src/HOL/Limits.thy
author hoelzl
Mon Dec 03 18:18:59 2012 +0100 (2012-12-03)
changeset 50322 b06b95a5fda2
parent 50247 491c5c81c2e8
child 50323 3764d4620fb3
permissions -rw-r--r--
rename filter_lim to filterlim to be consistent with filtermap
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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 '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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lemma eventually_subst:
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  assumes "eventually (\<lambda>n. P n = Q n) F"
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  shows "eventually P F = eventually Q F" (is "?L = ?R")
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proof -
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  from assms have "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
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      and "eventually (\<lambda>x. Q x \<longrightarrow> P x) F"
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    by (auto elim: eventually_elim1)
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  then show ?thesis by (auto elim: eventually_elim2)
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qed
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ML {*
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  fun eventually_elim_tac ctxt thms thm =
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    let
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      val thy = Proof_Context.theory_of ctxt
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      val mp_thms = thms RL [@{thm eventually_rev_mp}]
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      val raw_elim_thm =
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        (@{thm allI} RS @{thm always_eventually})
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        |> fold (fn thm1 => fn thm2 => thm2 RS thm1) mp_thms
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        |> fold (fn _ => fn thm => @{thm impI} RS thm) thms
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      val cases_prop = prop_of (raw_elim_thm RS thm)
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      val cases = (Rule_Cases.make_common (thy, cases_prop) [(("elim", []), [])])
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    in
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      CASES cases (rtac raw_elim_thm 1) thm
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    end
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*}
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method_setup eventually_elim = {*
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  Scan.succeed (fn ctxt => METHOD_CASES (eventually_elim_tac ctxt))
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*} "elimination of eventually quantifiers"
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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 {* Order filters *}
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definition at_top :: "('a::order) filter"
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  where "at_top = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
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lemma eventually_at_top_linorder:
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  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_top \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
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  unfolding at_top_def
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proof (rule eventually_Abs_filter, rule is_filter.intro)
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  fix P Q :: "'a \<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 eventually_at_top_dense:
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  fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_top \<longleftrightarrow> (\<exists>N. \<forall>n>N. P n)"
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  unfolding eventually_at_top_linorder
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proof safe
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  fix N assume "\<forall>n\<ge>N. P n" then show "\<exists>N. \<forall>n>N. P n" by (auto intro!: exI[of _ N])
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next
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  fix N assume "\<forall>n>N. P n" 
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  moreover from gt_ex[of N] guess y ..
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  ultimately show "\<exists>N. \<forall>n\<ge>N. P n" by (auto intro!: exI[of _ y])
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qed
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definition at_bot :: "('a::order) filter"
hoelzl@50247
   311
  where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
hoelzl@50247
   312
hoelzl@50247
   313
lemma eventually_at_bot_linorder:
hoelzl@50247
   314
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@50247
   315
  unfolding at_bot_def
hoelzl@50247
   316
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@50247
   317
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@50247
   318
  assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
hoelzl@50247
   319
  then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
hoelzl@50247
   320
  then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
hoelzl@50247
   321
  then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
hoelzl@50247
   322
qed auto
hoelzl@50247
   323
hoelzl@50247
   324
lemma eventually_at_bot_dense:
hoelzl@50247
   325
  fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
hoelzl@50247
   326
  unfolding eventually_at_bot_linorder
hoelzl@50247
   327
proof safe
hoelzl@50247
   328
  fix N assume "\<forall>n\<le>N. P n" then show "\<exists>N. \<forall>n<N. P n" by (auto intro!: exI[of _ N])
hoelzl@50247
   329
next
hoelzl@50247
   330
  fix N assume "\<forall>n<N. P n" 
hoelzl@50247
   331
  moreover from lt_ex[of N] guess y ..
hoelzl@50247
   332
  ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
hoelzl@50247
   333
qed
hoelzl@50247
   334
hoelzl@50247
   335
subsection {* Sequentially *}
hoelzl@50247
   336
hoelzl@50247
   337
abbreviation sequentially :: "nat filter"
hoelzl@50247
   338
  where "sequentially == at_top"
hoelzl@50247
   339
hoelzl@50247
   340
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
hoelzl@50247
   341
  unfolding at_top_def by simp
hoelzl@50247
   342
hoelzl@50247
   343
lemma eventually_sequentially:
hoelzl@50247
   344
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@50247
   345
  by (rule eventually_at_top_linorder)
hoelzl@50247
   346
huffman@44342
   347
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
huffman@44081
   348
  unfolding filter_eq_iff eventually_sequentially by auto
huffman@36662
   349
huffman@44342
   350
lemmas trivial_limit_sequentially = sequentially_bot
huffman@44342
   351
huffman@36662
   352
lemma eventually_False_sequentially [simp]:
huffman@36662
   353
  "\<not> eventually (\<lambda>n. False) sequentially"
huffman@44081
   354
  by (simp add: eventually_False)
huffman@36662
   355
huffman@36662
   356
lemma le_sequentially:
huffman@44195
   357
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
huffman@44081
   358
  unfolding le_filter_def eventually_sequentially
huffman@44081
   359
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
huffman@36662
   360
noschinl@45892
   361
lemma eventually_sequentiallyI:
noschinl@45892
   362
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
noschinl@45892
   363
  shows "eventually P sequentially"
noschinl@45892
   364
using assms by (auto simp: eventually_sequentially)
noschinl@45892
   365
huffman@36662
   366
huffman@44081
   367
subsection {* Standard filters *}
huffman@36662
   368
huffman@44081
   369
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
huffman@44195
   370
  where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
huffman@31392
   371
huffman@44206
   372
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
huffman@44081
   373
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   374
huffman@44206
   375
definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
huffman@44081
   376
  where "at a = nhds a within - {a}"
huffman@31447
   377
huffman@31392
   378
lemma eventually_within:
huffman@44195
   379
  "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
huffman@44081
   380
  unfolding within_def
huffman@44081
   381
  by (rule eventually_Abs_filter, rule is_filter.intro)
huffman@44081
   382
     (auto elim!: eventually_rev_mp)
huffman@31392
   383
huffman@45031
   384
lemma within_UNIV [simp]: "F within UNIV = F"
huffman@45031
   385
  unfolding filter_eq_iff eventually_within by simp
huffman@45031
   386
huffman@45031
   387
lemma within_empty [simp]: "F within {} = bot"
huffman@44081
   388
  unfolding filter_eq_iff eventually_within by simp
huffman@36360
   389
hoelzl@50247
   390
lemma within_le: "F within S \<le> F"
hoelzl@50247
   391
  unfolding le_filter_def eventually_within by (auto elim: eventually_elim1)
hoelzl@50247
   392
huffman@36654
   393
lemma eventually_nhds:
huffman@36654
   394
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   395
unfolding nhds_def
huffman@44081
   396
proof (rule eventually_Abs_filter, rule is_filter.intro)
huffman@36654
   397
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
huffman@36654
   398
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
huffman@36358
   399
next
huffman@36358
   400
  fix P Q
huffman@36654
   401
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   402
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
huffman@36358
   403
  then obtain S T where
huffman@36654
   404
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   405
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
huffman@36654
   406
  hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
huffman@36358
   407
    by (simp add: open_Int)
huffman@36654
   408
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
huffman@36358
   409
qed auto
huffman@31447
   410
huffman@36656
   411
lemma eventually_nhds_metric:
huffman@36656
   412
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
huffman@36656
   413
unfolding eventually_nhds open_dist
huffman@31447
   414
apply safe
huffman@31447
   415
apply fast
huffman@31492
   416
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
huffman@31447
   417
apply clarsimp
huffman@31447
   418
apply (rule_tac x="d - dist x a" in exI, clarsimp)
huffman@31447
   419
apply (simp only: less_diff_eq)
huffman@31447
   420
apply (erule le_less_trans [OF dist_triangle])
huffman@31447
   421
done
huffman@31447
   422
huffman@44571
   423
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
huffman@44571
   424
  unfolding trivial_limit_def eventually_nhds by simp
huffman@44571
   425
huffman@36656
   426
lemma eventually_at_topological:
huffman@36656
   427
  "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
huffman@36656
   428
unfolding at_def eventually_within eventually_nhds by simp
huffman@36656
   429
huffman@36656
   430
lemma eventually_at:
huffman@36656
   431
  fixes a :: "'a::metric_space"
huffman@36656
   432
  shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
huffman@36656
   433
unfolding at_def eventually_within eventually_nhds_metric by auto
huffman@36656
   434
huffman@44571
   435
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
huffman@44571
   436
  unfolding trivial_limit_def eventually_at_topological
huffman@44571
   437
  by (safe, case_tac "S = {a}", simp, fast, fast)
huffman@44571
   438
huffman@44571
   439
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
huffman@44571
   440
  by (simp add: at_eq_bot_iff not_open_singleton)
huffman@44571
   441
huffman@31392
   442
huffman@31355
   443
subsection {* Boundedness *}
huffman@31355
   444
huffman@44081
   445
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   446
  where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
huffman@31355
   447
huffman@31487
   448
lemma BfunI:
huffman@44195
   449
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
huffman@31355
   450
unfolding Bfun_def
huffman@31355
   451
proof (intro exI conjI allI)
huffman@31355
   452
  show "0 < max K 1" by simp
huffman@31355
   453
next
huffman@44195
   454
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
huffman@31355
   455
    using K by (rule eventually_elim1, simp)
huffman@31355
   456
qed
huffman@31355
   457
huffman@31355
   458
lemma BfunE:
huffman@44195
   459
  assumes "Bfun f F"
huffman@44195
   460
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
huffman@31355
   461
using assms unfolding Bfun_def by fast
huffman@31355
   462
huffman@31355
   463
huffman@31349
   464
subsection {* Convergence to Zero *}
huffman@31349
   465
huffman@44081
   466
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   467
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   468
huffman@31349
   469
lemma ZfunI:
huffman@44195
   470
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   471
  unfolding Zfun_def by simp
huffman@31349
   472
huffman@31349
   473
lemma ZfunD:
huffman@44195
   474
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   475
  unfolding Zfun_def by simp
huffman@31349
   476
huffman@31355
   477
lemma Zfun_ssubst:
huffman@44195
   478
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   479
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   480
huffman@44195
   481
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   482
  unfolding Zfun_def by simp
huffman@31349
   483
huffman@44195
   484
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   485
  unfolding Zfun_def by simp
huffman@31349
   486
huffman@31349
   487
lemma Zfun_imp_Zfun:
huffman@44195
   488
  assumes f: "Zfun f F"
huffman@44195
   489
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   490
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   491
proof (cases)
huffman@31349
   492
  assume K: "0 < K"
huffman@31349
   493
  show ?thesis
huffman@31349
   494
  proof (rule ZfunI)
huffman@31349
   495
    fix r::real assume "0 < r"
huffman@31349
   496
    hence "0 < r / K"
huffman@31349
   497
      using K by (rule divide_pos_pos)
huffman@44195
   498
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
huffman@31487
   499
      using ZfunD [OF f] by fast
huffman@44195
   500
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   501
    proof eventually_elim
noschinl@46887
   502
      case (elim x)
huffman@31487
   503
      hence "norm (f x) * K < r"
huffman@31349
   504
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   505
      thus ?case
noschinl@46887
   506
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   507
    qed
huffman@31349
   508
  qed
huffman@31349
   509
next
huffman@31349
   510
  assume "\<not> 0 < K"
huffman@31349
   511
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   512
  show ?thesis
huffman@31355
   513
  proof (rule ZfunI)
huffman@31355
   514
    fix r :: real
huffman@31355
   515
    assume "0 < r"
huffman@44195
   516
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   517
    proof eventually_elim
noschinl@46887
   518
      case (elim x)
noschinl@46887
   519
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   520
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   521
      finally show ?case
huffman@31355
   522
        using `0 < r` by simp
huffman@31355
   523
    qed
huffman@31355
   524
  qed
huffman@31349
   525
qed
huffman@31349
   526
huffman@44195
   527
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   528
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   529
huffman@31349
   530
lemma Zfun_add:
huffman@44195
   531
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   532
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   533
proof (rule ZfunI)
huffman@31349
   534
  fix r::real assume "0 < r"
huffman@31349
   535
  hence r: "0 < r / 2" by simp
huffman@44195
   536
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   537
    using f r by (rule ZfunD)
huffman@31349
   538
  moreover
huffman@44195
   539
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   540
    using g r by (rule ZfunD)
huffman@31349
   541
  ultimately
huffman@44195
   542
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   543
  proof eventually_elim
noschinl@46887
   544
    case (elim x)
huffman@31487
   545
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   546
      by (rule norm_triangle_ineq)
huffman@31349
   547
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   548
      using elim by (rule add_strict_mono)
noschinl@46887
   549
    finally show ?case
huffman@31349
   550
      by simp
huffman@31349
   551
  qed
huffman@31349
   552
qed
huffman@31349
   553
huffman@44195
   554
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   555
  unfolding Zfun_def by simp
huffman@31349
   556
huffman@44195
   557
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
huffman@44081
   558
  by (simp only: diff_minus Zfun_add Zfun_minus)
huffman@31349
   559
huffman@31349
   560
lemma (in bounded_linear) Zfun:
huffman@44195
   561
  assumes g: "Zfun g F"
huffman@44195
   562
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   563
proof -
huffman@31349
   564
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
huffman@31349
   565
    using bounded by fast
huffman@44195
   566
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   567
    by simp
huffman@31487
   568
  with g show ?thesis
huffman@31349
   569
    by (rule Zfun_imp_Zfun)
huffman@31349
   570
qed
huffman@31349
   571
huffman@31349
   572
lemma (in bounded_bilinear) Zfun:
huffman@44195
   573
  assumes f: "Zfun f F"
huffman@44195
   574
  assumes g: "Zfun g F"
huffman@44195
   575
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   576
proof (rule ZfunI)
huffman@31349
   577
  fix r::real assume r: "0 < r"
huffman@31349
   578
  obtain K where K: "0 < K"
huffman@31349
   579
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   580
    using pos_bounded by fast
huffman@31349
   581
  from K have K': "0 < inverse K"
huffman@31349
   582
    by (rule positive_imp_inverse_positive)
huffman@44195
   583
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   584
    using f r by (rule ZfunD)
huffman@31349
   585
  moreover
huffman@44195
   586
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   587
    using g K' by (rule ZfunD)
huffman@31349
   588
  ultimately
huffman@44195
   589
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   590
  proof eventually_elim
noschinl@46887
   591
    case (elim x)
huffman@31487
   592
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   593
      by (rule norm_le)
huffman@31487
   594
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   595
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   596
    also from K have "r * inverse K * K = r"
huffman@31349
   597
      by simp
noschinl@46887
   598
    finally show ?case .
huffman@31349
   599
  qed
huffman@31349
   600
qed
huffman@31349
   601
huffman@31349
   602
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   603
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   604
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   605
huffman@31349
   606
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   607
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   608
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   609
huffman@44282
   610
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   611
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   612
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   613
huffman@31349
   614
wenzelm@31902
   615
subsection {* Limits *}
huffman@31349
   616
hoelzl@50322
   617
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@50322
   618
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@50247
   619
hoelzl@50247
   620
syntax
hoelzl@50247
   621
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@50247
   622
hoelzl@50247
   623
translations
hoelzl@50322
   624
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@50247
   625
hoelzl@50322
   626
lemma filterlimE: "(LIM x F1. f x :> F2) \<Longrightarrow> eventually P F2 \<Longrightarrow> eventually (\<lambda>x. P (f x)) F1"
hoelzl@50322
   627
  by (auto simp: filterlim_def eventually_filtermap le_filter_def)
hoelzl@50247
   628
hoelzl@50322
   629
lemma filterlimI: "(\<And>P. eventually P F2 \<Longrightarrow> eventually (\<lambda>x. P (f x)) F1) \<Longrightarrow> (LIM x F1. f x :> F2)"
hoelzl@50322
   630
  by (auto simp: filterlim_def eventually_filtermap le_filter_def)
hoelzl@50247
   631
hoelzl@50247
   632
abbreviation (in topological_space)
huffman@44206
   633
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
hoelzl@50322
   634
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
noschinl@45892
   635
wenzelm@31902
   636
ML {*
wenzelm@31902
   637
structure Tendsto_Intros = Named_Thms
wenzelm@31902
   638
(
wenzelm@45294
   639
  val name = @{binding tendsto_intros}
wenzelm@31902
   640
  val description = "introduction rules for tendsto"
wenzelm@31902
   641
)
huffman@31565
   642
*}
huffman@31565
   643
wenzelm@31902
   644
setup Tendsto_Intros.setup
huffman@31565
   645
hoelzl@50247
   646
lemma tendsto_def: "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
hoelzl@50322
   647
  unfolding filterlim_def
hoelzl@50247
   648
proof safe
hoelzl@50247
   649
  fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
hoelzl@50247
   650
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@50247
   651
    unfolding eventually_nhds eventually_filtermap le_filter_def
hoelzl@50247
   652
    by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
hoelzl@50247
   653
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
hoelzl@50247
   654
huffman@44195
   655
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
huffman@44081
   656
  unfolding tendsto_def le_filter_def by fast
huffman@36656
   657
huffman@31488
   658
lemma topological_tendstoI:
huffman@44195
   659
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
huffman@44195
   660
    \<Longrightarrow> (f ---> l) F"
huffman@31349
   661
  unfolding tendsto_def by auto
huffman@31349
   662
huffman@31488
   663
lemma topological_tendstoD:
huffman@44195
   664
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
huffman@31488
   665
  unfolding tendsto_def by auto
huffman@31488
   666
huffman@31488
   667
lemma tendstoI:
huffman@44195
   668
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44195
   669
  shows "(f ---> l) F"
huffman@44081
   670
  apply (rule topological_tendstoI)
huffman@44081
   671
  apply (simp add: open_dist)
huffman@44081
   672
  apply (drule (1) bspec, clarify)
huffman@44081
   673
  apply (drule assms)
huffman@44081
   674
  apply (erule eventually_elim1, simp)
huffman@44081
   675
  done
huffman@31488
   676
huffman@31349
   677
lemma tendstoD:
huffman@44195
   678
  "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44081
   679
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
huffman@44081
   680
  apply (clarsimp simp add: open_dist)
huffman@44081
   681
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
huffman@44081
   682
  apply (simp only: less_diff_eq)
huffman@44081
   683
  apply (erule le_less_trans [OF dist_triangle])
huffman@44081
   684
  apply simp
huffman@44081
   685
  apply simp
huffman@44081
   686
  done
huffman@31488
   687
huffman@31488
   688
lemma tendsto_iff:
huffman@44195
   689
  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
huffman@44081
   690
  using tendstoI tendstoD by fast
huffman@31349
   691
huffman@44195
   692
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   693
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   694
huffman@45031
   695
lemma tendsto_bot [simp]: "(f ---> a) bot"
huffman@45031
   696
  unfolding tendsto_def by simp
huffman@45031
   697
huffman@31565
   698
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
huffman@44081
   699
  unfolding tendsto_def eventually_at_topological by auto
huffman@31565
   700
huffman@31565
   701
lemma tendsto_ident_at_within [tendsto_intros]:
huffman@36655
   702
  "((\<lambda>x. x) ---> a) (at a within S)"
huffman@44081
   703
  unfolding tendsto_def eventually_within eventually_at_topological by auto
huffman@31565
   704
huffman@44195
   705
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
huffman@44081
   706
  by (simp add: tendsto_def)
huffman@31349
   707
huffman@44205
   708
lemma tendsto_unique:
huffman@44205
   709
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
huffman@44205
   710
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
huffman@44205
   711
  shows "a = b"
huffman@44205
   712
proof (rule ccontr)
huffman@44205
   713
  assume "a \<noteq> b"
huffman@44205
   714
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
huffman@44205
   715
    using hausdorff [OF `a \<noteq> b`] by fast
huffman@44205
   716
  have "eventually (\<lambda>x. f x \<in> U) F"
huffman@44205
   717
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
huffman@44205
   718
  moreover
huffman@44205
   719
  have "eventually (\<lambda>x. f x \<in> V) F"
huffman@44205
   720
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
huffman@44205
   721
  ultimately
huffman@44205
   722
  have "eventually (\<lambda>x. False) F"
noschinl@46887
   723
  proof eventually_elim
noschinl@46887
   724
    case (elim x)
huffman@44205
   725
    hence "f x \<in> U \<inter> V" by simp
noschinl@46887
   726
    with `U \<inter> V = {}` show ?case by simp
huffman@44205
   727
  qed
huffman@44205
   728
  with `\<not> trivial_limit F` show "False"
huffman@44205
   729
    by (simp add: trivial_limit_def)
huffman@44205
   730
qed
huffman@44205
   731
huffman@36662
   732
lemma tendsto_const_iff:
huffman@44205
   733
  fixes a b :: "'a::t2_space"
huffman@44205
   734
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
huffman@44205
   735
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
huffman@44205
   736
huffman@44218
   737
lemma tendsto_compose:
huffman@44218
   738
  assumes g: "(g ---> g l) (at l)"
huffman@44218
   739
  assumes f: "(f ---> l) F"
huffman@44218
   740
  shows "((\<lambda>x. g (f x)) ---> g l) F"
huffman@44218
   741
proof (rule topological_tendstoI)
huffman@44218
   742
  fix B assume B: "open B" "g l \<in> B"
huffman@44218
   743
  obtain A where A: "open A" "l \<in> A"
huffman@44218
   744
    and gB: "\<forall>y. y \<in> A \<longrightarrow> g y \<in> B"
huffman@44218
   745
    using topological_tendstoD [OF g B] B(2)
huffman@44218
   746
    unfolding eventually_at_topological by fast
huffman@44218
   747
  hence "\<forall>x. f x \<in> A \<longrightarrow> g (f x) \<in> B" by simp
huffman@44218
   748
  from this topological_tendstoD [OF f A]
huffman@44218
   749
  show "eventually (\<lambda>x. g (f x) \<in> B) F"
huffman@44218
   750
    by (rule eventually_mono)
huffman@44218
   751
qed
huffman@44218
   752
huffman@44253
   753
lemma tendsto_compose_eventually:
huffman@44253
   754
  assumes g: "(g ---> m) (at l)"
huffman@44253
   755
  assumes f: "(f ---> l) F"
huffman@44253
   756
  assumes inj: "eventually (\<lambda>x. f x \<noteq> l) F"
huffman@44253
   757
  shows "((\<lambda>x. g (f x)) ---> m) F"
huffman@44253
   758
proof (rule topological_tendstoI)
huffman@44253
   759
  fix B assume B: "open B" "m \<in> B"
huffman@44253
   760
  obtain A where A: "open A" "l \<in> A"
huffman@44253
   761
    and gB: "\<And>y. y \<in> A \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> B"
huffman@44253
   762
    using topological_tendstoD [OF g B]
huffman@44253
   763
    unfolding eventually_at_topological by fast
huffman@44253
   764
  show "eventually (\<lambda>x. g (f x) \<in> B) F"
huffman@44253
   765
    using topological_tendstoD [OF f A] inj
huffman@44253
   766
    by (rule eventually_elim2) (simp add: gB)
huffman@44253
   767
qed
huffman@44253
   768
huffman@44251
   769
lemma metric_tendsto_imp_tendsto:
huffman@44251
   770
  assumes f: "(f ---> a) F"
huffman@44251
   771
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
huffman@44251
   772
  shows "(g ---> b) F"
huffman@44251
   773
proof (rule tendstoI)
huffman@44251
   774
  fix e :: real assume "0 < e"
huffman@44251
   775
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
huffman@44251
   776
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
huffman@44251
   777
    using le_less_trans by (rule eventually_elim2)
huffman@44251
   778
qed
huffman@44251
   779
huffman@44205
   780
subsubsection {* Distance and norms *}
huffman@36662
   781
huffman@31565
   782
lemma tendsto_dist [tendsto_intros]:
huffman@44195
   783
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
huffman@44195
   784
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
huffman@31565
   785
proof (rule tendstoI)
huffman@31565
   786
  fix e :: real assume "0 < e"
huffman@31565
   787
  hence e2: "0 < e/2" by simp
huffman@31565
   788
  from tendstoD [OF f e2] tendstoD [OF g e2]
huffman@44195
   789
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
noschinl@46887
   790
  proof (eventually_elim)
noschinl@46887
   791
    case (elim x)
huffman@31565
   792
    then show "dist (dist (f x) (g x)) (dist l m) < e"
huffman@31565
   793
      unfolding dist_real_def
huffman@31565
   794
      using dist_triangle2 [of "f x" "g x" "l"]
huffman@31565
   795
      using dist_triangle2 [of "g x" "l" "m"]
huffman@31565
   796
      using dist_triangle3 [of "l" "m" "f x"]
huffman@31565
   797
      using dist_triangle [of "f x" "m" "g x"]
huffman@31565
   798
      by arith
huffman@31565
   799
  qed
huffman@31565
   800
qed
huffman@31565
   801
huffman@36662
   802
lemma norm_conv_dist: "norm x = dist x 0"
huffman@44081
   803
  unfolding dist_norm by simp
huffman@36662
   804
huffman@31565
   805
lemma tendsto_norm [tendsto_intros]:
huffman@44195
   806
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
huffman@44081
   807
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   808
huffman@36662
   809
lemma tendsto_norm_zero:
huffman@44195
   810
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
huffman@44081
   811
  by (drule tendsto_norm, simp)
huffman@36662
   812
huffman@36662
   813
lemma tendsto_norm_zero_cancel:
huffman@44195
   814
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
huffman@44081
   815
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   816
huffman@36662
   817
lemma tendsto_norm_zero_iff:
huffman@44195
   818
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
huffman@44081
   819
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   820
huffman@44194
   821
lemma tendsto_rabs [tendsto_intros]:
huffman@44195
   822
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
huffman@44194
   823
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   824
huffman@44194
   825
lemma tendsto_rabs_zero:
huffman@44195
   826
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
huffman@44194
   827
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   828
huffman@44194
   829
lemma tendsto_rabs_zero_cancel:
huffman@44195
   830
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
huffman@44194
   831
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   832
huffman@44194
   833
lemma tendsto_rabs_zero_iff:
huffman@44195
   834
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
huffman@44194
   835
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   836
huffman@44194
   837
subsubsection {* Addition and subtraction *}
huffman@44194
   838
huffman@31565
   839
lemma tendsto_add [tendsto_intros]:
huffman@31349
   840
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   841
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   842
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   843
huffman@44194
   844
lemma tendsto_add_zero:
huffman@44194
   845
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   846
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   847
  by (drule (1) tendsto_add, simp)
huffman@44194
   848
huffman@31565
   849
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   850
  fixes a :: "'a::real_normed_vector"
huffman@44195
   851
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   852
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   853
huffman@31349
   854
lemma tendsto_minus_cancel:
huffman@31349
   855
  fixes a :: "'a::real_normed_vector"
huffman@44195
   856
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   857
  by (drule tendsto_minus, simp)
huffman@31349
   858
huffman@31565
   859
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   860
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   861
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
huffman@44081
   862
  by (simp add: diff_minus tendsto_add tendsto_minus)
huffman@31349
   863
huffman@31588
   864
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   865
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   866
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   867
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   868
proof (cases "finite S")
huffman@31588
   869
  assume "finite S" thus ?thesis using assms
huffman@44194
   870
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
huffman@31588
   871
next
huffman@31588
   872
  assume "\<not> finite S" thus ?thesis
huffman@31588
   873
    by (simp add: tendsto_const)
huffman@31588
   874
qed
huffman@31588
   875
noschinl@45892
   876
lemma real_tendsto_sandwich:
noschinl@45892
   877
  fixes f g h :: "'a \<Rightarrow> real"
noschinl@45892
   878
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
noschinl@45892
   879
  assumes lim: "(f ---> c) net" "(h ---> c) net"
noschinl@45892
   880
  shows "(g ---> c) net"
noschinl@45892
   881
proof -
noschinl@45892
   882
  have "((\<lambda>n. g n - f n) ---> 0) net"
noschinl@45892
   883
  proof (rule metric_tendsto_imp_tendsto)
noschinl@45892
   884
    show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
noschinl@45892
   885
      using ev by (rule eventually_elim2) (simp add: dist_real_def)
noschinl@45892
   886
    show "((\<lambda>n. h n - f n) ---> 0) net"
noschinl@45892
   887
      using tendsto_diff[OF lim(2,1)] by simp
noschinl@45892
   888
  qed
noschinl@45892
   889
  from tendsto_add[OF this lim(1)] show ?thesis by simp
noschinl@45892
   890
qed
noschinl@45892
   891
huffman@44194
   892
subsubsection {* Linear operators and multiplication *}
huffman@44194
   893
huffman@44282
   894
lemma (in bounded_linear) tendsto:
huffman@44195
   895
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   896
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   897
huffman@44194
   898
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   899
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   900
  by (drule tendsto, simp only: zero)
huffman@44194
   901
huffman@44282
   902
lemma (in bounded_bilinear) tendsto:
huffman@44195
   903
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   904
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   905
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   906
huffman@44194
   907
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   908
  assumes f: "(f ---> 0) F"
huffman@44195
   909
  assumes g: "(g ---> 0) F"
huffman@44195
   910
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   911
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   912
huffman@44194
   913
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   914
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   915
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   916
huffman@44194
   917
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   918
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   919
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   920
huffman@44282
   921
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   922
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   923
huffman@44282
   924
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   925
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   926
huffman@44282
   927
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   928
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   929
huffman@44568
   930
lemmas tendsto_mult_zero =
huffman@44568
   931
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   932
huffman@44568
   933
lemmas tendsto_mult_left_zero =
huffman@44568
   934
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   935
huffman@44568
   936
lemmas tendsto_mult_right_zero =
huffman@44568
   937
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   938
huffman@44194
   939
lemma tendsto_power [tendsto_intros]:
huffman@44194
   940
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   941
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
huffman@44194
   942
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
huffman@44194
   943
huffman@44194
   944
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   945
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   946
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   947
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   948
proof (cases "finite S")
huffman@44194
   949
  assume "finite S" thus ?thesis using assms
huffman@44194
   950
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
huffman@44194
   951
next
huffman@44194
   952
  assume "\<not> finite S" thus ?thesis
huffman@44194
   953
    by (simp add: tendsto_const)
huffman@44194
   954
qed
huffman@44194
   955
huffman@44194
   956
subsubsection {* Inverse and division *}
huffman@31355
   957
huffman@31355
   958
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   959
  assumes f: "Zfun f F"
huffman@44195
   960
  assumes g: "Bfun g F"
huffman@44195
   961
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   962
proof -
huffman@31355
   963
  obtain K where K: "0 \<le> K"
huffman@31355
   964
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   965
    using nonneg_bounded by fast
huffman@31355
   966
  obtain B where B: "0 < B"
huffman@44195
   967
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   968
    using g by (rule BfunE)
huffman@44195
   969
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   970
  using norm_g proof eventually_elim
noschinl@46887
   971
    case (elim x)
huffman@31487
   972
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   973
      by (rule norm_le)
huffman@31487
   974
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   975
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   976
                mult_nonneg_nonneg K elim)
huffman@31487
   977
    also have "\<dots> = norm (f x) * (B * K)"
huffman@31355
   978
      by (rule mult_assoc)
huffman@31487
   979
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   980
  qed
huffman@31487
   981
  with f show ?thesis
huffman@31487
   982
    by (rule Zfun_imp_Zfun)
huffman@31355
   983
qed
huffman@31355
   984
huffman@31355
   985
lemma (in bounded_bilinear) flip:
huffman@31355
   986
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@44081
   987
  apply default
huffman@44081
   988
  apply (rule add_right)
huffman@44081
   989
  apply (rule add_left)
huffman@44081
   990
  apply (rule scaleR_right)
huffman@44081
   991
  apply (rule scaleR_left)
huffman@44081
   992
  apply (subst mult_commute)
huffman@44081
   993
  using bounded by fast
huffman@31355
   994
huffman@31355
   995
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   996
  assumes f: "Bfun f F"
huffman@44195
   997
  assumes g: "Zfun g F"
huffman@44195
   998
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   999
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
  1000
huffman@31355
  1001
lemma Bfun_inverse_lemma:
huffman@31355
  1002
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
  1003
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
  1004
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
  1005
  apply (erule (1) le_imp_inverse_le)
huffman@44081
  1006
  done
huffman@31355
  1007
huffman@31355
  1008
lemma Bfun_inverse:
huffman@31355
  1009
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
  1010
  assumes f: "(f ---> a) F"
huffman@31355
  1011
  assumes a: "a \<noteq> 0"
huffman@44195
  1012
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
  1013
proof -
huffman@31355
  1014
  from a have "0 < norm a" by simp
huffman@31355
  1015
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
  1016
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
  1017
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
  1018
    using tendstoD [OF f r1] by fast
huffman@44195
  1019
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
  1020
  proof eventually_elim
noschinl@46887
  1021
    case (elim x)
huffman@31487
  1022
    hence 1: "norm (f x - a) < r"
huffman@31355
  1023
      by (simp add: dist_norm)
huffman@31487
  1024
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
  1025
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
  1026
      by (rule nonzero_norm_inverse)
huffman@31355
  1027
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
  1028
    proof (rule le_imp_inverse_le)
huffman@31355
  1029
      show "0 < norm a - r" using r2 by simp
huffman@31355
  1030
    next
huffman@31487
  1031
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
  1032
        by (rule norm_triangle_ineq2)
huffman@31487
  1033
      also have "\<dots> = norm (f x - a)"
huffman@31355
  1034
        by (rule norm_minus_commute)
huffman@31355
  1035
      also have "\<dots> < r" using 1 .
huffman@31487
  1036
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
  1037
    qed
huffman@31487
  1038
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
  1039
  qed
huffman@31355
  1040
  thus ?thesis by (rule BfunI)
huffman@31355
  1041
qed
huffman@31355
  1042
huffman@31565
  1043
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
  1044
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
  1045
  assumes f: "(f ---> a) F"
huffman@31355
  1046
  assumes a: "a \<noteq> 0"
huffman@44195
  1047
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
  1048
proof -
huffman@31355
  1049
  from a have "0 < norm a" by simp
huffman@44195
  1050
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
  1051
    by (rule tendstoD)
huffman@44195
  1052
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
  1053
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
  1054
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
  1055
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
  1056
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
  1057
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
  1058
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
  1059
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
  1060
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
  1061
  ultimately show ?thesis
huffman@44627
  1062
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
  1063
qed
huffman@31355
  1064
huffman@31565
  1065
lemma tendsto_divide [tendsto_intros]:
huffman@31355
  1066
  fixes a b :: "'a::real_normed_field"
huffman@44195
  1067
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
  1068
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
  1069
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
  1070
huffman@44194
  1071
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
  1072
  fixes l :: "'a::real_normed_vector"
huffman@44195
  1073
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
  1074
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
  1075
hoelzl@50247
  1076
subsection {* Limits to @{const at_top} and @{const at_bot} *}
hoelzl@50247
  1077
hoelzl@50322
  1078
lemma filterlim_at_top:
hoelzl@50247
  1079
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
hoelzl@50247
  1080
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@50322
  1081
  by (safe elim!: filterlimE intro!: filterlimI)
hoelzl@50247
  1082
     (auto simp: eventually_at_top_dense elim!: eventually_elim1)
hoelzl@50247
  1083
hoelzl@50322
  1084
lemma filterlim_at_bot: 
hoelzl@50247
  1085
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
hoelzl@50247
  1086
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
hoelzl@50322
  1087
  by (safe elim!: filterlimE intro!: filterlimI)
hoelzl@50247
  1088
     (auto simp: eventually_at_bot_dense elim!: eventually_elim1)
hoelzl@50247
  1089
hoelzl@50322
  1090
lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
hoelzl@50322
  1091
  unfolding filterlim_at_top
hoelzl@50247
  1092
  apply (intro allI)
hoelzl@50247
  1093
  apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
hoelzl@50247
  1094
  apply (auto simp: natceiling_le_eq)
hoelzl@50247
  1095
  done
hoelzl@50247
  1096
huffman@31349
  1097
end