 |
assumes cont: "continuous (at_left (Sup S)) f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2331 |
assumes S: "S \<noteq> {}" "bdd_above S" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2332 |
shows "f (Sup S) = (INF s:S. f s)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2333 |
proof (rule antisym) |
61973 | 2334 |
have f: "(f \<longlongrightarrow> f (Sup S)) (at_left (Sup S))" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2335 |
using cont unfolding continuous_within . |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2336 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2337 |
show "(INF s:S. f s) \<le> f (Sup S)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2338 |
proof cases |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2339 |
assume "Sup S \<in> S" then show ?thesis |
60758 | 2340 |
by (intro cINF_lower) (auto intro: bdd_below_image_antimono S \<open>antimono f\<close>) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2341 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2342 |
assume "Sup S \<notin> S" |
60758 | 2343 |
from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2344 |
by auto |
60758 | 2345 |
with \<open>Sup S \<notin> S\<close> S have "s < Sup S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2346 |
unfolding less_le by (blast intro: cSup_upper) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2347 |
show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2348 |
proof (rule ccontr) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2349 |
assume "\<not> ?thesis" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2350 |
with order_tendstoD(2)[OF f, of "INF s:S. f s"] obtain b where "b < Sup S" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2351 |
and *: "\<And>y. b < y \<Longrightarrow> y < Sup S \<Longrightarrow> f y < (INF s:S. f s)" |
60758 | 2352 |
by (auto simp: not_le eventually_at_left[OF \<open>s < Sup S\<close>]) |
2353 |
with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "b < c" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2354 |
using less_cSupD[of S b] by auto |
60758 | 2355 |
with \<open>Sup S \<notin> S\<close> S have "c < Sup S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2356 |
unfolding less_le by (blast intro: cSup_upper) |
60758 | 2357 |
from *[OF \<open>b < c\<close> \<open>c < Sup S\<close>] cINF_lower[OF bdd_below_image_antimono, of f S c] \<open>c \<in> S\<close> |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2358 |
show False |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2359 |
by (auto simp: assms) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2360 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2361 |
qed |
60758 | 2362 |
qed (intro cINF_greatest \<open>antimono f\<close>[THEN antimonoD] cSup_upper S) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2363 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2364 |
lemma continuous_at_Inf_mono: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2365 |
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2366 |
assumes "mono f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2367 |
assumes cont: "continuous (at_right (Inf S)) f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2368 |
assumes S: "S \<noteq> {}" "bdd_below S" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2369 |
shows "f (Inf S) = (INF s:S. f s)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2370 |
proof (rule antisym) |
61973 | 2371 |
have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2372 |
using cont unfolding continuous_within . |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2373 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2374 |
show "(INF s:S. f s) \<le> f (Inf S)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2375 |
proof cases |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2376 |
assume "Inf S \<in> S" then show ?thesis |
60758 | 2377 |
by (rule cINF_lower[rotated]) (auto intro: bdd_below_image_mono S \<open>mono f\<close>) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2378 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2379 |
assume "Inf S \<notin> S" |
60758 | 2380 |
from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2381 |
by auto |
60758 | 2382 |
with \<open>Inf S \<notin> S\<close> S have "Inf S < s" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2383 |
unfolding less_le by (blast intro: cInf_lower) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2384 |
show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2385 |
proof (rule ccontr) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2386 |
assume "\<not> ?thesis" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2387 |
with order_tendstoD(2)[OF f, of "INF s:S. f s"] obtain b where "Inf S < b" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2388 |
and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> f y < (INF s:S. f s)" |
60758 | 2389 |
by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>]) |
2390 |
with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2391 |
using cInf_lessD[of S b] by auto |
60758 | 2392 |
with \<open>Inf S \<notin> S\<close> S have "Inf S < c" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2393 |
unfolding less_le by (blast intro: cInf_lower) |
60758 | 2394 |
from *[OF \<open>Inf S < c\<close> \<open>c < b\<close>] cINF_lower[OF bdd_below_image_mono[of f] \<open>c \<in> S\<close>] |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2395 |
show False |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2396 |
by (auto simp: assms) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2397 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2398 |
qed |
60758 | 2399 |
qed (intro cINF_greatest \<open>mono f\<close>[THEN monoD] cInf_lower \<open>bdd_below S\<close> \<open>S \<noteq> {}\<close>) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2400 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2401 |
lemma continuous_at_Inf_antimono: |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2402 |
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder} \<Rightarrow> 'b :: {linorder_topology, conditionally_complete_linorder}" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2403 |
assumes "antimono f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2404 |
assumes cont: "continuous (at_right (Inf S)) f" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2405 |
assumes S: "S \<noteq> {}" "bdd_below S" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2406 |
shows "f (Inf S) = (SUP s:S. f s)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2407 |
proof (rule antisym) |
61973 | 2408 |
have f: "(f \<longlongrightarrow> f (Inf S)) (at_right (Inf S))" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2409 |
using cont unfolding continuous_within . |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2410 |
|
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2411 |
show "f (Inf S) \<le> (SUP s:S. f s)" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2412 |
proof cases |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2413 |
assume "Inf S \<in> S" then show ?thesis |
60758 | 2414 |
by (rule cSUP_upper) (auto intro: bdd_above_image_antimono S \<open>antimono f\<close>) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2415 |
next |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2416 |
assume "Inf S \<notin> S" |
60758 | 2417 |
from \<open>S \<noteq> {}\<close> obtain s where "s \<in> S" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2418 |
by auto |
60758 | 2419 |
with \<open>Inf S \<notin> S\<close> S have "Inf S < s" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2420 |
unfolding less_le by (blast intro: cInf_lower) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2421 |
show ?thesis |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2422 |
proof (rule ccontr) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2423 |
assume "\<not> ?thesis" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2424 |
with order_tendstoD(1)[OF f, of "SUP s:S. f s"] obtain b where "Inf S < b" |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2425 |
and *: "\<And>y. Inf S < y \<Longrightarrow> y < b \<Longrightarrow> (SUP s:S. f s) < f y" |
60758 | 2426 |
by (auto simp: not_le eventually_at_right[OF \<open>Inf S < s\<close>]) |
2427 |
with \<open>S \<noteq> {}\<close> obtain c where "c \<in> S" "c < b" |
|
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2428 |
using cInf_lessD[of S b] by auto |
60758 | 2429 |
with \<open>Inf S \<notin> S\<close> S have "Inf S < c" |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2430 |
unfolding less_le by (blast intro: cInf_lower) |
60758 | 2431 |
from *[OF \<open>Inf S < c\<close> \<open>c < b\<close>] cSUP_upper[OF \<open>c \<in> S\<close> bdd_above_image_antimono[of f]] |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2432 |
show False |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2433 |
by (auto simp: assms) |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2434 |
qed |
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2435 |
qed |
60758 | 2436 |
qed (intro cSUP_least \<open>antimono f\<close>[THEN antimonoD] cInf_lower S) |
59452
2538b2c51769
ereal: tuned proofs concerning continuity and suprema
hoelzl
parents:
59106
diff
changeset
|
2437 |
|
51518
6a56b7088a6a
separate SupInf into Conditional_Complete_Lattice, move instantiation of real to RealDef
hoelzl
parents:
51481
diff
changeset
|
2438 |
end |