src/HOL/Limits.thy
author hoelzl
Mon Dec 03 18:19:02 2012 +0100 (2012-12-03)
changeset 50324 0a1242d5e7d4
parent 50323 3764d4620fb3
child 50325 5e40ad9f212a
permissions -rw-r--r--
add filterlim rules for diverging multiplication and addition; move at_infinity to the HOL image
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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: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::linorder. \<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: "eventually P at_top \<longleftrightarrow> (\<exists>N::'a::dense_linorder. \<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"
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   309
  where "at_bot = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<le>k. P n)"
hoelzl@50247
   310
hoelzl@50247
   311
lemma eventually_at_bot_linorder:
hoelzl@50247
   312
  fixes P :: "'a::linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n\<le>N. P n)"
hoelzl@50247
   313
  unfolding at_bot_def
hoelzl@50247
   314
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@50247
   315
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@50247
   316
  assume "\<exists>i. \<forall>n\<le>i. P n" and "\<exists>j. \<forall>n\<le>j. Q n"
hoelzl@50247
   317
  then obtain i j where "\<forall>n\<le>i. P n" and "\<forall>n\<le>j. Q n" by auto
hoelzl@50247
   318
  then have "\<forall>n\<le>min i j. P n \<and> Q n" by simp
hoelzl@50247
   319
  then show "\<exists>k. \<forall>n\<le>k. P n \<and> Q n" ..
hoelzl@50247
   320
qed auto
hoelzl@50247
   321
hoelzl@50247
   322
lemma eventually_at_bot_dense:
hoelzl@50247
   323
  fixes P :: "'a::dense_linorder \<Rightarrow> bool" shows "eventually P at_bot \<longleftrightarrow> (\<exists>N. \<forall>n<N. P n)"
hoelzl@50247
   324
  unfolding eventually_at_bot_linorder
hoelzl@50247
   325
proof safe
hoelzl@50247
   326
  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
   327
next
hoelzl@50247
   328
  fix N assume "\<forall>n<N. P n" 
hoelzl@50247
   329
  moreover from lt_ex[of N] guess y ..
hoelzl@50247
   330
  ultimately show "\<exists>N. \<forall>n\<le>N. P n" by (auto intro!: exI[of _ y])
hoelzl@50247
   331
qed
hoelzl@50247
   332
hoelzl@50247
   333
subsection {* Sequentially *}
hoelzl@50247
   334
hoelzl@50247
   335
abbreviation sequentially :: "nat filter"
hoelzl@50247
   336
  where "sequentially == at_top"
hoelzl@50247
   337
hoelzl@50247
   338
lemma sequentially_def: "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
hoelzl@50247
   339
  unfolding at_top_def by simp
hoelzl@50247
   340
hoelzl@50247
   341
lemma eventually_sequentially:
hoelzl@50247
   342
  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
hoelzl@50247
   343
  by (rule eventually_at_top_linorder)
hoelzl@50247
   344
huffman@44342
   345
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
huffman@44081
   346
  unfolding filter_eq_iff eventually_sequentially by auto
huffman@36662
   347
huffman@44342
   348
lemmas trivial_limit_sequentially = sequentially_bot
huffman@44342
   349
huffman@36662
   350
lemma eventually_False_sequentially [simp]:
huffman@36662
   351
  "\<not> eventually (\<lambda>n. False) sequentially"
huffman@44081
   352
  by (simp add: eventually_False)
huffman@36662
   353
huffman@36662
   354
lemma le_sequentially:
huffman@44195
   355
  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
huffman@44081
   356
  unfolding le_filter_def eventually_sequentially
huffman@44081
   357
  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
huffman@36662
   358
noschinl@45892
   359
lemma eventually_sequentiallyI:
noschinl@45892
   360
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
noschinl@45892
   361
  shows "eventually P sequentially"
noschinl@45892
   362
using assms by (auto simp: eventually_sequentially)
noschinl@45892
   363
huffman@36662
   364
huffman@44081
   365
subsection {* Standard filters *}
huffman@36662
   366
huffman@44081
   367
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
huffman@44195
   368
  where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
huffman@31392
   369
huffman@44206
   370
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
huffman@44081
   371
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   372
huffman@44206
   373
definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
huffman@44081
   374
  where "at a = nhds a within - {a}"
huffman@31447
   375
hoelzl@50324
   376
definition at_infinity :: "'a::real_normed_vector filter" where
hoelzl@50324
   377
  "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
hoelzl@50324
   378
huffman@31392
   379
lemma eventually_within:
huffman@44195
   380
  "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
huffman@44081
   381
  unfolding within_def
huffman@44081
   382
  by (rule eventually_Abs_filter, rule is_filter.intro)
huffman@44081
   383
     (auto elim!: eventually_rev_mp)
huffman@31392
   384
huffman@45031
   385
lemma within_UNIV [simp]: "F within UNIV = F"
huffman@45031
   386
  unfolding filter_eq_iff eventually_within by simp
huffman@45031
   387
huffman@45031
   388
lemma within_empty [simp]: "F within {} = bot"
huffman@44081
   389
  unfolding filter_eq_iff eventually_within by simp
huffman@36360
   390
hoelzl@50247
   391
lemma within_le: "F within S \<le> F"
hoelzl@50247
   392
  unfolding le_filter_def eventually_within by (auto elim: eventually_elim1)
hoelzl@50247
   393
hoelzl@50323
   394
lemma le_withinI: "F \<le> F' \<Longrightarrow> eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S"
hoelzl@50323
   395
  unfolding le_filter_def eventually_within by (auto elim: eventually_elim2)
hoelzl@50323
   396
hoelzl@50323
   397
lemma le_within_iff: "eventually (\<lambda>x. x \<in> S) F \<Longrightarrow> F \<le> F' within S \<longleftrightarrow> F \<le> F'"
hoelzl@50323
   398
  by (blast intro: within_le le_withinI order_trans)
hoelzl@50323
   399
huffman@36654
   400
lemma eventually_nhds:
huffman@36654
   401
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   402
unfolding nhds_def
huffman@44081
   403
proof (rule eventually_Abs_filter, rule is_filter.intro)
huffman@36654
   404
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
hoelzl@50324
   405
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
huffman@36358
   406
next
huffman@36358
   407
  fix P Q
huffman@36654
   408
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   409
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
huffman@36358
   410
  then obtain S T where
huffman@36654
   411
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   412
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
huffman@36654
   413
  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
   414
    by (simp add: open_Int)
hoelzl@50324
   415
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" ..
huffman@36358
   416
qed auto
huffman@31447
   417
huffman@36656
   418
lemma eventually_nhds_metric:
huffman@36656
   419
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
huffman@36656
   420
unfolding eventually_nhds open_dist
huffman@31447
   421
apply safe
huffman@31447
   422
apply fast
huffman@31492
   423
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
huffman@31447
   424
apply clarsimp
huffman@31447
   425
apply (rule_tac x="d - dist x a" in exI, clarsimp)
huffman@31447
   426
apply (simp only: less_diff_eq)
huffman@31447
   427
apply (erule le_less_trans [OF dist_triangle])
huffman@31447
   428
done
huffman@31447
   429
huffman@44571
   430
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
huffman@44571
   431
  unfolding trivial_limit_def eventually_nhds by simp
huffman@44571
   432
huffman@36656
   433
lemma eventually_at_topological:
huffman@36656
   434
  "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
   435
unfolding at_def eventually_within eventually_nhds by simp
huffman@36656
   436
huffman@36656
   437
lemma eventually_at:
huffman@36656
   438
  fixes a :: "'a::metric_space"
huffman@36656
   439
  shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
huffman@36656
   440
unfolding at_def eventually_within eventually_nhds_metric by auto
huffman@36656
   441
huffman@44571
   442
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
huffman@44571
   443
  unfolding trivial_limit_def eventually_at_topological
huffman@44571
   444
  by (safe, case_tac "S = {a}", simp, fast, fast)
huffman@44571
   445
huffman@44571
   446
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
huffman@44571
   447
  by (simp add: at_eq_bot_iff not_open_singleton)
huffman@44571
   448
hoelzl@50324
   449
lemma eventually_at_infinity:
hoelzl@50324
   450
  "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
hoelzl@50324
   451
unfolding at_infinity_def
hoelzl@50324
   452
proof (rule eventually_Abs_filter, rule is_filter.intro)
hoelzl@50324
   453
  fix P Q :: "'a \<Rightarrow> bool"
hoelzl@50324
   454
  assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
hoelzl@50324
   455
  then obtain r s where
hoelzl@50324
   456
    "\<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<forall>x. s \<le> norm x \<longrightarrow> Q x" by auto
hoelzl@50324
   457
  then have "\<forall>x. max r s \<le> norm x \<longrightarrow> P x \<and> Q x" by simp
hoelzl@50324
   458
  then show "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x \<and> Q x" ..
hoelzl@50324
   459
qed auto
huffman@31392
   460
huffman@31355
   461
subsection {* Boundedness *}
huffman@31355
   462
huffman@44081
   463
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   464
  where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
huffman@31355
   465
huffman@31487
   466
lemma BfunI:
huffman@44195
   467
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
huffman@31355
   468
unfolding Bfun_def
huffman@31355
   469
proof (intro exI conjI allI)
huffman@31355
   470
  show "0 < max K 1" by simp
huffman@31355
   471
next
huffman@44195
   472
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
huffman@31355
   473
    using K by (rule eventually_elim1, simp)
huffman@31355
   474
qed
huffman@31355
   475
huffman@31355
   476
lemma BfunE:
huffman@44195
   477
  assumes "Bfun f F"
huffman@44195
   478
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
huffman@31355
   479
using assms unfolding Bfun_def by fast
huffman@31355
   480
huffman@31355
   481
huffman@31349
   482
subsection {* Convergence to Zero *}
huffman@31349
   483
huffman@44081
   484
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   485
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   486
huffman@31349
   487
lemma ZfunI:
huffman@44195
   488
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   489
  unfolding Zfun_def by simp
huffman@31349
   490
huffman@31349
   491
lemma ZfunD:
huffman@44195
   492
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   493
  unfolding Zfun_def by simp
huffman@31349
   494
huffman@31355
   495
lemma Zfun_ssubst:
huffman@44195
   496
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   497
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   498
huffman@44195
   499
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   500
  unfolding Zfun_def by simp
huffman@31349
   501
huffman@44195
   502
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   503
  unfolding Zfun_def by simp
huffman@31349
   504
huffman@31349
   505
lemma Zfun_imp_Zfun:
huffman@44195
   506
  assumes f: "Zfun f F"
huffman@44195
   507
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   508
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   509
proof (cases)
huffman@31349
   510
  assume K: "0 < K"
huffman@31349
   511
  show ?thesis
huffman@31349
   512
  proof (rule ZfunI)
huffman@31349
   513
    fix r::real assume "0 < r"
huffman@31349
   514
    hence "0 < r / K"
huffman@31349
   515
      using K by (rule divide_pos_pos)
huffman@44195
   516
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
huffman@31487
   517
      using ZfunD [OF f] by fast
huffman@44195
   518
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   519
    proof eventually_elim
noschinl@46887
   520
      case (elim x)
huffman@31487
   521
      hence "norm (f x) * K < r"
huffman@31349
   522
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   523
      thus ?case
noschinl@46887
   524
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   525
    qed
huffman@31349
   526
  qed
huffman@31349
   527
next
huffman@31349
   528
  assume "\<not> 0 < K"
huffman@31349
   529
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   530
  show ?thesis
huffman@31355
   531
  proof (rule ZfunI)
huffman@31355
   532
    fix r :: real
huffman@31355
   533
    assume "0 < r"
huffman@44195
   534
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   535
    proof eventually_elim
noschinl@46887
   536
      case (elim x)
noschinl@46887
   537
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   538
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   539
      finally show ?case
huffman@31355
   540
        using `0 < r` by simp
huffman@31355
   541
    qed
huffman@31355
   542
  qed
huffman@31349
   543
qed
huffman@31349
   544
huffman@44195
   545
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   546
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   547
huffman@31349
   548
lemma Zfun_add:
huffman@44195
   549
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   550
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   551
proof (rule ZfunI)
huffman@31349
   552
  fix r::real assume "0 < r"
huffman@31349
   553
  hence r: "0 < r / 2" by simp
huffman@44195
   554
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   555
    using f r by (rule ZfunD)
huffman@31349
   556
  moreover
huffman@44195
   557
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   558
    using g r by (rule ZfunD)
huffman@31349
   559
  ultimately
huffman@44195
   560
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   561
  proof eventually_elim
noschinl@46887
   562
    case (elim x)
huffman@31487
   563
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   564
      by (rule norm_triangle_ineq)
huffman@31349
   565
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   566
      using elim by (rule add_strict_mono)
noschinl@46887
   567
    finally show ?case
huffman@31349
   568
      by simp
huffman@31349
   569
  qed
huffman@31349
   570
qed
huffman@31349
   571
huffman@44195
   572
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   573
  unfolding Zfun_def by simp
huffman@31349
   574
huffman@44195
   575
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
huffman@44081
   576
  by (simp only: diff_minus Zfun_add Zfun_minus)
huffman@31349
   577
huffman@31349
   578
lemma (in bounded_linear) Zfun:
huffman@44195
   579
  assumes g: "Zfun g F"
huffman@44195
   580
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   581
proof -
huffman@31349
   582
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
huffman@31349
   583
    using bounded by fast
huffman@44195
   584
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   585
    by simp
huffman@31487
   586
  with g show ?thesis
huffman@31349
   587
    by (rule Zfun_imp_Zfun)
huffman@31349
   588
qed
huffman@31349
   589
huffman@31349
   590
lemma (in bounded_bilinear) Zfun:
huffman@44195
   591
  assumes f: "Zfun f F"
huffman@44195
   592
  assumes g: "Zfun g F"
huffman@44195
   593
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   594
proof (rule ZfunI)
huffman@31349
   595
  fix r::real assume r: "0 < r"
huffman@31349
   596
  obtain K where K: "0 < K"
huffman@31349
   597
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   598
    using pos_bounded by fast
huffman@31349
   599
  from K have K': "0 < inverse K"
huffman@31349
   600
    by (rule positive_imp_inverse_positive)
huffman@44195
   601
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   602
    using f r by (rule ZfunD)
huffman@31349
   603
  moreover
huffman@44195
   604
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   605
    using g K' by (rule ZfunD)
huffman@31349
   606
  ultimately
huffman@44195
   607
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   608
  proof eventually_elim
noschinl@46887
   609
    case (elim x)
huffman@31487
   610
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   611
      by (rule norm_le)
huffman@31487
   612
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   613
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   614
    also from K have "r * inverse K * K = r"
huffman@31349
   615
      by simp
noschinl@46887
   616
    finally show ?case .
huffman@31349
   617
  qed
huffman@31349
   618
qed
huffman@31349
   619
huffman@31349
   620
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   621
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   622
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   623
huffman@31349
   624
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   625
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   626
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   627
huffman@44282
   628
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   629
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   630
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   631
huffman@31349
   632
wenzelm@31902
   633
subsection {* Limits *}
huffman@31349
   634
hoelzl@50322
   635
definition filterlim :: "('a \<Rightarrow> 'b) \<Rightarrow> 'b filter \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@50322
   636
  "filterlim f F2 F1 \<longleftrightarrow> filtermap f F1 \<le> F2"
hoelzl@50247
   637
hoelzl@50247
   638
syntax
hoelzl@50247
   639
  "_LIM" :: "pttrns \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> bool" ("(3LIM (_)/ (_)./ (_) :> (_))" [1000, 10, 0, 10] 10)
hoelzl@50247
   640
hoelzl@50247
   641
translations
hoelzl@50322
   642
  "LIM x F1. f :> F2"   == "CONST filterlim (%x. f) F2 F1"
hoelzl@50247
   643
hoelzl@50322
   644
lemma filterlimE: "(LIM x F1. f x :> F2) \<Longrightarrow> eventually P F2 \<Longrightarrow> eventually (\<lambda>x. P (f x)) F1"
hoelzl@50322
   645
  by (auto simp: filterlim_def eventually_filtermap le_filter_def)
hoelzl@50247
   646
hoelzl@50322
   647
lemma filterlimI: "(\<And>P. eventually P F2 \<Longrightarrow> eventually (\<lambda>x. P (f x)) F1) \<Longrightarrow> (LIM x F1. f x :> F2)"
hoelzl@50322
   648
  by (auto simp: filterlim_def eventually_filtermap le_filter_def)
hoelzl@50247
   649
hoelzl@50323
   650
lemma filterlim_compose: 
hoelzl@50323
   651
  "filterlim g F3 F2 \<Longrightarrow> filterlim f F2 F1 \<Longrightarrow> filterlim (\<lambda>x. g (f x)) F3 F1"
hoelzl@50323
   652
  unfolding filterlim_def filtermap_filtermap[symmetric] by (metis filtermap_mono order_trans)
hoelzl@50323
   653
hoelzl@50323
   654
lemma filterlim_mono: 
hoelzl@50323
   655
  "filterlim f F2 F1 \<Longrightarrow> F2 \<le> F2' \<Longrightarrow> F1' \<le> F1 \<Longrightarrow> filterlim f F2' F1'"
hoelzl@50323
   656
  unfolding filterlim_def by (metis filtermap_mono order_trans)
hoelzl@50323
   657
hoelzl@50247
   658
abbreviation (in topological_space)
huffman@44206
   659
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
hoelzl@50322
   660
  "(f ---> l) F \<equiv> filterlim f (nhds l) F"
noschinl@45892
   661
wenzelm@31902
   662
ML {*
wenzelm@31902
   663
structure Tendsto_Intros = Named_Thms
wenzelm@31902
   664
(
wenzelm@45294
   665
  val name = @{binding tendsto_intros}
wenzelm@31902
   666
  val description = "introduction rules for tendsto"
wenzelm@31902
   667
)
huffman@31565
   668
*}
huffman@31565
   669
wenzelm@31902
   670
setup Tendsto_Intros.setup
huffman@31565
   671
hoelzl@50247
   672
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
   673
  unfolding filterlim_def
hoelzl@50247
   674
proof safe
hoelzl@50247
   675
  fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
hoelzl@50247
   676
  then show "eventually (\<lambda>x. f x \<in> S) F"
hoelzl@50247
   677
    unfolding eventually_nhds eventually_filtermap le_filter_def
hoelzl@50247
   678
    by (auto elim!: allE[of _ "\<lambda>x. x \<in> S"] eventually_rev_mp)
hoelzl@50247
   679
qed (auto elim!: eventually_rev_mp simp: eventually_nhds eventually_filtermap le_filter_def)
hoelzl@50247
   680
huffman@44195
   681
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
huffman@44081
   682
  unfolding tendsto_def le_filter_def by fast
huffman@36656
   683
huffman@31488
   684
lemma topological_tendstoI:
huffman@44195
   685
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
huffman@44195
   686
    \<Longrightarrow> (f ---> l) F"
huffman@31349
   687
  unfolding tendsto_def by auto
huffman@31349
   688
huffman@31488
   689
lemma topological_tendstoD:
huffman@44195
   690
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
huffman@31488
   691
  unfolding tendsto_def by auto
huffman@31488
   692
huffman@31488
   693
lemma tendstoI:
huffman@44195
   694
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44195
   695
  shows "(f ---> l) F"
huffman@44081
   696
  apply (rule topological_tendstoI)
huffman@44081
   697
  apply (simp add: open_dist)
huffman@44081
   698
  apply (drule (1) bspec, clarify)
huffman@44081
   699
  apply (drule assms)
huffman@44081
   700
  apply (erule eventually_elim1, simp)
huffman@44081
   701
  done
huffman@31488
   702
huffman@31349
   703
lemma tendstoD:
huffman@44195
   704
  "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44081
   705
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
huffman@44081
   706
  apply (clarsimp simp add: open_dist)
huffman@44081
   707
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
huffman@44081
   708
  apply (simp only: less_diff_eq)
huffman@44081
   709
  apply (erule le_less_trans [OF dist_triangle])
huffman@44081
   710
  apply simp
huffman@44081
   711
  apply simp
huffman@44081
   712
  done
huffman@31488
   713
huffman@31488
   714
lemma tendsto_iff:
huffman@44195
   715
  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
huffman@44081
   716
  using tendstoI tendstoD by fast
huffman@31349
   717
huffman@44195
   718
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   719
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   720
huffman@45031
   721
lemma tendsto_bot [simp]: "(f ---> a) bot"
huffman@45031
   722
  unfolding tendsto_def by simp
huffman@45031
   723
huffman@31565
   724
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
huffman@44081
   725
  unfolding tendsto_def eventually_at_topological by auto
huffman@31565
   726
huffman@31565
   727
lemma tendsto_ident_at_within [tendsto_intros]:
huffman@36655
   728
  "((\<lambda>x. x) ---> a) (at a within S)"
huffman@44081
   729
  unfolding tendsto_def eventually_within eventually_at_topological by auto
huffman@31565
   730
huffman@44195
   731
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
huffman@44081
   732
  by (simp add: tendsto_def)
huffman@31349
   733
huffman@44205
   734
lemma tendsto_unique:
huffman@44205
   735
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
huffman@44205
   736
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
huffman@44205
   737
  shows "a = b"
huffman@44205
   738
proof (rule ccontr)
huffman@44205
   739
  assume "a \<noteq> b"
huffman@44205
   740
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
huffman@44205
   741
    using hausdorff [OF `a \<noteq> b`] by fast
huffman@44205
   742
  have "eventually (\<lambda>x. f x \<in> U) F"
huffman@44205
   743
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
huffman@44205
   744
  moreover
huffman@44205
   745
  have "eventually (\<lambda>x. f x \<in> V) F"
huffman@44205
   746
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
huffman@44205
   747
  ultimately
huffman@44205
   748
  have "eventually (\<lambda>x. False) F"
noschinl@46887
   749
  proof eventually_elim
noschinl@46887
   750
    case (elim x)
huffman@44205
   751
    hence "f x \<in> U \<inter> V" by simp
noschinl@46887
   752
    with `U \<inter> V = {}` show ?case by simp
huffman@44205
   753
  qed
huffman@44205
   754
  with `\<not> trivial_limit F` show "False"
huffman@44205
   755
    by (simp add: trivial_limit_def)
huffman@44205
   756
qed
huffman@44205
   757
huffman@36662
   758
lemma tendsto_const_iff:
huffman@44205
   759
  fixes a b :: "'a::t2_space"
huffman@44205
   760
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
huffman@44205
   761
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
huffman@44205
   762
hoelzl@50323
   763
lemma tendsto_at_iff_tendsto_nhds:
hoelzl@50323
   764
  "(g ---> g l) (at l) \<longleftrightarrow> (g ---> g l) (nhds l)"
hoelzl@50323
   765
  unfolding tendsto_def at_def eventually_within
hoelzl@50323
   766
  by (intro ext all_cong imp_cong) (auto elim!: eventually_elim1)
hoelzl@50323
   767
huffman@44218
   768
lemma tendsto_compose:
hoelzl@50323
   769
  "(g ---> g l) (at l) \<Longrightarrow> (f ---> l) F \<Longrightarrow> ((\<lambda>x. g (f x)) ---> g l) F"
hoelzl@50323
   770
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])
huffman@44218
   771
huffman@44253
   772
lemma tendsto_compose_eventually:
huffman@44253
   773
  assumes g: "(g ---> m) (at l)"
huffman@44253
   774
  assumes f: "(f ---> l) F"
huffman@44253
   775
  assumes inj: "eventually (\<lambda>x. f x \<noteq> l) F"
huffman@44253
   776
  shows "((\<lambda>x. g (f x)) ---> m) F"
hoelzl@50323
   777
proof -
hoelzl@50323
   778
  from f inj have "LIM x F. f x :> at l"
hoelzl@50323
   779
    unfolding filterlim_def at_def by (simp add: le_within_iff eventually_filtermap)
hoelzl@50323
   780
  from filterlim_compose[OF g this] show ?thesis .
huffman@44253
   781
qed
huffman@44253
   782
huffman@44251
   783
lemma metric_tendsto_imp_tendsto:
huffman@44251
   784
  assumes f: "(f ---> a) F"
huffman@44251
   785
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
huffman@44251
   786
  shows "(g ---> b) F"
huffman@44251
   787
proof (rule tendstoI)
huffman@44251
   788
  fix e :: real assume "0 < e"
huffman@44251
   789
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
huffman@44251
   790
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
huffman@44251
   791
    using le_less_trans by (rule eventually_elim2)
huffman@44251
   792
qed
huffman@44251
   793
huffman@44205
   794
subsubsection {* Distance and norms *}
huffman@36662
   795
huffman@31565
   796
lemma tendsto_dist [tendsto_intros]:
huffman@44195
   797
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
huffman@44195
   798
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
huffman@31565
   799
proof (rule tendstoI)
huffman@31565
   800
  fix e :: real assume "0 < e"
huffman@31565
   801
  hence e2: "0 < e/2" by simp
huffman@31565
   802
  from tendstoD [OF f e2] tendstoD [OF g e2]
huffman@44195
   803
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
noschinl@46887
   804
  proof (eventually_elim)
noschinl@46887
   805
    case (elim x)
huffman@31565
   806
    then show "dist (dist (f x) (g x)) (dist l m) < e"
huffman@31565
   807
      unfolding dist_real_def
huffman@31565
   808
      using dist_triangle2 [of "f x" "g x" "l"]
huffman@31565
   809
      using dist_triangle2 [of "g x" "l" "m"]
huffman@31565
   810
      using dist_triangle3 [of "l" "m" "f x"]
huffman@31565
   811
      using dist_triangle [of "f x" "m" "g x"]
huffman@31565
   812
      by arith
huffman@31565
   813
  qed
huffman@31565
   814
qed
huffman@31565
   815
huffman@36662
   816
lemma norm_conv_dist: "norm x = dist x 0"
huffman@44081
   817
  unfolding dist_norm by simp
huffman@36662
   818
huffman@31565
   819
lemma tendsto_norm [tendsto_intros]:
huffman@44195
   820
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
huffman@44081
   821
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   822
huffman@36662
   823
lemma tendsto_norm_zero:
huffman@44195
   824
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
huffman@44081
   825
  by (drule tendsto_norm, simp)
huffman@36662
   826
huffman@36662
   827
lemma tendsto_norm_zero_cancel:
huffman@44195
   828
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
huffman@44081
   829
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   830
huffman@36662
   831
lemma tendsto_norm_zero_iff:
huffman@44195
   832
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
huffman@44081
   833
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   834
huffman@44194
   835
lemma tendsto_rabs [tendsto_intros]:
huffman@44195
   836
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
huffman@44194
   837
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   838
huffman@44194
   839
lemma tendsto_rabs_zero:
huffman@44195
   840
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
huffman@44194
   841
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   842
huffman@44194
   843
lemma tendsto_rabs_zero_cancel:
huffman@44195
   844
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
huffman@44194
   845
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   846
huffman@44194
   847
lemma tendsto_rabs_zero_iff:
huffman@44195
   848
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
huffman@44194
   849
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   850
huffman@44194
   851
subsubsection {* Addition and subtraction *}
huffman@44194
   852
huffman@31565
   853
lemma tendsto_add [tendsto_intros]:
huffman@31349
   854
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   855
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   856
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   857
huffman@44194
   858
lemma tendsto_add_zero:
huffman@44194
   859
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   860
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   861
  by (drule (1) tendsto_add, simp)
huffman@44194
   862
huffman@31565
   863
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   864
  fixes a :: "'a::real_normed_vector"
huffman@44195
   865
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   866
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   867
huffman@31349
   868
lemma tendsto_minus_cancel:
huffman@31349
   869
  fixes a :: "'a::real_normed_vector"
huffman@44195
   870
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   871
  by (drule tendsto_minus, simp)
huffman@31349
   872
huffman@31565
   873
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   874
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   875
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
huffman@44081
   876
  by (simp add: diff_minus tendsto_add tendsto_minus)
huffman@31349
   877
huffman@31588
   878
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   879
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   880
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   881
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   882
proof (cases "finite S")
huffman@31588
   883
  assume "finite S" thus ?thesis using assms
huffman@44194
   884
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
huffman@31588
   885
next
huffman@31588
   886
  assume "\<not> finite S" thus ?thesis
huffman@31588
   887
    by (simp add: tendsto_const)
huffman@31588
   888
qed
huffman@31588
   889
noschinl@45892
   890
lemma real_tendsto_sandwich:
noschinl@45892
   891
  fixes f g h :: "'a \<Rightarrow> real"
noschinl@45892
   892
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
noschinl@45892
   893
  assumes lim: "(f ---> c) net" "(h ---> c) net"
noschinl@45892
   894
  shows "(g ---> c) net"
noschinl@45892
   895
proof -
noschinl@45892
   896
  have "((\<lambda>n. g n - f n) ---> 0) net"
noschinl@45892
   897
  proof (rule metric_tendsto_imp_tendsto)
noschinl@45892
   898
    show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
noschinl@45892
   899
      using ev by (rule eventually_elim2) (simp add: dist_real_def)
noschinl@45892
   900
    show "((\<lambda>n. h n - f n) ---> 0) net"
noschinl@45892
   901
      using tendsto_diff[OF lim(2,1)] by simp
noschinl@45892
   902
  qed
noschinl@45892
   903
  from tendsto_add[OF this lim(1)] show ?thesis by simp
noschinl@45892
   904
qed
noschinl@45892
   905
huffman@44194
   906
subsubsection {* Linear operators and multiplication *}
huffman@44194
   907
huffman@44282
   908
lemma (in bounded_linear) tendsto:
huffman@44195
   909
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   910
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   911
huffman@44194
   912
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   913
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   914
  by (drule tendsto, simp only: zero)
huffman@44194
   915
huffman@44282
   916
lemma (in bounded_bilinear) tendsto:
huffman@44195
   917
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   918
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   919
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   920
huffman@44194
   921
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   922
  assumes f: "(f ---> 0) F"
huffman@44195
   923
  assumes g: "(g ---> 0) F"
huffman@44195
   924
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   925
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   926
huffman@44194
   927
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   928
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   929
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   930
huffman@44194
   931
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   932
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   933
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   934
huffman@44282
   935
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   936
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   937
huffman@44282
   938
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   939
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   940
huffman@44282
   941
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   942
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   943
huffman@44568
   944
lemmas tendsto_mult_zero =
huffman@44568
   945
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   946
huffman@44568
   947
lemmas tendsto_mult_left_zero =
huffman@44568
   948
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   949
huffman@44568
   950
lemmas tendsto_mult_right_zero =
huffman@44568
   951
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   952
huffman@44194
   953
lemma tendsto_power [tendsto_intros]:
huffman@44194
   954
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   955
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
huffman@44194
   956
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
huffman@44194
   957
huffman@44194
   958
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   959
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   960
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   961
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   962
proof (cases "finite S")
huffman@44194
   963
  assume "finite S" thus ?thesis using assms
huffman@44194
   964
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
huffman@44194
   965
next
huffman@44194
   966
  assume "\<not> finite S" thus ?thesis
huffman@44194
   967
    by (simp add: tendsto_const)
huffman@44194
   968
qed
huffman@44194
   969
huffman@44194
   970
subsubsection {* Inverse and division *}
huffman@31355
   971
huffman@31355
   972
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   973
  assumes f: "Zfun f F"
huffman@44195
   974
  assumes g: "Bfun g F"
huffman@44195
   975
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   976
proof -
huffman@31355
   977
  obtain K where K: "0 \<le> K"
huffman@31355
   978
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   979
    using nonneg_bounded by fast
huffman@31355
   980
  obtain B where B: "0 < B"
huffman@44195
   981
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   982
    using g by (rule BfunE)
huffman@44195
   983
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   984
  using norm_g proof eventually_elim
noschinl@46887
   985
    case (elim x)
huffman@31487
   986
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   987
      by (rule norm_le)
huffman@31487
   988
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   989
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   990
                mult_nonneg_nonneg K elim)
huffman@31487
   991
    also have "\<dots> = norm (f x) * (B * K)"
huffman@31355
   992
      by (rule mult_assoc)
huffman@31487
   993
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   994
  qed
huffman@31487
   995
  with f show ?thesis
huffman@31487
   996
    by (rule Zfun_imp_Zfun)
huffman@31355
   997
qed
huffman@31355
   998
huffman@31355
   999
lemma (in bounded_bilinear) flip:
huffman@31355
  1000
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@44081
  1001
  apply default
huffman@44081
  1002
  apply (rule add_right)
huffman@44081
  1003
  apply (rule add_left)
huffman@44081
  1004
  apply (rule scaleR_right)
huffman@44081
  1005
  apply (rule scaleR_left)
huffman@44081
  1006
  apply (subst mult_commute)
huffman@44081
  1007
  using bounded by fast
huffman@31355
  1008
huffman@31355
  1009
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
  1010
  assumes f: "Bfun f F"
huffman@44195
  1011
  assumes g: "Zfun g F"
huffman@44195
  1012
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
  1013
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
  1014
huffman@31355
  1015
lemma Bfun_inverse_lemma:
huffman@31355
  1016
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
  1017
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
  1018
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
  1019
  apply (erule (1) le_imp_inverse_le)
huffman@44081
  1020
  done
huffman@31355
  1021
huffman@31355
  1022
lemma Bfun_inverse:
huffman@31355
  1023
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
  1024
  assumes f: "(f ---> a) F"
huffman@31355
  1025
  assumes a: "a \<noteq> 0"
huffman@44195
  1026
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
  1027
proof -
huffman@31355
  1028
  from a have "0 < norm a" by simp
huffman@31355
  1029
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
  1030
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
  1031
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
  1032
    using tendstoD [OF f r1] by fast
huffman@44195
  1033
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
  1034
  proof eventually_elim
noschinl@46887
  1035
    case (elim x)
huffman@31487
  1036
    hence 1: "norm (f x - a) < r"
huffman@31355
  1037
      by (simp add: dist_norm)
huffman@31487
  1038
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
  1039
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
  1040
      by (rule nonzero_norm_inverse)
huffman@31355
  1041
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
  1042
    proof (rule le_imp_inverse_le)
huffman@31355
  1043
      show "0 < norm a - r" using r2 by simp
huffman@31355
  1044
    next
huffman@31487
  1045
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
  1046
        by (rule norm_triangle_ineq2)
huffman@31487
  1047
      also have "\<dots> = norm (f x - a)"
huffman@31355
  1048
        by (rule norm_minus_commute)
huffman@31355
  1049
      also have "\<dots> < r" using 1 .
huffman@31487
  1050
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
  1051
    qed
huffman@31487
  1052
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
  1053
  qed
huffman@31355
  1054
  thus ?thesis by (rule BfunI)
huffman@31355
  1055
qed
huffman@31355
  1056
huffman@31565
  1057
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
  1058
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
  1059
  assumes f: "(f ---> a) F"
huffman@31355
  1060
  assumes a: "a \<noteq> 0"
huffman@44195
  1061
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
  1062
proof -
huffman@31355
  1063
  from a have "0 < norm a" by simp
huffman@44195
  1064
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
  1065
    by (rule tendstoD)
huffman@44195
  1066
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
  1067
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
  1068
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
  1069
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
  1070
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
  1071
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
  1072
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
  1073
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
  1074
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
  1075
  ultimately show ?thesis
huffman@44627
  1076
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
  1077
qed
huffman@31355
  1078
huffman@31565
  1079
lemma tendsto_divide [tendsto_intros]:
huffman@31355
  1080
  fixes a b :: "'a::real_normed_field"
huffman@44195
  1081
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
  1082
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
  1083
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
  1084
huffman@44194
  1085
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
  1086
  fixes l :: "'a::real_normed_vector"
huffman@44195
  1087
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
  1088
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
  1089
hoelzl@50247
  1090
subsection {* Limits to @{const at_top} and @{const at_bot} *}
hoelzl@50247
  1091
hoelzl@50322
  1092
lemma filterlim_at_top:
hoelzl@50247
  1093
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
hoelzl@50247
  1094
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. Z < f x) F)"
hoelzl@50322
  1095
  by (safe elim!: filterlimE intro!: filterlimI)
hoelzl@50247
  1096
     (auto simp: eventually_at_top_dense elim!: eventually_elim1)
hoelzl@50247
  1097
hoelzl@50323
  1098
lemma filterlim_at_top_gt:
hoelzl@50323
  1099
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
hoelzl@50323
  1100
  shows "(LIM x F. f x :> at_top) \<longleftrightarrow> (\<forall>Z>c. eventually (\<lambda>x. Z < f x) F)"
hoelzl@50323
  1101
  unfolding filterlim_at_top
hoelzl@50323
  1102
proof safe
hoelzl@50323
  1103
  fix Z assume *: "\<forall>Z>c. eventually (\<lambda>x. Z < f x) F"
hoelzl@50323
  1104
  from gt_ex[of "max Z c"] guess x ..
hoelzl@50323
  1105
  with *[THEN spec, of x] show "eventually (\<lambda>x. Z < f x) F"
hoelzl@50323
  1106
    by (auto elim!: eventually_elim1)
hoelzl@50323
  1107
qed simp
hoelzl@50323
  1108
hoelzl@50322
  1109
lemma filterlim_at_bot: 
hoelzl@50247
  1110
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)"
hoelzl@50247
  1111
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z. eventually (\<lambda>x. f x < Z) F)"
hoelzl@50322
  1112
  by (safe elim!: filterlimE intro!: filterlimI)
hoelzl@50247
  1113
     (auto simp: eventually_at_bot_dense elim!: eventually_elim1)
hoelzl@50247
  1114
hoelzl@50323
  1115
lemma filterlim_at_bot_lt:
hoelzl@50323
  1116
  fixes f :: "'a \<Rightarrow> ('b::dense_linorder)" and c :: "'b"
hoelzl@50323
  1117
  shows "(LIM x F. f x :> at_bot) \<longleftrightarrow> (\<forall>Z<c. eventually (\<lambda>x. Z > f x) F)"
hoelzl@50323
  1118
  unfolding filterlim_at_bot
hoelzl@50323
  1119
proof safe
hoelzl@50323
  1120
  fix Z assume *: "\<forall>Z<c. eventually (\<lambda>x. Z > f x) F"
hoelzl@50323
  1121
  from lt_ex[of "min Z c"] guess x ..
hoelzl@50323
  1122
  with *[THEN spec, of x] show "eventually (\<lambda>x. Z > f x) F"
hoelzl@50323
  1123
    by (auto elim!: eventually_elim1)
hoelzl@50323
  1124
qed simp
hoelzl@50323
  1125
hoelzl@50322
  1126
lemma filterlim_real_sequentially: "LIM x sequentially. real x :> at_top"
hoelzl@50322
  1127
  unfolding filterlim_at_top
hoelzl@50247
  1128
  apply (intro allI)
hoelzl@50247
  1129
  apply (rule_tac c="natceiling (Z + 1)" in eventually_sequentiallyI)
hoelzl@50247
  1130
  apply (auto simp: natceiling_le_eq)
hoelzl@50247
  1131
  done
hoelzl@50247
  1132
hoelzl@50323
  1133
lemma filterlim_inverse_at_top_pos:
hoelzl@50323
  1134
  "LIM x (nhds 0 within {0::real <..}). inverse x :> at_top"
hoelzl@50323
  1135
  unfolding filterlim_at_top_gt[where c=0] eventually_within
hoelzl@50323
  1136
proof safe
hoelzl@50323
  1137
  fix Z :: real assume [arith]: "0 < Z"
hoelzl@50323
  1138
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
hoelzl@50323
  1139
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@50323
  1140
  then show "eventually (\<lambda>x. x \<in> {0<..} \<longrightarrow> Z < inverse x) (nhds 0)"
hoelzl@50323
  1141
    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
hoelzl@50323
  1142
qed
hoelzl@50323
  1143
hoelzl@50323
  1144
lemma filterlim_inverse_at_top:
hoelzl@50323
  1145
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
hoelzl@50323
  1146
  by (intro filterlim_compose[OF filterlim_inverse_at_top_pos])
hoelzl@50323
  1147
     (simp add: filterlim_def eventually_filtermap le_within_iff)
hoelzl@50323
  1148
hoelzl@50323
  1149
lemma filterlim_inverse_at_bot_neg:
hoelzl@50323
  1150
  "LIM x (nhds 0 within {..< 0::real}). inverse x :> at_bot"
hoelzl@50323
  1151
  unfolding filterlim_at_bot_lt[where c=0] eventually_within
hoelzl@50323
  1152
proof safe
hoelzl@50323
  1153
  fix Z :: real assume [arith]: "Z < 0"
hoelzl@50323
  1154
  have "eventually (\<lambda>x. inverse Z < x) (nhds 0)"
hoelzl@50323
  1155
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@50323
  1156
  then show "eventually (\<lambda>x. x \<in> {..< 0} \<longrightarrow> inverse x < Z) (nhds 0)"
hoelzl@50323
  1157
    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
hoelzl@50323
  1158
qed
hoelzl@50323
  1159
hoelzl@50323
  1160
lemma filterlim_inverse_at_bot:
hoelzl@50323
  1161
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
hoelzl@50323
  1162
  by (intro filterlim_compose[OF filterlim_inverse_at_bot_neg])
hoelzl@50323
  1163
     (simp add: filterlim_def eventually_filtermap le_within_iff)
hoelzl@50323
  1164
hoelzl@50323
  1165
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
hoelzl@50323
  1166
  unfolding filterlim_at_top eventually_at_bot_dense
hoelzl@50323
  1167
  by (blast intro: less_minus_iff[THEN iffD1])
hoelzl@50323
  1168
hoelzl@50323
  1169
lemma filterlim_uminus_at_top: "LIM x F. f x :> at_bot \<Longrightarrow> LIM x F. - (f x) :: real :> at_top"
hoelzl@50323
  1170
  by (rule filterlim_compose[OF filterlim_uminus_at_top_at_bot])
hoelzl@50323
  1171
hoelzl@50323
  1172
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
hoelzl@50323
  1173
  unfolding filterlim_at_bot eventually_at_top_dense
hoelzl@50323
  1174
  by (blast intro: minus_less_iff[THEN iffD1])
hoelzl@50323
  1175
hoelzl@50323
  1176
lemma filterlim_uminus_at_bot: "LIM x F. f x :> at_top \<Longrightarrow> LIM x F. - (f x) :: real :> at_bot"
hoelzl@50323
  1177
  by (rule filterlim_compose[OF filterlim_uminus_at_bot_at_top])
hoelzl@50323
  1178
hoelzl@50324
  1179
text {*
hoelzl@50324
  1180
hoelzl@50324
  1181
We only show rules for multiplication and addition when the functions are either against a real
hoelzl@50324
  1182
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
hoelzl@50324
  1183
hoelzl@50324
  1184
*}
hoelzl@50324
  1185
hoelzl@50324
  1186
lemma filterlim_tendsto_pos_mult_at_top: 
hoelzl@50324
  1187
  assumes f: "(f ---> c) F" and c: "0 < c"
hoelzl@50324
  1188
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1189
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1190
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1191
proof safe
hoelzl@50324
  1192
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1193
  from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
hoelzl@50324
  1194
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
hoelzl@50324
  1195
             simp: dist_real_def abs_real_def split: split_if_asm)
hoelzl@50324
  1196
  moreover from g have "eventually (\<lambda>x. (Z / c * 2) < g x) F"
hoelzl@50324
  1197
    unfolding filterlim_at_top by auto
hoelzl@50324
  1198
  ultimately show "eventually (\<lambda>x. Z < f x * g x) F"
hoelzl@50324
  1199
  proof eventually_elim
hoelzl@50324
  1200
    fix x assume "c / 2 < f x" "Z / c * 2 < g x"
hoelzl@50324
  1201
    with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) < f x * g x"
hoelzl@50324
  1202
      by (intro mult_strict_mono) (auto simp: zero_le_divide_iff)
hoelzl@50324
  1203
    with `0 < c` show "Z < f x * g x"
hoelzl@50324
  1204
       by simp
hoelzl@50324
  1205
  qed
hoelzl@50324
  1206
qed
hoelzl@50324
  1207
hoelzl@50324
  1208
lemma filterlim_at_top_mult_at_top: 
hoelzl@50324
  1209
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1210
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1211
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1212
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1213
proof safe
hoelzl@50324
  1214
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1215
  from f have "eventually (\<lambda>x. 1 < f x) F"
hoelzl@50324
  1216
    unfolding filterlim_at_top by auto
hoelzl@50324
  1217
  moreover from g have "eventually (\<lambda>x. Z < g x) F"
hoelzl@50324
  1218
    unfolding filterlim_at_top by auto
hoelzl@50324
  1219
  ultimately show "eventually (\<lambda>x. Z < f x * g x) F"
hoelzl@50324
  1220
  proof eventually_elim
hoelzl@50324
  1221
    fix x assume "1 < f x" "Z < g x"
hoelzl@50324
  1222
    with `0 < Z` have "1 * Z < f x * g x"
hoelzl@50324
  1223
      by (intro mult_strict_mono) (auto simp: zero_le_divide_iff)
hoelzl@50324
  1224
    then show "Z < f x * g x"
hoelzl@50324
  1225
       by simp
hoelzl@50324
  1226
  qed
hoelzl@50324
  1227
qed
hoelzl@50324
  1228
hoelzl@50324
  1229
lemma filterlim_tendsto_add_at_top: 
hoelzl@50324
  1230
  assumes f: "(f ---> c) F"
hoelzl@50324
  1231
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1232
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1233
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1234
proof safe
hoelzl@50324
  1235
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1236
  from f have "eventually (\<lambda>x. c - 1 < f x) F"
hoelzl@50324
  1237
    by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50324
  1238
  moreover from g have "eventually (\<lambda>x. Z - (c - 1) < g x) F"
hoelzl@50324
  1239
    unfolding filterlim_at_top by auto
hoelzl@50324
  1240
  ultimately show "eventually (\<lambda>x. Z < f x + g x) F"
hoelzl@50324
  1241
    by eventually_elim simp
hoelzl@50324
  1242
qed
hoelzl@50324
  1243
hoelzl@50324
  1244
lemma filterlim_at_top_add_at_top: 
hoelzl@50324
  1245
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1246
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1247
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1248
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1249
proof safe
hoelzl@50324
  1250
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1251
  from f have "eventually (\<lambda>x. 0 < f x) F"
hoelzl@50324
  1252
    unfolding filterlim_at_top by auto
hoelzl@50324
  1253
  moreover from g have "eventually (\<lambda>x. Z < g x) F"
hoelzl@50324
  1254
    unfolding filterlim_at_top by auto
hoelzl@50324
  1255
  ultimately show "eventually (\<lambda>x. Z < f x + g x) F"
hoelzl@50324
  1256
    by eventually_elim simp
hoelzl@50324
  1257
qed
hoelzl@50324
  1258
huffman@31349
  1259
end
hoelzl@50324
  1260