src/HOL/Limits.thy
author noschinl
Mon Mar 12 21:28:10 2012 +0100 (2012-03-12)
changeset 46886 4cd29473c65d
parent 45892 8dcf6692433f
child 46887 cb891d9a23c1
permissions -rw-r--r--
add eventually_elim method
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(*  Title       : Limits.thy
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    Author      : Brian Huffman
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*)
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header {* Filters and Limits *}
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theory Limits
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imports RealVector
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begin
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subsection {* Filters *}
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text {*
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  This definition also allows non-proper filters.
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*}
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locale is_filter =
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  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
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  assumes True: "F (\<lambda>x. True)"
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  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
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  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
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typedef (open) 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
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proof
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  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
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qed
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lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
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  using Rep_filter [of F] by simp
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lemma Abs_filter_inverse':
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  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
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  using assms by (simp add: Abs_filter_inverse)
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subsection {* Eventually *}
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definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
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  where "eventually P F \<longleftrightarrow> Rep_filter F P"
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lemma eventually_Abs_filter:
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  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
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  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
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lemma filter_eq_iff:
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  shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
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  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
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lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
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  unfolding eventually_def
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  by (rule is_filter.True [OF is_filter_Rep_filter])
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lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
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proof -
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  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
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  thus "eventually P F" by simp
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qed
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lemma eventually_mono:
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  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
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  unfolding eventually_def
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  by (rule is_filter.mono [OF is_filter_Rep_filter])
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lemma eventually_conj:
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  assumes P: "eventually (\<lambda>x. P x) F"
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  assumes Q: "eventually (\<lambda>x. Q x) F"
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  shows "eventually (\<lambda>x. P x \<and> Q x) F"
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  using assms unfolding eventually_def
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  by (rule is_filter.conj [OF is_filter_Rep_filter])
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lemma eventually_mp:
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  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
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  assumes "eventually (\<lambda>x. P x) F"
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  shows "eventually (\<lambda>x. Q x) F"
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proof (rule eventually_mono)
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  show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
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  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) F"
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    using assms by (rule eventually_conj)
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qed
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lemma eventually_rev_mp:
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  assumes "eventually (\<lambda>x. P x) F"
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  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
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  shows "eventually (\<lambda>x. Q x) F"
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using assms(2) assms(1) by (rule eventually_mp)
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lemma eventually_conj_iff:
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  "eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
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  by (auto intro: eventually_conj elim: eventually_rev_mp)
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lemma eventually_elim1:
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  assumes "eventually (\<lambda>i. P i) F"
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  assumes "\<And>i. P i \<Longrightarrow> Q i"
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  shows "eventually (\<lambda>i. Q i) F"
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  using assms by (auto elim!: eventually_rev_mp)
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lemma eventually_elim2:
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  assumes "eventually (\<lambda>i. P i) F"
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  assumes "eventually (\<lambda>i. Q i) F"
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  assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
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  shows "eventually (\<lambda>i. R i) F"
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  using assms by (auto elim!: eventually_rev_mp)
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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 ev_elim_tac ctxt thms thm = 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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  fun eventually_elim_setup name =
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    Method.setup name (Scan.succeed (fn ctxt => METHOD_CASES (ev_elim_tac ctxt)))
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      "elimination of eventually quantifiers"
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*}
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setup {* eventually_elim_setup @{binding "eventually_elim"} *}
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subsection {* Finer-than relation *}
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text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
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filter @{term F'}. *}
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instantiation filter :: (type) complete_lattice
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begin
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definition le_filter_def:
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  "F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
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definition
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  "(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
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definition
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  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
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definition
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  "bot = Abs_filter (\<lambda>P. True)"
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definition
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  "sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
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definition
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  "inf F F' = Abs_filter
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      (\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
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definition
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  "Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
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definition
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  "Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
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lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
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  unfolding top_filter_def
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  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
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lemma eventually_bot [simp]: "eventually P bot"
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  unfolding bot_filter_def
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  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
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lemma eventually_sup:
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  "eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
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  unfolding sup_filter_def
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  by (rule eventually_Abs_filter, rule is_filter.intro)
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     (auto elim!: eventually_rev_mp)
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lemma eventually_inf:
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  "eventually P (inf F F') \<longleftrightarrow>
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   (\<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
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  unfolding inf_filter_def
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  apply (rule eventually_Abs_filter, rule is_filter.intro)
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  apply (fast intro: eventually_True)
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  apply clarify
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  apply (intro exI conjI)
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  apply (erule (1) eventually_conj)
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  apply (erule (1) eventually_conj)
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  apply simp
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  apply auto
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  done
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lemma eventually_Sup:
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  "eventually P (Sup S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
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  unfolding Sup_filter_def
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  apply (rule eventually_Abs_filter, rule is_filter.intro)
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  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
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  done
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instance proof
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  fix F F' F'' :: "'a filter" and S :: "'a filter set"
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  { show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
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    by (rule less_filter_def) }
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  { show "F \<le> F"
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    unfolding le_filter_def by simp }
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  { assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
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    unfolding le_filter_def by simp }
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  { assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
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    unfolding le_filter_def filter_eq_iff by fast }
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  { show "F \<le> top"
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    unfolding le_filter_def eventually_top by (simp add: always_eventually) }
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  { show "bot \<le> F"
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    unfolding le_filter_def by simp }
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  { show "F \<le> sup F F'" and "F' \<le> sup F F'"
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    unfolding le_filter_def eventually_sup by simp_all }
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  { assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
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    unfolding le_filter_def eventually_sup by simp }
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  { show "inf F F' \<le> F" and "inf F F' \<le> F'"
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    unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
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  { assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
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    unfolding le_filter_def eventually_inf
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    by (auto elim!: eventually_mono intro: eventually_conj) }
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  { assume "F \<in> S" thus "F \<le> Sup S"
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    unfolding le_filter_def eventually_Sup by simp }
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  { assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
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    unfolding le_filter_def eventually_Sup by simp }
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  { assume "F'' \<in> S" thus "Inf S \<le> F''"
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    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
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  { assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
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    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
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qed
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end
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lemma filter_leD:
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  "F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
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  unfolding le_filter_def by simp
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lemma filter_leI:
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  "(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
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  unfolding le_filter_def by simp
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lemma eventually_False:
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  "eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
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  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
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abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
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  where "trivial_limit F \<equiv> F = bot"
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lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
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  by (rule eventually_False [symmetric])
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subsection {* Map function for filters *}
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definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
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  where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
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lemma eventually_filtermap:
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  "eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
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  unfolding filtermap_def
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  apply (rule eventually_Abs_filter)
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  apply (rule is_filter.intro)
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  apply (auto elim!: eventually_rev_mp)
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  done
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lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
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  by (simp add: filter_eq_iff eventually_filtermap)
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lemma filtermap_filtermap:
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  "filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
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  by (simp add: filter_eq_iff eventually_filtermap)
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lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
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  unfolding le_filter_def eventually_filtermap by simp
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lemma filtermap_bot [simp]: "filtermap f bot = bot"
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  by (simp add: filter_eq_iff eventually_filtermap)
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subsection {* Sequentially *}
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definition sequentially :: "nat filter"
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  where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
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lemma eventually_sequentially:
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  "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
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unfolding sequentially_def
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proof (rule eventually_Abs_filter, rule is_filter.intro)
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  fix P Q :: "nat \<Rightarrow> bool"
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  assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
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  then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
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  then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
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  then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
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qed auto
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lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
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  unfolding filter_eq_iff eventually_sequentially by auto
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lemmas trivial_limit_sequentially = sequentially_bot
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lemma eventually_False_sequentially [simp]:
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  "\<not> eventually (\<lambda>n. False) sequentially"
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  by (simp add: eventually_False)
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lemma le_sequentially:
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  "F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
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  unfolding le_filter_def eventually_sequentially
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  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
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lemma eventually_sequentiallyI:
noschinl@45892
   316
  assumes "\<And>x. c \<le> x \<Longrightarrow> P x"
noschinl@45892
   317
  shows "eventually P sequentially"
noschinl@45892
   318
using assms by (auto simp: eventually_sequentially)
noschinl@45892
   319
huffman@36662
   320
huffman@44081
   321
subsection {* Standard filters *}
huffman@36662
   322
huffman@44081
   323
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
huffman@44195
   324
  where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
huffman@31392
   325
huffman@44206
   326
definition (in topological_space) nhds :: "'a \<Rightarrow> 'a filter"
huffman@44081
   327
  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   328
huffman@44206
   329
definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
huffman@44081
   330
  where "at a = nhds a within - {a}"
huffman@31447
   331
huffman@31392
   332
lemma eventually_within:
huffman@44195
   333
  "eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
huffman@44081
   334
  unfolding within_def
huffman@44081
   335
  by (rule eventually_Abs_filter, rule is_filter.intro)
huffman@44081
   336
     (auto elim!: eventually_rev_mp)
huffman@31392
   337
huffman@45031
   338
lemma within_UNIV [simp]: "F within UNIV = F"
huffman@45031
   339
  unfolding filter_eq_iff eventually_within by simp
huffman@45031
   340
huffman@45031
   341
lemma within_empty [simp]: "F within {} = bot"
huffman@44081
   342
  unfolding filter_eq_iff eventually_within by simp
huffman@36360
   343
huffman@36654
   344
lemma eventually_nhds:
huffman@36654
   345
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
huffman@36654
   346
unfolding nhds_def
huffman@44081
   347
proof (rule eventually_Abs_filter, rule is_filter.intro)
huffman@36654
   348
  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
huffman@36654
   349
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
huffman@36358
   350
next
huffman@36358
   351
  fix P Q
huffman@36654
   352
  assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   353
     and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
huffman@36358
   354
  then obtain S T where
huffman@36654
   355
    "open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
huffman@36654
   356
    "open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
huffman@36654
   357
  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
   358
    by (simp add: open_Int)
huffman@36654
   359
  thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
huffman@36358
   360
qed auto
huffman@31447
   361
huffman@36656
   362
lemma eventually_nhds_metric:
huffman@36656
   363
  "eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
huffman@36656
   364
unfolding eventually_nhds open_dist
huffman@31447
   365
apply safe
huffman@31447
   366
apply fast
huffman@31492
   367
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
huffman@31447
   368
apply clarsimp
huffman@31447
   369
apply (rule_tac x="d - dist x a" in exI, clarsimp)
huffman@31447
   370
apply (simp only: less_diff_eq)
huffman@31447
   371
apply (erule le_less_trans [OF dist_triangle])
huffman@31447
   372
done
huffman@31447
   373
huffman@44571
   374
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
huffman@44571
   375
  unfolding trivial_limit_def eventually_nhds by simp
huffman@44571
   376
huffman@36656
   377
lemma eventually_at_topological:
huffman@36656
   378
  "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
   379
unfolding at_def eventually_within eventually_nhds by simp
huffman@36656
   380
huffman@36656
   381
lemma eventually_at:
huffman@36656
   382
  fixes a :: "'a::metric_space"
huffman@36656
   383
  shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
huffman@36656
   384
unfolding at_def eventually_within eventually_nhds_metric by auto
huffman@36656
   385
huffman@44571
   386
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
huffman@44571
   387
  unfolding trivial_limit_def eventually_at_topological
huffman@44571
   388
  by (safe, case_tac "S = {a}", simp, fast, fast)
huffman@44571
   389
huffman@44571
   390
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
huffman@44571
   391
  by (simp add: at_eq_bot_iff not_open_singleton)
huffman@44571
   392
huffman@31392
   393
huffman@31355
   394
subsection {* Boundedness *}
huffman@31355
   395
huffman@44081
   396
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   397
  where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
huffman@31355
   398
huffman@31487
   399
lemma BfunI:
huffman@44195
   400
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
huffman@31355
   401
unfolding Bfun_def
huffman@31355
   402
proof (intro exI conjI allI)
huffman@31355
   403
  show "0 < max K 1" by simp
huffman@31355
   404
next
huffman@44195
   405
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
huffman@31355
   406
    using K by (rule eventually_elim1, simp)
huffman@31355
   407
qed
huffman@31355
   408
huffman@31355
   409
lemma BfunE:
huffman@44195
   410
  assumes "Bfun f F"
huffman@44195
   411
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
huffman@31355
   412
using assms unfolding Bfun_def by fast
huffman@31355
   413
huffman@31355
   414
huffman@31349
   415
subsection {* Convergence to Zero *}
huffman@31349
   416
huffman@44081
   417
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   418
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   419
huffman@31349
   420
lemma ZfunI:
huffman@44195
   421
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   422
  unfolding Zfun_def by simp
huffman@31349
   423
huffman@31349
   424
lemma ZfunD:
huffman@44195
   425
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   426
  unfolding Zfun_def by simp
huffman@31349
   427
huffman@31355
   428
lemma Zfun_ssubst:
huffman@44195
   429
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   430
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   431
huffman@44195
   432
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   433
  unfolding Zfun_def by simp
huffman@31349
   434
huffman@44195
   435
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   436
  unfolding Zfun_def by simp
huffman@31349
   437
huffman@31349
   438
lemma Zfun_imp_Zfun:
huffman@44195
   439
  assumes f: "Zfun f F"
huffman@44195
   440
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   441
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   442
proof (cases)
huffman@31349
   443
  assume K: "0 < K"
huffman@31349
   444
  show ?thesis
huffman@31349
   445
  proof (rule ZfunI)
huffman@31349
   446
    fix r::real assume "0 < r"
huffman@31349
   447
    hence "0 < r / K"
huffman@31349
   448
      using K by (rule divide_pos_pos)
huffman@44195
   449
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
huffman@31487
   450
      using ZfunD [OF f] by fast
huffman@44195
   451
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
huffman@31355
   452
    proof (rule eventually_elim2)
huffman@31487
   453
      fix x
huffman@31487
   454
      assume *: "norm (g x) \<le> norm (f x) * K"
huffman@31487
   455
      assume "norm (f x) < r / K"
huffman@31487
   456
      hence "norm (f x) * K < r"
huffman@31349
   457
        by (simp add: pos_less_divide_eq K)
huffman@31487
   458
      thus "norm (g x) < r"
huffman@31355
   459
        by (simp add: order_le_less_trans [OF *])
huffman@31349
   460
    qed
huffman@31349
   461
  qed
huffman@31349
   462
next
huffman@31349
   463
  assume "\<not> 0 < K"
huffman@31349
   464
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   465
  show ?thesis
huffman@31355
   466
  proof (rule ZfunI)
huffman@31355
   467
    fix r :: real
huffman@31355
   468
    assume "0 < r"
huffman@44195
   469
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
huffman@31355
   470
    proof (rule eventually_elim1)
huffman@31487
   471
      fix x
huffman@31487
   472
      assume "norm (g x) \<le> norm (f x) * K"
huffman@31487
   473
      also have "\<dots> \<le> norm (f x) * 0"
huffman@31355
   474
        using K norm_ge_zero by (rule mult_left_mono)
huffman@31487
   475
      finally show "norm (g x) < r"
huffman@31355
   476
        using `0 < r` by simp
huffman@31355
   477
    qed
huffman@31355
   478
  qed
huffman@31349
   479
qed
huffman@31349
   480
huffman@44195
   481
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   482
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   483
huffman@31349
   484
lemma Zfun_add:
huffman@44195
   485
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   486
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   487
proof (rule ZfunI)
huffman@31349
   488
  fix r::real assume "0 < r"
huffman@31349
   489
  hence r: "0 < r / 2" by simp
huffman@44195
   490
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   491
    using f r by (rule ZfunD)
huffman@31349
   492
  moreover
huffman@44195
   493
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   494
    using g r by (rule ZfunD)
huffman@31349
   495
  ultimately
huffman@44195
   496
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
huffman@31349
   497
  proof (rule eventually_elim2)
huffman@31487
   498
    fix x
huffman@31487
   499
    assume *: "norm (f x) < r/2" "norm (g x) < r/2"
huffman@31487
   500
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   501
      by (rule norm_triangle_ineq)
huffman@31349
   502
    also have "\<dots> < r/2 + r/2"
huffman@31349
   503
      using * by (rule add_strict_mono)
huffman@31487
   504
    finally show "norm (f x + g x) < r"
huffman@31349
   505
      by simp
huffman@31349
   506
  qed
huffman@31349
   507
qed
huffman@31349
   508
huffman@44195
   509
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   510
  unfolding Zfun_def by simp
huffman@31349
   511
huffman@44195
   512
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
huffman@44081
   513
  by (simp only: diff_minus Zfun_add Zfun_minus)
huffman@31349
   514
huffman@31349
   515
lemma (in bounded_linear) Zfun:
huffman@44195
   516
  assumes g: "Zfun g F"
huffman@44195
   517
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   518
proof -
huffman@31349
   519
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
huffman@31349
   520
    using bounded by fast
huffman@44195
   521
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   522
    by simp
huffman@31487
   523
  with g show ?thesis
huffman@31349
   524
    by (rule Zfun_imp_Zfun)
huffman@31349
   525
qed
huffman@31349
   526
huffman@31349
   527
lemma (in bounded_bilinear) Zfun:
huffman@44195
   528
  assumes f: "Zfun f F"
huffman@44195
   529
  assumes g: "Zfun g F"
huffman@44195
   530
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   531
proof (rule ZfunI)
huffman@31349
   532
  fix r::real assume r: "0 < r"
huffman@31349
   533
  obtain K where K: "0 < K"
huffman@31349
   534
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   535
    using pos_bounded by fast
huffman@31349
   536
  from K have K': "0 < inverse K"
huffman@31349
   537
    by (rule positive_imp_inverse_positive)
huffman@44195
   538
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   539
    using f r by (rule ZfunD)
huffman@31349
   540
  moreover
huffman@44195
   541
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   542
    using g K' by (rule ZfunD)
huffman@31349
   543
  ultimately
huffman@44195
   544
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
huffman@31349
   545
  proof (rule eventually_elim2)
huffman@31487
   546
    fix x
huffman@31487
   547
    assume *: "norm (f x) < r" "norm (g x) < inverse K"
huffman@31487
   548
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   549
      by (rule norm_le)
huffman@31487
   550
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
huffman@31349
   551
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
huffman@31349
   552
    also from K have "r * inverse K * K = r"
huffman@31349
   553
      by simp
huffman@31487
   554
    finally show "norm (f x ** g x) < r" .
huffman@31349
   555
  qed
huffman@31349
   556
qed
huffman@31349
   557
huffman@31349
   558
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   559
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   560
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   561
huffman@31349
   562
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   563
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   564
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   565
huffman@44282
   566
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   567
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   568
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   569
huffman@31349
   570
wenzelm@31902
   571
subsection {* Limits *}
huffman@31349
   572
huffman@44206
   573
definition (in topological_space)
huffman@44206
   574
  tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
huffman@44195
   575
  "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
huffman@31349
   576
noschinl@45892
   577
definition real_tendsto_inf :: "('a \<Rightarrow> real) \<Rightarrow> 'a filter \<Rightarrow> bool" where
noschinl@45892
   578
  "real_tendsto_inf f F \<equiv> \<forall>x. eventually (\<lambda>y. x < f y) F"
noschinl@45892
   579
wenzelm@31902
   580
ML {*
wenzelm@31902
   581
structure Tendsto_Intros = Named_Thms
wenzelm@31902
   582
(
wenzelm@45294
   583
  val name = @{binding tendsto_intros}
wenzelm@31902
   584
  val description = "introduction rules for tendsto"
wenzelm@31902
   585
)
huffman@31565
   586
*}
huffman@31565
   587
wenzelm@31902
   588
setup Tendsto_Intros.setup
huffman@31565
   589
huffman@44195
   590
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
huffman@44081
   591
  unfolding tendsto_def le_filter_def by fast
huffman@36656
   592
huffman@31488
   593
lemma topological_tendstoI:
huffman@44195
   594
  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
huffman@44195
   595
    \<Longrightarrow> (f ---> l) F"
huffman@31349
   596
  unfolding tendsto_def by auto
huffman@31349
   597
huffman@31488
   598
lemma topological_tendstoD:
huffman@44195
   599
  "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
huffman@31488
   600
  unfolding tendsto_def by auto
huffman@31488
   601
huffman@31488
   602
lemma tendstoI:
huffman@44195
   603
  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44195
   604
  shows "(f ---> l) F"
huffman@44081
   605
  apply (rule topological_tendstoI)
huffman@44081
   606
  apply (simp add: open_dist)
huffman@44081
   607
  apply (drule (1) bspec, clarify)
huffman@44081
   608
  apply (drule assms)
huffman@44081
   609
  apply (erule eventually_elim1, simp)
huffman@44081
   610
  done
huffman@31488
   611
huffman@31349
   612
lemma tendstoD:
huffman@44195
   613
  "(f ---> l) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
huffman@44081
   614
  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
huffman@44081
   615
  apply (clarsimp simp add: open_dist)
huffman@44081
   616
  apply (rule_tac x="e - dist x l" in exI, clarsimp)
huffman@44081
   617
  apply (simp only: less_diff_eq)
huffman@44081
   618
  apply (erule le_less_trans [OF dist_triangle])
huffman@44081
   619
  apply simp
huffman@44081
   620
  apply simp
huffman@44081
   621
  done
huffman@31488
   622
huffman@31488
   623
lemma tendsto_iff:
huffman@44195
   624
  "(f ---> l) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
huffman@44081
   625
  using tendstoI tendstoD by fast
huffman@31349
   626
huffman@44195
   627
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   628
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   629
huffman@45031
   630
lemma tendsto_bot [simp]: "(f ---> a) bot"
huffman@45031
   631
  unfolding tendsto_def by simp
huffman@45031
   632
huffman@31565
   633
lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
huffman@44081
   634
  unfolding tendsto_def eventually_at_topological by auto
huffman@31565
   635
huffman@31565
   636
lemma tendsto_ident_at_within [tendsto_intros]:
huffman@36655
   637
  "((\<lambda>x. x) ---> a) (at a within S)"
huffman@44081
   638
  unfolding tendsto_def eventually_within eventually_at_topological by auto
huffman@31565
   639
huffman@44195
   640
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
huffman@44081
   641
  by (simp add: tendsto_def)
huffman@31349
   642
huffman@44205
   643
lemma tendsto_unique:
huffman@44205
   644
  fixes f :: "'a \<Rightarrow> 'b::t2_space"
huffman@44205
   645
  assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
huffman@44205
   646
  shows "a = b"
huffman@44205
   647
proof (rule ccontr)
huffman@44205
   648
  assume "a \<noteq> b"
huffman@44205
   649
  obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
huffman@44205
   650
    using hausdorff [OF `a \<noteq> b`] by fast
huffman@44205
   651
  have "eventually (\<lambda>x. f x \<in> U) F"
huffman@44205
   652
    using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
huffman@44205
   653
  moreover
huffman@44205
   654
  have "eventually (\<lambda>x. f x \<in> V) F"
huffman@44205
   655
    using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
huffman@44205
   656
  ultimately
huffman@44205
   657
  have "eventually (\<lambda>x. False) F"
huffman@44205
   658
  proof (rule eventually_elim2)
huffman@44205
   659
    fix x
huffman@44205
   660
    assume "f x \<in> U" "f x \<in> V"
huffman@44205
   661
    hence "f x \<in> U \<inter> V" by simp
huffman@44205
   662
    with `U \<inter> V = {}` show "False" by simp
huffman@44205
   663
  qed
huffman@44205
   664
  with `\<not> trivial_limit F` show "False"
huffman@44205
   665
    by (simp add: trivial_limit_def)
huffman@44205
   666
qed
huffman@44205
   667
huffman@36662
   668
lemma tendsto_const_iff:
huffman@44205
   669
  fixes a b :: "'a::t2_space"
huffman@44205
   670
  assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
huffman@44205
   671
  by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
huffman@44205
   672
huffman@44218
   673
lemma tendsto_compose:
huffman@44218
   674
  assumes g: "(g ---> g l) (at l)"
huffman@44218
   675
  assumes f: "(f ---> l) F"
huffman@44218
   676
  shows "((\<lambda>x. g (f x)) ---> g l) F"
huffman@44218
   677
proof (rule topological_tendstoI)
huffman@44218
   678
  fix B assume B: "open B" "g l \<in> B"
huffman@44218
   679
  obtain A where A: "open A" "l \<in> A"
huffman@44218
   680
    and gB: "\<forall>y. y \<in> A \<longrightarrow> g y \<in> B"
huffman@44218
   681
    using topological_tendstoD [OF g B] B(2)
huffman@44218
   682
    unfolding eventually_at_topological by fast
huffman@44218
   683
  hence "\<forall>x. f x \<in> A \<longrightarrow> g (f x) \<in> B" by simp
huffman@44218
   684
  from this topological_tendstoD [OF f A]
huffman@44218
   685
  show "eventually (\<lambda>x. g (f x) \<in> B) F"
huffman@44218
   686
    by (rule eventually_mono)
huffman@44218
   687
qed
huffman@44218
   688
huffman@44253
   689
lemma tendsto_compose_eventually:
huffman@44253
   690
  assumes g: "(g ---> m) (at l)"
huffman@44253
   691
  assumes f: "(f ---> l) F"
huffman@44253
   692
  assumes inj: "eventually (\<lambda>x. f x \<noteq> l) F"
huffman@44253
   693
  shows "((\<lambda>x. g (f x)) ---> m) F"
huffman@44253
   694
proof (rule topological_tendstoI)
huffman@44253
   695
  fix B assume B: "open B" "m \<in> B"
huffman@44253
   696
  obtain A where A: "open A" "l \<in> A"
huffman@44253
   697
    and gB: "\<And>y. y \<in> A \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> B"
huffman@44253
   698
    using topological_tendstoD [OF g B]
huffman@44253
   699
    unfolding eventually_at_topological by fast
huffman@44253
   700
  show "eventually (\<lambda>x. g (f x) \<in> B) F"
huffman@44253
   701
    using topological_tendstoD [OF f A] inj
huffman@44253
   702
    by (rule eventually_elim2) (simp add: gB)
huffman@44253
   703
qed
huffman@44253
   704
huffman@44251
   705
lemma metric_tendsto_imp_tendsto:
huffman@44251
   706
  assumes f: "(f ---> a) F"
huffman@44251
   707
  assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
huffman@44251
   708
  shows "(g ---> b) F"
huffman@44251
   709
proof (rule tendstoI)
huffman@44251
   710
  fix e :: real assume "0 < e"
huffman@44251
   711
  with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
huffman@44251
   712
  with le show "eventually (\<lambda>x. dist (g x) b < e) F"
huffman@44251
   713
    using le_less_trans by (rule eventually_elim2)
huffman@44251
   714
qed
huffman@44251
   715
noschinl@45892
   716
lemma real_tendsto_inf_real: "real_tendsto_inf real sequentially"
noschinl@45892
   717
proof (unfold real_tendsto_inf_def, rule allI)
noschinl@45892
   718
  fix x show "eventually (\<lambda>y. x < real y) sequentially"
noschinl@45892
   719
    by (rule eventually_sequentiallyI[of "natceiling (x + 1)"])
noschinl@45892
   720
        (simp add: natceiling_le_eq)
noschinl@45892
   721
qed
noschinl@45892
   722
noschinl@45892
   723
noschinl@45892
   724
huffman@44205
   725
subsubsection {* Distance and norms *}
huffman@36662
   726
huffman@31565
   727
lemma tendsto_dist [tendsto_intros]:
huffman@44195
   728
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
huffman@44195
   729
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
huffman@31565
   730
proof (rule tendstoI)
huffman@31565
   731
  fix e :: real assume "0 < e"
huffman@31565
   732
  hence e2: "0 < e/2" by simp
huffman@31565
   733
  from tendstoD [OF f e2] tendstoD [OF g e2]
huffman@44195
   734
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
huffman@31565
   735
  proof (rule eventually_elim2)
huffman@31565
   736
    fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
huffman@31565
   737
    then show "dist (dist (f x) (g x)) (dist l m) < e"
huffman@31565
   738
      unfolding dist_real_def
huffman@31565
   739
      using dist_triangle2 [of "f x" "g x" "l"]
huffman@31565
   740
      using dist_triangle2 [of "g x" "l" "m"]
huffman@31565
   741
      using dist_triangle3 [of "l" "m" "f x"]
huffman@31565
   742
      using dist_triangle [of "f x" "m" "g x"]
huffman@31565
   743
      by arith
huffman@31565
   744
  qed
huffman@31565
   745
qed
huffman@31565
   746
huffman@36662
   747
lemma norm_conv_dist: "norm x = dist x 0"
huffman@44081
   748
  unfolding dist_norm by simp
huffman@36662
   749
huffman@31565
   750
lemma tendsto_norm [tendsto_intros]:
huffman@44195
   751
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
huffman@44081
   752
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   753
huffman@36662
   754
lemma tendsto_norm_zero:
huffman@44195
   755
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
huffman@44081
   756
  by (drule tendsto_norm, simp)
huffman@36662
   757
huffman@36662
   758
lemma tendsto_norm_zero_cancel:
huffman@44195
   759
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
huffman@44081
   760
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   761
huffman@36662
   762
lemma tendsto_norm_zero_iff:
huffman@44195
   763
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
huffman@44081
   764
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   765
huffman@44194
   766
lemma tendsto_rabs [tendsto_intros]:
huffman@44195
   767
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
huffman@44194
   768
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   769
huffman@44194
   770
lemma tendsto_rabs_zero:
huffman@44195
   771
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
huffman@44194
   772
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   773
huffman@44194
   774
lemma tendsto_rabs_zero_cancel:
huffman@44195
   775
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
huffman@44194
   776
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   777
huffman@44194
   778
lemma tendsto_rabs_zero_iff:
huffman@44195
   779
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
huffman@44194
   780
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   781
huffman@44194
   782
subsubsection {* Addition and subtraction *}
huffman@44194
   783
huffman@31565
   784
lemma tendsto_add [tendsto_intros]:
huffman@31349
   785
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   786
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   787
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   788
huffman@44194
   789
lemma tendsto_add_zero:
huffman@44194
   790
  fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   791
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   792
  by (drule (1) tendsto_add, simp)
huffman@44194
   793
huffman@31565
   794
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   795
  fixes a :: "'a::real_normed_vector"
huffman@44195
   796
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   797
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   798
huffman@31349
   799
lemma tendsto_minus_cancel:
huffman@31349
   800
  fixes a :: "'a::real_normed_vector"
huffman@44195
   801
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   802
  by (drule tendsto_minus, simp)
huffman@31349
   803
huffman@31565
   804
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   805
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   806
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
huffman@44081
   807
  by (simp add: diff_minus tendsto_add tendsto_minus)
huffman@31349
   808
huffman@31588
   809
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   810
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   811
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   812
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   813
proof (cases "finite S")
huffman@31588
   814
  assume "finite S" thus ?thesis using assms
huffman@44194
   815
    by (induct, simp add: tendsto_const, simp add: tendsto_add)
huffman@31588
   816
next
huffman@31588
   817
  assume "\<not> finite S" thus ?thesis
huffman@31588
   818
    by (simp add: tendsto_const)
huffman@31588
   819
qed
huffman@31588
   820
noschinl@45892
   821
lemma real_tendsto_sandwich:
noschinl@45892
   822
  fixes f g h :: "'a \<Rightarrow> real"
noschinl@45892
   823
  assumes ev: "eventually (\<lambda>n. f n \<le> g n) net" "eventually (\<lambda>n. g n \<le> h n) net"
noschinl@45892
   824
  assumes lim: "(f ---> c) net" "(h ---> c) net"
noschinl@45892
   825
  shows "(g ---> c) net"
noschinl@45892
   826
proof -
noschinl@45892
   827
  have "((\<lambda>n. g n - f n) ---> 0) net"
noschinl@45892
   828
  proof (rule metric_tendsto_imp_tendsto)
noschinl@45892
   829
    show "eventually (\<lambda>n. dist (g n - f n) 0 \<le> dist (h n - f n) 0) net"
noschinl@45892
   830
      using ev by (rule eventually_elim2) (simp add: dist_real_def)
noschinl@45892
   831
    show "((\<lambda>n. h n - f n) ---> 0) net"
noschinl@45892
   832
      using tendsto_diff[OF lim(2,1)] by simp
noschinl@45892
   833
  qed
noschinl@45892
   834
  from tendsto_add[OF this lim(1)] show ?thesis by simp
noschinl@45892
   835
qed
noschinl@45892
   836
huffman@44194
   837
subsubsection {* Linear operators and multiplication *}
huffman@44194
   838
huffman@44282
   839
lemma (in bounded_linear) tendsto:
huffman@44195
   840
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   841
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   842
huffman@44194
   843
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   844
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   845
  by (drule tendsto, simp only: zero)
huffman@44194
   846
huffman@44282
   847
lemma (in bounded_bilinear) tendsto:
huffman@44195
   848
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   849
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   850
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   851
huffman@44194
   852
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   853
  assumes f: "(f ---> 0) F"
huffman@44195
   854
  assumes g: "(g ---> 0) F"
huffman@44195
   855
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   856
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   857
huffman@44194
   858
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   859
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   860
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   861
huffman@44194
   862
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   863
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   864
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   865
huffman@44282
   866
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   867
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   868
huffman@44282
   869
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   870
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   871
huffman@44282
   872
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   873
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   874
huffman@44568
   875
lemmas tendsto_mult_zero =
huffman@44568
   876
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   877
huffman@44568
   878
lemmas tendsto_mult_left_zero =
huffman@44568
   879
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   880
huffman@44568
   881
lemmas tendsto_mult_right_zero =
huffman@44568
   882
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   883
huffman@44194
   884
lemma tendsto_power [tendsto_intros]:
huffman@44194
   885
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   886
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
huffman@44194
   887
  by (induct n) (simp_all add: tendsto_const tendsto_mult)
huffman@44194
   888
huffman@44194
   889
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   890
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   891
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   892
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   893
proof (cases "finite S")
huffman@44194
   894
  assume "finite S" thus ?thesis using assms
huffman@44194
   895
    by (induct, simp add: tendsto_const, simp add: tendsto_mult)
huffman@44194
   896
next
huffman@44194
   897
  assume "\<not> finite S" thus ?thesis
huffman@44194
   898
    by (simp add: tendsto_const)
huffman@44194
   899
qed
huffman@44194
   900
huffman@44194
   901
subsubsection {* Inverse and division *}
huffman@31355
   902
huffman@31355
   903
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   904
  assumes f: "Zfun f F"
huffman@44195
   905
  assumes g: "Bfun g F"
huffman@44195
   906
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   907
proof -
huffman@31355
   908
  obtain K where K: "0 \<le> K"
huffman@31355
   909
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   910
    using nonneg_bounded by fast
huffman@31355
   911
  obtain B where B: "0 < B"
huffman@44195
   912
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   913
    using g by (rule BfunE)
huffman@44195
   914
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
huffman@31487
   915
  using norm_g proof (rule eventually_elim1)
huffman@31487
   916
    fix x
huffman@31487
   917
    assume *: "norm (g x) \<le> B"
huffman@31487
   918
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   919
      by (rule norm_le)
huffman@31487
   920
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   921
      by (intro mult_mono' order_refl norm_g norm_ge_zero
huffman@31355
   922
                mult_nonneg_nonneg K *)
huffman@31487
   923
    also have "\<dots> = norm (f x) * (B * K)"
huffman@31355
   924
      by (rule mult_assoc)
huffman@31487
   925
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   926
  qed
huffman@31487
   927
  with f show ?thesis
huffman@31487
   928
    by (rule Zfun_imp_Zfun)
huffman@31355
   929
qed
huffman@31355
   930
huffman@31355
   931
lemma (in bounded_bilinear) flip:
huffman@31355
   932
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@44081
   933
  apply default
huffman@44081
   934
  apply (rule add_right)
huffman@44081
   935
  apply (rule add_left)
huffman@44081
   936
  apply (rule scaleR_right)
huffman@44081
   937
  apply (rule scaleR_left)
huffman@44081
   938
  apply (subst mult_commute)
huffman@44081
   939
  using bounded by fast
huffman@31355
   940
huffman@31355
   941
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   942
  assumes f: "Bfun f F"
huffman@44195
   943
  assumes g: "Zfun g F"
huffman@44195
   944
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   945
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   946
huffman@31355
   947
lemma Bfun_inverse_lemma:
huffman@31355
   948
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   949
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
   950
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
   951
  apply (erule (1) le_imp_inverse_le)
huffman@44081
   952
  done
huffman@31355
   953
huffman@31355
   954
lemma Bfun_inverse:
huffman@31355
   955
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   956
  assumes f: "(f ---> a) F"
huffman@31355
   957
  assumes a: "a \<noteq> 0"
huffman@44195
   958
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
   959
proof -
huffman@31355
   960
  from a have "0 < norm a" by simp
huffman@31355
   961
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
   962
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
   963
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
   964
    using tendstoD [OF f r1] by fast
huffman@44195
   965
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
huffman@31355
   966
  proof (rule eventually_elim1)
huffman@31487
   967
    fix x
huffman@31487
   968
    assume "dist (f x) a < r"
huffman@31487
   969
    hence 1: "norm (f x - a) < r"
huffman@31355
   970
      by (simp add: dist_norm)
huffman@31487
   971
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   972
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   973
      by (rule nonzero_norm_inverse)
huffman@31355
   974
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   975
    proof (rule le_imp_inverse_le)
huffman@31355
   976
      show "0 < norm a - r" using r2 by simp
huffman@31355
   977
    next
huffman@31487
   978
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   979
        by (rule norm_triangle_ineq2)
huffman@31487
   980
      also have "\<dots> = norm (f x - a)"
huffman@31355
   981
        by (rule norm_minus_commute)
huffman@31355
   982
      also have "\<dots> < r" using 1 .
huffman@31487
   983
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   984
    qed
huffman@31487
   985
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   986
  qed
huffman@31355
   987
  thus ?thesis by (rule BfunI)
huffman@31355
   988
qed
huffman@31355
   989
huffman@31565
   990
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
   991
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   992
  assumes f: "(f ---> a) F"
huffman@31355
   993
  assumes a: "a \<noteq> 0"
huffman@44195
   994
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
   995
proof -
huffman@31355
   996
  from a have "0 < norm a" by simp
huffman@44195
   997
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
   998
    by (rule tendstoD)
huffman@44195
   999
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
  1000
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
  1001
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
  1002
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
  1003
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
  1004
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
  1005
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
  1006
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
  1007
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
  1008
  ultimately show ?thesis
huffman@44627
  1009
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
  1010
qed
huffman@31355
  1011
huffman@31565
  1012
lemma tendsto_divide [tendsto_intros]:
huffman@31355
  1013
  fixes a b :: "'a::real_normed_field"
huffman@44195
  1014
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
  1015
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
  1016
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
  1017
huffman@44194
  1018
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
  1019
  fixes l :: "'a::real_normed_vector"
huffman@44195
  1020
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
  1021
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
  1022
huffman@31349
  1023
end