src/HOL/Limits.thy
author huffman
Wed, 31 Aug 2011 07:51:55 -0700
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child 45031 9583f2b56f85
permissions -rw-r--r--
convert to Isar-style proof
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(*  Title       : Limits.thy
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    Author      : Brian Huffman
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*)
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header {* Filters and Limits *}
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theory Limits
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imports RealVector
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begin
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subsection {* Filters *}
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text {*
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  This definition also allows non-proper filters.
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*}
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locale is_filter =
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  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
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  assumes True: "F (\<lambda>x. True)"
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  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
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  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
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typedef (open) 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
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proof
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  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
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qed
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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
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  using Rep_filter [of F] by simp
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lemma Abs_filter_inverse':
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  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
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  using assms by (simp add: Abs_filter_inverse)
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subsection {* Eventually *}
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
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  where "eventually P F \<longleftrightarrow> Rep_filter F P"
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lemma eventually_Abs_filter:
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  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
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  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
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lemma filter_eq_iff:
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  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
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  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
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  unfolding eventually_def
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  by (rule is_filter.True [OF is_filter_Rep_filter])
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
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proof -
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  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
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  thus "eventually P F" by simp
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qed
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lemma eventually_mono:
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  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
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  unfolding eventually_def
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  by (rule is_filter.mono [OF is_filter_Rep_filter])
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lemma eventually_conj:
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  assumes P: "eventually (\<lambda>x. P x) F"
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  assumes Q: "eventually (\<lambda>x. Q x) F"
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  shows "eventually (\<lambda>x. P x \<and> Q x) F"
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  using assms unfolding eventually_def
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  by (rule is_filter.conj [OF is_filter_Rep_filter])
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lemma eventually_mp:
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  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
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  assumes "eventually (\<lambda>x. P x) F"
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  shows "eventually (\<lambda>x. Q x) F"
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proof (rule eventually_mono)
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  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
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  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
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    using assms by (rule eventually_conj)
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qed
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lemma eventually_rev_mp:
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  assumes "eventually (\<lambda>x. P x) F"
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  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
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  shows "eventually (\<lambda>x. Q x) F"
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using assms(2) assms(1) by (rule eventually_mp)
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lemma eventually_conj_iff:
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  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
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  by (auto intro: eventually_conj elim: eventually_rev_mp)
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lemma eventually_elim1:
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  assumes "eventually (\<lambda>i. P i) F"
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  assumes "\<And>i. P i \<Longrightarrow> Q i"
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  shows "eventually (\<lambda>i. Q i) F"
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  using assms by (auto elim!: eventually_rev_mp)
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lemma eventually_elim2:
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  assumes "eventually (\<lambda>i. P i) F"
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  assumes "eventually (\<lambda>i. Q i) F"
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  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
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  shows "eventually (\<lambda>i. R i) F"
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  using assms by (auto elim!: eventually_rev_mp)
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subsection {* Finer-than relation *}
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text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
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filter @{term F'}. *}
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instantiation filter :: (type) complete_lattice
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begin
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definition le_filter_def:
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  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
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definition
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  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
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definition
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  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
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definition
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  "bot = Abs_filter (\<lambda>P. True)"
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definition
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  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
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definition
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  "inf F F' = Abs_filter
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      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
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definition
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  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
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definition
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  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
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lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
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  unfolding top_filter_def
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  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
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lemma eventually_bot [simp]: "eventually P bot"
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  unfolding bot_filter_def
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  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
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lemma eventually_sup:
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  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
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  unfolding sup_filter_def
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  by (rule eventually_Abs_filter, rule is_filter.intro)
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     (auto elim!: eventually_rev_mp)
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lemma eventually_inf:
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  "eventually P (inf F F') \<longleftrightarrow>
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   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
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  unfolding inf_filter_def
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  apply (rule eventually_Abs_filter, rule is_filter.intro)
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  apply (fast intro: eventually_True)
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  apply clarify
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  apply (intro exI conjI)
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  apply (erule (1) eventually_conj)
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  apply (erule (1) eventually_conj)
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  apply simp
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  apply auto
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  done
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lemma eventually_Sup:
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  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
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  unfolding Sup_filter_def
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  apply (rule eventually_Abs_filter, rule is_filter.intro)
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  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
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  done
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instance proof
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  fix F F' F'' :: "'a filter" and S :: "'a filter set"
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  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
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    by (rule less_filter_def) }
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  { show "F \<le> F"
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    unfolding le_filter_def by simp }
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  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
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    unfolding le_filter_def by simp }
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  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
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    unfolding le_filter_def filter_eq_iff by fast }
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  { show "F \<le> top"
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    unfolding le_filter_def eventually_top by (simp add: always_eventually) }
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  { show "bot \<le> F"
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    unfolding le_filter_def by simp }
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  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
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    unfolding le_filter_def eventually_sup by simp_all }
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  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
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    unfolding le_filter_def eventually_sup by simp }
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  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
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    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
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  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
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    unfolding le_filter_def eventually_inf
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    by (auto elim!: eventually_mono intro: eventually_conj) }
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  { assume "F \<in> S" thus "F \<le> Sup S"
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    unfolding le_filter_def eventually_Sup by simp }
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  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
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    unfolding le_filter_def eventually_Sup by simp }
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  { assume "F'' \<in> S" thus "Inf S \<le> F''"
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    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
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  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
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    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
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qed
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end
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lemma filter_leD:
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  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
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  unfolding le_filter_def by simp
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lemma filter_leI:
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  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
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  unfolding le_filter_def by simp
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lemma eventually_False:
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  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
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  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
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abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
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  where "trivial_limit F \<equiv> F = bot"
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lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
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  by (rule eventually_False [symmetric])
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subsection {* Map function for filters *}
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definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
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  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
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lemma eventually_filtermap:
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  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
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  unfolding filtermap_def
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  apply (rule eventually_Abs_filter)
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  apply (rule is_filter.intro)
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  apply (auto elim!: eventually_rev_mp)
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  done
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lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
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  by (simp add: filter_eq_iff eventually_filtermap)
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lemma filtermap_filtermap:
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  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
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  by (simp add: filter_eq_iff eventually_filtermap)
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lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
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  unfolding le_filter_def eventually_filtermap by simp
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lemma filtermap_bot [simp]: "filtermap f bot = bot"
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  by (simp add: filter_eq_iff eventually_filtermap)
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subsection {* Sequentially *}
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definition sequentially :: "nat filter"
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  where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
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lemma eventually_sequentially:
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  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
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unfolding sequentially_def
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proof (rule eventually_Abs_filter, rule is_filter.intro)
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  fix P Q :: "nat \<Rightarrow> bool"
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  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
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  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
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  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
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  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
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qed auto
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lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
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  unfolding filter_eq_iff eventually_sequentially by auto
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lemmas trivial_limit_sequentially = sequentially_bot
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lemma eventually_False_sequentially [simp]:
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  "\<not> eventually (\<lambda>n. False) sequentially"
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  by (simp add: eventually_False)
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lemma le_sequentially:
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  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
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  unfolding le_filter_def eventually_sequentially
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  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
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subsection {* Standard filters *}
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definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
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  where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
31392
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parents: 31357
diff changeset
   288
44206
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parents: 44205
diff changeset
   289
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
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parents: 44079
diff changeset
   290
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
36654
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diff changeset
   291
44206
5e4a1664106e locale-ize some constant definitions, so complete_space can inherit from metric_space
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parents: 44205
diff changeset
   292
definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
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parents: 44079
diff changeset
   293
  where "at a = nhds a within - {a}"
31447
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diff changeset
   294
31392
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parents: 31357
diff changeset
   295
lemma eventually_within:
44195
f5363511b212 consistently use variable name 'F' for filters
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diff changeset
   296
  "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
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parents: 44079
diff changeset
   297
  unfolding within_def
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
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diff changeset
   298
  by (rule eventually_Abs_filter, rule is_filter.intro)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
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parents: 44079
diff changeset
   299
     (auto elim!: eventually_rev_mp)
31392
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parents: 31357
diff changeset
   300
44195
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diff changeset
   301
lemma within_UNIV: "F within UNIV = F"
44081
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huffman
parents: 44079
diff changeset
   302
  unfolding filter_eq_iff eventually_within by simp
36360
9d8f7efd9289 define finer-than ordering on net type; move some theorems into Limits.thy
huffman
parents: 36358
diff changeset
   303
36654
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diff changeset
   304
lemma eventually_nhds:
7c8eb32724ce add constants netmap and nhds
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parents: 36630
diff changeset
   305
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
7c8eb32724ce add constants netmap and nhds
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parents: 36630
diff changeset
   306
unfolding nhds_def
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   307
proof (rule eventually_Abs_filter, rule is_filter.intro)
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   308
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   309
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   310
next
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   311
  fix P Q
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   312
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   313
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   314
  then obtain S T where
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   315
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   316
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   317
  hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   318
    by (simp add: open_Int)
36654
7c8eb32724ce add constants netmap and nhds
huffman
parents: 36630
diff changeset
   319
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
36358
246493d61204 define nets directly as filters, instead of as filter bases
huffman
parents: 31902
diff changeset
   320
qed auto
31447
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huffman
parents: 31392
diff changeset
   321
36656
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huffman
parents: 36655
diff changeset
   322
lemma eventually_nhds_metric:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   323
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   324
unfolding eventually_nhds open_dist
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   325
apply safe
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   326
apply fast
31492
5400beeddb55 replace 'topo' with 'open'; add extra type constraint for 'open'
huffman
parents: 31488
diff changeset
   327
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
31447
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   328
apply clarsimp
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   329
apply (rule_tac x="d - dist x a" in exI, clarsimp)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   330
apply (simp only: less_diff_eq)
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   331
apply (erule le_less_trans [OF dist_triangle])
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   332
done
97bab1ac463e generalize type of 'at' to topological_space; generalize some lemmas
huffman
parents: 31392
diff changeset
   333
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   334
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   335
  unfolding trivial_limit_def eventually_nhds by simp
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   336
36656
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   337
lemma eventually_at_topological:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   338
  "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   339
unfolding at_def eventually_within eventually_nhds by simp
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   340
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   341
lemma eventually_at:
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   342
  fixes a :: "'a::metric_space"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   343
  shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   344
unfolding at_def eventually_within eventually_nhds_metric by auto
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   345
44571
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   346
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   347
  unfolding trivial_limit_def eventually_at_topological
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   348
  by (safe, case_tac "S = {a}", simp, fast, fast)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   349
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   350
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   351
  by (simp add: at_eq_bot_iff not_open_singleton)
bd91b77c4cd6 move class perfect_space into RealVector.thy;
huffman
parents: 44568
diff changeset
   352
31392
69570155ddf8 replace filters with filter bases
huffman
parents: 31357
diff changeset
   353
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   354
subsection {* Boundedness *}
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   355
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   356
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   357
  where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   358
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   359
lemma BfunI:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   360
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   361
unfolding Bfun_def
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   362
proof (intro exI conjI allI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   363
  show "0 < max K 1" by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   364
next
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   365
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   366
    using K by (rule eventually_elim1, simp)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   367
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   368
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   369
lemma BfunE:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   370
  assumes "Bfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   371
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   372
using assms unfolding Bfun_def by fast
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   373
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   374
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   375
subsection {* Convergence to Zero *}
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   376
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   377
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   378
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   379
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   380
lemma ZfunI:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   381
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   382
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   383
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   384
lemma ZfunD:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   385
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   386
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   387
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   388
lemma Zfun_ssubst:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   389
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   390
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   391
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   392
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   393
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   394
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   395
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   396
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   397
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   398
lemma Zfun_imp_Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   399
  assumes f: "Zfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   400
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   401
  shows "Zfun (\<lambda>x. g x) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   402
proof (cases)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   403
  assume K: "0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   404
  show ?thesis
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   405
  proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   406
    fix r::real assume "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   407
    hence "0 < r / K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   408
      using K by (rule divide_pos_pos)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   409
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   410
      using ZfunD [OF f] by fast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   411
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   412
    proof (rule eventually_elim2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   413
      fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   414
      assume *: "norm (g x) \<le> norm (f x) * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   415
      assume "norm (f x) < r / K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   416
      hence "norm (f x) * K < r"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   417
        by (simp add: pos_less_divide_eq K)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   418
      thus "norm (g x) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   419
        by (simp add: order_le_less_trans [OF *])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   420
    qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   421
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   422
next
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   423
  assume "\<not> 0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   424
  hence K: "K \<le> 0" by (simp only: not_less)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   425
  show ?thesis
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   426
  proof (rule ZfunI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   427
    fix r :: real
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   428
    assume "0 < r"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   429
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   430
    proof (rule eventually_elim1)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   431
      fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   432
      assume "norm (g x) \<le> norm (f x) * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   433
      also have "\<dots> \<le> norm (f x) * 0"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   434
        using K norm_ge_zero by (rule mult_left_mono)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   435
      finally show "norm (g x) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   436
        using `0 < r` by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   437
    qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   438
  qed
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   439
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   440
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   441
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   442
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   443
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   444
lemma Zfun_add:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   445
  assumes f: "Zfun f F" and g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   446
  shows "Zfun (\<lambda>x. f x + g x) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   447
proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   448
  fix r::real assume "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   449
  hence r: "0 < r / 2" by simp
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   450
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   451
    using f r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   452
  moreover
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   453
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   454
    using g r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   455
  ultimately
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   456
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   457
  proof (rule eventually_elim2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   458
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   459
    assume *: "norm (f x) < r/2" "norm (g x) < r/2"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   460
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   461
      by (rule norm_triangle_ineq)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   462
    also have "\<dots> < r/2 + r/2"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   463
      using * by (rule add_strict_mono)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   464
    finally show "norm (f x + g x) < r"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   465
      by simp
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   466
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   467
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   468
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   469
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   470
  unfolding Zfun_def by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   471
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   472
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   473
  by (simp only: diff_minus Zfun_add Zfun_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   474
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   475
lemma (in bounded_linear) Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   476
  assumes g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   477
  shows "Zfun (\<lambda>x. f (g x)) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   478
proof -
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   479
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   480
    using bounded by fast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   481
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   482
    by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   483
  with g show ?thesis
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   484
    by (rule Zfun_imp_Zfun)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   485
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   486
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   487
lemma (in bounded_bilinear) Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   488
  assumes f: "Zfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   489
  assumes g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   490
  shows "Zfun (\<lambda>x. f x ** g x) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   491
proof (rule ZfunI)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   492
  fix r::real assume r: "0 < r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   493
  obtain K where K: "0 < K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   494
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   495
    using pos_bounded by fast
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   496
  from K have K': "0 < inverse K"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   497
    by (rule positive_imp_inverse_positive)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   498
  have "eventually (\<lambda>x. norm (f x) < r) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   499
    using f r by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   500
  moreover
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   501
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   502
    using g K' by (rule ZfunD)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   503
  ultimately
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   504
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   505
  proof (rule eventually_elim2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   506
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   507
    assume *: "norm (f x) < r" "norm (g x) < inverse K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   508
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   509
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   510
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   511
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   512
    also from K have "r * inverse K * K = r"
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   513
      by simp
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   514
    finally show "norm (f x ** g x) < r" .
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   515
  qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   516
qed
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   517
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   518
lemma (in bounded_bilinear) Zfun_left:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   519
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   520
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   521
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   522
lemma (in bounded_bilinear) Zfun_right:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   523
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   524
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   525
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   526
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   527
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   528
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   529
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   530
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   531
subsection {* Limits *}
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   532
44206
5e4a1664106e locale-ize some constant definitions, so complete_space can inherit from metric_space
huffman
parents: 44205
diff changeset
   533
definition (in topological_space)
5e4a1664106e locale-ize some constant definitions, so complete_space can inherit from metric_space
huffman
parents: 44205
diff changeset
   534
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   535
  "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   536
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   537
ML {*
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   538
structure Tendsto_Intros = Named_Thms
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   539
(
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   540
  val name = "tendsto_intros"
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   541
  val description = "introduction rules for tendsto"
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   542
)
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   543
*}
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   544
31902
862ae16a799d renamed NamedThmsFun to Named_Thms;
wenzelm
parents: 31588
diff changeset
   545
setup Tendsto_Intros.setup
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   546
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   547
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   548
  unfolding tendsto_def le_filter_def by fast
36656
fec55067ae9b add lemmas eventually_nhds_metric and tendsto_mono
huffman
parents: 36655
diff changeset
   549
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   550
lemma topological_tendstoI:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   551
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   552
    \<Longrightarrow> (f ---> l) F"
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   553
  unfolding tendsto_def by auto
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   554
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   555
lemma topological_tendstoD:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   556
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   557
  unfolding tendsto_def by auto
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   558
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   559
lemma tendstoI:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   560
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   561
  shows "(f ---> l) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   562
  apply (rule topological_tendstoI)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   563
  apply (simp add: open_dist)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   564
  apply (drule (1) bspec, clarify)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   565
  apply (drule assms)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   566
  apply (erule eventually_elim1, simp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   567
  done
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   568
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   569
lemma tendstoD:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   570
  "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   571
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   572
  apply (clarsimp simp add: open_dist)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   573
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   574
  apply (simp only: less_diff_eq)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   575
  apply (erule le_less_trans [OF dist_triangle])
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   576
  apply simp
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   577
  apply simp
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   578
  done
31488
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   579
5691ccb8d6b5 generalize tendsto to class topological_space
huffman
parents: 31487
diff changeset
   580
lemma tendsto_iff:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   581
  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   582
  using tendstoI tendstoD by fast
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   583
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   584
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   585
  by (simp only: tendsto_iff Zfun_def dist_norm)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   586
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   587
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   588
  unfolding tendsto_def eventually_at_topological by auto
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   589
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   590
lemma tendsto_ident_at_within [tendsto_intros]:
36655
88f0125c3bd2 remove unneeded premise
huffman
parents: 36654
diff changeset
   591
  "((\<lambda>x. x) ---> a) (at a within S)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   592
  unfolding tendsto_def eventually_within eventually_at_topological by auto
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   593
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   594
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   595
  by (simp add: tendsto_def)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   596
44205
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   597
lemma tendsto_unique:
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   598
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   599
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   600
  shows "a = b"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   601
proof (rule ccontr)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   602
  assume "a \<noteq> b"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   603
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   604
    using hausdorff [OF `a \<noteq> b`] by fast
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   605
  have "eventually (\<lambda>x. f x \<in> U) F"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   606
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   607
  moreover
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   608
  have "eventually (\<lambda>x. f x \<in> V) F"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   609
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   610
  ultimately
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   611
  have "eventually (\<lambda>x. False) F"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   612
  proof (rule eventually_elim2)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   613
    fix x
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   614
    assume "f x \<in> U" "f x \<in> V"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   615
    hence "f x \<in> U \<inter> V" by simp
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   616
    with `U \<inter> V = {}` show "False" by simp
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   617
  qed
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   618
  with `\<not> trivial_limit F` show "False"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   619
    by (simp add: trivial_limit_def)
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   620
qed
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   621
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   622
lemma tendsto_const_iff:
44205
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   623
  fixes a b :: "'a::t2_space"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   624
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   625
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   626
44218
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   627
lemma tendsto_compose:
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   628
  assumes g: "(g ---> g l) (at l)"
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   629
  assumes f: "(f ---> l) F"
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   630
  shows "((\<lambda>x. g (f x)) ---> g l) F"
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   631
proof (rule topological_tendstoI)
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   632
  fix B assume B: "open B" "g l \<in> B"
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   633
  obtain A where A: "open A" "l \<in> A"
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   634
    and gB: "\<forall>y. y \<in> A \<longrightarrow> g y \<in> B"
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   635
    using topological_tendstoD [OF g B] B(2)
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   636
    unfolding eventually_at_topological by fast
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   637
  hence "\<forall>x. f x \<in> A \<longrightarrow> g (f x) \<in> B" by simp
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   638
  from this topological_tendstoD [OF f A]
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   639
  show "eventually (\<lambda>x. g (f x) \<in> B) F"
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   640
    by (rule eventually_mono)
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   641
qed
f0e442e24816 add lemma tendsto_compose
huffman
parents: 44206
diff changeset
   642
44253
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   643
lemma tendsto_compose_eventually:
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   644
  assumes g: "(g ---> m) (at l)"
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   645
  assumes f: "(f ---> l) F"
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   646
  assumes inj: "eventually (\<lambda>x. f x \<noteq> l) F"
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   647
  shows "((\<lambda>x. g (f x)) ---> m) F"
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   648
proof (rule topological_tendstoI)
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   649
  fix B assume B: "open B" "m \<in> B"
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   650
  obtain A where A: "open A" "l \<in> A"
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   651
    and gB: "\<And>y. y \<in> A \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> B"
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   652
    using topological_tendstoD [OF g B]
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   653
    unfolding eventually_at_topological by fast
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   654
  show "eventually (\<lambda>x. g (f x) \<in> B) F"
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   655
    using topological_tendstoD [OF f A] inj
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   656
    by (rule eventually_elim2) (simp add: gB)
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   657
qed
c073a0bd8458 add lemma tendsto_compose_eventually; use it to shorten some proofs
huffman
parents: 44251
diff changeset
   658
44251
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   659
lemma metric_tendsto_imp_tendsto:
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   660
  assumes f: "(f ---> a) F"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   661
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   662
  shows "(g ---> b) F"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   663
proof (rule tendstoI)
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   664
  fix e :: real assume "0 < e"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   665
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   666
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   667
    using le_less_trans by (rule eventually_elim2)
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   668
qed
d101ed3177b6 add lemma metric_tendsto_imp_tendsto
huffman
parents: 44218
diff changeset
   669
44205
18da2a87421c generalize constant 'lim' and limit uniqueness theorems to class t2_space
huffman
parents: 44195
diff changeset
   670
subsubsection {* Distance and norms *}
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   671
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   672
lemma tendsto_dist [tendsto_intros]:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   673
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   674
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   675
proof (rule tendstoI)
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   676
  fix e :: real assume "0 < e"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   677
  hence e2: "0 < e/2" by simp
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   678
  from tendstoD [OF f e2] tendstoD [OF g e2]
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   679
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   680
  proof (rule eventually_elim2)
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   681
    fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   682
    then show "dist (dist (f x) (g x)) (dist l m) < e"
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   683
      unfolding dist_real_def
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   684
      using dist_triangle2 [of "f x" "g x" "l"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   685
      using dist_triangle2 [of "g x" "l" "m"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   686
      using dist_triangle3 [of "l" "m" "f x"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   687
      using dist_triangle [of "f x" "m" "g x"]
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   688
      by arith
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   689
  qed
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   690
qed
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   691
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   692
lemma norm_conv_dist: "norm x = dist x 0"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   693
  unfolding dist_norm by simp
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   694
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   695
lemma tendsto_norm [tendsto_intros]:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   696
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   697
  unfolding norm_conv_dist by (intro tendsto_intros)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   698
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   699
lemma tendsto_norm_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   700
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   701
  by (drule tendsto_norm, simp)
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   702
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   703
lemma tendsto_norm_zero_cancel:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   704
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   705
  unfolding tendsto_iff dist_norm by simp
36662
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   706
621122eeb138 generalize types of LIMSEQ and LIM; generalize many lemmas
huffman
parents: 36656
diff changeset
   707
lemma tendsto_norm_zero_iff:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   708
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   709
  unfolding tendsto_iff dist_norm by simp
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   710
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   711
lemma tendsto_rabs [tendsto_intros]:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   712
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   713
  by (fold real_norm_def, rule tendsto_norm)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   714
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   715
lemma tendsto_rabs_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   716
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   717
  by (fold real_norm_def, rule tendsto_norm_zero)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   718
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   719
lemma tendsto_rabs_zero_cancel:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   720
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   721
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   722
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   723
lemma tendsto_rabs_zero_iff:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   724
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   725
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   726
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   727
subsubsection {* Addition and subtraction *}
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   728
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   729
lemma tendsto_add [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   730
  fixes a b :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   731
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   732
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   733
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   734
lemma tendsto_add_zero:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   735
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   736
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   737
  by (drule (1) tendsto_add, simp)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   738
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   739
lemma tendsto_minus [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   740
  fixes a :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   741
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   742
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   743
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   744
lemma tendsto_minus_cancel:
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   745
  fixes a :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   746
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   747
  by (drule tendsto_minus, simp)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   748
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   749
lemma tendsto_diff [tendsto_intros]:
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   750
  fixes a b :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   751
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   752
  by (simp add: diff_minus tendsto_add tendsto_minus)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   753
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   754
lemma tendsto_setsum [tendsto_intros]:
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   755
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   756
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   757
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   758
proof (cases "finite S")
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   759
  assume "finite S" thus ?thesis using assms
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   760
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
31588
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   761
next
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   762
  assume "\<not> finite S" thus ?thesis
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   763
    by (simp add: tendsto_const)
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   764
qed
2651f172c38b add lemma tendsto_setsum
huffman
parents: 31565
diff changeset
   765
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   766
subsubsection {* Linear operators and multiplication *}
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   767
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   768
lemma (in bounded_linear) tendsto:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   769
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   770
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   771
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   772
lemma (in bounded_linear) tendsto_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   773
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   774
  by (drule tendsto, simp only: zero)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   775
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   776
lemma (in bounded_bilinear) tendsto:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   777
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   778
  by (simp only: tendsto_Zfun_iff prod_diff_prod
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   779
                 Zfun_add Zfun Zfun_left Zfun_right)
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   780
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   781
lemma (in bounded_bilinear) tendsto_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   782
  assumes f: "(f ---> 0) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   783
  assumes g: "(g ---> 0) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   784
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   785
  using tendsto [OF f g] by (simp add: zero_left)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   786
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   787
lemma (in bounded_bilinear) tendsto_left_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   788
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   789
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   790
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   791
lemma (in bounded_bilinear) tendsto_right_zero:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   792
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   793
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   794
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   795
lemmas tendsto_of_real [tendsto_intros] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   796
  bounded_linear.tendsto [OF bounded_linear_of_real]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   797
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   798
lemmas tendsto_scaleR [tendsto_intros] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   799
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   800
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   801
lemmas tendsto_mult [tendsto_intros] =
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   802
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   803
44568
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   804
lemmas tendsto_mult_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   805
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   806
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   807
lemmas tendsto_mult_left_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   808
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   809
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   810
lemmas tendsto_mult_right_zero =
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   811
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
e6f291cb5810 discontinue many legacy theorems about LIM and LIMSEQ, in favor of tendsto theorems
huffman
parents: 44342
diff changeset
   812
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   813
lemma tendsto_power [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   814
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   815
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   816
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   817
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   818
lemma tendsto_setprod [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   819
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   820
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   821
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   822
proof (cases "finite S")
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   823
  assume "finite S" thus ?thesis using assms
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   824
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   825
next
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   826
  assume "\<not> finite S" thus ?thesis
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   827
    by (simp add: tendsto_const)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   828
qed
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   829
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   830
subsubsection {* Inverse and division *}
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   831
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   832
lemma (in bounded_bilinear) Zfun_prod_Bfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   833
  assumes f: "Zfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   834
  assumes g: "Bfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   835
  shows "Zfun (\<lambda>x. f x ** g x) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   836
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   837
  obtain K where K: "0 \<le> K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   838
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   839
    using nonneg_bounded by fast
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   840
  obtain B where B: "0 < B"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   841
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   842
    using g by (rule BfunE)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   843
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   844
  using norm_g proof (rule eventually_elim1)
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   845
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   846
    assume *: "norm (g x) \<le> B"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   847
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   848
      by (rule norm_le)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   849
    also have "\<dots> \<le> norm (f x) * B * K"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   850
      by (intro mult_mono' order_refl norm_g norm_ge_zero
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   851
                mult_nonneg_nonneg K *)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   852
    also have "\<dots> = norm (f x) * (B * K)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   853
      by (rule mult_assoc)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   854
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   855
  qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   856
  with f show ?thesis
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   857
    by (rule Zfun_imp_Zfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   858
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   859
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   860
lemma (in bounded_bilinear) flip:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   861
  "bounded_bilinear (\<lambda>x y. y ** x)"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   862
  apply default
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   863
  apply (rule add_right)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   864
  apply (rule add_left)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   865
  apply (rule scaleR_right)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   866
  apply (rule scaleR_left)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   867
  apply (subst mult_commute)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   868
  using bounded by fast
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   869
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   870
lemma (in bounded_bilinear) Bfun_prod_Zfun:
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   871
  assumes f: "Bfun f F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   872
  assumes g: "Zfun g F"
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   873
  shows "Zfun (\<lambda>x. f x ** g x) F"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   874
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   875
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   876
lemma Bfun_inverse_lemma:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   877
  fixes x :: "'a::real_normed_div_algebra"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   878
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
44081
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   879
  apply (subst nonzero_norm_inverse, clarsimp)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   880
  apply (erule (1) le_imp_inverse_le)
730f7cced3a6 rename type 'a net to 'a filter, following standard mathematical terminology
huffman
parents: 44079
diff changeset
   881
  done
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   882
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   883
lemma Bfun_inverse:
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   884
  fixes a :: "'a::real_normed_div_algebra"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   885
  assumes f: "(f ---> a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   886
  assumes a: "a \<noteq> 0"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   887
  shows "Bfun (\<lambda>x. inverse (f x)) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   888
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   889
  from a have "0 < norm a" by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   890
  hence "\<exists>r>0. r < norm a" by (rule dense)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   891
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   892
  have "eventually (\<lambda>x. dist (f x) a < r) F"
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   893
    using tendstoD [OF f r1] by fast
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   894
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   895
  proof (rule eventually_elim1)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   896
    fix x
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   897
    assume "dist (f x) a < r"
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   898
    hence 1: "norm (f x - a) < r"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   899
      by (simp add: dist_norm)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   900
    hence 2: "f x \<noteq> 0" using r2 by auto
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   901
    hence "norm (inverse (f x)) = inverse (norm (f x))"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   902
      by (rule nonzero_norm_inverse)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   903
    also have "\<dots> \<le> inverse (norm a - r)"
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   904
    proof (rule le_imp_inverse_le)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   905
      show "0 < norm a - r" using r2 by simp
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   906
    next
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   907
      have "norm a - norm (f x) \<le> norm (a - f x)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   908
        by (rule norm_triangle_ineq2)
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   909
      also have "\<dots> = norm (f x - a)"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   910
        by (rule norm_minus_commute)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   911
      also have "\<dots> < r" using 1 .
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   912
      finally show "norm a - r \<le> norm (f x)" by simp
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   913
    qed
31487
93938cafc0e6 put syntax for tendsto in Limits.thy; rename variables
huffman
parents: 31447
diff changeset
   914
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   915
  qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   916
  thus ?thesis by (rule BfunI)
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   917
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   918
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   919
lemma tendsto_inverse [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   920
  fixes a :: "'a::real_normed_div_algebra"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   921
  assumes f: "(f ---> a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   922
  assumes a: "a \<noteq> 0"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   923
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   924
proof -
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   925
  from a have "0 < norm a" by simp
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   926
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   927
    by (rule tendstoD)
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   928
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   929
    unfolding dist_norm by (auto elim!: eventually_elim1)
44627
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   930
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   931
    - (inverse (f x) * (f x - a) * inverse a)) F"
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   932
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   933
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   934
    by (intro Zfun_minus Zfun_mult_left
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   935
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   936
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   937
  ultimately show ?thesis
134c06282ae6 convert to Isar-style proof
huffman
parents: 44571
diff changeset
   938
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   939
qed
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   940
31565
da5a5589418e theorem attribute [tendsto_intros]
huffman
parents: 31492
diff changeset
   941
lemma tendsto_divide [tendsto_intros]:
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   942
  fixes a b :: "'a::real_normed_field"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   943
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   944
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
44282
f0de18b62d63 remove bounded_(bi)linear locale interpretations, to avoid duplicating so many lemmas
huffman
parents: 44253
diff changeset
   945
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
31355
3d18766ddc4b limits of inverse using filters
huffman
parents: 31353
diff changeset
   946
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   947
lemma tendsto_sgn [tendsto_intros]:
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   948
  fixes l :: "'a::real_normed_vector"
44195
f5363511b212 consistently use variable name 'F' for filters
huffman
parents: 44194
diff changeset
   949
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
44194
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   950
  unfolding sgn_div_norm by (simp add: tendsto_intros)
0639898074ae generalize lemmas about LIM and LIMSEQ to tendsto
huffman
parents: 44081
diff changeset
   951
31349
2261c8781f73 new theory of filters and limits; prove LIMSEQ and LIM lemmas using filters
huffman
parents:
diff changeset
   952
end